**Meet the editor**

Dr. Farzad Ebrahimi was born in Qazvin, Iran, in 1979. He graduated in mechanical engineering, from the University of Tehran, Iran, in 2002. He received his Msc and PhD in mechanical engineering, with a specialization in applied design from the University of Tehran, Iran, in 2009. Since 2002, he has been working at the "Smart Materials and Structures Lab" Research Center of the fac-

ulty of mechanical engineering at the University of Tehran. Now he is an assistant professor in the department of mechanical engineering at Imam Khomeini International University, Qazvin. He has authored more than 30 papers and also authored/edited 6 books in the fields of "Vibration Analysis", "functionally graded structures" and "Piezoelectric Transducers".

Contents

**Preface IX** 

**"Bio-Engineering" 1** 

Josipa Borcic and Alen Braut

Chapter 2 **Application of Finite Element Analysis in Implant Dentistry 21** 

Chapter 3 **Finite Element Analysis of the Stress on** 

Liangjian Chen

Chapter 5 **Finite Element Analysis** 

Chapter 6 **Finite Element Analysis** 

Govindaraj Thangavel

Chapter 7 **The IEEE Model for a Ground Rod in** 

Chapter 4 **Finite Element Modelling of** 

**the Implant-Bone Interface of Dental Implants with Different Structures 55** 

**a Multi-Bone Joint: The Human Wrist 77**  Magnús Kjartan Gíslason and David H. Nash

**Section 2 Recent Advances of Finite Element Analysis in "Electrical Engineering" 99** 

> **of Stationary Magnetic Field 101**  Elena Otilia Virjoghe, Diana Enescu, Mihail-Florin Stan and Marcel Ionel

**of the Direct Drive PMLOM 131** 

**a Two Layer Soil – A FEM Approach 143** 

António Martins, Sílvio Mariano and Maria do Rosário Calado

**Section 1 Recent Advances of Finite Element Analysis in** 

Chapter 1 **Finite Element Analysis in Dental Medicine 3** 

B. Alper Gultekin, Pinar Gultekin and Serdar Yalcin

## Contents

### **Preface XI**



Preface

multiplied rapidly.

Finite Element Analysis (FEA) was developed as a numerical method of stress analysis, but now it has been extended as a general method of solution to many complex engineering and physical science problems. In the past few decades, the FEA has been developed into a key indispensable technology in the modeling and simulation of various engineering systems. In the development of an advanced engineering system, engineers have to go through a very rigorous process of modeling, simulation, visualization, analysis, designing, prototyping, testing, and finally, fabrication/construction. As such, techniques related to modeling and simulation in a rapid and effective way play an increasingly important role in building advanced engineering systems, and therefore the application of the FEA has

This book reports on the state of the art research and development findings on this very broad matter through original and innovative research studies exhibiting various investigation directions. The book has been grouped into three major domains: Biomedical engineering, electrical engineering, civil engineering. It is meant to provide a small but valuable sample of contemporary research activities around the world in this field and it is expected to be useful to a large number of researchers. Through its 17 chapters the reader will have access to works related to Dental Medicine, Implants, Sandwich Panels, Tunnel excavation, Stiffener run-outs, Tubular Footbridges, DC circuit breaker, Permanent Magnet Motors, MEMS and several other exciting topics.

The present book is a result of contributions of experts from international scientific community working in different aspects of Finite Element Analysis. The introductions, data, and references in this book will help the readers know more about this topic and help them explore this exciting and fast-evolving field. The text is addressed not only to researchers, but also to professional engineers, students and other experts in a variety of disciplines, both academic and industrial seeking to gain a better understanding of what has been done in the field recently, and what kind of open problems are in this area. It has been written at a level suitable for the use in a graduate course on applications of finite element modeling and analysis (Electrical,

civil and biomedical engineering studies, for instance).


## Preface

VI Contents

Chapter 8 **FEA in Micro-Electro-Mechanical Systems**

**Finite Element Technique 183**

Hao Ren and Jun Yao

Kazumi Kurihara

Marco Riva

Simona Roatesi

Chapter 15 **Finite Element Modelling of** 

Jian Sun and Jürgen Kosel

**(MEMS) Applications: A Micromachined Spatial Light Modulator (μSLM) 161**

Chapter 9 **Steady-State and Transient Performance Analysis of** 

Chapter 10 **Finite-Element Modelling and Analysis of Hall Effect** 

Chapter 12 **Integrated FEA and Raytracing in Instrumentation for Astronomy: Smart Structures Evaluation** 

**in "Civil & Structural Engineering" 275**

**and Structural Optimization 253**

**Section 3 Recent Advances of Finite Element Analysis**

Chapter 13 **Damage-Tolerant Design of Stiffener Run-Outs: A Finite Element Approach 277**  S. Psarras, S.T. Pinho and B.G. Falzon

Chapter 14 **Finite Element Analysis for the Problem of Tunnel** 

and Luciano Rodrigues Ornelas de Lima

**Spots in Interconnects and Packages 377** 

Chapter 16 **Finite Element Analysis of Loading Area Effect**

Salih Akour and Hussein Maaitah

Chapter 17 **The Finite Element Analysis of Weak** 

Kirsten Weide-Zaage

**Permanent-Magnet Machines Using Time-Stepping** 

**and Extraordinary Magnetoresistance Effect 201** 

**by Introducing Stress Concentration Region (SCR) 225** 

**Excavation Successive Phases and Lining Mounting 301** 

**the Dynamic Behaviour of Tubular Footbridges 333** 

José Guilherme Santos da Silva, Ana Cristina Castro Fontenla Sieira, Gilvan Lunz Debona, Pedro Colmar Gonçalves da Silva Vellasco

**on Sandwich Panel Behaviour Beyond the Yield Limit 353** 

Chapter 11 **High Sensitive Piezoresistive Cantilever MEMS Based Sensor**

Sh Mohd Firdaus, Husna Omar and Ishak Abd Azid

Finite Element Analysis (FEA) was developed as a numerical method of stress analysis, but now it has been extended as a general method of solution to many complex engineering and physical science problems. In the past few decades, the FEA has been developed into a key indispensable technology in the modeling and simulation of various engineering systems. In the development of an advanced engineering system, engineers have to go through a very rigorous process of modeling, simulation, visualization, analysis, designing, prototyping, testing, and finally, fabrication/construction. As such, techniques related to modeling and simulation in a rapid and effective way play an increasingly important role in building advanced engineering systems, and therefore the application of the FEA has multiplied rapidly.

This book reports on the state of the art research and development findings on this very broad matter through original and innovative research studies exhibiting various investigation directions. The book has been grouped into three major domains: Biomedical engineering, electrical engineering, civil engineering. It is meant to provide a small but valuable sample of contemporary research activities around the world in this field and it is expected to be useful to a large number of researchers. Through its 17 chapters the reader will have access to works related to Dental Medicine, Implants, Sandwich Panels, Tunnel excavation, Stiffener run-outs, Tubular Footbridges, DC circuit breaker, Permanent Magnet Motors, MEMS and several other exciting topics.

The present book is a result of contributions of experts from international scientific community working in different aspects of Finite Element Analysis. The introductions, data, and references in this book will help the readers know more about this topic and help them explore this exciting and fast-evolving field. The text is addressed not only to researchers, but also to professional engineers, students and other experts in a variety of disciplines, both academic and industrial seeking to gain a better understanding of what has been done in the field recently, and what kind of open problems are in this area. It has been written at a level suitable for the use in a graduate course on applications of finite element modeling and analysis (Electrical, civil and biomedical engineering studies, for instance).

#### XII Preface

I am honored to be editing such a valuable book, which contains contributions of a selected group of researchers describing the best of their work. I would like to express my sincere gratitude to all of them for their outstanding chapters. I also wish to acknowledge the InTech editorial staff, in particular Oliver Kurelic, for indispensable technical assistance in book preparation and publishing.

#### **Dr. Farzad Ebrahimi**

**Section 1** 

**Recent Advances of Finite Element Analysis** 

**in "Bio-Engineering"** 

Faculty of Engineering, Mechanical Engineering Department, International University of Imam Khomeini, Qazvin, I.R.Iran

**Recent Advances of Finite Element Analysis in "Bio-Engineering"** 

X Preface

I am honored to be editing such a valuable book, which contains contributions of a selected group of researchers describing the best of their work. I would like to express my sincere gratitude to all of them for their outstanding chapters. I also wish to acknowledge the InTech editorial staff, in particular Oliver Kurelic, for indispensable

> **Dr. Farzad Ebrahimi** Faculty of Engineering,

> > Qazvin, I.R.Iran

Mechanical Engineering Department, International University of Imam Khomeini,

technical assistance in book preparation and publishing.

**Chapter 1** 

© 2012 Borcic and Braut, licensee InTech. This is an open access chapter distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

© 2012 Borcic and Braut, licensee InTech. This is a paper distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

**Finite Element Analysis in Dental Medicine** 

Studying dental structures and surrounding tissues in the oral cavity presents the basis for understanding the occurrence of pathological process and enables the correct approach and treatment. Oral rehabilitation is inherently difficult, due to the functional and parafunctional forces within the mouth that result in extremely complex structural responses by the oral tissue [1]. The success of restorative materials depends on their properties to withstand and resist occlusal forces and successfully support the remaining oral structure [2]. Studies examining the biomechanical behavior of oral structures require sophisticated simulations

There were numerous ways and attempts of experimental research, but due to the complexity of dental structures, composed of various tissue materials mechanically and chemically interconnected, and due to complex tooth morphology and surrounding structures, these attempts failed to obtain precise and reliable results. Researches have used photoelastic methods, computer simulation methods and finite element analysis to conduct stress analyses of sound and restored teeth in order to predict their fracture resistance. Conventional methods such as photoelasticity and the strain-gauge methods are inadequate to predict reliable stress distribution in the tooth [4]. The use of traditional load-to-failure bench-top testing is unable to recreate the failure mechanisms seen clinically; hence the use of FEA is gaining popularity because of its ability to accurately asses the complex biomechanical behavior of irregular prosthetic structures and heterogeneous material in a

Finite element analysis (FEA) is a numerical method of analyzing stresses and deformations in structures which originated from the need for solving complex structural problems in civil and aeronautical engineering. In order to achieve this goal, the structures are broken

Josipa Borcic and Alen Braut

http://dx.doi.org/10.5772/50038

**1. Introduction** 

Additional information is available at the end of the chapter

of the fundaments of the stomatognathic system [3].

non-destructive, repeatable manner [5].

**2. Finite element analysis** 

## **Finite Element Analysis in Dental Medicine**

Josipa Borcic and Alen Braut

Additional information is available at the end of the chapter

http://dx.doi.org/10.5772/50038

## **1. Introduction**

Studying dental structures and surrounding tissues in the oral cavity presents the basis for understanding the occurrence of pathological process and enables the correct approach and treatment. Oral rehabilitation is inherently difficult, due to the functional and parafunctional forces within the mouth that result in extremely complex structural responses by the oral tissue [1]. The success of restorative materials depends on their properties to withstand and resist occlusal forces and successfully support the remaining oral structure [2]. Studies examining the biomechanical behavior of oral structures require sophisticated simulations of the fundaments of the stomatognathic system [3].

There were numerous ways and attempts of experimental research, but due to the complexity of dental structures, composed of various tissue materials mechanically and chemically interconnected, and due to complex tooth morphology and surrounding structures, these attempts failed to obtain precise and reliable results. Researches have used photoelastic methods, computer simulation methods and finite element analysis to conduct stress analyses of sound and restored teeth in order to predict their fracture resistance. Conventional methods such as photoelasticity and the strain-gauge methods are inadequate to predict reliable stress distribution in the tooth [4]. The use of traditional load-to-failure bench-top testing is unable to recreate the failure mechanisms seen clinically; hence the use of FEA is gaining popularity because of its ability to accurately asses the complex biomechanical behavior of irregular prosthetic structures and heterogeneous material in a non-destructive, repeatable manner [5].

## **2. Finite element analysis**

Finite element analysis (FEA) is a numerical method of analyzing stresses and deformations in structures which originated from the need for solving complex structural problems in civil and aeronautical engineering. In order to achieve this goal, the structures are broken

down into many small simple segments or elements, each with specific physical properties (figure 1). Than, an operator uses a computer program in order to obtain a model of stresses produced by various loads [6,7]. A major advantage of finite element analysis (FEA) is its ability to solve complex biomechanical problems for witch other study methods are inadequate. Stress, strain and some other qualities can be calculated in every point throughout the structure. FEA is also being used as part of the design process to simulate possible structure failure, as a mean to reduce the need for making prototypes, and reducing a need for performing actual experiments, that are usually expensive and time-consuming [8]. This method allows researches to overcome some ethical and methodological limitations and enables them to verify how the stresses are transferred throughout the materials [9].

Finite Element Analysis in Dental Medicine 5

The decision to use 2D or 3D models to investigate biomechanical behavior of complex structures, by FEA, depends on many inter-related factors, such as the complexity of the geometry, material, properties, mode of analysis, etc. Although 2D models are simpler, easier to build and less time consuming, they do not represent the complexity of the real problem. 2D model might be considered when studying the qualitative biomechanical behavior, but for the quantitative stress analysis the 2D models overestimate stress magnitudes and do not represent the realistic model. 3D model may provide more reliable data that more accurately represent non-linear and anisotropic materials. 3D models should be carefully created with appropriate mesh density [3]. Khera et al. were the pioneers in the utilization of 3D models. The models were obtained from sectional images of human mandible, but this is no longer required due to the use of a computerized tomography (CT)

The 3D geometry of the tooth (figure 2) can be reconstructed in two ways. The old traditional method consists of embedding the tooth in red epoxy and sectioning it perpendicularly to the long axis by a precise saw (figure 3). Each section is than digitally photographed and the 3D geometry of the tooth is being constructed from these crosssections using specialized computer program. The solid model is transferred into a finite element analysis program, where a 3D mesh is being created, and subsequently the stress

**2.1. Finite element model** 

distribution analysis performed (figure 4) [4, 25].

[10].

**Figure 2.** Natural tooth

In the area of dentistry, FEA has been used to simulate the bone remodeling process, to study internal stresses in teeth and different dental materials, and to optimize the shape of restorations. Because of the large inherent variations in biological material properties and anatomy, mechanical testing involving biomaterials usually require a large number of samples. With FEA the necessity of traditional specimens can be avoided, and by using a mathematical model it also eliminates the need for large number of experimental teeth. It has been used to represent simulated tooth mechanical behavior under occlusal loads in details [8].

**Figure 1.** Elements of an FEA model.

## **2.1. Finite element model**

4 Finite Element Analysis – New Trends and Developments

**Figure 1.** Elements of an FEA model.

down into many small simple segments or elements, each with specific physical properties (figure 1). Than, an operator uses a computer program in order to obtain a model of stresses produced by various loads [6,7]. A major advantage of finite element analysis (FEA) is its ability to solve complex biomechanical problems for witch other study methods are inadequate. Stress, strain and some other qualities can be calculated in every point throughout the structure. FEA is also being used as part of the design process to simulate possible structure failure, as a mean to reduce the need for making prototypes, and reducing a need for performing actual experiments, that are usually expensive and time-consuming [8]. This method allows researches to overcome some ethical and methodological limitations and enables them to verify how the stresses are transferred throughout the materials [9].

In the area of dentistry, FEA has been used to simulate the bone remodeling process, to study internal stresses in teeth and different dental materials, and to optimize the shape of restorations. Because of the large inherent variations in biological material properties and anatomy, mechanical testing involving biomaterials usually require a large number of samples. With FEA the necessity of traditional specimens can be avoided, and by using a mathematical model it also eliminates the need for large number of experimental teeth. It has been used to represent simulated tooth mechanical behavior under occlusal loads in details [8].

The decision to use 2D or 3D models to investigate biomechanical behavior of complex structures, by FEA, depends on many inter-related factors, such as the complexity of the geometry, material, properties, mode of analysis, etc. Although 2D models are simpler, easier to build and less time consuming, they do not represent the complexity of the real problem. 2D model might be considered when studying the qualitative biomechanical behavior, but for the quantitative stress analysis the 2D models overestimate stress magnitudes and do not represent the realistic model. 3D model may provide more reliable data that more accurately represent non-linear and anisotropic materials. 3D models should be carefully created with appropriate mesh density [3]. Khera et al. were the pioneers in the utilization of 3D models. The models were obtained from sectional images of human mandible, but this is no longer required due to the use of a computerized tomography (CT) [10].

The 3D geometry of the tooth (figure 2) can be reconstructed in two ways. The old traditional method consists of embedding the tooth in red epoxy and sectioning it perpendicularly to the long axis by a precise saw (figure 3). Each section is than digitally photographed and the 3D geometry of the tooth is being constructed from these crosssections using specialized computer program. The solid model is transferred into a finite element analysis program, where a 3D mesh is being created, and subsequently the stress distribution analysis performed (figure 4) [4, 25].

**Figure 2.** Natural tooth

Finite Element Analysis in Dental Medicine 7

before and after the performed therapy procedures, and periodical follow-ups of the therapy success. Technologies such as micro-CT scanning open up the possibility for complex 3D modeling [11]. However, the process of going from image to mesh involves a number of

The results obtained from a FEA on the restored system contain information about the stress distribution of each component of the restoration, instead of only a single value of failure load typical of in vitro results. A correct interpretation of FEA results should be based on the stresses and strength of each component of the system. To obtain accurate conclusions from these interpretations, three conditions must be fulfilled. First, FEA should adequately represent the real stress values; second, strength of the different materials must be known;

It is not possible to implement the results from FEA directly into a clinical situation, but it has to design the model in such a way that is mimics the real situation as closely as possible. FEA analysis must be interpreted with a certain amount of caution. Most of the researches modeled dental structures as isotropic and not othotropic. The finite element model represents a static situation at the moment of load application and not an actual clinical situation. In reality, the loading of the structure is more dynamic and cyclic. The materials of the various tooth structures were assumed to be isotropic, homogenous and elastic, and that they remain such under applied loads. More precise measurements can be obtained if the material properties are set as anizotrophic and non-homogeneous, but such setup requires much more complex mathematical calculations. It is better to use a non-linear elastic-plastic

The values from finite element analysis are presented as maximum and minimum principal stresses. Most of the previously published studies have analyzed the results from Von Mises maximal stress [15-19]. This is probably associated to the fact that this is the normal criterion for the most engineering analyses, which usually deal with ductile materials such as steel and aluminum [13]. It is known that the Von Mises criterion is only valid for the ductile materials with equal compressive or tensile strength, but materials exhibiting brittle behavior such as ceramics, cements or resin composites presents reported values of compressive strength significantly greater than tensile strength [20]. Positive and negative values indicate that the corresponding regions are subjected to tensile or compressive

The response of the structure is different if asymmetrical loading is considered. When the tooth is compressively loaded, displacements do not appear to be significant because of the rather large compressive yield strength. The situation is different if the asymmetrical loading is considered, when the tensile stress occurs. The dental tissues are more resilient to compressive than tensile forces. Any occlusal contact that can create tensile stress, also creates the possibility to create a lesion in tooth structure. When lateral loads are applied, tensile stresses generated in the areas are of higher values than when vertical loads are

material model than the linear models that are used in most FEA studies [14].

processing steps, each with potential geometric errors [12].

third, an adequate failure criterion must be used [13].

**2.2. Interpretation of the FEA results** 

stresses (figure 5) [21].

**Figure 3.** Embedded tooth in red epoxy.

**Figure 4.** 3D model of the sound tooth

The second, latest method of reconstructing a 3D tooth model is performed with the aid of CT. It facilitates and speeds up the acquisition and produces more accurate model. With this method the surrounding soft structures can be also included, larger areas scanned and reconstructed, while the structures itself still remain in the patient mouth. The next big advantage of CT model rendering consists of the possibility to scan the same structure before and after the performed therapy procedures, and periodical follow-ups of the therapy success. Technologies such as micro-CT scanning open up the possibility for complex 3D modeling [11]. However, the process of going from image to mesh involves a number of processing steps, each with potential geometric errors [12].

## **2.2. Interpretation of the FEA results**

6 Finite Element Analysis – New Trends and Developments

**Figure 3.** Embedded tooth in red epoxy.

**Figure 4.** 3D model of the sound tooth

The second, latest method of reconstructing a 3D tooth model is performed with the aid of CT. It facilitates and speeds up the acquisition and produces more accurate model. With this method the surrounding soft structures can be also included, larger areas scanned and reconstructed, while the structures itself still remain in the patient mouth. The next big advantage of CT model rendering consists of the possibility to scan the same structure The results obtained from a FEA on the restored system contain information about the stress distribution of each component of the restoration, instead of only a single value of failure load typical of in vitro results. A correct interpretation of FEA results should be based on the stresses and strength of each component of the system. To obtain accurate conclusions from these interpretations, three conditions must be fulfilled. First, FEA should adequately represent the real stress values; second, strength of the different materials must be known; third, an adequate failure criterion must be used [13].

It is not possible to implement the results from FEA directly into a clinical situation, but it has to design the model in such a way that is mimics the real situation as closely as possible. FEA analysis must be interpreted with a certain amount of caution. Most of the researches modeled dental structures as isotropic and not othotropic. The finite element model represents a static situation at the moment of load application and not an actual clinical situation. In reality, the loading of the structure is more dynamic and cyclic. The materials of the various tooth structures were assumed to be isotropic, homogenous and elastic, and that they remain such under applied loads. More precise measurements can be obtained if the material properties are set as anizotrophic and non-homogeneous, but such setup requires much more complex mathematical calculations. It is better to use a non-linear elastic-plastic material model than the linear models that are used in most FEA studies [14].

The values from finite element analysis are presented as maximum and minimum principal stresses. Most of the previously published studies have analyzed the results from Von Mises maximal stress [15-19]. This is probably associated to the fact that this is the normal criterion for the most engineering analyses, which usually deal with ductile materials such as steel and aluminum [13]. It is known that the Von Mises criterion is only valid for the ductile materials with equal compressive or tensile strength, but materials exhibiting brittle behavior such as ceramics, cements or resin composites presents reported values of compressive strength significantly greater than tensile strength [20]. Positive and negative values indicate that the corresponding regions are subjected to tensile or compressive stresses (figure 5) [21].

The response of the structure is different if asymmetrical loading is considered. When the tooth is compressively loaded, displacements do not appear to be significant because of the rather large compressive yield strength. The situation is different if the asymmetrical loading is considered, when the tensile stress occurs. The dental tissues are more resilient to compressive than tensile forces. Any occlusal contact that can create tensile stress, also creates the possibility to create a lesion in tooth structure. When lateral loads are applied, tensile stresses generated in the areas are of higher values than when vertical loads are

applied onto the same areas. The increase in the load does not cause a change in the overall stress pattern, but increases the values. The loading, that the tooth is subjected to, may cause cracks in the tooth, but not necessarily its immediate failure. Most of the failures of dental materials used for tooth restorations are caused by tensile stress. Precise occlusal adjustments of teeth occlusal surfaces should be performed to prevent such events. The average chewing force varies between 11 and 150 N, whereas force peaks are 200N in the anterior, 350N in the posterior and 1000N with bruxism [22].

Finite Element Analysis in Dental Medicine 9

changes can occur depending upon the magnitude of the force, which can affect the tooth

The properties of tooth are not homogenous, but are anisptropic like dentin (due to its capillary morphological structure) or enamel (due to its prismatic structure) [23]. Various studies have shown that the failure was confined mostly to the occlusal walls and margins, and was usually seen on the buccal surfaces of lower molars and premolars (figure 6 and 7) [24,25]. Excursive mandibular movements place the buccal cusps in tension or in compression and open up the occlusal margins (figure 8). Enamel near the cemento-enamel junction is highly stressed because the reactive forces have to flow into and through this thin wedge of tissue for it to be transmitted into the root of the tooth and subsequently into the supporting alveolus bone [2]. This is the reason why the restorations inserted into the cervical region can be subjected to high compressive stresses even though these areas are

morphology in extreme (premature contacts) or repetitive cases (fatigue) [11].

not susceptible to direct contact during mastication [26,4].

**Figure 6.** FEA analysis in sound tooth in normal occlusion looking from outside

**3.1. Natural tooth** 

**Figure 5.** FEA model of a restored apicotomysed tooth

## **3. Materials and types of reconstructions in dental medicine**

The use of different materials for restoration substantially modifies the stress distribution of an originally healthy tooth. The difference between the elastic modulus of tooth and restorative material may be a source of stress in the dental structures. If the stress exceeds the yield strength of the materials, fracture of the restorative materials or the tooth may occur. The occlusal force leaning against the tooth or dental implant axis causes the structure to bend, and the higher tensile stresses are produced. The oblique force loading on the dental structure is the major cause of dental damage and the further attention should be paid to the importance of the occlusal adjustment [4, 7, 25].

The way the chewing force application is much more important than the dentine and the enamel properties, or even the properties of the restorative materials. The consequences of the same chewing force for different teeth also need to be highlighted because structural changes can occur depending upon the magnitude of the force, which can affect the tooth morphology in extreme (premature contacts) or repetitive cases (fatigue) [11].

#### **3.1. Natural tooth**

8 Finite Element Analysis – New Trends and Developments

anterior, 350N in the posterior and 1000N with bruxism [22].

**Figure 5.** FEA model of a restored apicotomysed tooth

importance of the occlusal adjustment [4, 7, 25].

**3. Materials and types of reconstructions in dental medicine** 

The use of different materials for restoration substantially modifies the stress distribution of an originally healthy tooth. The difference between the elastic modulus of tooth and restorative material may be a source of stress in the dental structures. If the stress exceeds the yield strength of the materials, fracture of the restorative materials or the tooth may occur. The occlusal force leaning against the tooth or dental implant axis causes the structure to bend, and the higher tensile stresses are produced. The oblique force loading on the dental structure is the major cause of dental damage and the further attention should be paid to the

The way the chewing force application is much more important than the dentine and the enamel properties, or even the properties of the restorative materials. The consequences of the same chewing force for different teeth also need to be highlighted because structural

applied onto the same areas. The increase in the load does not cause a change in the overall stress pattern, but increases the values. The loading, that the tooth is subjected to, may cause cracks in the tooth, but not necessarily its immediate failure. Most of the failures of dental materials used for tooth restorations are caused by tensile stress. Precise occlusal adjustments of teeth occlusal surfaces should be performed to prevent such events. The average chewing force varies between 11 and 150 N, whereas force peaks are 200N in the

The properties of tooth are not homogenous, but are anisptropic like dentin (due to its capillary morphological structure) or enamel (due to its prismatic structure) [23]. Various studies have shown that the failure was confined mostly to the occlusal walls and margins, and was usually seen on the buccal surfaces of lower molars and premolars (figure 6 and 7) [24,25]. Excursive mandibular movements place the buccal cusps in tension or in compression and open up the occlusal margins (figure 8). Enamel near the cemento-enamel junction is highly stressed because the reactive forces have to flow into and through this thin wedge of tissue for it to be transmitted into the root of the tooth and subsequently into the supporting alveolus bone [2]. This is the reason why the restorations inserted into the cervical region can be subjected to high compressive stresses even though these areas are not susceptible to direct contact during mastication [26,4].

**Figure 6.** FEA analysis in sound tooth in normal occlusion looking from outside

Finite Element Analysis in Dental Medicine 11

restoration may also influence the retention of class V restoration because of the tooth flexure theory. Breakdown of the margins of class V restorations may be the result of

**Figure 8.** FEA analysis in the sound tooth in the case of malocclusion

A growing interest in aesthetic dental restorations has led to the development of innovative materials for aesthetic restorations of teeth. These new systems have focused on physical properties, such as modulus of elasticity, that are more closely matched to natural tissue, in order to decrease stress concentrations within the dental structure and reduce the incidence of failure. The development of adhesives has created a need to measure the adhesive bond strength of restorative materials to mineralized tissue. Several methods for studying have been developed but FEA capable of quantifying the effect of each tested parameter on bond

Composite occlusal restorations have been shown to reduce cuspal flexure compared to an occlusal amalgam restoration [26]. Composite resin in combination with the acid-etch technique and adhesive systems have been used for the restoration of tooth caries and cervical lesions that we commonly call "dental composite restoration". Evaluation of

occlusal loading [2].

strength [30].

**Figure 7.** FEA analysis of the sound tooth (cross section)

## **3.2. Dental restorations**

Many detrimental effects during restorative procedures are reported to be produced because of lack of understanding of biomechanical principles underlying treatment. Biomechanical studies are crucial in order to highlight the behavior of restored tooth to functional forces [27].

It was earlier thought that the only forces that dislodge the tooth restorations were the pulling forces of sticky foods, while little thought was given to the biomechanics of the tooth structure. Later, it was seen that forces applied on the occlusal surface of the tooth could induce stresses in a restoration remote from the point of application of the force [2]. Heymann suggested that two mechanisms operate and cause failure. One is the lateral excursive movements resulting in lateral cuspal movements which generate tensile stresses along tooth restoration interface, and the other one are heavy forces in centric occlusion which cause vertical deformation on the tooth leading to compressive and shear stresses [24]. The presence of an occlusal restoration weakens the tooth structure and increases the stresses. Especially the depth is more critical than the width [26,28,29]. This restoration may also influence the retention of class V restoration because of the tooth flexure theory. Breakdown of the margins of class V restorations may be the result of occlusal loading [2].

10 Finite Element Analysis – New Trends and Developments

**Figure 7.** FEA analysis of the sound tooth (cross section)

Many detrimental effects during restorative procedures are reported to be produced because of lack of understanding of biomechanical principles underlying treatment. Biomechanical studies are crucial in order to highlight the behavior of restored tooth to functional forces [27].

It was earlier thought that the only forces that dislodge the tooth restorations were the pulling forces of sticky foods, while little thought was given to the biomechanics of the tooth structure. Later, it was seen that forces applied on the occlusal surface of the tooth could induce stresses in a restoration remote from the point of application of the force [2]. Heymann suggested that two mechanisms operate and cause failure. One is the lateral excursive movements resulting in lateral cuspal movements which generate tensile stresses along tooth restoration interface, and the other one are heavy forces in centric occlusion which cause vertical deformation on the tooth leading to compressive and shear stresses [24]. The presence of an occlusal restoration weakens the tooth structure and increases the stresses. Especially the depth is more critical than the width [26,28,29]. This

**3.2. Dental restorations** 

**Figure 8.** FEA analysis in the sound tooth in the case of malocclusion

A growing interest in aesthetic dental restorations has led to the development of innovative materials for aesthetic restorations of teeth. These new systems have focused on physical properties, such as modulus of elasticity, that are more closely matched to natural tissue, in order to decrease stress concentrations within the dental structure and reduce the incidence of failure. The development of adhesives has created a need to measure the adhesive bond strength of restorative materials to mineralized tissue. Several methods for studying have been developed but FEA capable of quantifying the effect of each tested parameter on bond strength [30].

Composite occlusal restorations have been shown to reduce cuspal flexure compared to an occlusal amalgam restoration [26]. Composite resin in combination with the acid-etch technique and adhesive systems have been used for the restoration of tooth caries and cervical lesions that we commonly call "dental composite restoration". Evaluation of

marginal integrity at the composite resin-tooth interface is required for clinically successful restorations. Polymerization contraction occurs during light curing and may cause marginal disintegration [31]. The maximum stresses due to the shrinkage of the cement layer may cause debonding of the cement layer. This debonding on one side will cause relaxation of stresses at the other side of the restoration and will cause (micro) leakage with all its detrimental effects [14].

Finite Element Analysis in Dental Medicine 13

was a key element since the elasticity modulus of porcelains is matched well with enamel

The widely used method for treatment of structurally weakened teeth is the post and core system. This system can be classified into two basic core system, metal posts and cores that are custom cast as a single piece, and two element designs composing a prefabricated post to which other materials core is subsequently adapted [34]. The difference between the elastic modulus of dentine and the post material may be a source of stress for root structures. Debonding of posts because of contraction stress of the cement was found as the most common mode of failure [27]. The effect of post design is also very important for dentinal stress distribution since the placement of a post can create stresses that lead to root fracture (figure 9) [1]. Increased intracanal stresses below the level of crestal bone would explain the higher incidence of deep root fractures in teeth restored with post-retained crowns Horizontal loads generate more dentinal stress than vertical loads. Shorter posts are associated with more dentinal stress concentration around the post apex. Consequently, extending the apical post beyond the level of alveolar bone is essential to avoid stress concentration in the region of the post apex. However, very long posts are associated with higher intracanal stress values. A higher amount of radicular dentin around the post is important in order to reduce dentinal stress concentration within the root [35]. The use of post materials conflicts with the mechanical resistance of teeth because of mismatch in the stiffness with the residual dental structure [36]. Many studies have shown that fiberglass posts give better biomechanical performance. Titanium posts concentrate stress close to the post-cement interface, promoting weakness of restored tooth. Akkayan [37] observed that the fractures occurring with the use of fiberglass and quartz posts systems could be repaired, whereas this was not the case with zirconium and titanium posts. Thus, fiberglass post can be considered a very good choice because they offer good biomechanical performance, provide excellent aesthetics, and exhibit good adhesion to cementing agents

Clinicians generally agree that NiTi rotary files have good properties to produce desirable tapered root canal forms, but also have a risk of fracture during instrumentation. These instruments have been developed to overcome the rigidity of stainless steel instruments [39]. Design of an instrument is the main factor in their mechanical behavior. Cyclic fatigue, which is a failure process associated with repetitive stressing, and torsion have been

With the application of adhesive technology to endodontics, the term monoblock has become familiar. Monoblock units can be created in a root canal system either by adhesive root sealers in combination with a bondable root filling material or adhesive post systems. The concept of creating mechanically homogenous units within the root dentine is excellent in theory, but accomplishing these ideal monoblock in the canal space is challenging because bonding to dentine is compromised by volumetric changes in resin-based materials, high cavity configuration factors, debris on canal walls, and differences in regional bond

reported as dominant factors in file fracture [40].

[11].

[38].

strengths [27].

The fracture load of the final restoration is the result of the combined effects of bonding between the underlying tooth, the ceramic restoration, and the resin composite cement. Clinical stress distribution in ceramic dental restorations may be quite complex. Several factors are associated with crack initiation and propagation, including the shape, microstructural no homogeneities, the size and distribution of surface flows, residual stresses, ceramic-cement interfacial features, thickness of restorations, different elastic modulus and the magnitude and orientation of the applied load. On the structural factors, the connector areas are the most influential in failure [22]. Traditional load-to–failure testing has proved irrelevant in predicting the clinical performance of ceramics, largely because they cannot recreate the failure mechanisms seen in clinical specimens [5]. The FEA was used to determine the optimal stress distribution in the ceramics bridges that would reduce the risk of connector fracture. The points of greater stress were found within, or near the connector [22].

The FEA demonstrated that with the use of an idealized inlay preparation form and an optimized bridge design emphasizing a broadening of the gingival embrasure, the forces derived from mastication can be adequately distributed to levels which are within the fracture strength of current ceramics [5]. Tensile stresses tend to be more critical than compressive stresses for ceramic materials. The strength of ceramic restorations is significantly affected by the presence of flows or other microscopic defects [32]. Tensile stress concentration at cementation surface of the ceramic layer was suggested to be the predominant factor controlling ceramic failure [33]. Fea showed lower tensile stress levels at the cementation surface than in the area under and between the load points, which could explain the occlusal to cervical direction of fracture seen in the fractographic analysis. Although the polymer crown had a higher fracture resistance than ceramics, a larger amount of the occlusal load was transferred through the tooth, resulting in catastrophic fracture of the tooth. This fracture behavior can limit the use of polymer crowns when compared to ceramic systems [32]. Molar crowns made of stiffer materials are less prone to debonding and crowns made of higher elastic modulus material protect the tooth structures from damage [23].

Veneers used in restorative rehabilitations for anterior teeth are retained by the adhesive systems and resin cements. These restorations are mechanically not strong, because they are made up of a brittle material, but they have good retention due to the resin-dentine bonding. The remaining tooth tissue is the most important factor for the longevity of the veneers where the buccal, cervical region is the most critical region. Teeth totally recover their properties when veneers are placed as a partial enamel substitute. The use of ceramic was a key element since the elasticity modulus of porcelains is matched well with enamel [11].

12 Finite Element Analysis – New Trends and Developments

detrimental effects [14].

from damage [23].

marginal integrity at the composite resin-tooth interface is required for clinically successful restorations. Polymerization contraction occurs during light curing and may cause marginal disintegration [31]. The maximum stresses due to the shrinkage of the cement layer may cause debonding of the cement layer. This debonding on one side will cause relaxation of stresses at the other side of the restoration and will cause (micro) leakage with all its

The fracture load of the final restoration is the result of the combined effects of bonding between the underlying tooth, the ceramic restoration, and the resin composite cement. Clinical stress distribution in ceramic dental restorations may be quite complex. Several factors are associated with crack initiation and propagation, including the shape, microstructural no homogeneities, the size and distribution of surface flows, residual stresses, ceramic-cement interfacial features, thickness of restorations, different elastic modulus and the magnitude and orientation of the applied load. On the structural factors, the connector areas are the most influential in failure [22]. Traditional load-to–failure testing has proved irrelevant in predicting the clinical performance of ceramics, largely because they cannot recreate the failure mechanisms seen in clinical specimens [5]. The FEA was used to determine the optimal stress distribution in the ceramics bridges that would reduce the risk of connector fracture. The

The FEA demonstrated that with the use of an idealized inlay preparation form and an optimized bridge design emphasizing a broadening of the gingival embrasure, the forces derived from mastication can be adequately distributed to levels which are within the fracture strength of current ceramics [5]. Tensile stresses tend to be more critical than compressive stresses for ceramic materials. The strength of ceramic restorations is significantly affected by the presence of flows or other microscopic defects [32]. Tensile stress concentration at cementation surface of the ceramic layer was suggested to be the predominant factor controlling ceramic failure [33]. Fea showed lower tensile stress levels at the cementation surface than in the area under and between the load points, which could explain the occlusal to cervical direction of fracture seen in the fractographic analysis. Although the polymer crown had a higher fracture resistance than ceramics, a larger amount of the occlusal load was transferred through the tooth, resulting in catastrophic fracture of the tooth. This fracture behavior can limit the use of polymer crowns when compared to ceramic systems [32]. Molar crowns made of stiffer materials are less prone to debonding and crowns made of higher elastic modulus material protect the tooth structures

Veneers used in restorative rehabilitations for anterior teeth are retained by the adhesive systems and resin cements. These restorations are mechanically not strong, because they are made up of a brittle material, but they have good retention due to the resin-dentine bonding. The remaining tooth tissue is the most important factor for the longevity of the veneers where the buccal, cervical region is the most critical region. Teeth totally recover their properties when veneers are placed as a partial enamel substitute. The use of ceramic

points of greater stress were found within, or near the connector [22].

The widely used method for treatment of structurally weakened teeth is the post and core system. This system can be classified into two basic core system, metal posts and cores that are custom cast as a single piece, and two element designs composing a prefabricated post to which other materials core is subsequently adapted [34]. The difference between the elastic modulus of dentine and the post material may be a source of stress for root structures. Debonding of posts because of contraction stress of the cement was found as the most common mode of failure [27]. The effect of post design is also very important for dentinal stress distribution since the placement of a post can create stresses that lead to root fracture (figure 9) [1]. Increased intracanal stresses below the level of crestal bone would explain the higher incidence of deep root fractures in teeth restored with post-retained crowns Horizontal loads generate more dentinal stress than vertical loads. Shorter posts are associated with more dentinal stress concentration around the post apex. Consequently, extending the apical post beyond the level of alveolar bone is essential to avoid stress concentration in the region of the post apex. However, very long posts are associated with higher intracanal stress values. A higher amount of radicular dentin around the post is important in order to reduce dentinal stress concentration within the root [35]. The use of post materials conflicts with the mechanical resistance of teeth because of mismatch in the stiffness with the residual dental structure [36]. Many studies have shown that fiberglass posts give better biomechanical performance. Titanium posts concentrate stress close to the post-cement interface, promoting weakness of restored tooth. Akkayan [37] observed that the fractures occurring with the use of fiberglass and quartz posts systems could be repaired, whereas this was not the case with zirconium and titanium posts. Thus, fiberglass post can be considered a very good choice because they offer good biomechanical performance, provide excellent aesthetics, and exhibit good adhesion to cementing agents [38].

Clinicians generally agree that NiTi rotary files have good properties to produce desirable tapered root canal forms, but also have a risk of fracture during instrumentation. These instruments have been developed to overcome the rigidity of stainless steel instruments [39]. Design of an instrument is the main factor in their mechanical behavior. Cyclic fatigue, which is a failure process associated with repetitive stressing, and torsion have been reported as dominant factors in file fracture [40].

With the application of adhesive technology to endodontics, the term monoblock has become familiar. Monoblock units can be created in a root canal system either by adhesive root sealers in combination with a bondable root filling material or adhesive post systems. The concept of creating mechanically homogenous units within the root dentine is excellent in theory, but accomplishing these ideal monoblock in the canal space is challenging because bonding to dentine is compromised by volumetric changes in resin-based materials, high cavity configuration factors, debris on canal walls, and differences in regional bond strengths [27].

Finite Element Analysis in Dental Medicine 15

in the peri-impant bone interfaces. Different heights and the use of soft liners were relevant in the stress distribution to the bone adjacent to the implants. Better distribution of the stresses

Prosthesis retention remains a much debated topic in the implant literature. Clinical studies comparing cement- and screw- retained implant restorations reveal no differences in outcomes. There is evidence from laboratory and FEA studies that implants with an internal-type connection exhibit better stress distribution with off-axis loading [43]. The combined use of implants and teeth has been questioned because of the differences of mobility between the abutments. Several authors have concluded that the tooth-implant bond does not have a negative influence on the marginal bone and soft tissues, but special care must be taken in planning in order to compensate for the differences in biomechanical

The biomechanical background of orthodontic tooth movement has been explored by many authors, and orthodontic movement principally depends on stress and strain in periodontal ligament (PDL). PDL is a thin connective tissue between the root and bone and play a key role in tooth mobility [12]. Accurate FEA model creation of a tooth and PDL is possible due to the use of micro-CT. Anchorage control in orthodontic treatment is an important factor in treatments outcome. Miniscrews and miniplates are being widely used because of their small size and superiority over endosseous implants due to the fact that they can be immediately loaded. Miniplates have the same features with the plates used in maxillofacial surgery [45]. Good treatment results have been reported by using miniscrews for orthodontic anchorage in various malocclusions, but major problem is their high failure rate. Unlike dental implants, mechanical interdigitation at the cortical bone rather than osseointegration is required for the stability of miniscrews. The placement angle, the type of miniscrews, and the direction of forces significantly affect the distribution area and the amount of stress [46]. Inadequate design and non-homogenous force distribution can cause stress directly effecting on the screws and may impair screws stability. Mobile plates can irritate the surrounding tissue and may cause inflammation. The FEA study revealed that

the new miniplates are highly efficient in reducing stress on the fixation screws [45].

Fractures of the mandibular angle are the most problematic in the facial region because of the high frequency of complications and difficult surgical access to the site [47]. Infection and nonunion are commonly reported after rigid internal fixation of these fractures [48]. The stress analyses obtained from FEA modeling can provide information regarding interactions between hardware and bone during normal patient functioning. A single tension band on the superior borders provided more angle fracture stability than a single bicortical plate placed inferiorly. This results support the use of the single tension band configuration as a

The results of the finite element analysis must be interpreted with a certain amount of caution. Most of the researches modeled dental structures as isotropic and not othotropic.

will provide a more predictable osseointegration [9].

responses between the implant and the tooth [44].

less invasive fixation approach to fractures [47].

**4. General guidelines** 

**Figure 9.** 3D modeling of post and core system.

## **3.3. Dental implants and anchorage systems for tooth movements and bone fracture**

Dental implants are widely used to replace decayed teeth or to support prostheses. The failure is associated with bone loss around an implant neck. Bone loss can be activated by excessive implant loading, as by bacterial infection or trauma [41]. Mechanical stress can have positive and negative consequences for bone tissue and for maintaining osseointegration of an oral implant [9]. The prognosis for stress concentration at the bone-implant interface is of the utmost importance in dental implant research [41]. FEA has been widely used in the field of oral implantology to estimate peri-implant stress and strain [42]. The relation between implant design and load distribution at implant-bone interface is important in the search for optimal implant configuration with minimum stress peaks. Another significant factor is bone quality, in mechanical terms; this is determined by bone strength. Increase in implant length and diameter leads to reduction of stress magnitudes within the cortical bone [41]. FEA study shows that non-submerged implants showed higher stress values in the peri-implant bone than submerged ones and the use of soft liner materials considerably reduces the stress levels in the peri-impant bone interfaces. Different heights and the use of soft liners were relevant in the stress distribution to the bone adjacent to the implants. Better distribution of the stresses will provide a more predictable osseointegration [9].

Prosthesis retention remains a much debated topic in the implant literature. Clinical studies comparing cement- and screw- retained implant restorations reveal no differences in outcomes. There is evidence from laboratory and FEA studies that implants with an internal-type connection exhibit better stress distribution with off-axis loading [43]. The combined use of implants and teeth has been questioned because of the differences of mobility between the abutments. Several authors have concluded that the tooth-implant bond does not have a negative influence on the marginal bone and soft tissues, but special care must be taken in planning in order to compensate for the differences in biomechanical responses between the implant and the tooth [44].

The biomechanical background of orthodontic tooth movement has been explored by many authors, and orthodontic movement principally depends on stress and strain in periodontal ligament (PDL). PDL is a thin connective tissue between the root and bone and play a key role in tooth mobility [12]. Accurate FEA model creation of a tooth and PDL is possible due to the use of micro-CT. Anchorage control in orthodontic treatment is an important factor in treatments outcome. Miniscrews and miniplates are being widely used because of their small size and superiority over endosseous implants due to the fact that they can be immediately loaded. Miniplates have the same features with the plates used in maxillofacial surgery [45]. Good treatment results have been reported by using miniscrews for orthodontic anchorage in various malocclusions, but major problem is their high failure rate. Unlike dental implants, mechanical interdigitation at the cortical bone rather than osseointegration is required for the stability of miniscrews. The placement angle, the type of miniscrews, and the direction of forces significantly affect the distribution area and the amount of stress [46]. Inadequate design and non-homogenous force distribution can cause stress directly effecting on the screws and may impair screws stability. Mobile plates can irritate the surrounding tissue and may cause inflammation. The FEA study revealed that the new miniplates are highly efficient in reducing stress on the fixation screws [45].

Fractures of the mandibular angle are the most problematic in the facial region because of the high frequency of complications and difficult surgical access to the site [47]. Infection and nonunion are commonly reported after rigid internal fixation of these fractures [48]. The stress analyses obtained from FEA modeling can provide information regarding interactions between hardware and bone during normal patient functioning. A single tension band on the superior borders provided more angle fracture stability than a single bicortical plate placed inferiorly. This results support the use of the single tension band configuration as a less invasive fixation approach to fractures [47].

## **4. General guidelines**

14 Finite Element Analysis – New Trends and Developments

**Figure 9.** 3D modeling of post and core system.

**fracture** 

**3.3. Dental implants and anchorage systems for tooth movements and bone** 

Dental implants are widely used to replace decayed teeth or to support prostheses. The failure is associated with bone loss around an implant neck. Bone loss can be activated by excessive implant loading, as by bacterial infection or trauma [41]. Mechanical stress can have positive and negative consequences for bone tissue and for maintaining osseointegration of an oral implant [9]. The prognosis for stress concentration at the bone-implant interface is of the utmost importance in dental implant research [41]. FEA has been widely used in the field of oral implantology to estimate peri-implant stress and strain [42]. The relation between implant design and load distribution at implant-bone interface is important in the search for optimal implant configuration with minimum stress peaks. Another significant factor is bone quality, in mechanical terms; this is determined by bone strength. Increase in implant length and diameter leads to reduction of stress magnitudes within the cortical bone [41]. FEA study shows that non-submerged implants showed higher stress values in the peri-implant bone than submerged ones and the use of soft liner materials considerably reduces the stress levels

The results of the finite element analysis must be interpreted with a certain amount of caution. Most of the researches modeled dental structures as isotropic and not othotropic.

The finite element model represented a static situation at the moment of load application and not an actual clinical situation. In reality, the loading of the structure is more dynamic and cyclic. More precise measurements could be obtained if the material properties are set as anisotropic and non-homogeneous, but such setup requires much more complex mathematical calculations.

Finite Element Analysis in Dental Medicine 17

mathematical model it eliminates the need of large number of teeth. The use of more detailed 3D models is helpful in understand critical problems related to the restorative material choice and optimal application procedures. Improved computer and modeling techniques render

When the tooth is compressively loaded, displacements do not appear to be significant because of its rather large compressive yield strength. The situation is different when asymmetrical loading is considered and tensile stress occurs. The dental tissues are more resilient to compressive than tensile forces. Any occlusal contact that can create tensile stress, also creates the possibility to create a lesion in tooth structure. Most of the failures of dental materials used for tooth restorations are caused by tensile stress. Precise occlusal adjustments of teeth occlusal surfaces should be performed to prevent such events. The difference between the elastic modulus of tooth and restorative material may be a source of stress in the dental structures. If the stress exceeds the yield strength of the materials,

The FEA helps to improve preparation designs, indicates the right material or combination of materials to be used in various load and stress conditions in order to reduce material

*Department of Oral and Maxillofacial Surgery, Medical Faculty, School of Dental Medicine,* 

*Department of Restorative Dentistry and Endodontics, Medical Faculty, School of Dental Medicine,* 

[1] Silva NR, Castro CG, Santos-Filho PCF, Silva GR, Campos RE, Soares OV, Soares CJ. Influence of different post design and composition on stress distribution in maxillary

[2] Vasudeva G, Bogra P, Nikhil V, Singh V. Effect of occlusal restoration on stresses around class V restoration interface: A finite-element study. Indian J Dent Res

[3] Poiate IAVP, Vasconcellos AB, Mori M, Poiate E Jr. 2D and 3D finite element analysis of central incisor generated by computerized tomography. Computer method and

[4] Borcic J, Antonic R, Muhvic Urek M, Petricevic N, Nola-Fuchs P, Catic A , Smojver I. 3-

central incisor: Finite element analysis. Indian J Dent Res 2009;20:153-158.

D Stress Analysis in Premolar, Coll. Antropol. 31 (2007) 4: 315–319.

the FEA a very reliable and accurate approach in biomechanical applications.

fracture of the restorative materials or the tooth may occur.

and/or tooth failure in clinical practice.

*University of Rijeka, Rijeka, Croatia* 

*University of Rijeka, Rijeka, Croatia* 

**Author details** 

Josipa Borcic\*

Alen Braut

 \*

**6. References** 

2011;22:295-302.

Corresponding Author

programs in biomedicine 2011;104:292-299.

To obtain better understanding of the tooth lesions, which is important for the clinical treatment and restoration of damage, analyses of stress distribution in the oral cavity under various loading condition are highly desirable. FEA is a valuable tool for investigation of stress distribution within various types of reconstructions and prosthodontic appliances in dental medicine.

The dental profession is influenced by various sources of information, which may be considered as "evidence-based" (controlled clinical studies) and "expert opinion". A realistic approach is to identify the strengths and weaknesses of the available clinical data and combine it with clinical experience [43]. Most researchers in FEA assumed that all materials used were homogenous, isotropic and linearly elastic. However, this assumption does not reflect the exact situation. The periodontal ligament has nonlinear mechanical properties and the bone is inhomogeneous [9,35]. The 3D analysis permits high efficiency when the biomechanical behavior of the structure needs to be evaluated under different loading conditions. In the last four decades many studies have shown how the Finite Element Analysis applied to dental mechanics has become a popular numerical method to investigate the critical aspects related to stress distribution. The use of more detailed 3D models could be helpful in understanding critical problems related to the restorative material choice and optimal application procedures. Improved computer and modeling techniques render the FEA a very reliable and accurate approach in biomechanical applications [9].

The results from FEA confirm the concept that the interfaces of materials with different module of elasticity represent the weak point of restorative systems. Restorations with material having a similar elastic modulus to tooth can save and strengthen the remaining tooth structure [27]. Combining fatigue experiments with FEA may eliminate, or at least minimize, experimental limitations by correlating fatigue failure to stress instead of specific testing configuration.

## **5. Conclusions**

There are numerous ways and attempts of experimental research, but due to complexity of dental structures, composed of various tissue materials mechanically and chemically interconnected, and due to complex tooth morphology and surrounding structures, most of these attempts fail to present precise and reliable results.

The 3D analysis permits high efficiency when the biomechanical behavior of the structure should be evaluated under different loading conditions. In the biomedical fields, the FEA is an important tool since it can avoid the necessity of traditional specimens, and by using a mathematical model it eliminates the need of large number of teeth. The use of more detailed 3D models is helpful in understand critical problems related to the restorative material choice and optimal application procedures. Improved computer and modeling techniques render the FEA a very reliable and accurate approach in biomechanical applications.

When the tooth is compressively loaded, displacements do not appear to be significant because of its rather large compressive yield strength. The situation is different when asymmetrical loading is considered and tensile stress occurs. The dental tissues are more resilient to compressive than tensile forces. Any occlusal contact that can create tensile stress, also creates the possibility to create a lesion in tooth structure. Most of the failures of dental materials used for tooth restorations are caused by tensile stress. Precise occlusal adjustments of teeth occlusal surfaces should be performed to prevent such events. The difference between the elastic modulus of tooth and restorative material may be a source of stress in the dental structures. If the stress exceeds the yield strength of the materials, fracture of the restorative materials or the tooth may occur.

The FEA helps to improve preparation designs, indicates the right material or combination of materials to be used in various load and stress conditions in order to reduce material and/or tooth failure in clinical practice.

## **Author details**

Josipa Borcic\*

16 Finite Element Analysis – New Trends and Developments

mathematical calculations.

dental medicine.

applications [9].

testing configuration.

**5. Conclusions** 

The finite element model represented a static situation at the moment of load application and not an actual clinical situation. In reality, the loading of the structure is more dynamic and cyclic. More precise measurements could be obtained if the material properties are set as anisotropic and non-homogeneous, but such setup requires much more complex

To obtain better understanding of the tooth lesions, which is important for the clinical treatment and restoration of damage, analyses of stress distribution in the oral cavity under various loading condition are highly desirable. FEA is a valuable tool for investigation of stress distribution within various types of reconstructions and prosthodontic appliances in

The dental profession is influenced by various sources of information, which may be considered as "evidence-based" (controlled clinical studies) and "expert opinion". A realistic approach is to identify the strengths and weaknesses of the available clinical data and combine it with clinical experience [43]. Most researchers in FEA assumed that all materials used were homogenous, isotropic and linearly elastic. However, this assumption does not reflect the exact situation. The periodontal ligament has nonlinear mechanical properties and the bone is inhomogeneous [9,35]. The 3D analysis permits high efficiency when the biomechanical behavior of the structure needs to be evaluated under different loading conditions. In the last four decades many studies have shown how the Finite Element Analysis applied to dental mechanics has become a popular numerical method to investigate the critical aspects related to stress distribution. The use of more detailed 3D models could be helpful in understanding critical problems related to the restorative material choice and optimal application procedures. Improved computer and modeling techniques render the FEA a very reliable and accurate approach in biomechanical

The results from FEA confirm the concept that the interfaces of materials with different module of elasticity represent the weak point of restorative systems. Restorations with material having a similar elastic modulus to tooth can save and strengthen the remaining tooth structure [27]. Combining fatigue experiments with FEA may eliminate, or at least minimize, experimental limitations by correlating fatigue failure to stress instead of specific

There are numerous ways and attempts of experimental research, but due to complexity of dental structures, composed of various tissue materials mechanically and chemically interconnected, and due to complex tooth morphology and surrounding structures, most of

The 3D analysis permits high efficiency when the biomechanical behavior of the structure should be evaluated under different loading conditions. In the biomedical fields, the FEA is an important tool since it can avoid the necessity of traditional specimens, and by using a

these attempts fail to present precise and reliable results.

*Department of Oral and Maxillofacial Surgery, Medical Faculty, School of Dental Medicine, University of Rijeka, Rijeka, Croatia* 

## Alen Braut

*Department of Restorative Dentistry and Endodontics, Medical Faculty, School of Dental Medicine, University of Rijeka, Rijeka, Croatia* 

## **6. References**


<sup>\*</sup> Corresponding Author

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Finite Element Analysis in Dental Medicine 19

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[22] Rezaei SMM, Heidarifar H, Arezodar FF, Azary A, Mokhtarykhoee S. Influence of Connector Width on the Stress Distribution of Posterior Bridges under Loading. Journal

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[27] Belli S, Eraslan O, Eskitascioglu G, Karbhari V. Monoblocks in root canals: a finite elemental stress analysis study. International Endodontic Journal 2011;44:817-826. [28] Hood JA. Biomechanical of the intact, prepared and restored tooth. Some implication an adaptive finite-element approach fo the analysis of dental restorations. International

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18 Finite Element Analysis – New Trends and Developments

Dental Journal 2011;56:301-311.

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finite element analysis. J Prosthodont 2009;18:393-402.

[5] Thompson MC, Field CJ, Swain MV. The all-ceramic, inlay supported fixed partial denture. Part 2. Fixed partial denture design: a finite element analysis. Australian

[6] Ding X, Zhu XH, Liao SH, Zhang XH, Chen H. Implant-bone interfaces stress distribution in immediately loaded implants of different diameters: a three-dimensional

[7] Zienkiewicz OC, Taylor RL. The Finite Element Method. 5th ed. Oxford, England:

[8] Haiyan L, Jianying L, Zhenmin Z, Fok ASL. Fracture simulation of restored teeth using a continuum damage mechanics failure model. Dental Materials 2011;27:e125-e133. [9] Santos MBF, Silva Neto JP, Consani RLX, Mesquita MF. Three-dimensional finite element analysis of stress distribution in peri-implant bone with relined dentures and

[11] Matson MR, Lewgoy HR, Barros Filho DA, Amore R, Anido-Anido A, Alonso RCB, Carrilho MRO, Ansuate-Netto C. Finite element analysis of stress distribution in intact and porcelain veneer restored teeth. Computer Methods in Biomechanics and

[12] Hohmann A, Kober C, Young P, Dorow C, Geiger M, Boryor A, Sander FM, Sander C, Sander FG. Influence of different modeling strategies for the periodontal ligament on finite element simulation results. American Journal of Orthodontics and Dentofacial

[13] Perez-Gonzalez A, Iserte-Vilar JL, Gonzalez-Lluch C. Biomedical Engineering http://www.biomedical-engineering-online.com/content/10/1/44 (accessed 10 March

[14] Jongsma LA, Jager Ir. N, Kleverlaan CJ, Feilzer AJ. Reduced contraction stress formation obtained by a two-step cementation procedure for fiber posts. Dental

[15] Pegoretti A, Fambri L, Zappini G, Biachetti M. Finite element analysis of glass fibre

[16] Asmussen E, Peutzfeldt A, Sahafi A. Finite element analysis of stresses in endodontically treated, dowel-restored teeth. Journal of Prosthetics Dentistry

[17] Genocese K, lamberti L, pappalettere C. Finite element analysis of a new customized composite post system for endodontically treated teeth. Journal of Biomechanics

[18] Sorrentino R, Aversa R, Ferro V, Auriemma T, Zarone F, Ferrari M et al. Threedimensional finite element analysis of strain and stress distributions in endodontically treated maxillary central incisors restored with different post, core and crown materials.

[19] Gonzalez-Lluch C, Rodriguez-Cervantes PJ, Sancho-Bru JL, perez-Gonzalez A, barjau-Escribano A, Vergara-Monedero M et al. Influence of material and diameter of pre-

reinforced composite endodontic post. Biomaterials 2002;23:2667-2682.

different heights of healing caps. Journal of Oral Rehabilitation 2011;38:691-6. [10] Khera SC, Goel VK, Chen RCS, Gurusami SA. A three-dimensional element model.

	- [35] Al-Omiri MK, Rayyan MR, Abu-Hammad O. Stress analysis of endodontically treated teeth restored with post-retained crowns: A finite element analysis study. The Journal of the American Dental Association 2011;142:289-300.

**Chapter 2** 

© 2012 Gultekin et al., licensee InTech. This is an open access chapter distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

© 2012 Gultekin et al., licensee InTech. This is a paper distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

**Application of Finite Element Analysis in** 

Since Brånemark's discovery, dental implants have become the most common restorative technique for the rehabilitation of edentulism. Many factors can impact the survival of implant-supported restorations. The most important factor for determining the long-term success of osseointegration is the state of the peri-implant bone [1-3]. Ideal biomechanical conditions directly affect bone remodeling and help to maintain the integrity of non-living structures such as the implant, abutment, and superstructures (Figures 1-7). Oral dental implant interventions involving surgical and restorative procedures for the rehabilitation of various causes of edentulism are associated with several risks. In particular, mechanical and technical risks plays a major role in implant dentistry, resulting in increased rates of repairs, unnecessary costs and lost time, and even complications that may not be easily corrected (Figures 8-10) [4-7]. Therefore, the potential mechanical and technical risks of failure or associated complications need to be evaluated before undertaking such interventions, since the application of necessary precautions may improve the survival of implant-supported restorations. Consequently, the number of biomechanical studies in the field of implant

Several methods based on photoelastic, strain-gauge, and finite element analysis (FEA) based studies have been used to investigate stress in the peri-implant region and in the components of implant-supported restorations [8-11]. FEA is a numerical stress analysis technique that is widely used to assess engineering and biomechanical problems before they occur [12,13]. A finite element model is constructed by dividing solid objects into several elements that are connected at a common nodal point. Each element is assigned appropriate material properties corresponding to the properties of the object being modeled. The first step is to subdivide the complex object geometry into a suitable set of smaller 'elements' of 'finite' dimensions. When combined with the 'mesh' model of the investigated structures,

dentistry has dramatically increased in an effort to reduce failure rates.

B. Alper Gultekin, Pinar Gultekin and Serdar Yalcin

Additional information is available at the end of the chapter

**Implant Dentistry** 

http://dx.doi.org/10.5772/48339

**1. Introduction** 


## **Application of Finite Element Analysis in Implant Dentistry**

B. Alper Gultekin, Pinar Gultekin and Serdar Yalcin

Additional information is available at the end of the chapter

http://dx.doi.org/10.5772/48339

## **1. Introduction**

20 Finite Element Analysis – New Trends and Developments

2011;37:1152-1157.

Dentistry 2012;6:9-15.

Orthopedics 2011;140:e273-e280.

192.

3014.

of the American Dental Association 2011;142:289-300.

[35] Al-Omiri MK, Rayyan MR, Abu-Hammad O. Stress analysis of endodontically treated teeth restored with post-retained crowns: A finite element analysis study. The Journal

[36] Ausiello P, Franciosa P, Martorelli M, Watts D. Mechanical behaviour of post-restored

[37] Akkayan B, Gulmez T. Resistance to fracture of endodontically treated teeth restored with different post systems. Journal of Prosthetic Dentistry 2002;23:2667-2682. [38] Cooney JP, Caputo AA, Trabert KC. Retention and stress distribution of tapered-end

[39] LeeMH, Versluis A, Kim BM, Lee CJ, Hur B, Kom HC. Journal of Endodontics

[40] Sattapan B, Nervo GJ, palamara JE, Messer HH. Defects in rotary nickel-titanium files

[41] Demenko V, Linetskiy I, Nesvit K, Shevchenko A. Ulitmate masticatory force as a

[42] Geng JP, Tan KB, Liu GR. Application of finite element analysis in implant dentistry: a

[43] Lewis MB, Klineberg I. Prosthodontic considerations designed to optimize outcomes for single-tooth implants. A review of the literature. Australian Dental Journal 2011;56:181-

[44] Lanza MDS, Seraidarian PI, Jansen WC, Lanza MD. Stress analysis of a fixed implantsupported denture by the finite element method (FEM) when varying the number of

[45] Nalbantgil D, Tozlu M, Ozdemir F, Oztoprak MO, Arun T. FEM analysis of a new miniplate: stress distribution on the plate, screws and the bone. European Journal of

[46] Suzuki A, Masuda T, Takahashi I, Deguchi T, Suzuki O, Takana-Yamamoto T. Changes in stress distribution of orthodontic miniscrews and surrounding bone evaluated by 3 dimensional finite element analysis. Americam Journal of Orthodontics and Dentofacial

[47] Kimsal J, Baack B, Candelaria L, Khraishi T, Lovald S. Biomechanical analysis of mandibular angle fractures. Journal of Oral and Maxillofacial Surgery 2011;69:3010-

[48] Mathog RH, Toma V, Clayman L et al. Nonunion of the mandible: An analysis of

contributing factors. Journal of Oral and Maxillofacial Surgery 2000;59:746.

criterion in implant selection. Journal of Dental Research 2011;90:1211-1215.

review of the literature. Journal of Prosthetic Dentistry 2001;85:585-598.

upper canine teeth: A 3D FE analysis. Dental Materials 2011;27:1285-1294.

endodontic posts. Journal of Prosthetics Dentistry 1986;55:540-546.

after clinical use. Jounal of Endodontics 2000;26:161-165.

teeth used as abutments. J Appl Oral Sci 2011;19:655-661.

Since Brånemark's discovery, dental implants have become the most common restorative technique for the rehabilitation of edentulism. Many factors can impact the survival of implant-supported restorations. The most important factor for determining the long-term success of osseointegration is the state of the peri-implant bone [1-3]. Ideal biomechanical conditions directly affect bone remodeling and help to maintain the integrity of non-living structures such as the implant, abutment, and superstructures (Figures 1-7). Oral dental implant interventions involving surgical and restorative procedures for the rehabilitation of various causes of edentulism are associated with several risks. In particular, mechanical and technical risks plays a major role in implant dentistry, resulting in increased rates of repairs, unnecessary costs and lost time, and even complications that may not be easily corrected (Figures 8-10) [4-7]. Therefore, the potential mechanical and technical risks of failure or associated complications need to be evaluated before undertaking such interventions, since the application of necessary precautions may improve the survival of implant-supported restorations. Consequently, the number of biomechanical studies in the field of implant dentistry has dramatically increased in an effort to reduce failure rates.

Several methods based on photoelastic, strain-gauge, and finite element analysis (FEA) based studies have been used to investigate stress in the peri-implant region and in the components of implant-supported restorations [8-11]. FEA is a numerical stress analysis technique that is widely used to assess engineering and biomechanical problems before they occur [12,13]. A finite element model is constructed by dividing solid objects into several elements that are connected at a common nodal point. Each element is assigned appropriate material properties corresponding to the properties of the object being modeled. The first step is to subdivide the complex object geometry into a suitable set of smaller 'elements' of 'finite' dimensions. When combined with the 'mesh' model of the investigated structures,

© 2012 Gultekin et al., licensee InTech. This is an open access chapter distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. © 2012 Gultekin et al., licensee InTech. This is a paper distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

each element can adopt a specific geometric shape (i.e., triangle, square, tetrahedron, etc.) with a specific internal strain function. Using these functions and the actual geometry of the element, the equilibrium equations between the external forces acting on the element and the displacement occuring at each node can be determined [9].

Application of Finite Element Analysis in Implant Dentistry 23

**Figure 2.** After flap elevation, the cortical bone is visible

**Figure 3.** Dental implant with an abutment to be placed in the ridge created by the missing molar

**Figure 1.** Missing molar in the mandible, to be treated with a dental implant-supported restoration

In implant dentistry literature, commonly used materials in FEA studies can be classified as either implant, peri-implant bone (cortical and cancellous bone), and restoration (Figure 11). This method allows application of simulated forces at specific points in the system and stress analysis in the peri-implant region and surrounding structures. 2-D and 3-D models can be created and models for every treatment alternative can be explored. However, 2-D models cannot simulate the behavior of 3-D structures as realistically as 3-D models, so most recent studies have focused on 3-D modeling [14-17].

**Figure 2.** After flap elevation, the cortical bone is visible

the displacement occuring at each node can be determined [9].

each element can adopt a specific geometric shape (i.e., triangle, square, tetrahedron, etc.) with a specific internal strain function. Using these functions and the actual geometry of the element, the equilibrium equations between the external forces acting on the element and

**Figure 1.** Missing molar in the mandible, to be treated with a dental implant-supported restoration

recent studies have focused on 3-D modeling [14-17].

In implant dentistry literature, commonly used materials in FEA studies can be classified as either implant, peri-implant bone (cortical and cancellous bone), and restoration (Figure 11). This method allows application of simulated forces at specific points in the system and stress analysis in the peri-implant region and surrounding structures. 2-D and 3-D models can be created and models for every treatment alternative can be explored. However, 2-D models cannot simulate the behavior of 3-D structures as realistically as 3-D models, so most

**Figure 3.** Dental implant with an abutment to be placed in the ridge created by the missing molar

Application of Finite Element Analysis in Implant Dentistry 25

**Figure 6.** Abutment is prepared and attached to the implant

**Figure 7.** Porcelain-fused metal implant-supported restoration in use with optimum treatment planning

**Figure 4.** Implant is placed in the ridge

**Figure 5.** Occlusal view of the implant after 2 months of healing

**Figure 6.** Abutment is prepared and attached to the implant

**Figure 4.** Implant is placed in the ridge

**Figure 5.** Occlusal view of the implant after 2 months of healing

**Figure 7.** Porcelain-fused metal implant-supported restoration in use with optimum treatment planning

Application of Finite Element Analysis in Implant Dentistry 27

**Figure 10.** Severe bone defect is seen after implant removal; advanced bone regeneration techniques are

needed to replace the implant

**Figure 11.** Modeling of bone, implant, abutment, and restoration

**Figure 8.** Intraoral picture of a broken implant due to excessive loading after 1 year of use

**Figure 9.** Severe bone resorption after 2 years of loading; implant and superstructure have no mechanical failure, but peri-implant bone could not resist excessive loading (biomechanical failure because of improper occlusal adjustment)

Application of Finite Element Analysis in Implant Dentistry 27

26 Finite Element Analysis – New Trends and Developments

**Figure 8.** Intraoral picture of a broken implant due to excessive loading after 1 year of use

**Figure 9.** Severe bone resorption after 2 years of loading; implant and superstructure have no mechanical failure, but peri-implant bone could not resist excessive loading (biomechanical failure

because of improper occlusal adjustment)

**Figure 10.** Severe bone defect is seen after implant removal; advanced bone regeneration techniques are needed to replace the implant

**Figure 11.** Modeling of bone, implant, abutment, and restoration

## **2. Modelization of living structure (bone)**

To improve the quality of FEA research, strict attention should be paid to the modelization procedure as one of the most important part of FEA studies. The features of the model should resemble the physical properties of the actual structure as closely as possible, with respect to dimension and material properties. The most difficult and complex part of the modelization process involves capturing the detailed properties of living structures. Therefore, in general, specifications drawn from chapters of a detailed anatomy book or from tomographic scans of a jaw from a cadaveric human specimen can be used for the modeling procedure. Volumetric data obtained from tomography devices or magnetic resonance imaging are digitally reconstructed [18,19]. Then, the material properties applied to the elements can be varied according to the modeling requirements of a particular situation. Computed tomography offers another advantage for realistic modeling in not only the development of anatomic structures, but also the inclusion of material properties according to different bone density values [20,21]. In some studies, the bone is totally or partially modeled as a simple rectangle, elipsoid, or U-shape [18]. In detailed studies, especially with data obtained from scanners, bone can be modeled in a very realistical manner; however, this increased level of geometric detail will result in increased working and computing time. According to the treatment alternatives being investigated, cortical bone can be layered in milimeters or can be neglected altogether in order to simulate weak bone properties similar to those found in the posterior maxilla (Figure 12). Bone properties related to density can be calibrated to range from very soft to dense bone, according to the individual research protocol. If only a specific area and/or condition of the mandible or maxilla is being investigated, there is no need to visualize and construct a model of entire jaw. Limiting the scope or features of the model will distinctly decrease the working time and costs, as previously discussed. A region of interest can be extracted using a number of techniques, such as a Boolean process (Figures 13-15), and any implant design can be adopted for the study. Regions of interest may change according to the study protocol. Portions of the mandible or maxilla, maxillary sinus region, and temporomandibular joint are the most common anatomical areas used in studies related to implantology. In the existing literature 2-D FEA bone models are generally simplified as a rectangular shape [14]. However, recent studies have used 3-D bone modeling to better represent the realistic anatomy of these complex structures [22-24].

Application of Finite Element Analysis in Implant Dentistry 29

element model (FEM). In the study mentioned above, sections were digitized into the DICOM 3.0 format and visualized using 3-D Doctor software (Able Software Corp., Lexington, MA, USA). Cortical bone of 2 mm uniform thickness, and cancellous bone were also modeled (Figure 17). In this study, cortical and cancellous bone model components were considered homogenous. However, in fact, cancellous bone in particular has widely variable density properties. The non-uniform nature of the density of this anatomic structure may affect the magnitude and distribution of stress concentration after loading. These simplifications are common in studies that employ FEA and are aimed at limiting the computing difficulties associated with performance of these studies [18,26,27]. To develop more realistic models of living structures, future studies may include variable density properties obtained from bone density values measured in Hounsfield Units or from other advanced data obtained from computed tomography scans performed with individual

**Figure 12.** Cortical thickness of the posterior maxilla is neglected; only cancellous bone properties are

patients (Figure 18) [28-30].

modeled

In a previous FEA study, the human mandible model was based on a cadaveric mandible obtained from the anatomy department [25]. The edentulous cadaver mandible was scanned using a dental volumetric computed tomography device (ILUMA, Orthocad, CBCT scanner, 3M ESPE, St. Paul, MN, USA) (Figure 16). Volumetric data were reconstructed in 0.2 mm thick sections. The mandibular height and width were at least 10 mm and 5 mm, respectively. More detailed anatomic representations could be created in future studies through the use of computed tomography scanners that can slice objects into thinner sections, but this may increase the working time and development cost of the final finite element model (FEM). In the study mentioned above, sections were digitized into the DICOM 3.0 format and visualized using 3-D Doctor software (Able Software Corp., Lexington, MA, USA). Cortical bone of 2 mm uniform thickness, and cancellous bone were also modeled (Figure 17). In this study, cortical and cancellous bone model components were considered homogenous. However, in fact, cancellous bone in particular has widely variable density properties. The non-uniform nature of the density of this anatomic structure may affect the magnitude and distribution of stress concentration after loading. These simplifications are common in studies that employ FEA and are aimed at limiting the computing difficulties associated with performance of these studies [18,26,27]. To develop more realistic models of living structures, future studies may include variable density properties obtained from bone density values measured in Hounsfield Units or from other advanced data obtained from computed tomography scans performed with individual patients (Figure 18) [28-30].

28 Finite Element Analysis – New Trends and Developments

anatomy of these complex structures [22-24].

**2. Modelization of living structure (bone)** 

To improve the quality of FEA research, strict attention should be paid to the modelization procedure as one of the most important part of FEA studies. The features of the model should resemble the physical properties of the actual structure as closely as possible, with respect to dimension and material properties. The most difficult and complex part of the modelization process involves capturing the detailed properties of living structures. Therefore, in general, specifications drawn from chapters of a detailed anatomy book or from tomographic scans of a jaw from a cadaveric human specimen can be used for the modeling procedure. Volumetric data obtained from tomography devices or magnetic resonance imaging are digitally reconstructed [18,19]. Then, the material properties applied to the elements can be varied according to the modeling requirements of a particular situation. Computed tomography offers another advantage for realistic modeling in not only the development of anatomic structures, but also the inclusion of material properties according to different bone density values [20,21]. In some studies, the bone is totally or partially modeled as a simple rectangle, elipsoid, or U-shape [18]. In detailed studies, especially with data obtained from scanners, bone can be modeled in a very realistical manner; however, this increased level of geometric detail will result in increased working and computing time. According to the treatment alternatives being investigated, cortical bone can be layered in milimeters or can be neglected altogether in order to simulate weak bone properties similar to those found in the posterior maxilla (Figure 12). Bone properties related to density can be calibrated to range from very soft to dense bone, according to the individual research protocol. If only a specific area and/or condition of the mandible or maxilla is being investigated, there is no need to visualize and construct a model of entire jaw. Limiting the scope or features of the model will distinctly decrease the working time and costs, as previously discussed. A region of interest can be extracted using a number of techniques, such as a Boolean process (Figures 13-15), and any implant design can be adopted for the study. Regions of interest may change according to the study protocol. Portions of the mandible or maxilla, maxillary sinus region, and temporomandibular joint are the most common anatomical areas used in studies related to implantology. In the existing literature 2-D FEA bone models are generally simplified as a rectangular shape [14]. However, recent studies have used 3-D bone modeling to better represent the realistic

In a previous FEA study, the human mandible model was based on a cadaveric mandible obtained from the anatomy department [25]. The edentulous cadaver mandible was scanned using a dental volumetric computed tomography device (ILUMA, Orthocad, CBCT scanner, 3M ESPE, St. Paul, MN, USA) (Figure 16). Volumetric data were reconstructed in 0.2 mm thick sections. The mandibular height and width were at least 10 mm and 5 mm, respectively. More detailed anatomic representations could be created in future studies through the use of computed tomography scanners that can slice objects into thinner sections, but this may increase the working time and development cost of the final finite

**Figure 12.** Cortical thickness of the posterior maxilla is neglected; only cancellous bone properties are modeled

Application of Finite Element Analysis in Implant Dentistry 31

**Figure 15.** Part of the mandible modeled with superstructure, implant, and surrounding bone

**Figure 16.** The edentulous mandible obtained from a cadaver was scanned using a dental volumetric

tomography device

**Figure 13.** Mandible is modeled and region of interest is selected

**Figure 14.** Region of interest is extracted by Boolean process

Application of Finite Element Analysis in Implant Dentistry 31

30 Finite Element Analysis – New Trends and Developments

**Figure 13.** Mandible is modeled and region of interest is selected

**Figure 14.** Region of interest is extracted by Boolean process

**Figure 15.** Part of the mandible modeled with superstructure, implant, and surrounding bone

**Figure 16.** The edentulous mandible obtained from a cadaver was scanned using a dental volumetric tomography device

Application of Finite Element Analysis in Implant Dentistry 33

**3. Modelization of non-living structure (materials)** 

the Poisson effect [9,13,32,33].

with the goal of the modeling exercise.

Non-living mechanical structures such as implants, abutments, and restorations can be simulated in detail and can substantially influence the calculated stress and strain values, similar to living structures. These materials can be digitally modeled in FEA studies using previously determined isotropic, transversely isotropic, orthotropic, and/or anisotropic properties [31]. In an isotropic material, the relevant material properties are the same in all directions, resulting in only 2 independent material constants, such as Young's modulus and Poisson's ratio [9,13,31]. Young's modulus (MPa), also known as the tensile modulus, is a quantity used to characterize materials and is a measure of the stiffness of an elastic material. Young's modulus is also called the elastic modulus or modulus of elasticity, because Young's modulus is the most commonly used elastic modulus [9,13,32,33]. When a sample object is stretched, Poisson's ratio is the ratio of the contraction or transverse strain (perpendicular to the applied load), to the extension or axial strain (in the direction of the applied load). When a material is compressed in 1 direction, it tends to expand in the other 2 directions perpendicular to the direction of compression. This phenomenon is called the Poisson effect. Poisson's ratio is a measure of

An anisotropic material has material properties that vary by direction [31]. Isotropic material properties are used in most FEA studies related to implant dentistry [18,25,34]. For instance, the material properties of living bone are anisotropic, and inhomogeneous. These properties of real bone greatly affect stress and strain patterns. In addition, bone density may differ among various regions of the same jaw and areas of differing densities may only be separated by milimeters. For simplification and to overcome computing difficulties, in most cases, the materials are modeled as homogenous, isotropic, and linearly elastic [35-39]. However, some studies have modeled the bone block using anisotropic properties (i.e., the material properties differ with respect to direction) [26]. The material properties of both living and non-living structures are chosen in accordance

In some studies, implants are modeled using a screw design but without threads (Figure 19). This may simplify the computing process, but does not reflect the reality of implant geometry. If one or more study parameters are related to implant dimensions, there is little doubt that inclusion of implant threads in the model is quite important to the quality of the research. Most clinicians are interested in the magnitude and distribution of stress that may induce microdamage to the bone and result in crestal bone resorption; therefore, macro and micro threads are crucial in the modeling stage of an implant study. The implant thread design influences the induced bone stress around the implant, which contributes to crestal bone loss, and can jeopardize the maintenance of osseointegration [40-43]. In recent FEA studies, implant threads are modeled in detail (Figure 20,21). There are 2 ways to model implant and abutment materials. One way is to obtain all of the geometric information (e.g., length, diameter, macro-micro thread configuration) in

**Figure 17.** Volumetric data were reconstructed in 0.2 mm thick sections

**Figure 18.** Bone density values can be measured according to gray scale using advanced 3-D radiographic techniques

## **3. Modelization of non-living structure (materials)**

32 Finite Element Analysis – New Trends and Developments

**Figure 17.** Volumetric data were reconstructed in 0.2 mm thick sections

**Figure 18.** Bone density values can be measured according to gray scale using advanced 3-D

radiographic techniques

Non-living mechanical structures such as implants, abutments, and restorations can be simulated in detail and can substantially influence the calculated stress and strain values, similar to living structures. These materials can be digitally modeled in FEA studies using previously determined isotropic, transversely isotropic, orthotropic, and/or anisotropic properties [31]. In an isotropic material, the relevant material properties are the same in all directions, resulting in only 2 independent material constants, such as Young's modulus and Poisson's ratio [9,13,31]. Young's modulus (MPa), also known as the tensile modulus, is a quantity used to characterize materials and is a measure of the stiffness of an elastic material. Young's modulus is also called the elastic modulus or modulus of elasticity, because Young's modulus is the most commonly used elastic modulus [9,13,32,33]. When a sample object is stretched, Poisson's ratio is the ratio of the contraction or transverse strain (perpendicular to the applied load), to the extension or axial strain (in the direction of the applied load). When a material is compressed in 1 direction, it tends to expand in the other 2 directions perpendicular to the direction of compression. This phenomenon is called the Poisson effect. Poisson's ratio is a measure of the Poisson effect [9,13,32,33].

An anisotropic material has material properties that vary by direction [31]. Isotropic material properties are used in most FEA studies related to implant dentistry [18,25,34]. For instance, the material properties of living bone are anisotropic, and inhomogeneous. These properties of real bone greatly affect stress and strain patterns. In addition, bone density may differ among various regions of the same jaw and areas of differing densities may only be separated by milimeters. For simplification and to overcome computing difficulties, in most cases, the materials are modeled as homogenous, isotropic, and linearly elastic [35-39]. However, some studies have modeled the bone block using anisotropic properties (i.e., the material properties differ with respect to direction) [26]. The material properties of both living and non-living structures are chosen in accordance with the goal of the modeling exercise.

In some studies, implants are modeled using a screw design but without threads (Figure 19). This may simplify the computing process, but does not reflect the reality of implant geometry. If one or more study parameters are related to implant dimensions, there is little doubt that inclusion of implant threads in the model is quite important to the quality of the research. Most clinicians are interested in the magnitude and distribution of stress that may induce microdamage to the bone and result in crestal bone resorption; therefore, macro and micro threads are crucial in the modeling stage of an implant study. The implant thread design influences the induced bone stress around the implant, which contributes to crestal bone loss, and can jeopardize the maintenance of osseointegration [40-43]. In recent FEA studies, implant threads are modeled in detail (Figure 20,21). There are 2 ways to model implant and abutment materials. One way is to obtain all of the geometric information (e.g., length, diameter, macro-micro thread configuration) in

milimeters from the manufacturer. The second option is to scan implants and abutment materials and digitally reconstruct them. Efficient and realistic models can be obtained by using either option. In general, for the digital preparation of crown models, an anatomy atlas of the tooth can be used as a reference to calculate the form and both mesiodistal and buccolingual dimensions [44]. The prosthetic superstructure can be simulated according to various treatment protocols. Superstructure can also be modeled as a geometric figure, such as a simple rectangular shape, but this may interfere with the realism of the model (Figure 22).

Application of Finite Element Analysis in Implant Dentistry 35

bonded [9,18,25,31,34]. Implant, abutment, abutment screw, framework, and porcelain structures are considered to be a single unit (Figure 24). In contrast, there are some studies that use a contact condition between the abutment and implant set as a frictional coefficient [26]. In these studies, the corresponding material properties are used and modeled separately. Most studies also model the implant as rigidly anchored in the bone model along its entire interface and with total osseointegration. It is impossible to visualize these interface conditions in real life, but simplifications in interface conditions will inevitably result in considerable inaccuracy. The most common drawback of FEA from the clinical perspective is that many features that directly affect model accuracy, such as loading conditions, material properties, and interface conditions are neglected or ignored. In most cases, researchers neglect one or more features in their studies. Moreover, bias may result from interpretation of data obtained from an FEA study to that obtained from another. Within a single study, these simplifications are consistent for all the simulated models; therefore, the accuracy of the analysis from the stress distribution viewpoint is not affected, as long as the models are compared with each other in the same

study [9,18,25,31,34].

**Figure 20.** Implants are modeled with micro and macro threads

**Figure 19.** Implants are modeled without threads

In a previous study, the crown model was simulated as porcelain fused to metal restoration. To calculate the mesiodistal width of the second premolar and first molar, Wheeler's Atlas of Anatomical Natural Tooth Morphology was used (Figure 23) [44]. The atlas was used again for digital preparation of the crown models. Properties of chromium-cobalt alloy were used for the framework and feldspatic porcelain as used to simulate the second premolar and the first molar of a mandibular model. The metal thickness of the framework was 0.8 mm and the porcelain thickness was at least 2.0 mm. The thickness of porcelain changes with the creation of pits and trabeculae of the tooth surface. In most FEA studies, not only the cement thickness but also the interface between the materials is assumed to be 100% bonded [9,18,25,31,34]. Implant, abutment, abutment screw, framework, and porcelain structures are considered to be a single unit (Figure 24). In contrast, there are some studies that use a contact condition between the abutment and implant set as a frictional coefficient [26]. In these studies, the corresponding material properties are used and modeled separately. Most studies also model the implant as rigidly anchored in the bone model along its entire interface and with total osseointegration. It is impossible to visualize these interface conditions in real life, but simplifications in interface conditions will inevitably result in considerable inaccuracy. The most common drawback of FEA from the clinical perspective is that many features that directly affect model accuracy, such as loading conditions, material properties, and interface conditions are neglected or ignored. In most cases, researchers neglect one or more features in their studies. Moreover, bias may result from interpretation of data obtained from an FEA study to that obtained from another. Within a single study, these simplifications are consistent for all the simulated models; therefore, the accuracy of the analysis from the stress distribution viewpoint is not affected, as long as the models are compared with each other in the same study [9,18,25,31,34].

34 Finite Element Analysis – New Trends and Developments

**Figure 19.** Implants are modeled without threads

(Figure 22).

milimeters from the manufacturer. The second option is to scan implants and abutment materials and digitally reconstruct them. Efficient and realistic models can be obtained by using either option. In general, for the digital preparation of crown models, an anatomy atlas of the tooth can be used as a reference to calculate the form and both mesiodistal and buccolingual dimensions [44]. The prosthetic superstructure can be simulated according to various treatment protocols. Superstructure can also be modeled as a geometric figure, such as a simple rectangular shape, but this may interfere with the realism of the model

In a previous study, the crown model was simulated as porcelain fused to metal restoration. To calculate the mesiodistal width of the second premolar and first molar, Wheeler's Atlas of Anatomical Natural Tooth Morphology was used (Figure 23) [44]. The atlas was used again for digital preparation of the crown models. Properties of chromium-cobalt alloy were used for the framework and feldspatic porcelain as used to simulate the second premolar and the first molar of a mandibular model. The metal thickness of the framework was 0.8 mm and the porcelain thickness was at least 2.0 mm. The thickness of porcelain changes with the creation of pits and trabeculae of the tooth surface. In most FEA studies, not only the cement thickness but also the interface between the materials is assumed to be 100%

**Figure 20.** Implants are modeled with micro and macro threads

Application of Finite Element Analysis in Implant Dentistry 37

**Figure 23.** Digital preparation of crown models

**Figure 24.** Implant, abutment, abutment screw, framework, and porcelain structure are modeled as 1 unit

**Figure 21.** Implants are modeled with threads and abutments

**Figure 22.** Superstructure modeled into a rectangular shape

Application of Finite Element Analysis in Implant Dentistry 37

**Figure 23.** Digital preparation of crown models

36 Finite Element Analysis – New Trends and Developments

**Figure 21.** Implants are modeled with threads and abutments

**Figure 22.** Superstructure modeled into a rectangular shape

**Figure 24.** Implant, abutment, abutment screw, framework, and porcelain structure are modeled as 1 unit

Almost all of the elastic properties of selected living and non-living materials are available in the literature [9,25,31,34]. Young's modulus and Poisson's ratio are used in models to simulate reality as closely as possible. For example, alveolar bone (both cortical and cancellous portions), implant, abutment, metal framework, and porcelain can be included in the model properties.

Application of Finite Element Analysis in Implant Dentistry 39

between marginal bone loss and occlusal forces; including the engineering principles, biomechanical relationships to living tissues, and the mechanical properties of bone surrounding implants [50]. In recent years, a greater amount of materials used for oral implantology are fabricated from titanium and titanium alloy. The Young's modulus of titanium is 5-10 times greater than that of cortical ridge bone surrounding implants [51]. The fundamental engineering principle, composite beam analysis, expresses the concept that when 2 materials of different Young's modulus are placed in direct contact with no intervening material and 1 material loaded, a stress contour will be described at the point where the 2 materials come into contact [52]. For oral implantology, these stress contours are of greater magnitude at the crestal bone. Therefore, the loading condition is another important part of FEA studies. Each component modelization stage contributes to the final analysis after loading. In other words, from the beginning to the end, all procedures and FEA stages add to the ability to extrapolate the results of bite forces surrounding the peri-

Bite forces may be defined as compressive, tensile, or shear forces. Compressive forces attempt to push materials toward each other. Tensile forces pull objects apart. Shear forces on implants cause sliding. The most detrimental forces that can increase the stress around the implant-bone interface and prosthetic assembly are tensile and shear forces. These forces tend to harm material integrity and cause stress build-up. In general, the implant-prosthetic unit can adapt to compressive forces [51]. In actual mastication, the repeated pattern of cyclic forces transmits loading via the restoration and dental implants to peri-impant bone. This generates different amounts of stress around the ridge and also in the prosthetic structure. However, randomized cyclic forces are not easily simulated. Therefore, most FEA studies use static axial and/or non-axial forces. Non-axial loads generate distinctive stress in the ridge especially in the cortical bone. The main remodeling differences between axial and non-axial loading are affected mostly by the horizontal component of the resultant stresses [53]. Therefore, for realistic simulation, combined oblique loads (axial and non-axial) are generally used. One study, comparing dynamic with static loading, revealed that dynamic loading resulted in greater stress levels than static loading [54]. Dynamic loading has consistently been found to have more osteogenic potential than static loading [55]. Sagat et al. investigated the influence of static force on peri-implant stress. In varied models, 100 N static forces were applied vertically and separately to the anterior and posterior parts of a bridge [18]. In another study, static forces of 100 N were applied at 30 degrees obliquely and separately to the lingual inclination of the buccal cusps of a crown (Figures 25,26) [25]. In another study, loading was simulated by applying an oblique load (vertical load of 100 N and horizontal load of 20 N) from buccal to palatal region at 4 different locations. An equivalent load of 200 N was applied in the vertical direction and 40 N in the buccal-palatal direction. The application point of the force was on the central and distal fossae of the crown [48]. Eskitascioglu et al. used an average occlusal force of 300 N applied to a missing second premolar implant-supported crown. Three-point vertical loads were applied to the tip of the buccal cusp (150 N) and distal fossa (150 N); the tip of the buccal cusp (100 N), distal fossa

implant region and prosthetic structures.

(100 N), and mesial fossa (100 N) [56].

## **4. Boundary conditions**

A boundary condition is the application of force and constraint. The different ways to apply force and moment include a concentrated load (at a point or single node), force on a line or edge, a distributed load (force varying as an equation), bending moments, and torque [45]. In structural analysis, boundary conditions are applied to those regions of the model where the displacements and/or rotations are known. Such regions may be constrained to remain fixed (have zero displacement and/or rotation) during the simulation or may have specified, non-zero displacements and/or rotations. The directions in which motion is possible are called degrees of freedom (DOF). Zero-displacement constraints must be placed on some boundaries of the model to ensure an equilibrium solution. The constraints should be placed on nodes that are located far from the region of interest to prevent overlap of the stress or strain fields associated with reaction forces with the bone-implant interface. In maxillary FEA models, the nodes along the external lines of the cortical bone of the oral and nasopharyngeal cavities were fixed in all directions [46].

In most FEA studies that include models of the mandible, the boundary conditions are set as a fixed boundary [9]. Zhou et al. developed a more realistic 3-D mandibular FEA model from transversely scanned computed tomography imaging data. The functions of the muscles of mastication and the ligamentous and functional movements of the temporomandibular joints (TMJs) were simulated by means of cable elements and compressive gap elements, respectively. Using this mandibular FEA model, it was concluded that cable and gap elements could be used to set boundary conditions, improving the model mimicry and accuracy [47]. Chang et al. used a technique in which only half of the model was meshed, thus symmetry boundary conditions were prescribed at the nodes on the symmetry plane. Models were constrained in all directions at the nodes on the mesial and symmetrical distal bone surfaces [48]. Expanding the domain of the model can reduce the influence of inaccurate modeling of the boundary conditions. This, however, will be at the expense of computing and modeling time. Teixera et al. concluded that in a 3-D mandibular model, modeling the mandible at distances greater than 4.2 mm mesial or distal from the implant did not result in any significant increase in FEA accuracy [49]. Use of infinite elements is another potential method for modeling boundary conditions [9].

## **5. Loading conditions**

Marginal bone loss in the peri-implant region may be the result of excessive occlusal force [50]. Extensive investigations are needed to establish and understand the correlation between marginal bone loss and occlusal forces; including the engineering principles, biomechanical relationships to living tissues, and the mechanical properties of bone surrounding implants [50]. In recent years, a greater amount of materials used for oral implantology are fabricated from titanium and titanium alloy. The Young's modulus of titanium is 5-10 times greater than that of cortical ridge bone surrounding implants [51]. The fundamental engineering principle, composite beam analysis, expresses the concept that when 2 materials of different Young's modulus are placed in direct contact with no intervening material and 1 material loaded, a stress contour will be described at the point where the 2 materials come into contact [52]. For oral implantology, these stress contours are of greater magnitude at the crestal bone. Therefore, the loading condition is another important part of FEA studies. Each component modelization stage contributes to the final analysis after loading. In other words, from the beginning to the end, all procedures and FEA stages add to the ability to extrapolate the results of bite forces surrounding the periimplant region and prosthetic structures.

38 Finite Element Analysis – New Trends and Developments

nasopharyngeal cavities were fixed in all directions [46].

the model properties.

**4. Boundary conditions** 

**5. Loading conditions** 

Almost all of the elastic properties of selected living and non-living materials are available in the literature [9,25,31,34]. Young's modulus and Poisson's ratio are used in models to simulate reality as closely as possible. For example, alveolar bone (both cortical and cancellous portions), implant, abutment, metal framework, and porcelain can be included in

A boundary condition is the application of force and constraint. The different ways to apply force and moment include a concentrated load (at a point or single node), force on a line or edge, a distributed load (force varying as an equation), bending moments, and torque [45]. In structural analysis, boundary conditions are applied to those regions of the model where the displacements and/or rotations are known. Such regions may be constrained to remain fixed (have zero displacement and/or rotation) during the simulation or may have specified, non-zero displacements and/or rotations. The directions in which motion is possible are called degrees of freedom (DOF). Zero-displacement constraints must be placed on some boundaries of the model to ensure an equilibrium solution. The constraints should be placed on nodes that are located far from the region of interest to prevent overlap of the stress or strain fields associated with reaction forces with the bone-implant interface. In maxillary FEA models, the nodes along the external lines of the cortical bone of the oral and

In most FEA studies that include models of the mandible, the boundary conditions are set as a fixed boundary [9]. Zhou et al. developed a more realistic 3-D mandibular FEA model from transversely scanned computed tomography imaging data. The functions of the muscles of mastication and the ligamentous and functional movements of the temporomandibular joints (TMJs) were simulated by means of cable elements and compressive gap elements, respectively. Using this mandibular FEA model, it was concluded that cable and gap elements could be used to set boundary conditions, improving the model mimicry and accuracy [47]. Chang et al. used a technique in which only half of the model was meshed, thus symmetry boundary conditions were prescribed at the nodes on the symmetry plane. Models were constrained in all directions at the nodes on the mesial and symmetrical distal bone surfaces [48]. Expanding the domain of the model can reduce the influence of inaccurate modeling of the boundary conditions. This, however, will be at the expense of computing and modeling time. Teixera et al. concluded that in a 3-D mandibular model, modeling the mandible at distances greater than 4.2 mm mesial or distal from the implant did not result in any significant increase in FEA accuracy [49]. Use of

infinite elements is another potential method for modeling boundary conditions [9].

Marginal bone loss in the peri-implant region may be the result of excessive occlusal force [50]. Extensive investigations are needed to establish and understand the correlation Bite forces may be defined as compressive, tensile, or shear forces. Compressive forces attempt to push materials toward each other. Tensile forces pull objects apart. Shear forces on implants cause sliding. The most detrimental forces that can increase the stress around the implant-bone interface and prosthetic assembly are tensile and shear forces. These forces tend to harm material integrity and cause stress build-up. In general, the implant-prosthetic unit can adapt to compressive forces [51]. In actual mastication, the repeated pattern of cyclic forces transmits loading via the restoration and dental implants to peri-impant bone. This generates different amounts of stress around the ridge and also in the prosthetic structure. However, randomized cyclic forces are not easily simulated. Therefore, most FEA studies use static axial and/or non-axial forces. Non-axial loads generate distinctive stress in the ridge especially in the cortical bone. The main remodeling differences between axial and non-axial loading are affected mostly by the horizontal component of the resultant stresses [53]. Therefore, for realistic simulation, combined oblique loads (axial and non-axial) are generally used. One study, comparing dynamic with static loading, revealed that dynamic loading resulted in greater stress levels than static loading [54]. Dynamic loading has consistently been found to have more osteogenic potential than static loading [55]. Sagat et al. investigated the influence of static force on peri-implant stress. In varied models, 100 N static forces were applied vertically and separately to the anterior and posterior parts of a bridge [18]. In another study, static forces of 100 N were applied at 30 degrees obliquely and separately to the lingual inclination of the buccal cusps of a crown (Figures 25,26) [25]. In another study, loading was simulated by applying an oblique load (vertical load of 100 N and horizontal load of 20 N) from buccal to palatal region at 4 different locations. An equivalent load of 200 N was applied in the vertical direction and 40 N in the buccal-palatal direction. The application point of the force was on the central and distal fossae of the crown [48]. Eskitascioglu et al. used an average occlusal force of 300 N applied to a missing second premolar implant-supported crown. Three-point vertical loads were applied to the tip of the buccal cusp (150 N) and distal fossa (150 N); the tip of the buccal cusp (100 N), distal fossa (100 N), and mesial fossa (100 N) [56].

Application of Finite Element Analysis in Implant Dentistry 41

**Figure 26.** Force application to the region of restoration

The 'osseointegration' concept was described as the direct contact between living bone and a loaded dental implant surface by Brånemark et al. [58]. The most widely used material for dental implant manufacture is pure titanium (Grade 4), titanium alloy (Grade 5), and rarely zirconia [59-62]. These materials have good biocompatibility with surrounding tissues, are resistant to deformation, and are easily manipulated for shaping as a natural tooth root forms by Computer Numerical Control (CNC) machines [59-62]. Titanium alloy has mechanical advantages over pure titanium in implant manufacture. With increases in grade number, the alloy becomes much stronger and more resistant to fractures or wearing of the components [59-62]. However biocompatability may be reduced in inverse proportion the increase in grade number. Implant companies use Grade 4 or Grade 5 titanium for the implant body and generally choose Grade 5 titanium for implant abutment manufacture. Recently, to increase the strength of implant bodies, new materials have also been introduced into the market, such as roxolid (a zirconium and titanium combination) [63]. The use of zirconium and titanium combination material as an implant body has limited

**6. Bone-implant interface** 

**Figure 25.** Static forces were applied at 30 degrees obliquely and separately to the lingual inclination of the buccal cusps of the crown

As mentioned before, oblique loads are more destructive to the peri-implant bone region and clinically disruptive to prosthetic structures. The magnitude of bite force may change according to age, sex, edentulism, parafunctional habits, and may differ from anterior to posterior in the same mouth [9,31]. In FEA literature, the locations for the application of bite force change according to the modeling of the restoration [9,31]. In advanced modeling studies, more realistic force application could be described including ridges of the cusp, labial or lingual surfaces of crown, occlusal surface, distal, and mesial fossa [9,27,31,57]. For realistic simulation of biting, loading forces should be applied to the restoration first, and then transmitted by the abutment to the implant and surrounding bone. Stress concentrations will then be generated, evaluated, and proper risk assessment will be considered.

**Figure 26.** Force application to the region of restoration

## **6. Bone-implant interface**

40 Finite Element Analysis – New Trends and Developments

the buccal cusps of the crown

considered.

**Figure 25.** Static forces were applied at 30 degrees obliquely and separately to the lingual inclination of

As mentioned before, oblique loads are more destructive to the peri-implant bone region and clinically disruptive to prosthetic structures. The magnitude of bite force may change according to age, sex, edentulism, parafunctional habits, and may differ from anterior to posterior in the same mouth [9,31]. In FEA literature, the locations for the application of bite force change according to the modeling of the restoration [9,31]. In advanced modeling studies, more realistic force application could be described including ridges of the cusp, labial or lingual surfaces of crown, occlusal surface, distal, and mesial fossa [9,27,31,57]. For realistic simulation of biting, loading forces should be applied to the restoration first, and then transmitted by the abutment to the implant and surrounding bone. Stress concentrations will then be generated, evaluated, and proper risk assessment will be The 'osseointegration' concept was described as the direct contact between living bone and a loaded dental implant surface by Brånemark et al. [58]. The most widely used material for dental implant manufacture is pure titanium (Grade 4), titanium alloy (Grade 5), and rarely zirconia [59-62]. These materials have good biocompatibility with surrounding tissues, are resistant to deformation, and are easily manipulated for shaping as a natural tooth root forms by Computer Numerical Control (CNC) machines [59-62]. Titanium alloy has mechanical advantages over pure titanium in implant manufacture. With increases in grade number, the alloy becomes much stronger and more resistant to fractures or wearing of the components [59-62]. However biocompatability may be reduced in inverse proportion the increase in grade number. Implant companies use Grade 4 or Grade 5 titanium for the implant body and generally choose Grade 5 titanium for implant abutment manufacture. Recently, to increase the strength of implant bodies, new materials have also been introduced into the market, such as roxolid (a zirconium and titanium combination) [63]. The use of zirconium and titanium combination material as an implant body has limited

scientific data and requires long-term investigations. Therefore, most FEA studies in the literature involve titanium and titanium alloys [9,18,24,31].

Application of Finite Element Analysis in Implant Dentistry 43

Newton (force) per square meter (unit area), that is N/m2. Stress is often reported in scientific publications as MPa. Stress is directly proportional to the force and inversely proportional to the area across which the force is applied. It is important to determine the area across which any force is applied. For example, the surface area of the occlusal pit restoration less than 4 mm. For this reason, the magnitude of stress in many restorations reaches hundreds of MPa

When the force is applied to mass, a deformation occurs as a result of this force. A strain is a normalized measure of deformation representing the displacement between particles in the body relative to a reference length [9,16,18,23,25,51,86,87,88]. There is no measurement unit

In FEA studies related to implant dentistry, frequently von Mises stress (equivalent tensile stress), minimum principal, and maximum principal are used to evaluate the effect of loading forces on the peri-implant region or prosthesis structure [9,16,18,23,25,89]. When a specific force is applied to the body, von Mises stress is the criterion used to determine the strain energy principles. Loading forces affecting the object can be evaluated 2 or 3 dimensionally. There are 3 "Principal Stresses" that can be calculated at any point, acting in the x, y, and z directions. The von Mises criteria refer to a formula for combining these 3 stresses into an equivalent stress, which is then compared to the yield stress of the material [25,90]. The major stress values are formed when all the components of the shear are zero. When an element is in this position, the normal stresses are called principal stresses. Principal stresses are classified as maximum, intermediate, and minimum principal stresses. The maximum principal stress is a positive value indicating the highest tension. The intermediate principal stress represents intermediate values. The minimum principal stress is a negative value indicating the highest compression [9,16,18,23,25,89]. If the data obtained from the analysis are positive values, then they are considered tensile stresses, negative

Frequently, different color figures are used according to the amount of stress around periimplant regions and prosthetic structures (Figure 27). Stresses on each model are evaluated according to the stress values from low to high. In other words, the most favorable model has the lowest stress values, and in contrast, the most deleterious model has the highest

**Figure 27.** Different colors indicate the amount of stress around the peri-implant region and prosthetic

of strain. Strain can be defined as the deformation ratio of the original length.

[9,16,18,23,25,51].

values indicate compression-type strains.

stress values (Figure 28).

structure

The most commonly used surfaces for implant bodies are rough surfaces. Different implant surface modifications (sandblasted, acid-etched, sandblasted and acid-etched, anodized, hydroxyapatite coatings, and plasma-sprayed) are proposed to change the characteristics of the surface from machined to rough, to increase the osteoblastic cell attachment level and also bone-implant contact (BIC) [64-68]. The influence of these surface modifications on BIC and cell attachment are still being investigated for a stronger osseointegration level between implant body and bone. Comparative studies show different BIC levels changing from 13% to 80% percent [69-79]. BIC values may change according to the jaw, placement of the region of the implant, healing time, implant design, and surface structure [64,69,70,72-74].

In most FEA studies, the bone-implant interface was assumed to be 100% bonded or completely osseointegrated [9,16,18,23,25]. As mentioned before, this is not proper modeling from a clinically realistic point of view. Cortical and cancellous bone also have different levels of BIC because of density and availability. Therefore, most studies use cortical bone of uniform thickness surrounding cancellous bone and proper material properties are chosen while modeling [9,16,18,23,25]. The degree of BIC distinctly affects the stress concentration value and distribution. In denser bone, there is less strain under loading compared with softer bone [80]. In some studies, BIC levels were assumed to be ≤100% for simulation of soft bone or immediate loading scenarios [9,81]. Evaluation of peri-implant stress in FEA studies is important for obtaining accurate treatment methods in implant dentistry. Implant and surrounding bone should be stressed within a certain range for dynamic physiologic remodeling. If ideal functional forces are placed on a restoration, the surrounding bone can adapt to the stresses and increase its density [82]. Overload may cause high stresses at the crest of the ridge and result in bone resorption. The direct opposite of this result is disuse atrophy of bone due to too little stress in the peri-implant region. Maintenance of bone density and stabilization is a direct result of the ideal stress distribution [80]. According to Frost studies, strains in the range of 50-1500 microstrain stimulates cortical bone mass and represents the physiological range. Strain beyond this range may cause overload and strain less than this range may not stimulate bone enough [80,83-85]. Most FEA studies, evaluate the risk assessment according to high stress values [9,16,18,23,25]. In other words, the most favorable modeling has the lowest stress values, and in contrast, the most deleterious modeling has the highest stress values [9,16,18,23,25]. However intensely lower stress values may also cause bone resorption because of inadequate bone stimulation.

## **7. Evaluation of stress**

Under bite force, localized stress occur at the prosthesis structure and bone. Stress is the magnitude of the internal forces acting within a deformable body. It is a measure of the average force per unit area of a surface within the body on which internal forces act. These internal forces appear as a response to external forces directed on the body [86-88]. Internal resistance after the application of the force applied on the body is not practically measurable. Therefore an easier process is to measure the applied force to a cross-sectional area. The dimension of stress is that of pressure, the Pascal (*Pa*), which is equivalent to 1 Newton (force) per square meter (unit area), that is N/m2. Stress is often reported in scientific publications as MPa. Stress is directly proportional to the force and inversely proportional to the area across which the force is applied. It is important to determine the area across which any force is applied. For example, the surface area of the occlusal pit restoration less than 4 mm. For this reason, the magnitude of stress in many restorations reaches hundreds of MPa [9,16,18,23,25,51].

42 Finite Element Analysis – New Trends and Developments

literature involve titanium and titanium alloys [9,18,24,31].

scientific data and requires long-term investigations. Therefore, most FEA studies in the

The most commonly used surfaces for implant bodies are rough surfaces. Different implant surface modifications (sandblasted, acid-etched, sandblasted and acid-etched, anodized, hydroxyapatite coatings, and plasma-sprayed) are proposed to change the characteristics of the surface from machined to rough, to increase the osteoblastic cell attachment level and also bone-implant contact (BIC) [64-68]. The influence of these surface modifications on BIC and cell attachment are still being investigated for a stronger osseointegration level between implant body and bone. Comparative studies show different BIC levels changing from 13% to 80% percent [69-79]. BIC values may change according to the jaw, placement of the region

In most FEA studies, the bone-implant interface was assumed to be 100% bonded or completely osseointegrated [9,16,18,23,25]. As mentioned before, this is not proper modeling from a clinically realistic point of view. Cortical and cancellous bone also have different levels of BIC because of density and availability. Therefore, most studies use cortical bone of uniform thickness surrounding cancellous bone and proper material properties are chosen while modeling [9,16,18,23,25]. The degree of BIC distinctly affects the stress concentration value and distribution. In denser bone, there is less strain under loading compared with softer bone [80]. In some studies, BIC levels were assumed to be ≤100% for simulation of soft bone or immediate loading scenarios [9,81]. Evaluation of peri-implant stress in FEA studies is important for obtaining accurate treatment methods in implant dentistry. Implant and surrounding bone should be stressed within a certain range for dynamic physiologic remodeling. If ideal functional forces are placed on a restoration, the surrounding bone can adapt to the stresses and increase its density [82]. Overload may cause high stresses at the crest of the ridge and result in bone resorption. The direct opposite of this result is disuse atrophy of bone due to too little stress in the peri-implant region. Maintenance of bone density and stabilization is a direct result of the ideal stress distribution [80]. According to Frost studies, strains in the range of 50-1500 microstrain stimulates cortical bone mass and represents the physiological range. Strain beyond this range may cause overload and strain less than this range may not stimulate bone enough [80,83-85]. Most FEA studies, evaluate the risk assessment according to high stress values [9,16,18,23,25]. In other words, the most favorable modeling has the lowest stress values, and in contrast, the most deleterious modeling has the highest stress values [9,16,18,23,25]. However intensely lower stress values

of the implant, healing time, implant design, and surface structure [64,69,70,72-74].

may also cause bone resorption because of inadequate bone stimulation.

Under bite force, localized stress occur at the prosthesis structure and bone. Stress is the magnitude of the internal forces acting within a deformable body. It is a measure of the average force per unit area of a surface within the body on which internal forces act. These internal forces appear as a response to external forces directed on the body [86-88]. Internal resistance after the application of the force applied on the body is not practically measurable. Therefore an easier process is to measure the applied force to a cross-sectional area. The dimension of stress is that of pressure, the Pascal (*Pa*), which is equivalent to 1

**7. Evaluation of stress** 

When the force is applied to mass, a deformation occurs as a result of this force. A strain is a normalized measure of deformation representing the displacement between particles in the body relative to a reference length [9,16,18,23,25,51,86,87,88]. There is no measurement unit of strain. Strain can be defined as the deformation ratio of the original length.

In FEA studies related to implant dentistry, frequently von Mises stress (equivalent tensile stress), minimum principal, and maximum principal are used to evaluate the effect of loading forces on the peri-implant region or prosthesis structure [9,16,18,23,25,89]. When a specific force is applied to the body, von Mises stress is the criterion used to determine the strain energy principles. Loading forces affecting the object can be evaluated 2 or 3 dimensionally. There are 3 "Principal Stresses" that can be calculated at any point, acting in the x, y, and z directions. The von Mises criteria refer to a formula for combining these 3 stresses into an equivalent stress, which is then compared to the yield stress of the material [25,90]. The major stress values are formed when all the components of the shear are zero. When an element is in this position, the normal stresses are called principal stresses. Principal stresses are classified as maximum, intermediate, and minimum principal stresses. The maximum principal stress is a positive value indicating the highest tension. The intermediate principal stress represents intermediate values. The minimum principal stress is a negative value indicating the highest compression [9,16,18,23,25,89]. If the data obtained from the analysis are positive values, then they are considered tensile stresses, negative values indicate compression-type strains.

Frequently, different color figures are used according to the amount of stress around periimplant regions and prosthetic structures (Figure 27). Stresses on each model are evaluated according to the stress values from low to high. In other words, the most favorable model has the lowest stress values, and in contrast, the most deleterious model has the highest stress values (Figure 28).

**Figure 27.** Different colors indicate the amount of stress around the peri-implant region and prosthetic structure

Application of Finite Element Analysis in Implant Dentistry 45

Implant manufacturers change their macro design and connections according to perceived clinical benefits. The aim of these improvements are less bone resorption around peri-implant regions, less micromotion at abutments, better loading distributions at dental implant structures, and good conical sealing. These properties are commonly related to biomechanics and should be investigated not only with clinical studies but also with FEA studies. All novel designs of implants or materials can be subject to investigation and can be compared with traditional structures. Another way of instituting FEA study is investigating treatment alternatives. New and old treatment modeling can be compared, limitations, and application areas can

2. *Computer stage*: This is the second part of FEA study. Generally clinicians have limited knowledge about modeling in computers and need help from computer engineers. It will be very wise to collaborate with friends at that field. Without a collaborator in computer engineering, too much time will be spent learning how to prepare models and developing the appropriate knowledge for the computational techniques necessary for model implementation. The clinician should manage the study and provide direction to the engineer. If the engineer does not have knowledge of the field of implant dentistry, seminars can be given to introduce the basic concepts of implantology. The seminars can include concepts such as indications for dental implants, dental implant parts, bone physiology, biting forces, connections of implants with bone, and the logic of implantology. As mentioned before, the shape of the materials can be scanned and converted digitally. Dental volumetric or computed tomography are good alternatives to scan and build bone structures. Devices used for routine treatments, can be found easily and are not expensive. For modeling of implant parts and superstructure, there are many sources, including manufacturers guidelines, scanning (advanced engineering 3-D scanning needed), and tooth atlas. The clinician should make every effort to maintain contact with their colleagues to allow frequent and efficient model evaluation and adaptation. The number of elements and nodes, can be increased to achieve more detailed modeling. However, this may be quiet timeconsuming and may implicate computing complications. Therefore, the engineer should clearly understand the aim of the research. Boundaries, limitations can be applied at modeling and element numbers can be increased only at the region of interest. These applications should not directly affect the results achieved. In the literature there are many software packages available for FEA study. The computer engineer can aid clinicians in choosing the appropriate software package for the specific application. In general, von Mises (equivalent stress), minimum, and maximum principal stress values are being used in FEA studies related to implant dentistry. These stress values are evaluated from low to high, and assessments are made according to these values. Higher values are considered more destructive and involve greater risk than low values. The most common material properties used in FEA studies of implant

be better understood.

dentistry are listed in Table 1 [9,27,48,56,57,91-109].

**Figure 28.** High and low stress values depicted in different colors in the models

In a previous study evaluating stress distribution, maximum von Mises (equivalent) stresses on each model are depicted around peri-implant region [18]. Eskitascioglu et al. evaluated maximum stresses (maximum von Mises) within the cortical bone surrounding the implant, framework of restoration, and occlusal surface material [56]. In a previous study, the FE model was used to calculate not only von Mises stress but also the principal stress. Authors explained their approach for this debate as follows: bone can sometimes be classified as brittle material; therefore, the principal stress was also implemented to evaluate the situation of cortical bone around implants [48].

## **8. Good FEA research development in implant dentistry**

This section is provided for clinicians and researchers who want to plan FEA studies related to implant dentistry and to provide a brief summary of research methodology.

1. *Planning a scenario*: The most important part of an FEA study is planning a unique model of treatment. There are countless FEA studies in the implantology literature; therefore, at the beginning of the study, it is highly recommended that you evaluate the available literature on your subject. Implant technology is improving rapidly. There is currently no perfect dental implant design or implant-abutment connection. Implant manufacturers change their macro design and connections according to perceived clinical benefits. The aim of these improvements are less bone resorption around peri-implant regions, less micromotion at abutments, better loading distributions at dental implant structures, and good conical sealing. These properties are commonly related to biomechanics and should be investigated not only with clinical studies but also with FEA studies. All novel designs of implants or materials can be subject to investigation and can be compared with traditional structures. Another way of instituting FEA study is investigating treatment alternatives. New and old treatment modeling can be compared, limitations, and application areas can be better understood.

44 Finite Element Analysis – New Trends and Developments

**Figure 28.** High and low stress values depicted in different colors in the models

**8. Good FEA research development in implant dentistry** 

to implant dentistry and to provide a brief summary of research methodology.

situation of cortical bone around implants [48].

In a previous study evaluating stress distribution, maximum von Mises (equivalent) stresses on each model are depicted around peri-implant region [18]. Eskitascioglu et al. evaluated maximum stresses (maximum von Mises) within the cortical bone surrounding the implant, framework of restoration, and occlusal surface material [56]. In a previous study, the FE model was used to calculate not only von Mises stress but also the principal stress. Authors explained their approach for this debate as follows: bone can sometimes be classified as brittle material; therefore, the principal stress was also implemented to evaluate the

This section is provided for clinicians and researchers who want to plan FEA studies related

1. *Planning a scenario*: The most important part of an FEA study is planning a unique model of treatment. There are countless FEA studies in the implantology literature; therefore, at the beginning of the study, it is highly recommended that you evaluate the available literature on your subject. Implant technology is improving rapidly. There is currently no perfect dental implant design or implant-abutment connection. 2. *Computer stage*: This is the second part of FEA study. Generally clinicians have limited knowledge about modeling in computers and need help from computer engineers. It will be very wise to collaborate with friends at that field. Without a collaborator in computer engineering, too much time will be spent learning how to prepare models and developing the appropriate knowledge for the computational techniques necessary for model implementation. The clinician should manage the study and provide direction to the engineer. If the engineer does not have knowledge of the field of implant dentistry, seminars can be given to introduce the basic concepts of implantology. The seminars can include concepts such as indications for dental implants, dental implant parts, bone physiology, biting forces, connections of implants with bone, and the logic of implantology. As mentioned before, the shape of the materials can be scanned and converted digitally. Dental volumetric or computed tomography are good alternatives to scan and build bone structures. Devices used for routine treatments, can be found easily and are not expensive. For modeling of implant parts and superstructure, there are many sources, including manufacturers guidelines, scanning (advanced engineering 3-D scanning needed), and tooth atlas. The clinician should make every effort to maintain contact with their colleagues to allow frequent and efficient model evaluation and adaptation. The number of elements and nodes, can be increased to achieve more detailed modeling. However, this may be quiet timeconsuming and may implicate computing complications. Therefore, the engineer should clearly understand the aim of the research. Boundaries, limitations can be applied at modeling and element numbers can be increased only at the region of interest. These applications should not directly affect the results achieved. In the literature there are many software packages available for FEA study. The computer engineer can aid clinicians in choosing the appropriate software package for the specific application. In general, von Mises (equivalent stress), minimum, and maximum principal stress values are being used in FEA studies related to implant dentistry. These stress values are evaluated from low to high, and assessments are made according to these values. Higher values are considered more destructive and involve greater risk than low values. The most common material properties used in FEA studies of implant dentistry are listed in Table 1 [9,27,48,56,57,91-109].


Application of Finite Element Analysis in Implant Dentistry 47

3. *Interpretation of results:* FEA studies have several advantages over clinical, pre-clinical, and in vitro studies. Most importantly, patients will not be harmed by the application of new materials and treatment modalities that have not been previously tested. Animals will not suffer from these biomechanical studies. However, clinicians should be aware that all of these applications are being performed on a computer, with critical limitations and assumptions that will clearly affect the applicability of the results to a real scenario. In the application of FEA studies, the most common drawback is overemphasis of the results. Simplifications are made for all simulated models; therefore, the models should be compared with each other within the same study. Other studies may use varied material properties and different planning scenarios. Confirming the FEA results with mechanical tests, conventional clinical model analysis, and preclinical tests are essential. It should not be forgotten that FEA studies are helpful for clinical trials but the results achieved from these studies are not valuable as clinical study results. However, before beginning biomechanical clinical trials, it will be wise to

FEA is a numerical stress analysis technique and is extensively used in implant dentistry to evaluate the risk factors from a biomechanical point of view. Simplifications and assumptions are the limitations of FEA studies. Although advanced computer technology is used to obtain results from simulated models, many factors affecting clinical features such as implant macro and micro design, material properties, loading conditions, and boundary conditions are neglected or ignored. Therefore, correlating FEA results with preclinical and

*Istanbul University Faculty of Dentistry, Department of Oral Implantology, Istanbul, Turkey* 

[1] Malevez C, Hermans M, Daelemans P (1996) Marginal bone levels at Brånemark system implants used for single tooth restoration. The influence of implant design and

[2] Hermann F, Lerner H, Palti A (2007) Factors influencing the preservation of the

*Istanbul University Faculty of Dentistry, Department of Prosthodontics, Istanbul, Turkey* 

long-term clinical studies may help to validate research models.

anatomical region. Clin Oral Implants Res Jun;7(2):162-9.

periimplant marginal bone. Implant Dent. Jun;16(2):165-75.

refer to FEA studies.

**9. Conclusion** 

**Author details** 

Pinar Gultekin\*

**10. References** 

Corresponding Author

 \*

B. Alper Gultekin and Serdar Yalcin

**Table 1.** Material properties used in finite element analysis studies of implant dentistry

3. *Interpretation of results:* FEA studies have several advantages over clinical, pre-clinical, and in vitro studies. Most importantly, patients will not be harmed by the application of new materials and treatment modalities that have not been previously tested. Animals will not suffer from these biomechanical studies. However, clinicians should be aware that all of these applications are being performed on a computer, with critical limitations and assumptions that will clearly affect the applicability of the results to a real scenario. In the application of FEA studies, the most common drawback is overemphasis of the results. Simplifications are made for all simulated models; therefore, the models should be compared with each other within the same study. Other studies may use varied material properties and different planning scenarios. Confirming the FEA results with mechanical tests, conventional clinical model analysis, and preclinical tests are essential. It should not be forgotten that FEA studies are helpful for clinical trials but the results achieved from these studies are not valuable as clinical study results. However, before beginning biomechanical clinical trials, it will be wise to refer to FEA studies.

## **9. Conclusion**

46 Finite Element Analysis – New Trends and Developments

110,000 100,000

100,000 80,000

13,400 10,000 15,000

1,500 1,370 150,000 250,000 790,000

67,700

46,890 82,500 84,000

80,000

*Periodontal ligament* 170 0.45 108 *Ni-Cr alloy* 204,000 0.3 108 *Dentin* 18,600 0.31 108

*Co-Cr alloy* 218,000 0.33 56 *Feldspathic porcelain* 82,800 0.35 56

*Mucosa* 10 0.40 103

*Resin* 2,700 0.35 31 *Resin composite* 7,000 0.2 31 *Gold alloy screw* 100,000 0.3 93 *Titanium abutment* 110,000 0.28 109 *Titanium abutment screw* 110,000 0.28 109 *Zirconia implant* 200,000 0.31 105, 107 *Zirconia abutment* 200,000 0.31 105, 107 *Zirconia core* 200,000 0.31 105, 107 *Zirconia veneer* 80,000 0.265 106, 107

**Table 1.** Material properties used in finite element analysis studies of implant dentistry

*Ti-6Al-4V* 110,000

*Type 3 gold alloy* 90,000

*Cortical bone* 13,700

*Trabecular bone* 1,370

*Porcelain* 66,900

*Enamel* 41,400

*Ag-Pd alloy* 95,000

**Material Young Modulus (MPa) Poisson Ratio Ref. No.** 

*Pure titanium* 117,000 0.3 9, 92, 93

0.35 0.33 0.35

0.3 0.3 0.33

0.3 0.3 0.3 0.3

0.3 0.3 0.31 0.3 0.3 0.3

0.29 0.28

0.3 0.3 0.33 0.33

0.33 0.33 27, 48, 57, 91

48, 94, 95

27, 56, 57, 96, 97

27, 56, 57, 98, 99

31, 48, 109

97, 100-102

109

FEA is a numerical stress analysis technique and is extensively used in implant dentistry to evaluate the risk factors from a biomechanical point of view. Simplifications and assumptions are the limitations of FEA studies. Although advanced computer technology is used to obtain results from simulated models, many factors affecting clinical features such as implant macro and micro design, material properties, loading conditions, and boundary conditions are neglected or ignored. Therefore, correlating FEA results with preclinical and long-term clinical studies may help to validate research models.

## **Author details**

B. Alper Gultekin and Serdar Yalcin *Istanbul University Faculty of Dentistry, Department of Oral Implantology, Istanbul, Turkey* 

Pinar Gultekin\* *Istanbul University Faculty of Dentistry, Department of Prosthodontics, Istanbul, Turkey* 

## **10. References**


<sup>\*</sup> Corresponding Author

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implant preload. J Prosthet Dent. Dec;90(6):539-46.

implant screws. Int J Oral Maxillofac Implants.10(3):295-302.

characteristics of porous dental implants. J Dent Res.61(8):1006-9.

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[79] Ivanoff CJ, Widmark G, Johansson C, Wennerberg A (2003) Histologic evaluation of bone response to oxidized and turned titanium micro-implants in human jawbone. Int J Oral Maxillofac Implants. May-Jun;18(3):341-8.

52 Finite Element Analysis – New Trends and Developments

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Sep;75(9):1262-8.

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Implantol. 26(3):163-9.

[64] Wennerberg A, Albrektsson T, Johansson C, Andersson B (1996) Experimental study of turned and grit-blasted screw-shaped implants with special emphasis on effects of

[65] Abron A, Hopfensperger M, Thompson J, Cooper LF (2001) Evaluation of a predictive model for implant surface topography effects on early osseointegration in the rat tibia

[66] Blumenthal NC, Cosma V (1989) Inhibition of apatite formation by titanium and

[67] Klokkevold PR, Johnson P, Dadgostari S, Caputo A, Davies JE, Nishimura RD (2001) Early endosseous integration enhanced by dual acid etching of titanium: a torque

[68] Weng D, Hoffmeyer M, Hürzeler MB, Richter EJ (2003) Osseotite vs. machined surface in poor bone quality. A study in dogs. Clin Oral Implants Res. Dec;14(6):703-8. [69] Piattelli A, Degidi M, Paolantonio M, Mangano C, Scarano A (2003) Residual aluminum oxide on the surface of titanium implants has no effect on osseointegration.

[70] Wennerberg A, Albrektsson T, Andersson B, Krol JJ (1995) A histomorphometric and removal torque study of screw-shaped titanium implants with three different surface

[71] Gotfredsen K, Nimb L, Hjörting-Hansen E, Jensen JS, Holmén A (1992) Histomorphometric and removal torque analysis for TiO2-blasted titanium implants.

[72] Ivanoff CJ, Hallgren C, Widmark G, Sennerby L, Wennerberg A (2001) Histologic evaluation of the bone integration of TiO(2) blasted and turned titanium microimplants

[73] van Steenberghe D, De Mars G, Quirynen M, Jacobs R, Naert I (2000) A prospective split-mouth comparative study of two screw-shaped self-tapping pure titanium implant

[74] Kohal RJ, Weng D, Bächle M, Strub JR (2004) Loaded custom-made zirconia and titanium implants show similar osseointegration: an animal experiment. J Periodontol.

[75] Xue W, Liu X, Zheng X, Ding C (2005) In vivo evaluation of plasma-sprayed titanium

[76] Piattelli A, Corigliano M, Scarano A, Costigliola G, Paolantonio M (1998) Immediate loading of titanium plasma-sprayed implants: an histologic analysis in monkeys. J

[77] Scarano A, Iezzi G, Petrone G, Marinho VC, Corigliano M, Piattelli A (2000) Immediate postextraction implants: a histologic and histometric analysis in monkeys. J Oral

[78] Galli C, Guizzardi S, Passeri G, Martini D, Tinti A, Mauro G, Macaluso GM (2005) Comparison of human mandibular osteoblasts grown on two commercially available

An experimental study on dogs. Clin Oral Implants Res. Jun;3(2):77-84.

coating after alkali modification. Biomaterials. Jun;26(16):3029-37.

titanium implant surfaces. J Periodontol. Mar;76(3):364-72.

blasting material and surface topography. Biomaterials. Jan;17(1):15-22.

vanadium ions. J Biomed Mater Res. Apr;23(A1 Suppl):13-22.

topographies. Clin Oral Implants Res. Mar;6(1):24-30.

in humans. Clin Oral Implants Res. Apr;12(2):128-34.

systems. Clin Oral Implants Res. Jun;11(3):202-9.

removal study in the rabbit. Clin Oral Implants Res. Aug;12(4):350-7.

	- [98] MacGregor AR, Miller TP, Farah JW (1980) Stress analysis of mandibular partial dentures with bounded and free-end saddles. J Dent.8(1):27-34.

**Chapter 3** 

© 2012 Chen, licensee InTech. This is an open access chapter distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use,

distribution, and reproduction in any medium, provided the original work is properly cited.

© 2012 Chen, licensee InTech. This is a paper distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

**Finite Element Analysis of the Stress on** 

Titanium and titanium alloys have become the preferred materials for dental implants owing to their good biocompatibility, excellent corrosion resistance and suitable mechanical properties. However, the existing titanium implants still have several drawbacks. Firstly, the bonding strength at the interface between the implant and the bone is not high enough and the biological fixation has not been achieved. Secondly, there exist mismatches between the elastic modulus of the implant and of the bone. A stress shielding or concentration can be easily induced on the interface and results in a potential risk to the long-term stability of the implant. The success or failure of an implant is determined by the manner how the stresses at the bone-implant interface are transferred to the surrounding bones [1,2]. The mandible has structural characteristic of an outer layer of dense cortical bone and an inner layer of porous cancellous bone. The elastic modulus and mechanical properties of cortical bones are different from those of cancellous bones. Nevertheless, current dental implants are mainly fabricated using dense titanium and titanium alloys, which have no features representing the difference between the inner and outer layers of the mandible or that between their elastic modulus. And therefore, the incompatibility of the mechanical properties between the implant and the bone was encountered. The use of porous metal implants for medical applications has two main advantages. One is the similar elastic modulus to the bone, which helps to prevent the stress shielding effect at the bone interfaces. The other is that it can provide a structural condition for the bone ingrowth to achieve biological fixation [3,4]. However, the low mechanical strength limits their further applications in the implanting industry. In this study, according to the structural characteristics of the mandible and the clinical requirements for the implant mechanical properties, a novel bio-mimetic design of implant is proposed for the titanium implants, which composes of a cortical bone zone with

**the Implant-Bone Interface of Dental** 

**Implants with Different Structures** 

Additional information is available at the end of the chapter

Liangjian Chen

**1. Introduction** 

http://dx.doi.org/10.5772/50699


## **Finite Element Analysis of the Stress on the Implant-Bone Interface of Dental Implants with Different Structures**

Liangjian Chen

54 Finite Element Analysis – New Trends and Developments

amalgam restorations. J Dent Res. 54(1):10-5.

beneath a complete denture. J Dent Res. 68(9):1370-3.

and implants. Int J Oral Maxillofac Implants. 26(5):961-9.

and supporting bone. Journal of Imp. 34(1):1-6.

number of teeth used as abutments. J Appl Oral Sci. 19(6):655-61.

screws. Int J Oral Maxillofac Implants. 10:295-302.

Biomech.10(3):159-66.

[98] MacGregor AR, Miller TP, Farah JW (1980) Stress analysis of mandibular partial

[99] Knoell AC (1977) A mathematical model of an in vitro human mandible. J

[100] Davy DT, Dilley GL, Krejci RF (1981) Determination of stress patterns in root-filled

[101] Wright KW, Yettram AL (1979)Reactive force distributions for teeth when loaded singly and when used as fixed partial denture abutments. J Prosthet Dent. 42(4):411-6. [102] Farah JW, Hood JA, Craig RG (1975) Effects of cement bases on the stresses in

[103] Maeda Y, Wood WW (1989) Finite element method simulation of bone resorption

[104] Ronald LS, Svenn EB (1995) Nonlinear contact analysis of preload in dental implant

[105] Kohal RJ, Papavasiliou G, Kamposiora P, Tripodakis A, Strub JR (2002) Threedimensional computerized stress analysis of commercially pure titanium and yttrium-

[108] Lanza MDS, Seraidarian PI, Jansen WC, Lanza MD (2011) Stress analysis of a fixed implant-supported denture by the finite element method (FEM) when varying the

[109] Quaresma SET, Cury PR, Sendyk WR, Sendyk C (2008) A finite element analysis of two different dental implants: Stress distribution in the prosthesis, abutment, implant,

partially stabilized zirconia implants. Int J Prosthodont. Mar-Apr;15(2):189-94. [106] White SN, Miklus VG, McLaren EA, Lang LA, Caputo AA (2005) Flexural strength of a layered zirconia and porcelain dental all-ceramic system. J Prosthet Dent. 94(2):125-31. [107] Caglar A, Bal BT, Karakoca S, Aydn C, Ylmaz H, Sarsoy S (2011) Three-dimensional finite element analysis of titanium and yttrium-stabilized zirconium dioxide abutments

dentures with bounded and free-end saddles. J Dent.8(1):27-34.

teeth incorporating various dowel designs. J Dent Res. 60(7):1301-10.

Additional information is available at the end of the chapter

http://dx.doi.org/10.5772/50699

## **1. Introduction**

Titanium and titanium alloys have become the preferred materials for dental implants owing to their good biocompatibility, excellent corrosion resistance and suitable mechanical properties. However, the existing titanium implants still have several drawbacks. Firstly, the bonding strength at the interface between the implant and the bone is not high enough and the biological fixation has not been achieved. Secondly, there exist mismatches between the elastic modulus of the implant and of the bone. A stress shielding or concentration can be easily induced on the interface and results in a potential risk to the long-term stability of the implant. The success or failure of an implant is determined by the manner how the stresses at the bone-implant interface are transferred to the surrounding bones [1,2]. The mandible has structural characteristic of an outer layer of dense cortical bone and an inner layer of porous cancellous bone. The elastic modulus and mechanical properties of cortical bones are different from those of cancellous bones. Nevertheless, current dental implants are mainly fabricated using dense titanium and titanium alloys, which have no features representing the difference between the inner and outer layers of the mandible or that between their elastic modulus. And therefore, the incompatibility of the mechanical properties between the implant and the bone was encountered. The use of porous metal implants for medical applications has two main advantages. One is the similar elastic modulus to the bone, which helps to prevent the stress shielding effect at the bone interfaces. The other is that it can provide a structural condition for the bone ingrowth to achieve biological fixation [3,4]. However, the low mechanical strength limits their further applications in the implanting industry. In this study, according to the structural characteristics of the mandible and the clinical requirements for the implant mechanical properties, a novel bio-mimetic design of implant is proposed for the titanium implants, which composes of a cortical bone zone with

© 2012 Chen, licensee InTech. This is an open access chapter distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. © 2012 Chen, licensee InTech. This is a paper distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

a dense structure and a cancellous bone zone with a porous outer layer and a dense core, as well as another three implants with different structures.

Finite Element Analysis of the Stress on

the Implant-Bone Interface of Dental Implants with Different Structures 57

**2.2. CAD and finite element modeling of the elements** 

spongy center surrounded by cortical bone of 2 mm.

elements of implant and bones were shown in Table 2.

**Figure 2.** Finite element models of bone and implant

A 3-D model of a mandibular section of bone with a missing second premolar and its superstructures were used in this study. A mandibular bone model was selected according to the classification system of Lekholm and Zarb. Trabecular bone was modeled as a solid structure in cortical bone. A bone block with dimensions of 20×14×35mm, representing the section of the mandible in the second premolar region, was modeled. It consisted of a

Four implants models with dimensions of d4.1 mm×12mm were selected in this study. Those implants and abutment were assumed to consist of the same material. Implant No.1 was dense with a high elastic modulus. Implant No.2 was a bio-mimetic with a high modulus in the cortical bone zone and low modulus-outer and high modulus-interior in the cancellous bone zone. Implant No.3 had a high modulus in the cortical bone zone and a low modulus in the cancellous bone zone. Implant No.4 had a whole lower elastic modulus. The elastic modulus of the dense titanium (high modulus) was set as 103.4GPa. The elastic modulus of implant No.1 (low modulus) was set as 40% of the dense titanium. To investigate the effect of elastic modulus on the interface stress, modulus in the low modulus zone varied in the range of 80%,40% ,10% and 1.3% of the modulus of the dense titanium,

The 3-D model of the implants was constructed by the CAD software Pro/E. The finite element analyses were carried out using Ansys Workbench 10.0. Tetrahedron elements in implant and bone corresponding to SOLID45 type elements in ANSYS element library with each node had three degrees of freedom. The finite element model is shown Fig.2 and Fig.3. The physical interactions at implant–bone interfaces during loading were taken into account through bonded surface-to-surface contact features of ANSYS. Numbers of nodes and

i.e.1370MPa. Mechanical properties of the implants were shown in Table 1.

The finite element method is one of the most frequently used methods in stress analysis in both industry and science[5]. Three-dimensional (3-D) finite element analysis (FEA) has been widely used for the quantitative evaluation of stresses on the implant and its surrounding bone[6,7]. Therefore, FEA was selected for use in this study to examine the effect of the structure and elastic modulus of dental implant on the stress distribution at implantbone interface. The 3-D models of the designed implants were constructed and the finite element analyses were carried out using Ansys Workbench 10.0. The stress distributions on implant-bone interface were investigated under static loading condition in order to provide design guidelines for the development of new implants. At the same time,the stress distributions on implant-bone interface were investigated in both dynamic and static loading conditions, and the fatigue behaviors of the bio-mimetic implant were analyzed based on fatigue theories and the formulas, in order to provide theoretical basis for the development of new implants.

## **2. Material and methods**

## **2.1. Structur of the biomimetic implant**

The biomimetic implant comprised of two layers, including the porous layer of open connected pores, which can provide the structure for bone ingrowth and has mechanical properties similar to the surrounding bones. The dense core ensures that the mechanical properties of implant meet the requirements of clinical applications (Fig.1).

**Figure 1.** Structure of the bio-mimetic porous titanium implant. A:section plane, B : cross section, 1: dense core, 2: porous layer

## **2.2. CAD and finite element modeling of the elements**

56 Finite Element Analysis – New Trends and Developments

development of new implants.

**2. Material and methods** 

dense core, 2: porous layer

**2.1. Structur of the biomimetic implant** 

well as another three implants with different structures.

a dense structure and a cancellous bone zone with a porous outer layer and a dense core, as

The finite element method is one of the most frequently used methods in stress analysis in both industry and science[5]. Three-dimensional (3-D) finite element analysis (FEA) has been widely used for the quantitative evaluation of stresses on the implant and its surrounding bone[6,7]. Therefore, FEA was selected for use in this study to examine the effect of the structure and elastic modulus of dental implant on the stress distribution at implantbone interface. The 3-D models of the designed implants were constructed and the finite element analyses were carried out using Ansys Workbench 10.0. The stress distributions on implant-bone interface were investigated under static loading condition in order to provide design guidelines for the development of new implants. At the same time,the stress distributions on implant-bone interface were investigated in both dynamic and static loading conditions, and the fatigue behaviors of the bio-mimetic implant were analyzed based on fatigue theories and the formulas, in order to provide theoretical basis for the

The biomimetic implant comprised of two layers, including the porous layer of open connected pores, which can provide the structure for bone ingrowth and has mechanical properties similar to the surrounding bones. The dense core ensures that the mechanical

**Figure 1.** Structure of the bio-mimetic porous titanium implant. A:section plane, B : cross section, 1:

(A) (B)

properties of implant meet the requirements of clinical applications (Fig.1).

A 3-D model of a mandibular section of bone with a missing second premolar and its superstructures were used in this study. A mandibular bone model was selected according to the classification system of Lekholm and Zarb. Trabecular bone was modeled as a solid structure in cortical bone. A bone block with dimensions of 20×14×35mm, representing the section of the mandible in the second premolar region, was modeled. It consisted of a spongy center surrounded by cortical bone of 2 mm.

Four implants models with dimensions of d4.1 mm×12mm were selected in this study. Those implants and abutment were assumed to consist of the same material. Implant No.1 was dense with a high elastic modulus. Implant No.2 was a bio-mimetic with a high modulus in the cortical bone zone and low modulus-outer and high modulus-interior in the cancellous bone zone. Implant No.3 had a high modulus in the cortical bone zone and a low modulus in the cancellous bone zone. Implant No.4 had a whole lower elastic modulus. The elastic modulus of the dense titanium (high modulus) was set as 103.4GPa. The elastic modulus of implant No.1 (low modulus) was set as 40% of the dense titanium. To investigate the effect of elastic modulus on the interface stress, modulus in the low modulus zone varied in the range of 80%,40% ,10% and 1.3% of the modulus of the dense titanium, i.e.1370MPa. Mechanical properties of the implants were shown in Table 1.

The 3-D model of the implants was constructed by the CAD software Pro/E. The finite element analyses were carried out using Ansys Workbench 10.0. Tetrahedron elements in implant and bone corresponding to SOLID45 type elements in ANSYS element library with each node had three degrees of freedom. The finite element model is shown Fig.2 and Fig.3. The physical interactions at implant–bone interfaces during loading were taken into account through bonded surface-to-surface contact features of ANSYS. Numbers of nodes and elements of implant and bones were shown in Table 2.

**Figure 2.** Finite element models of bone and implant

Finite Element Analysis of the Stress on

the Implant-Bone Interface of Dental Implants with Different Structures 59

cortical and cancellous bones were set to be completely constrained, and the boundary conditions were extended to the corresponding node. Multi-constraining was imposed on

Static loading was loaded to evaluate the implant-bone model. The implants were assumed to be under an axial force of 50-300N and a lingual force of 25 N in the angle of

Static and dynamic analyses of the implant need to consider and ensure the safety in the design. In the literature, implants are often worked according to the results of static analysis. Under the same masticatory forces, dynamic effects may add 10–20% more loads to implant than static effects. This must be taken into account to safeguard the fracture or fatigue failure of the implant. Therefore, using dynamic loading during the evaluation of a new implant is more reasonable. In the simulation of the normal chewing motion, forces close to the masticatory forces of normal adults were loaded to implant-bone model. Time dependent masticatory load was applied. Time history of the dynamic load components for 5 s is demonstrated in Fig. 4. These estimations were based on the assumption that an individual has three episodes of chewing per day, each 15 min in duration at a chewing rate of 60 cycles per minute (1 Hz). This is equivalent to 2700 chewing cycles per day or roughly

The von Mises stresses were used as the key indicators to measure stress levels and evaluate the stress distribution at implant-bone interface, as well as the maximum stress values on cortical bone. The main indicators are: 1) stress distribution in axial at the implant-bone

implant from bottom to top, in order to limit the freedom of the roots.

approximately 45° to the occlusal plane .

interface, and 2) the maximum von Misese stresses.

**Figure 4.** Dynamic loading in 5 seconds

106 cycles per year.

**Figure 3.** Finite element models of implant. A: dense body, B: porous layer, C: implant


**Table 1.** Mechanical properties of materials used in the study


**Table 2.** Numbers of nodes and elements of implant and bones

#### **2.3. Loads and boundary conditions**

All materials were assumed to be homogenous, isotropic and linearly elastic. The boneimplant interfaces were assumed to be 100% osseointegrated. The sides and bottom of cortical and cancellous bones were set to be completely constrained, and the boundary conditions were extended to the corresponding node. Multi-constraining was imposed on implant from bottom to top, in order to limit the freedom of the roots.

Static loading was loaded to evaluate the implant-bone model. The implants were assumed to be under an axial force of 50-300N and a lingual force of 25 N in the angle of approximately 45° to the occlusal plane .

Static and dynamic analyses of the implant need to consider and ensure the safety in the design. In the literature, implants are often worked according to the results of static analysis. Under the same masticatory forces, dynamic effects may add 10–20% more loads to implant than static effects. This must be taken into account to safeguard the fracture or fatigue failure of the implant. Therefore, using dynamic loading during the evaluation of a new implant is more reasonable. In the simulation of the normal chewing motion, forces close to the masticatory forces of normal adults were loaded to implant-bone model. Time dependent masticatory load was applied. Time history of the dynamic load components for 5 s is demonstrated in Fig. 4. These estimations were based on the assumption that an individual has three episodes of chewing per day, each 15 min in duration at a chewing rate of 60 cycles per minute (1 Hz). This is equivalent to 2700 chewing cycles per day or roughly 106 cycles per year.

The von Mises stresses were used as the key indicators to measure stress levels and evaluate the stress distribution at implant-bone interface, as well as the maximum stress values on cortical bone. The main indicators are: 1) stress distribution in axial at the implant-bone interface, and 2) the maximum von Misese stresses.

**Figure 4.** Dynamic loading in 5 seconds

58 Finite Element Analysis – New Trends and Developments

**Figure 3.** Finite element models of implant. A: dense body, B: porous layer, C: implant

**Table 1.** Mechanical properties of materials used in the study

**Table 2.** Numbers of nodes and elements of implant and bones

**2.3. Loads and boundary conditions** 

Lower modulus titanium 41.36 0.35 Dense titanium 103.4 0.35 Cortical bone 13.70 0.30 Cancellous bone 1.37 0.30

Material Elastic Modulus/GPa Poisson ratio ,υ

(A) (B) (C)

Implants Number of nodes Number of elements

No.2 8880 9186 45486 42926 No.3 3968 12313 20230 65703

No.1 and No.4 15835 - 84828 -

All materials were assumed to be homogenous, isotropic and linearly elastic. The boneimplant interfaces were assumed to be 100% osseointegrated. The sides and bottom of

Cortical bone 13329 - 65297 - Cancellous bone - 5395 - 16324

Dense Porous Dense Porous

#### **2.4. Fatigue analysis**

A good dental implant design should satisfy the maximum or an infinite fatigue life. This can only be ensured by physical testing or a fatigue analysis. In this study, the fatigue life of the dental implant was predicted using the finite element stress analysis with computer code of ANSYS/Workbench (ANSYS, 2003). Fatigue properties shown in Fig. 5 were used in fatigue calculations. Fig. 5 was known as S–N curves, showing fatigue properties of pure titanium in terms of alternating stress versus number of cycles. Fatigue life of prosthesis was calculated based on Goodman, Soderberg, Gerber and mean-stress fatigue theories which were illustrated in Table 3.

In Table 3, N indicates the safety factor for fatigue life in loading cycle, while Se is for endurance limit and Su is for ultimate tensile strength of the material. Mean stress m and alternating stress a are defined respectively as below, respectively.

$$
\sigma\_m = \frac{\sigma\_{\text{max}} + \sigma\_{\text{min}}}{2} \tag{1}
$$

Finite Element Analysis of the Stress on

the Implant-Bone Interface of Dental Implants with Different Structures 61

1 *a m e u S SN* 

1 *a m e y S SN*

1 *a m*

2

 

*e u N N S S* 

Fatigue theories Fatigue formulas

**3.1. Stress distribution on implant-bone interface under static loading condition** 

Table 4 shows the maximum von Mises stresses of different structure implants. It can be seen that the interface stresses of implant No.3 are much higher than those of other implants. There is no obvious difference in the maximum stress between implant No.1 and No.4. Implant No.2 has the lowest maximum stress at both cancellous bone and root zone comparing with other implants. After the transferring of stress to the surrounding bones, the maximum stress in cortical bone is larger than that of cancellous bone in the surrounding bone tissue. Implant No.1 has the largest stress in cortical bone and No.3 has the largest

Cancellous bone

No.1 23.434 12.553 11.668 1.456 No.2 23.451 8.261 9.685 1.525 No.3 33.532 15.77 8.419 4.845 No.4 23.453 14.482 9.012 1.799

*3.1.2. Stress distribution at implant-bone interface of implants under static loading.* 

Figure 6 represents the stress distribution at the implant-bone interface in an axial direction. It can be seen that the maximum stresses of the implant No.1, 2 and 4 show no difference in the cortical bone zone and the maximum stress zone is located at the marginal zone of cortical bone. The maximum stress zone of implant No.3 is located at the interface between cortical and cancellous bones. The area of the high stress zone and the value of interface stress of the implant No.2 are the smallest in both the cancellous bone and its root apex.

Stress/MPa

interface Cortical bone Cancellous

bone

Goodman

Soderberg

Gerber

stress in the root of cancellous bone.

**3. Results** 

Implants

**Table 3.** Fatigue theories and formulas used in fatigue life predictions

*3.1.1. The maximum stresses at implant-bone interface* 

Cortical bone interface

**Table 4.** Maximum von Mises stresses of implants with different structures

$$
\sigma\_a = \frac{\sigma\_{\text{max}} - \sigma\_{\text{min}}}{2} \tag{2}
$$

Von Misses stresses obtained from finite element analyses are utilized in fatigue life calculations. All fatigue analyses were performed according to the infinite life criteria (i.e. N = 109 cycles).

**Figure 5.** Fatigue curves (S-N curve) of pure titanium

#### Finite Element Analysis of the Stress on

the Implant-Bone Interface of Dental Implants with Different Structures 61


**Table 3.** Fatigue theories and formulas used in fatigue life predictions

## **3. Results**

60 Finite Element Analysis – New Trends and Developments

A good dental implant design should satisfy the maximum or an infinite fatigue life. This can only be ensured by physical testing or a fatigue analysis. In this study, the fatigue life of the dental implant was predicted using the finite element stress analysis with computer code of ANSYS/Workbench (ANSYS, 2003). Fatigue properties shown in Fig. 5 were used in fatigue calculations. Fig. 5 was known as S–N curves, showing fatigue properties of pure titanium in terms of alternating stress versus number of cycles. Fatigue life of prosthesis was calculated based on Goodman, Soderberg, Gerber and mean-stress fatigue theories which

In Table 3, N indicates the safety factor for fatigue life in loading cycle, while Se is for endurance limit and Su is for ultimate tensile strength of the material. Mean stress m and

> 2 *<sup>m</sup>*

2 *<sup>a</sup>* 

Von Misses stresses obtained from finite element analyses are utilized in fatigue life calculations. All fatigue analyses were performed according to the infinite life criteria (i.e. N

max min

max min

(1)

(2)

alternating stress a are defined respectively as below, respectively.

**Figure 5.** Fatigue curves (S-N curve) of pure titanium

**2.4. Fatigue analysis** 

were illustrated in Table 3.

= 109 cycles).

## **3.1. Stress distribution on implant-bone interface under static loading condition**

### *3.1.1. The maximum stresses at implant-bone interface*

Table 4 shows the maximum von Mises stresses of different structure implants. It can be seen that the interface stresses of implant No.3 are much higher than those of other implants. There is no obvious difference in the maximum stress between implant No.1 and No.4. Implant No.2 has the lowest maximum stress at both cancellous bone and root zone comparing with other implants. After the transferring of stress to the surrounding bones, the maximum stress in cortical bone is larger than that of cancellous bone in the surrounding bone tissue. Implant No.1 has the largest stress in cortical bone and No.3 has the largest stress in the root of cancellous bone.


**Table 4.** Maximum von Mises stresses of implants with different structures

#### *3.1.2. Stress distribution at implant-bone interface of implants under static loading.*

Figure 6 represents the stress distribution at the implant-bone interface in an axial direction. It can be seen that the maximum stresses of the implant No.1, 2 and 4 show no difference in the cortical bone zone and the maximum stress zone is located at the marginal zone of cortical bone. The maximum stress zone of implant No.3 is located at the interface between cortical and cancellous bones. The area of the high stress zone and the value of interface stress of the implant No.2 are the smallest in both the cancellous bone and its root apex.

Figure 7 represents the stress distribution in the cancellous bone zone of the implants. In all cases, there are high stress zones in the junction of the porous layer and the dense body. Among them, implant No.2 has the lowest interface stress. In the cancellous bone zone, the interface stress decreases from top to bottom, and increases at the root apex. And once again, No.2 has the lowest stress at the root apex, while No.3 has an obvious higher value than the others. The maximum stress exists at the bone interface of the implant No.1, which was 42.96% higher than that of implant No.2.

Finite Element Analysis of the Stress on

Cancellous bone root apex

the Implant-Bone Interface of Dental Implants with Different Structures 63

Cortical bone brink

**3.2. Effect of elastic modulus on the interface stress distribution of implant No.2** 

It was demonstrated that implant No.2 has the lowest interface stress. Thus, it is chosen to study the effect of elastic modulus of low modulus zone on the interface stress distribution at the interfaces. The elastic modulus in the low modulus zone varies in the range of 80%, 40%, 10% 1.3% of the modulus of the dense titanium, i.e.1370MPa. Table 8 shows that the interface stress in cancellous bone decreases with the decrease of the modulus of the low modulus layer, while there is no significant change in the cortical bone zone. For the interface stress of surrounding bones, it can be seen that the stress increases and that at the root apex of

Stress/MPa

cancellous bone decreases with the decrease of the modulus of the low modulus layer.

Cancellous bone interface

80% 23.452 12.725 9.172 1.739 40% 23.451 8.261 9.685 1.525 10% 23.451 3.733 11.224 1.094 1370MPa 23.443 2.216 12.304 1.351

**Figure 8.** Stress distribution of implant No.2 in axial direction at implant-bone interface: (a)80%; (b)40%;

(a) (b) (c) (d)

Figure 9 represents the stress distribution at the implant-bone interface in the axial direction. It can be seen that, under the same loading, a decrease of the modulus at low modulus layer has no significant influence on the interface stress of cortical bone. Figure 5 shows the stress distribution at the interface between implant No.2 and cancellous bone. The interface stress varies significantly with the change of the modulus of the low modulus layer. As the modulus of the low modulus layer decreases, the area of the high stress zone reduces, and the volume of the interface decreases dramatically. When the modulus of the low modulus layer reduces to 10% of the dense value, a uniform distribution of the interfacial stress without any high stress zone is obtained. For the specimens with the modulus of 1370MPa, the interface stress is 2.216MPa, 82.6% smaller than that of 80% ones. With the decrease of

Implants

(c)10%; (d)1.3%

Cortical bone interface

**Table 5.** Maximum von Mises stresses of implants

It was demonstrated that the structure of the implants has a predominate influence on the interface stress. Implant No.3 has a high trend to cause the stress concentration, while implant No.2 can efficiently reduce the interface stress, facilitating the transportation of the interface stress to the surrounding bones, avoiding the stress shielding and concentration, which is beneficial for the long time stability of the implants.

**Figure 6.** Stress distribution in axial direction at implant-bone interface of different structure implants

**Figure 7.** Stress distribution in spongy bone zone of different structure implants

## **3.2. Effect of elastic modulus on the interface stress distribution of implant No.2**

It was demonstrated that implant No.2 has the lowest interface stress. Thus, it is chosen to study the effect of elastic modulus of low modulus zone on the interface stress distribution at the interfaces. The elastic modulus in the low modulus zone varies in the range of 80%, 40%, 10% 1.3% of the modulus of the dense titanium, i.e.1370MPa. Table 8 shows that the interface stress in cancellous bone decreases with the decrease of the modulus of the low modulus layer, while there is no significant change in the cortical bone zone. For the interface stress of surrounding bones, it can be seen that the stress increases and that at the root apex of cancellous bone decreases with the decrease of the modulus of the low modulus layer.


**Table 5.** Maximum von Mises stresses of implants

62 Finite Element Analysis – New Trends and Developments

was 42.96% higher than that of implant No.2.

which is beneficial for the long time stability of the implants.

Figure 7 represents the stress distribution in the cancellous bone zone of the implants. In all cases, there are high stress zones in the junction of the porous layer and the dense body. Among them, implant No.2 has the lowest interface stress. In the cancellous bone zone, the interface stress decreases from top to bottom, and increases at the root apex. And once again, No.2 has the lowest stress at the root apex, while No.3 has an obvious higher value than the others. The maximum stress exists at the bone interface of the implant No.1, which

It was demonstrated that the structure of the implants has a predominate influence on the interface stress. Implant No.3 has a high trend to cause the stress concentration, while implant No.2 can efficiently reduce the interface stress, facilitating the transportation of the interface stress to the surrounding bones, avoiding the stress shielding and concentration,

**Figure 6.** Stress distribution in axial direction at implant-bone interface of different structure implants

No. 4

No. 1 No. 2 No. 3 No. 4

No. 1 No. 2 No. 3

**Figure 7.** Stress distribution in spongy bone zone of different structure implants

**Figure 8.** Stress distribution of implant No.2 in axial direction at implant-bone interface: (a)80%; (b)40%; (c)10%; (d)1.3%

Figure 9 represents the stress distribution at the implant-bone interface in the axial direction. It can be seen that, under the same loading, a decrease of the modulus at low modulus layer has no significant influence on the interface stress of cortical bone. Figure 5 shows the stress distribution at the interface between implant No.2 and cancellous bone. The interface stress varies significantly with the change of the modulus of the low modulus layer. As the modulus of the low modulus layer decreases, the area of the high stress zone reduces, and the volume of the interface decreases dramatically. When the modulus of the low modulus layer reduces to 10% of the dense value, a uniform distribution of the interfacial stress without any high stress zone is obtained. For the specimens with the modulus of 1370MPa, the interface stress is 2.216MPa, 82.6% smaller than that of 80% ones. With the decrease of

the modulus, the interface stress between the dense core and the porous layer increases. Figure 10 represents the stress distribution in dense body of implants No.2. It can be seen that the high stress zone is located at the interface between the cortical bone and cancellous bone. For the specimens with the modulus of 10% of the dense ones, the maximum interfacial stress at the porous-dense core interface is 18.556MPa. And it reduced to 13.752MPa for those of 80% specimens.

Finite Element Analysis of the Stress on

the Implant-Bone Interface of Dental Implants with Different Structures 65

varying the thickness of the low modulus zone from 0.5, 0.75, 1 to 1.25mm and maintaining the same implant diameter of 4.1mm and a constant modulus of low modulus zone, i.e.1370MPa. Figure 11 represents the stress distribution at the implant-bone interface in the axial direction. It can be seen that, in all cases, the cortical bone are in high stress zone while the cancellous bone are in low stress zone. The change of the thickness of low modulus zone affects the stress distribution of cancellous bone a lot while it has little influence on cortical bone, as shown in Fig.12. With the increase of the thickness, the interface stress decreases, especially in the root apex. Moreover, the distribution of the interface stress becomes more uniform. When it comes to an optimal thickness suitable for the clinical application, the strength and ingrowth of the bone tissues should be considered, which need further

**Figure 11.** Stress distribution in axial direction ata implant-bone interface of implants No.2 with

(a) (b) (c) (d)

**Figure 12.** Stress distribution at cancellous bone interface of implant No.2 with different thickness of

(a) (b) (c) (d)

low modulus layer:(a)0.5mm;(b)0.75mm;(c)1mm;(d)1.25mm

different thickness of low modulus layer: (a)0.5mm;(b)0.75mm;(c)1mm;(d)1.25mm

verification of MADIT experiments.

**Figure 9.** Stress distribution at interface between implant No.2 and cancellous bone: (a)80%; (b)40%; (c)10%; (d)1.3%

**Figure 10.** Stress distribution in dense body of different modulus implants No.2: (a)80%; (b)40%; (c)10%; (d)1.3%

## **3.3. Effect of thickness of low modulus zone on the interface stress distribution of implant No.2**

In order to further optimize the structure of the implant, the effect of thickness of low modulus zone on the interface stress distribution of implant No.2 was carried out, by varying the thickness of the low modulus zone from 0.5, 0.75, 1 to 1.25mm and maintaining the same implant diameter of 4.1mm and a constant modulus of low modulus zone, i.e.1370MPa. Figure 11 represents the stress distribution at the implant-bone interface in the axial direction. It can be seen that, in all cases, the cortical bone are in high stress zone while the cancellous bone are in low stress zone. The change of the thickness of low modulus zone affects the stress distribution of cancellous bone a lot while it has little influence on cortical bone, as shown in Fig.12. With the increase of the thickness, the interface stress decreases, especially in the root apex. Moreover, the distribution of the interface stress becomes more uniform. When it comes to an optimal thickness suitable for the clinical application, the strength and ingrowth of the bone tissues should be considered, which need further verification of MADIT experiments.

64 Finite Element Analysis – New Trends and Developments

13.752MPa for those of 80% specimens.

(c)10%; (d)1.3%

(c)10%; (d)1.3%

**of implant No.2** 

the modulus, the interface stress between the dense core and the porous layer increases. Figure 10 represents the stress distribution in dense body of implants No.2. It can be seen that the high stress zone is located at the interface between the cortical bone and cancellous bone. For the specimens with the modulus of 10% of the dense ones, the maximum interfacial stress at the porous-dense core interface is 18.556MPa. And it reduced to

**Figure 9.** Stress distribution at interface between implant No.2 and cancellous bone: (a)80%; (b)40%;

(a) (b) (c) (d)

**Figure 10.** Stress distribution in dense body of different modulus implants No.2: (a)80%; (b)40%;

**3.3. Effect of thickness of low modulus zone on the interface stress distribution** 

(a) (b) (c) (d)

In order to further optimize the structure of the implant, the effect of thickness of low modulus zone on the interface stress distribution of implant No.2 was carried out, by

**Figure 11.** Stress distribution in axial direction ata implant-bone interface of implants No.2 with different thickness of low modulus layer: (a)0.5mm;(b)0.75mm;(c)1mm;(d)1.25mm

**Figure 12.** Stress distribution at cancellous bone interface of implant No.2 with different thickness of low modulus layer:(a)0.5mm;(b)0.75mm;(c)1mm;(d)1.25mm

## **3.4. Stresses distribution on implant-bone interface under static and dynamic loading conditions**

Finite Element Analysis of the Stress on

the Implant-Bone Interface of Dental Implants with Different Structures 67

No.1 dynamic

and 2 showed no difference in the cortical bone area, while the high stress zone of the implant No.1 was greater than that of the implant No.2 in the spongy bone area and around the root apex. The yield strength of pure titanium was 462MPa. In static and dynamic loading conditions, the maximum stresses of the implant No.2 at the interface were 15.264MPa and 17.882MPa respectively, and they were only3.3% and 3.87% of the yield

Figure.15 represent the stress distribution in interface of spongy-bone implant. The stresses at the implant interface in dynamic loading condition were all higher than those in static loading condition. Both implant bodies had high stress zones in the junction of the cortical bone and spongy bone, and the stresses at the implant interface showed a declining trend from top to bottom but increased at the root apex. The interface stresses of the implant No.1 was higher than that of the implant No. 2, and the maximum stress at the bone interface of

**Figure 13.** Stress distribution in the cortical bone of the implant No.1 and No.2 under static and

No.2 static No.2 dynamic

the implant No.1 was 75.97% higher than that of the implant No.2.

No.1 static

strength of pure titanium.

dynamic loading conditions.

In order to compare the Von Mises stresses of the sense implant with that of the bio-mimetic implant under dynamic loading and static loading stations, the model structures of NO.1 and NO.2 are designed in the same way. The elasitc modulus of NO.1 and NO.2 dense body both are 103.4Gpa,that of NO.2 porous layer is 41.36Gpa, the poisson ratio of all three is 0.35.

## *3.4.1. Maximum stresses*

As shown in table 6, the maximum stresses under dynamic loading conditions were 17.15% higher than that under static loading conditions. The maximum stresses in cortical bone of two implants were similiar. However, the maximum stress of the dense implant was 75.79% high than that of the bio-mimetic implant in spongy bone, and 22.46% higher in the root region. The maximum stresses at implant-bone interface were much smaller than the yield strength of pure titanium (462MPa).


**Table 6.** Maximum Von Mises stresses of the dense implant and the bio-mimetic implant under static and dynamic loading conditions

## *3.4.2. Stress distribution within the cortical bone surrounding the implant neck and in implant-bone interface of implants No.1 and No.2 in static and dynamic loading*

Figure. 13 represent the stress distribution within the cortical bone surrounding the implant neck. The maximum stress occurred at the edge of the cervical cortical bone of implant No.2 were greater than those of implant No.1. For No.2, the maximum stresses were 7.192MPa in static loading condition and 8.428MPa in dynamic loading condition. For No.4, the maximum stresses were the maximum stresses were 6.67MPa in static loading condition and 7.814MPa in dynamic loading condition. The results indicated the implant No.2 had 7.85% and 7.67% higher stresses than the implant No.4 in dynamic and static loading conditions, respectively. The maximum stresses of the implant No.2 in static and dynamic loading conditions were only10.42% and 21.21% of the yield strength of cortical bone, 69MPa, respectively .

Figure.14 represent the stress distribution in the implant-bone interface in an axial direction. In both loading conditions, the maximum stresses at implant interfaces in the implant No.1 and 2 showed no difference in the cortical bone area, while the high stress zone of the implant No.1 was greater than that of the implant No.2 in the spongy bone area and around the root apex. The yield strength of pure titanium was 462MPa. In static and dynamic loading conditions, the maximum stresses of the implant No.2 at the interface were 15.264MPa and 17.882MPa respectively, and they were only3.3% and 3.87% of the yield strength of pure titanium.

Figure.15 represent the stress distribution in interface of spongy-bone implant. The stresses at the implant interface in dynamic loading condition were all higher than those in static loading condition. Both implant bodies had high stress zones in the junction of the cortical bone and spongy bone, and the stresses at the implant interface showed a declining trend from top to bottom but increased at the root apex. The interface stresses of the implant No.1 was higher than that of the implant No. 2, and the maximum stress at the bone interface of the implant No.1 was 75.97% higher than that of the implant No.2.

66 Finite Element Analysis – New Trends and Developments

**loading conditions** 

*3.4.1. Maximum stresses* 

Loading region

Cancellous bone region

and dynamic loading conditions

strength of pure titanium (462MPa).

**3.4. Stresses distribution on implant-bone interface under static and dynamic** 

In order to compare the Von Mises stresses of the sense implant with that of the bio-mimetic implant under dynamic loading and static loading stations, the model structures of NO.1 and NO.2 are designed in the same way. The elasitc modulus of NO.1 and NO.2 dense body both are 103.4Gpa,that of NO.2 porous layer is 41.36Gpa, the poisson ratio of all three is 0.35.

As shown in table 6, the maximum stresses under dynamic loading conditions were 17.15% higher than that under static loading conditions. The maximum stresses in cortical bone of two implants were similiar. However, the maximum stress of the dense implant was 75.79% high than that of the bio-mimetic implant in spongy bone, and 22.46% higher in the root region. The maximum stresses at implant-bone interface were much smaller than the yield

dynamic

No.1 static No.1

Cortical bone region 15.265 17.884 15.264 17.882

Root-end region 4.973 5.826 4.069 4.767 **Table 6.** Maximum Von Mises stresses of the dense implant and the bio-mimetic implant under static

*3.4.2. Stress distribution within the cortical bone surrounding the implant neck and in implant-bone interface of implants No.1 and No.2 in static and dynamic loading* 

only10.42% and 21.21% of the yield strength of cortical bone, 69MPa, respectively .

Figure. 13 represent the stress distribution within the cortical bone surrounding the implant neck. The maximum stress occurred at the edge of the cervical cortical bone of implant No.2 were greater than those of implant No.1. For No.2, the maximum stresses were 7.192MPa in static loading condition and 8.428MPa in dynamic loading condition. For No.4, the maximum stresses were the maximum stresses were 6.67MPa in static loading condition and 7.814MPa in dynamic loading condition. The results indicated the implant No.2 had 7.85% and 7.67% higher stresses than the implant No.4 in dynamic and static loading conditions, respectively. The maximum stresses of the implant No.2 in static and dynamic loading conditions were

Figure.14 represent the stress distribution in the implant-bone interface in an axial direction. In both loading conditions, the maximum stresses at implant interfaces in the implant No.1

Maximum Von Mises stresses (MPa)

9.962 11.671 5.661 6.632

No.2 static

No.2 dynamic

**Figure 13.** Stress distribution in the cortical bone of the implant No.1 and No.2 under static and dynamic loading conditions.

Figure. 16 represent the stress distribution in the dense body of the implants. There was a high stress zone of the dense body in the junction of the porous layer and the dense body of the implant No.2. The maximum stress of 12.306MPa in dynamic loading condition was higher than that of 10.504MPa in static loading condition. In the spongy bone area, the high stress zone of the dense body of the implant No.2 was greater than that of the implant No.1. The yield strength of pure titanium was 462MPa; the maximum stress at the interfaces of implant dense body did not reach the yield strength of pure titanium.

Finite Element Analysis of the Stress on

the Implant-Bone Interface of Dental Implants with Different Structures 69

**Figure 16.** Stress distribution in the dense body of the implant No.1 and No.2 under static and dynamic

No.1 static No.1 dynamic No.2 static No.2 dynamic

In the fatigue calculations, referring to the fatigue curves of pure titanium (S-N curves) shown in Figure 5, the fatigue life of implant was calculated based on Goodman, Soderberg, Gerber and Mean-Stess fatigue theories and formulas which were illustrated in Table 3. The endurance limit of pure titanium (Se) is 259.9MPa, and the yield strength (Sy) is 462MPa. The safety factors of different dense bodies of bio-mimetic implants with dynamic preload were calculated using the Soderberg formula in Table 3, as shown in Figure. 17. Figure. 17 represent the safety factors of different dense bodies of bio-mimetic implants when the dynamic preload was applied. Under an axial force of 50~300N and a lingual force of 45°25N in dynamic loading condition, the safety factors of dense body were all above 10.

Figure 18 showed the maximum stress at the interface of porous layer and the bone under different preloading conditions. With the increase of the loading, the interface stress of porous layer linearly increased. In dynamic loading condition with normal chewing force (axial 150N and lingual 45°25N), the maximum stress at the porous layer interface (max) was 6.632MPa, and the minimum (mim) was 1.038MPa. When an axial force of 300N and a lingual force of 45°25N were applied, the maximum stress at the porous layer interface (max) was 11.38MPa, and the minimum (mim) was 1.97MPa. According to the simulation results, it was predicted that the strength of the porous layer of the bio-mimetic implant and its bonding strength with the dense body interface should both be greater than the maximum interface stress (11.38MPa), which ensured the implant safety. The analyses of the implant interface stress provide a basis of mechanical properties for the preparation of

The results show that the bio-mimetic implant is safe against fatigue load.

loading conditions

*3.4.3. Fatigue analysis of bio-mimetic implant* 

porous layer of bio-mimetic implant.

**Figure 14.** Stress distribution in the bone-interface of the implant No.1 and No.2 under static and dynamic loading conditions

**Figure 15.** Stress distribution in the spongy bone-interface of the implant No.1 and No.2 under static and dynamic loading conditions

#### Finite Element Analysis of the Stress on the Implant-Bone Interface of Dental Implants with Different Structures 69

**Figure 16.** Stress distribution in the dense body of the implant No.1 and No.2 under static and dynamic loading conditions

## *3.4.3. Fatigue analysis of bio-mimetic implant*

68 Finite Element Analysis – New Trends and Developments

dynamic loading conditions

and dynamic loading conditions

Figure. 16 represent the stress distribution in the dense body of the implants. There was a high stress zone of the dense body in the junction of the porous layer and the dense body of the implant No.2. The maximum stress of 12.306MPa in dynamic loading condition was higher than that of 10.504MPa in static loading condition. In the spongy bone area, the high stress zone of the dense body of the implant No.2 was greater than that of the implant No.1. The yield strength of pure titanium was 462MPa; the maximum stress at the interfaces of

**Figure 14.** Stress distribution in the bone-interface of the implant No.1 and No.2 under static and

No.1 static No.1 dynamic No.2 static No.2 dynamic

**Figure 15.** Stress distribution in the spongy bone-interface of the implant No.1 and No.2 under static

No.1 static No.1 dynamic No.2 static No.2 dynamic

implant dense body did not reach the yield strength of pure titanium.

In the fatigue calculations, referring to the fatigue curves of pure titanium (S-N curves) shown in Figure 5, the fatigue life of implant was calculated based on Goodman, Soderberg, Gerber and Mean-Stess fatigue theories and formulas which were illustrated in Table 3. The endurance limit of pure titanium (Se) is 259.9MPa, and the yield strength (Sy) is 462MPa. The safety factors of different dense bodies of bio-mimetic implants with dynamic preload were calculated using the Soderberg formula in Table 3, as shown in Figure. 17. Figure. 17 represent the safety factors of different dense bodies of bio-mimetic implants when the dynamic preload was applied. Under an axial force of 50~300N and a lingual force of 45°25N in dynamic loading condition, the safety factors of dense body were all above 10. The results show that the bio-mimetic implant is safe against fatigue load.

Figure 18 showed the maximum stress at the interface of porous layer and the bone under different preloading conditions. With the increase of the loading, the interface stress of porous layer linearly increased. In dynamic loading condition with normal chewing force (axial 150N and lingual 45°25N), the maximum stress at the porous layer interface (max) was 6.632MPa, and the minimum (mim) was 1.038MPa. When an axial force of 300N and a lingual force of 45°25N were applied, the maximum stress at the porous layer interface (max) was 11.38MPa, and the minimum (mim) was 1.97MPa. According to the simulation results, it was predicted that the strength of the porous layer of the bio-mimetic implant and its bonding strength with the dense body interface should both be greater than the maximum interface stress (11.38MPa), which ensured the implant safety. The analyses of the implant interface stress provide a basis of mechanical properties for the preparation of porous layer of bio-mimetic implant.

Finite Element Analysis of the Stress on

the Implant-Bone Interface of Dental Implants with Different Structures 71

The functions of implant are mainly dependent on the direct bonding with the surrounding bones. The long-term success of an implant is determined by the reliability and stability of the implant bone interface, and the success or failure of an implant is determined by the manner that the stresses at the bone-implant interface transfer to the surrounding bones[1, 2]. The main factors contributing to the stability of implants include the structure of the implants, the distribution of the interface stress and the combination mode of the interface. In order to ensure the long-term stability of an implant, the implant should be designed according to two main principles. First, the load should be minimized to avoid exceeding its physiological tolerance as overloading can cause bone resorption or fatigue failure of the implant. On the other hand, underloading may lead to disuse atrophy and subsequent bone loss[3, 4]. Second, the contact zone with the bone should be increased to reduce the bone interface stress. The structural characteristic of the mandible shows an outer layer of dense cortical bone and an inner layer of loose cancellous bone. Both the elastic modulus and mechanical strength of cortical bone (10~18GPa) are higher than those of cancellous bone (1.3~4GPa). Current dense implants do not have the structure similar to that of the mandible, as well as modulus. As a result, the mechanical compatibility between the implant and the bone remains unresolved, and the modified active coating on the surface gets easily damaged in the implantation process. An implant with a low elastic modulus is believed to be beneficial to transferring the stress to the surrounding bones, resulting in a long-term stability[8,9]. The porous implant materials can tremendously improve the implant biocompatibility [10-12] by improving the adhesion and outgrowth of those osteoblasts, promoting the deposition of extracellular matrix, increasing the adsorption of nutrients and oxygen, and promoting the new bones' growth into pores to achieve biological fixation. The porosity can be changed to adjust the density, strength and elastic modulus of the material to achieve similar mechanical properties to the replaced hard tissues. Meanwhile, the porous structure can provide scaffold for the bioactive coating to promote osseointegration. In this study, according to the structural characteristics of the mandible and the advantages of the porous implant material, an idea of a bio-mimetic implant is proposed. It is a titanium implant composed of a cortical bone zone with a dense structure and a cancellous bone zone with a porous outer layer and a dense body. The cortical bone has a high modulus, and the porous outer layer of the cancellus bone zone has a low modulus. The dense body ensures the strength to meet the requirements of clinical applications. To optimize the structure of the bio-mimetic implants, the finite element analysis was carried out. The effects of implant structure, modulus and thickness of the low

modulus layer on the distribution of the interfacial stress were studied.

The interfacial stress of the implants is mainly located at the interface between the implants and the surrounding bones,affecting the interface biological reactions such as bone resorption and remodeling. Cortical bone loss and early implant failure after loading are usually accompanied by the excess stress at the implant bone interface while a low stress

**4. Discussion** 

**Figure 17.** Safety factor for dense body of bio-mimetic implant under different dynamic loading

**Figure 18.** Maximum Von-mises stress for bone-interface porous layer of implant under different dynamic loading conditions.

## **4. Discussion**

70 Finite Element Analysis – New Trends and Developments

**Figure 17.** Safety factor for dense body of bio-mimetic implant under different dynamic loading

**Figure 18.** Maximum Von-mises stress for bone-interface porous layer of implant under different

dynamic loading conditions.

The functions of implant are mainly dependent on the direct bonding with the surrounding bones. The long-term success of an implant is determined by the reliability and stability of the implant bone interface, and the success or failure of an implant is determined by the manner that the stresses at the bone-implant interface transfer to the surrounding bones[1, 2]. The main factors contributing to the stability of implants include the structure of the implants, the distribution of the interface stress and the combination mode of the interface. In order to ensure the long-term stability of an implant, the implant should be designed according to two main principles. First, the load should be minimized to avoid exceeding its physiological tolerance as overloading can cause bone resorption or fatigue failure of the implant. On the other hand, underloading may lead to disuse atrophy and subsequent bone loss[3, 4]. Second, the contact zone with the bone should be increased to reduce the bone interface stress. The structural characteristic of the mandible shows an outer layer of dense cortical bone and an inner layer of loose cancellous bone. Both the elastic modulus and mechanical strength of cortical bone (10~18GPa) are higher than those of cancellous bone (1.3~4GPa). Current dense implants do not have the structure similar to that of the mandible, as well as modulus. As a result, the mechanical compatibility between the implant and the bone remains unresolved, and the modified active coating on the surface gets easily damaged in the implantation process. An implant with a low elastic modulus is believed to be beneficial to transferring the stress to the surrounding bones, resulting in a long-term stability[8,9]. The porous implant materials can tremendously improve the implant biocompatibility [10-12] by improving the adhesion and outgrowth of those osteoblasts, promoting the deposition of extracellular matrix, increasing the adsorption of nutrients and oxygen, and promoting the new bones' growth into pores to achieve biological fixation. The porosity can be changed to adjust the density, strength and elastic modulus of the material to achieve similar mechanical properties to the replaced hard tissues. Meanwhile, the porous structure can provide scaffold for the bioactive coating to promote osseointegration. In this study, according to the structural characteristics of the mandible and the advantages of the porous implant material, an idea of a bio-mimetic implant is proposed. It is a titanium implant composed of a cortical bone zone with a dense structure and a cancellous bone zone with a porous outer layer and a dense body. The cortical bone has a high modulus, and the porous outer layer of the cancellus bone zone has a low modulus. The dense body ensures the strength to meet the requirements of clinical applications. To optimize the structure of the bio-mimetic implants, the finite element analysis was carried out. The effects of implant structure, modulus and thickness of the low modulus layer on the distribution of the interfacial stress were studied.

The interfacial stress of the implants is mainly located at the interface between the implants and the surrounding bones,affecting the interface biological reactions such as bone resorption and remodeling. Cortical bone loss and early implant failure after loading are usually accompanied by the excess stress at the implant bone interface while a low stress

may lead to disuse atrophy and subsequent bone loss [13,14]. It is indicated that, under the same situation, the smaller the bone surface area in contact with the implant body is, the greater the overall stress becomes [15]. Cortical bone, which has a higher modulus, higher strength and more resistance to deformation than cancellous bone [16], can bear more loading in masticatory movements [17-20]. In this study, it was supposed that the implantbone osseointegration was 100%. Under the same loading condition, the stress distributions at the interface of four different structure implants were compared and analyzed, showing the change of the implant structure and modulus in the cancellous bone had significant effects on the stress distribution. In all cases, there are high stress zone at the interface between cortical and cancellous bone. In cancellous bone, the interface stress decreases from top to bottom, and increases at the root apex.

Finite Element Analysis of the Stress on

the Implant-Bone Interface of Dental Implants with Different Structures 73

F = (πR2 + A1 )σ1 + 2πRHτ1 (1)

F =πR2σ2 + 2πRHτ2 (2)

vertical loading. In the model, R refers to the radius of the implant, H refers to the height, and F refers to the vertical loading. Assuming that the compressive stress and shear stress are uniform, and the compressive stress and shear stress on porous and dense implants are σ1, τ1 and σ2, τ2, respectively. The porous implants provide more contact area with the bone than the dense implants. Assuming that A1 is the added contact area, the equilibrium

Because the compressive strength at the interface is much larger than its shear strength, the values of σ1 similar to σ2, and the added zone A1 are larger, we can obtain τ1«τ2. It means that the shear force of porous implants is much smaller that that of dense ones, which is

In current industry, a screw structure is usually adopted to improve the bond strength between the bone and implants. The modulus of screw zone is higher than that of cortical bone, which has a high trend to cause stress shielding and concentration and thus bone absorption [21]. For a porous structure, when the bone tissue grows into the porous structure, the bond strength is improved and the modulus of implants is similar to that of the surrounding bones. No bone absorption occurs under loading because part of the stress can be borne by bone tissues in the pore. In summary, biomimetic style implant No.2, with a high modulus in the cortical bone and low modulus-outer and high modulus-interior in the

equations of forces for porous and dense implants can be expressed as:

beneficial for the stability of the low strength cancellous bone.

**Figure 19.** Stress analysis of implants

In the cortical bone zone, all implants present high stress values and the maximum stresses are in the same level. In the cancellous bone zone, the maximum stress of the dense implant interface was 75.58% higher than that of the bio-mimetic implant, and 22.21% higher than that in the root apex zone. The maximum stresses in cancellous bone and root region of implant No.2 are lower than those of other three implants. The maximum stress of implant No.4 is 42.96% higher than that of No.2. Implant No.3 has the highest stresses in root region. The stress distribution at bone-implant interface varied with elastic modulus of low elastic modulus layer. The maximum stresses of implant No.2 decreases with the decreasing of elastic modulus in cancellous bone region, while there is no significant difference in cortical bone region. When the modulus of the low modulus layer is reduced to 10% of the dense ones, a uniform distribution of interfacial stress without any high stress zone was obtained. With the increase of the thickness of the low modulus layer, the interface stress decreases, especially in the root apex. Moreover, the distribution of the interface stress becomes much uniform.

From the biomechanical point of view, a structure like implant No.2, a modulus matches the cancellous bone and a suitable thickness can effectively reduce the stress in the implantbone interface and be beneficial to the transfer of interfacial stress to surrounding bones, which is favorable to the long-term stability of the implant. The structural characteristics of this implant are in line with those of the mandible, so that the elastic modulus of the porous zone can be reduced to make the elastic modulus of the implant match with that of the cancellous bone and thus help the interface stress tranferring. The structural characeristics of mandible of implants No.1and No.4 are ingored, which results in the un-uniform interface stress distribution and stress concentration in cancellous bone. Although implant No.3 has a mandible –like structure, the cancellous bone is a whole low modulus structure, which leads to stress concentration at both interface and root apex.

Implant No.2 has a low modulus-outer and high modulus-interior in the cancellous bone zone. The low modulus-outer can be realized by adjusting the porosity and pore size to match the mechanical properties, especially the elastic modulus, with the surrounding bones. Figure. 19 illustrates the stress distribution of the porous and dense implants under vertical loading. In the model, R refers to the radius of the implant, H refers to the height, and F refers to the vertical loading. Assuming that the compressive stress and shear stress are uniform, and the compressive stress and shear stress on porous and dense implants are σ1, τ1 and σ2, τ2, respectively. The porous implants provide more contact area with the bone than the dense implants. Assuming that A1 is the added contact area, the equilibrium equations of forces for porous and dense implants can be expressed as:

$$\mathbf{F} = (\pi \mathbf{R}^2 + \mathbf{A} \mathbf{r}\_1) \sigma \mathbf{r}\_1 + 2\pi \mathbf{R} \mathbf{H} \mathbf{r}\_1 \tag{1}$$

$$\mathbf{F} = \pi \mathbf{R}^2 \sigma \mathbf{z} + 2\pi \mathbf{R} \mathbf{H} \mathbf{\tau} \mathbf{z} \tag{2}$$

Because the compressive strength at the interface is much larger than its shear strength, the values of σ1 similar to σ2, and the added zone A1 are larger, we can obtain τ1«τ2. It means that the shear force of porous implants is much smaller that that of dense ones, which is beneficial for the stability of the low strength cancellous bone.

**Figure 19.** Stress analysis of implants

72 Finite Element Analysis – New Trends and Developments

top to bottom, and increases at the root apex.

to stress concentration at both interface and root apex.

uniform.

may lead to disuse atrophy and subsequent bone loss [13,14]. It is indicated that, under the same situation, the smaller the bone surface area in contact with the implant body is, the greater the overall stress becomes [15]. Cortical bone, which has a higher modulus, higher strength and more resistance to deformation than cancellous bone [16], can bear more loading in masticatory movements [17-20]. In this study, it was supposed that the implantbone osseointegration was 100%. Under the same loading condition, the stress distributions at the interface of four different structure implants were compared and analyzed, showing the change of the implant structure and modulus in the cancellous bone had significant effects on the stress distribution. In all cases, there are high stress zone at the interface between cortical and cancellous bone. In cancellous bone, the interface stress decreases from

In the cortical bone zone, all implants present high stress values and the maximum stresses are in the same level. In the cancellous bone zone, the maximum stress of the dense implant interface was 75.58% higher than that of the bio-mimetic implant, and 22.21% higher than that in the root apex zone. The maximum stresses in cancellous bone and root region of implant No.2 are lower than those of other three implants. The maximum stress of implant No.4 is 42.96% higher than that of No.2. Implant No.3 has the highest stresses in root region. The stress distribution at bone-implant interface varied with elastic modulus of low elastic modulus layer. The maximum stresses of implant No.2 decreases with the decreasing of elastic modulus in cancellous bone region, while there is no significant difference in cortical bone region. When the modulus of the low modulus layer is reduced to 10% of the dense ones, a uniform distribution of interfacial stress without any high stress zone was obtained. With the increase of the thickness of the low modulus layer, the interface stress decreases, especially in the root apex. Moreover, the distribution of the interface stress becomes much

From the biomechanical point of view, a structure like implant No.2, a modulus matches the cancellous bone and a suitable thickness can effectively reduce the stress in the implantbone interface and be beneficial to the transfer of interfacial stress to surrounding bones, which is favorable to the long-term stability of the implant. The structural characteristics of this implant are in line with those of the mandible, so that the elastic modulus of the porous zone can be reduced to make the elastic modulus of the implant match with that of the cancellous bone and thus help the interface stress tranferring. The structural characeristics of mandible of implants No.1and No.4 are ingored, which results in the un-uniform interface stress distribution and stress concentration in cancellous bone. Although implant No.3 has a mandible –like structure, the cancellous bone is a whole low modulus structure, which leads

Implant No.2 has a low modulus-outer and high modulus-interior in the cancellous bone zone. The low modulus-outer can be realized by adjusting the porosity and pore size to match the mechanical properties, especially the elastic modulus, with the surrounding bones. Figure. 19 illustrates the stress distribution of the porous and dense implants under In current industry, a screw structure is usually adopted to improve the bond strength between the bone and implants. The modulus of screw zone is higher than that of cortical bone, which has a high trend to cause stress shielding and concentration and thus bone absorption [21]. For a porous structure, when the bone tissue grows into the porous structure, the bond strength is improved and the modulus of implants is similar to that of the surrounding bones. No bone absorption occurs under loading because part of the stress can be borne by bone tissues in the pore. In summary, biomimetic style implant No.2, with a high modulus in the cortical bone and low modulus-outer and high modulus-interior in the

cancellous bone is superior in the stress teansferring. The porous structure can effectively reduce the shear force at the bone-implant interface, providing a suitable environment for bone tissue ingrowth, which is benefit for the longtime stability of the implants.

Finite Element Analysis of the Stress on

the Implant-Bone Interface of Dental Implants with Different Structures 75

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## **5. Conclusions**


## **Author details**

Chen Liangjian

*The Third Xiangya Hospital of Central South University, ChangSha, China State Key Laboratory of Powder Metallurgy Central South University, ChangSha, China* 

## **6. References**


[6] Sato Y, Wadamoto M, Tsuga K, Teixeira ER. The effectiveness of element down sizing on a three-dimensional finite element model of bone trabeculae in implant biomechanics[J]. J Oral Rehabil ,1999;26:288–91.

74 Finite Element Analysis – New Trends and Developments

shielding and concentration.

without any high stress zone was obtained.

distribution of the interface stress becomes much uniform.

*The Third Xiangya Hospital of Central South University, ChangSha, China* 

review of the literature [J]. J Prosthet Dent, 2001, 85(6):585–598.

bone remodeling[J]. J Biomed Mater Res, 1991, 25(4):467–483.

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*State Key Laboratory of Powder Metallurgy Central South University, ChangSha, China* 

loading around oral implants [J]. Clin Oral Implants Res, 1998, 9(6):407-412.

**5. Conclusions** 

**Author details** 

Chen Liangjian

**6. References** 

p. 148–377.

cancellous bone is superior in the stress teansferring. The porous structure can effectively reduce the shear force at the bone-implant interface, providing a suitable environment for

1. The distribution of interface stress is strongly depended on the structure of the implants. The bio-mimetic implant No.2 is favorable to transferring the interface stress from the cancellous bone and root apex bone to surrounding bones, avoiding stress

2. It is demonstrated that the interface stress varies significantly with the change of the modulus of the low modulus layer. The area of the high stress zone is reduced, and the value of the interface decreases dramatically. When the modulus of the low modulus layer is reduced to 10% of the dense value, a uniform interface stress distribution

3. The change of the thickness of low modulus zone affects the stress distribution of cancellous bone, while it has no significant influence on cortical bone. With the increase of the thickness, the interface stress decreases, especially in the root apex. Moreover, the

[1] Van Osterwyck H, Duyck J, Vander S, Vander PG, Decoomans M, Lieven S, Puers R, Naert L. The influence of bone mechanical properties and implant fixation upon bone

[2] Geng J, Tan K B C, Liu G. Application of finite element analysis in implant dentistry: a

[3] Vaillancourt H, Pillar RM, McCammond D. Factors affecting cortical bone loss with dental implants partially covered with a porous coating: a finite element analysis[J]. Int

[4] Pilliar RM, Deporter DA, Watson PA, Valiquette N. Dental implant design effect on

[5] Bathe KJ. Finite element procedures[M]. Upper Saddle River (NJ): Prentice-Hall; 1996.

bone tissue ingrowth, which is benefit for the longtime stability of the implants.

	- [21] Gefen A. computational simulations of stress shielding and bone resorption around existing and computer-designed orthopaedic screws[J]. Medical and Biological Engineering and Computing, 2002,40(3): 311-322.

**Chapter 4** 

© 2012 Gíslason and Nash, licensee InTech. This is an open access chapter distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

© 2012 Gíslason and Nash, licensee InTech. This is a paper distributed under the terms of the Creative Commons

**Finite Element Modelling of** 

Magnús Kjartan Gíslason and David H. Nash

Additional information is available at the end of the chapter

http://dx.doi.org/10.5772/50560

**1. Introduction** 

finite element method.

relevant loading conditions acting on the joint.

**a Multi-Bone Joint: The Human Wrist** 

Computational models of biomechanical systems have been available for over 40 years. In the first issue of Journal of Biomechanics from 1968 there exists a paper by Marangoni and Glaser looking at the viscoelastic behaviour biological tissue and presented numerical results using a discrete model which can be thought of as a predecessor of the modern finite element models. In 1971 Rybicki et al published a paper on the mechanical stresses of the femur using the finite element method. Since then, published papers on finite element modelling increased yearly and now, 40 years later, the finite element method plays an important part on the analysis of geometrically complex structures. The hip has been researched extensively over these 40 year and numerous papers have been published from various different research groups on the mechanical response of the femur and total hip arthroplasty under various types of loading. What makes the hip an excellent candidate for finite element analysis is the fact that the geometry of the joint is well defined and can be easily extracted from CT or MRI scans but also the fact that the joint contact forces and musculoskeletal modelling of the hip joint has been extensively researched and measured (Bergmann et al 1993) giving a well defined loading condition during gait and other activities. The knee has also been researched using the finite element method where the joint geometry is well defined, but the loading conditions and the kinematics are more complex. Taylor et al (2003) have investigated the performance of total knee replacement using the

Modelling of those joints is more complicated than of the hip and knee, due to complex bone geometry, soft tissue modelling as well as difficulty determining the physiologically

The wrist and the ankle pose a challenge in biomechanical modelling due to the complex interactions between the many bones comprising the joint. Each bone will contribute

**Chapter 4** 

## **Finite Element Modelling of a Multi-Bone Joint: The Human Wrist**

Magnús Kjartan Gíslason and David H. Nash

Additional information is available at the end of the chapter

http://dx.doi.org/10.5772/50560

## **1. Introduction**

76 Finite Element Analysis – New Trends and Developments

Engineering and Computing, 2002,40(3): 311-322.

[21] Gefen A. computational simulations of stress shielding and bone resorption around existing and computer-designed orthopaedic screws[J]. Medical and Biological

> Computational models of biomechanical systems have been available for over 40 years. In the first issue of Journal of Biomechanics from 1968 there exists a paper by Marangoni and Glaser looking at the viscoelastic behaviour biological tissue and presented numerical results using a discrete model which can be thought of as a predecessor of the modern finite element models. In 1971 Rybicki et al published a paper on the mechanical stresses of the femur using the finite element method. Since then, published papers on finite element modelling increased yearly and now, 40 years later, the finite element method plays an important part on the analysis of geometrically complex structures. The hip has been researched extensively over these 40 year and numerous papers have been published from various different research groups on the mechanical response of the femur and total hip arthroplasty under various types of loading. What makes the hip an excellent candidate for finite element analysis is the fact that the geometry of the joint is well defined and can be easily extracted from CT or MRI scans but also the fact that the joint contact forces and musculoskeletal modelling of the hip joint has been extensively researched and measured (Bergmann et al 1993) giving a well defined loading condition during gait and other activities. The knee has also been researched using the finite element method where the joint geometry is well defined, but the loading conditions and the kinematics are more complex. Taylor et al (2003) have investigated the performance of total knee replacement using the finite element method.

> Modelling of those joints is more complicated than of the hip and knee, due to complex bone geometry, soft tissue modelling as well as difficulty determining the physiologically relevant loading conditions acting on the joint.

> The wrist and the ankle pose a challenge in biomechanical modelling due to the complex interactions between the many bones comprising the joint. Each bone will contribute

© 2012 Gíslason and Nash, licensee InTech. This is an open access chapter distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. © 2012 Gíslason and Nash, licensee InTech. This is a paper distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

uniquely to the high range of motion of the joint. The challenge in modelling of the multibone models is to capture the mechanism contributing to the stabilization of the joint. A stable joint is able to provide three-dimensional equilibrium under external loading which can also be interpreted as the ability of a joint to maintain a normal relationship between the articulating bones and soft tissue constraints under physiologic loads throughout the whole range of motion (Garcia-Elias et al. 1995). This implies that the joints need to be capable of distributing loads without generating abnormally high stresses on the articulating surface as well as being able to move within the joint's range of motion. Geometry of the bones also plays an important role in joint stability and the concavity or convexity of the articulating bones helps the bones to distribute stresses across the joint.

Finite Element Modelling of a Multi-Bone Joint: The Human Wrist 79

Carrigan et al published the first three dimensional wrist model where all the carpal bones were incorporated but not the metacarpals. Loading was applied onto the distal aspect of the capitate and was 15 N compressive force which is not representative of physiological in vivo loading on the wrist. Additionally the scaphoid needed to be constrained using unphysiological constraints in order to achieve convergence. In 2009 full three dimensional models of the wrist were published by Gislason et al and Guo et al incorporating the distal ends of the radius and ulna, all the carpal bones as well as the metacarpals. The Gislason model aimed to simulate load transfer behaviour of the wrist during gripping in three different subjects with the wrist in three different positions. The loading was determined on a subject specific basis where the forces and moments acting on the fingers were measured and by using a biomechanical model, the external forces were converted into joint contact forces acting on the metacarpals. The Guo model aimed to simulate the carpal bone behaviour after the transverse carpal ligament had been excised. The loading applied onto the Guo model was a combined 100 N compressive force acting on the the 2nd and 3rd metacarpal and some unphysiological constraints were applied to the model. Bajuri et al (2012) created a full three dimensional model simulating the

effects of rheumatoid arthritis on the stress behaviour of the carpal bones.

finite element model of a multi bone joint, whether it be the wrist or the ankle.

spring elements or as separate geometrical entities.

quality of the finite element models produced.

**2. Image segmenting** 

3D doctor, 3D slicer to name a few.

Finite element models of the ankle also exist through the research of Chen et al (2003) and Cheung (2004) and although the chapter mainly discusses the creation of a finite element model of the wrist, there are many similarities in the methodology of creating a high quality

The fundamental problems that researchers face in the creation of a finite element model of the wrist are the loading applied and the soft tissue constraints on the carpus. The wrist are a mechanically unstable joint so external constraints, in the form of ligaments, must be applied in order for the carpal bones to return to equilibrium whether they be modelled as

With increased computational power and more enhanced software, it is possible to simulate more detailed structures to a higher degree of detail than before. With the current rate of software and hardware development, the user will soon become the limiting factor on the

A fully representative geometrical model is integral for the quality of the finite element model. With enhanced scanners and software it is possible to achieve high degree of resolution for the geometrical model. There exist many different image processing software packages that are capable of carrying out image processing and segmenting the scans in order to create three dimensional surface such as Mimics (Materialise), Simplware, Amira,

Segmentation of the wrist bones requires close attention to details as the geometrical features of each carpal bone can be highly irregular and can vary between individuals.

Work on finite element modelling of the wrist started in the 1990s with the works of Miyake et al and Anderson and Daniel who modelled the stresses on the radiocarpal joint using a plain strain contact model. That model contained the radius, scaphoid and the lunate as well as the extrinsic ligaments and the scapholunate ligaments. The TFCC was modelled using a series of spring elements. Albeit a two-dimensional model, it marked a beginning of further research interest in the numerical modelling of the wrist. Miyake et al (1994) published around the same time, a finite element model simulating the stress distribution of a malunited Colle's fracture. That same group later published a paper on the stress distribution in the carpus following a lunate ceramic replacement for Kienböck's disease (Oda et al 2000).


Other wrist models were published shortly afterwards and can be summarised in the following table.

**Table 1.** Previously published finite element models of the wrist

Carrigan et al published the first three dimensional wrist model where all the carpal bones were incorporated but not the metacarpals. Loading was applied onto the distal aspect of the capitate and was 15 N compressive force which is not representative of physiological in vivo loading on the wrist. Additionally the scaphoid needed to be constrained using unphysiological constraints in order to achieve convergence. In 2009 full three dimensional models of the wrist were published by Gislason et al and Guo et al incorporating the distal ends of the radius and ulna, all the carpal bones as well as the metacarpals. The Gislason model aimed to simulate load transfer behaviour of the wrist during gripping in three different subjects with the wrist in three different positions. The loading was determined on a subject specific basis where the forces and moments acting on the fingers were measured and by using a biomechanical model, the external forces were converted into joint contact forces acting on the metacarpals. The Guo model aimed to simulate the carpal bone behaviour after the transverse carpal ligament had been excised. The loading applied onto the Guo model was a combined 100 N compressive force acting on the the 2nd and 3rd metacarpal and some unphysiological constraints were applied to the model. Bajuri et al (2012) created a full three dimensional model simulating the effects of rheumatoid arthritis on the stress behaviour of the carpal bones.

Finite element models of the ankle also exist through the research of Chen et al (2003) and Cheung (2004) and although the chapter mainly discusses the creation of a finite element model of the wrist, there are many similarities in the methodology of creating a high quality finite element model of a multi bone joint, whether it be the wrist or the ankle.

The fundamental problems that researchers face in the creation of a finite element model of the wrist are the loading applied and the soft tissue constraints on the carpus. The wrist are a mechanically unstable joint so external constraints, in the form of ligaments, must be applied in order for the carpal bones to return to equilibrium whether they be modelled as spring elements or as separate geometrical entities.

With increased computational power and more enhanced software, it is possible to simulate more detailed structures to a higher degree of detail than before. With the current rate of software and hardware development, the user will soon become the limiting factor on the quality of the finite element models produced.

## **2. Image segmenting**

78 Finite Element Analysis – New Trends and Developments

(Oda et al 2000).

following table.

bones helps the bones to distribute stresses across the joint.

uniquely to the high range of motion of the joint. The challenge in modelling of the multibone models is to capture the mechanism contributing to the stabilization of the joint. A stable joint is able to provide three-dimensional equilibrium under external loading which can also be interpreted as the ability of a joint to maintain a normal relationship between the articulating bones and soft tissue constraints under physiologic loads throughout the whole range of motion (Garcia-Elias et al. 1995). This implies that the joints need to be capable of distributing loads without generating abnormally high stresses on the articulating surface as well as being able to move within the joint's range of motion. Geometry of the bones also plays an important role in joint stability and the concavity or convexity of the articulating

Work on finite element modelling of the wrist started in the 1990s with the works of Miyake et al and Anderson and Daniel who modelled the stresses on the radiocarpal joint using a plain strain contact model. That model contained the radius, scaphoid and the lunate as well as the extrinsic ligaments and the scapholunate ligaments. The TFCC was modelled using a series of spring elements. Albeit a two-dimensional model, it marked a beginning of further research interest in the numerical modelling of the wrist. Miyake et al (1994) published around the same time, a finite element model simulating the stress distribution of a malunited Colle's fracture. That same group later published a paper on the stress distribution in the carpus following a lunate ceramic replacement for Kienböck's disease

Other wrist models were published shortly afterwards and can be summarised in the

**Author Year Type Modelled** 

Schuind et al 1995 Rigid body Whole carpus

Miyake et al 1994 Finite element Radius, scaphoid lunate Anderson & Daniel 1995, 2005 Finite element Radius, scaphoid, lunate, ulna

Ulrich et al 1999 Finite element Radius, scaphoid, lunate

Oda et al 2000 Finite element Whole carpus excluding

Carrigan et al 2003 Finite element Whole carpus excluding

Gislason et al 2009, 2010 Finite element Whole carpus Guo et al 2009 Finite element Whole carpus Bajuri et al 2012 Finite element Whole carpus

Nedoma et al 2003 Mathematical

**Table 1.** Previously published finite element models of the wrist

metacarpals

metacarpals

model Whole carpus

A fully representative geometrical model is integral for the quality of the finite element model. With enhanced scanners and software it is possible to achieve high degree of resolution for the geometrical model. There exist many different image processing software packages that are capable of carrying out image processing and segmenting the scans in order to create three dimensional surface such as Mimics (Materialise), Simplware, Amira, 3D doctor, 3D slicer to name a few.

Segmentation of the wrist bones requires close attention to details as the geometrical features of each carpal bone can be highly irregular and can vary between individuals.

Using an automated segmentation from the abovementioned software packages sometimes can not be enough to capture the full three-dimensional geometry of the bones so manual segmentation is at times necessary. The importance of a high quality segmentation can not be underestimated in multibone modelling as the congruence of the articulating surfaces will play an important role in the contact formulation. Any rough edges on the articulating surfaces will cause penetration of nodepoints causing numerical instability and convergence problems once the finite element model will be run. It is therefore critical to the success of the computational model that the segmentation be carried out in an accurate manner. Another reason why the segmentation is the most critical aspect of the modelling, is the fact that once the geometry has been constructed and meshed, it is very difficult for the user to make any changes to it without starting from the beginning again.

Finite Element Modelling of a Multi-Bone Joint: The Human Wrist 81

**Figure 2.** Distribution between cortical and cancellous bone in scans and on finite element model

how the radius bone will look like, before and after smoothing.

**Figure 3.** The radius bone before and after three dimensional smoothing

effects on the volume of the bone.

Most software packages now offer the option of smoothing the three dimensional object. It is inevitable that unsmooth edges will occur from the image segmentation and will be more visible if some degree of manual segmentation is required. Figure 3 shows an example of

The smoothing is easily done within the software packages, but the user must be aware of the possible implications of the smoothing as it is possible to be too aggressive in the smoothing and therefore Lose volume whilst trying to obtain a good looking picture of the bone. Each iteration of the smoothing causes some changes in the volume of the three dimensional object although some software packages allow to compensate for the volume changes. A possible solution to these volume changes would be to recalculate the mask based upon and carry out manual adjustments of the mask and recalculate the three dimensional object and creating an iterative cycle until the smoothing will have negligible

The plane in which the segmentation should be carried out in, would be the plane with the highest resolution, which is primarily the axial plane. Using the sagittal and the coronal plane (or the other two planes with lower resolution) can also be beneficial in order to fine tune the segmentation in order to get a full three dimensional representation of the segmentation. Figure 1 shows segmentation of the carpal bones in axial and coronal planes.

Using the masks can also be a helpful tool in determining the distribution between cortical and cancellous bone. By eroding the mask of a given number of pixels, it is possible to create a hoop in each slice representing the two stiffness layers. Previously published papers have suggested that the thickness of the cortical shell in carpal bones is on average 2.6 mm (Louis et al 1995). Figure 2 shows the distribution between cortical and cancellous bone on the scans and in the finite element model.

**Figure 1.** Segmentation of carpal bones in two planes

scans and in the finite element model.

**Figure 1.** Segmentation of carpal bones in two planes

planes.

make any changes to it without starting from the beginning again.

Using an automated segmentation from the abovementioned software packages sometimes can not be enough to capture the full three-dimensional geometry of the bones so manual segmentation is at times necessary. The importance of a high quality segmentation can not be underestimated in multibone modelling as the congruence of the articulating surfaces will play an important role in the contact formulation. Any rough edges on the articulating surfaces will cause penetration of nodepoints causing numerical instability and convergence problems once the finite element model will be run. It is therefore critical to the success of the computational model that the segmentation be carried out in an accurate manner. Another reason why the segmentation is the most critical aspect of the modelling, is the fact that once the geometry has been constructed and meshed, it is very difficult for the user to

The plane in which the segmentation should be carried out in, would be the plane with the highest resolution, which is primarily the axial plane. Using the sagittal and the coronal plane (or the other two planes with lower resolution) can also be beneficial in order to fine tune the segmentation in order to get a full three dimensional representation of the segmentation. Figure 1 shows segmentation of the carpal bones in axial and coronal

Using the masks can also be a helpful tool in determining the distribution between cortical and cancellous bone. By eroding the mask of a given number of pixels, it is possible to create a hoop in each slice representing the two stiffness layers. Previously published papers have suggested that the thickness of the cortical shell in carpal bones is on average 2.6 mm (Louis et al 1995). Figure 2 shows the distribution between cortical and cancellous bone on the

**Figure 2.** Distribution between cortical and cancellous bone in scans and on finite element model

Most software packages now offer the option of smoothing the three dimensional object. It is inevitable that unsmooth edges will occur from the image segmentation and will be more visible if some degree of manual segmentation is required. Figure 3 shows an example of how the radius bone will look like, before and after smoothing.

**Figure 3.** The radius bone before and after three dimensional smoothing

The smoothing is easily done within the software packages, but the user must be aware of the possible implications of the smoothing as it is possible to be too aggressive in the smoothing and therefore Lose volume whilst trying to obtain a good looking picture of the bone. Each iteration of the smoothing causes some changes in the volume of the three dimensional object although some software packages allow to compensate for the volume changes. A possible solution to these volume changes would be to recalculate the mask based upon and carry out manual adjustments of the mask and recalculate the three dimensional object and creating an iterative cycle until the smoothing will have negligible effects on the volume of the bone.

## **3. Meshing**

The quality of the mesh of the finite element model will determine the quality of the solution. The process of meshing the three-dimensional objects using an automated meshing tool, which many of the image processing softwares packages discussed in previous chapter have incorporated, has significantly decreased the time and effort to create high quality meshes. The software packages then give the option of importing the meshes into finite element programs such as Ansys, Abaqus and others.

Finite Element Modelling of a Multi-Bone Joint: The Human Wrist 83

Modelling the cartilage is one of the greatest challenges faced by researchers working on joint modelling. Cartilage is not visible from CT scans, but can be identified using MRI scans. In clinical 3 Tesla scans it can be difficult to determine exactly where the cartilage boundary layer is located in three-dimensional space, making it difficult to create the cartilage layer via masking of the scans. In doing so, the researcher will need to interpolate the shape of the cartilage layer often resulting in an irregular shape causing meshing problems. Another aspect regarding incorporating the cartilage layer into the bone model is the scattering of stiff cortical bone elements and soft cartilage elements. That could cause numerical instabilities in the solution phase. A more practical approach is to extrude the external surfaces of the bones at the articulation and creating a solid volume layer representing the cartilage. Using this method will give a distinct boundary between the bone and the cartilage layer. Another possibility

Most finite element models of joints have used elastic material properties for both the cortical shell and for the cancellous bone. Bone is a viscoelastic material and its properties will depend on the strain rate. All published multibone joint finite element models have focussed on a quasi static analysis of the joint and therefore applying the loads slowly. The material properties used

modulus [MPa] Poisson's ratio Density [g/cm3] Ultimate tensile

strength [MPa]

would be to extrude the elements directly creating a layer of wedge elements.

for bone material can be seen in Table 2 and are obtained from Rho et al (1997).

Cortical 17\*103-19\*103 0.25 2000 150 Cancellous 100-200 0.30 1500 20

viscoelastic properties would add a substantial amount of complexity to the model.

The simplified material values presented in Table 2 will give an idea about the parameters that can be applied to a macroscopical finite element model of a bone. A more refined material model incorporating bone mineral density, the orthotropical behaviour and

Many finite element studies have simulated the mechanical properties of the articular cartilage as elastic material which can be subjected to large errors. Articular cartilage is a complex material that has the properties of a fluid and a solid and has been researched extensively in the literature. Much of that research hasn't been applied into the finite element modelling of multibody joints, although many finite element models exist of cartilage only focussing on the material behaviour. Attempts have been made (Gislason 10 and Bajuri 12) to incorporate the non-linearities of the articular cartilage behaviour into the

**4.1. Cartilage modelling** 

**4.2. Material modelling** 

Bone type Young's

**Table 2.** Bone material properties

*4.2.2. Cartilage* 

*4.2.1. Bone* 

The versatility of the tetrahedral elements have made them popular candidates for the automatic meshing tools in the software packages. The tetrahedral elements are capable of capturing a high degree of geometric non-linearity and are the most popular elements used in biomechanical modelling research today. The problem with the tetrahedral elements is the stiffness of the 4 node tetrahedral element which can give too high stress values compared to the 10 node tetrahedral element. If using a 4 node tetrahedral element, the user must be confident that a sufficient number of elements is being used to capture the nonlinear geometry. For the presented models an average of roughly 430 thousand elements were used, resulting in an element density of about 10 elements/mm3.

Hexahedral elements can also be used in biomechanical finite element models. In 2005 Ramos and Simões compared the performance of first and second order hexahedral elements and tetrahedral elements on a femur model and reported that there was little difference in the accuracy of the two types of tetrahedral elements. The tetrahedral elements were closer to a theoretical result, also calculated than the hexahedral elements. The hexahedral elements though showed a higher degree of stability and were less influenced by the number of elements.

As with other finite element models, the mesh quality will play a significant role in the overall solution quality. In a multibody analysis needing contact formulation, obtaining high element quality at the articular surfaces is important, as cartilage elements are soft and tend to deform to a greater extent than the bone elements. Therefore an ill shaped cartilage element, undergoing large deformations, is likely to be excessively distorted and cause divergence of the solution.

With increased computing power, the automatic meshing tool have become extremely powerful and have made it possible that the user will not need to spend much time on producing a high quality mesh, making it possible to model larger numbers of models and incorporating subject specific models.

## **4. Creation of the finite element model**

During the creation of the finite element model, the best practice is to import each carpal bone individually allowing the user to keep control over whole assembly. Most of the image processing software packages will take into account the coordinates of individual pixels from the MRI or CT scans. Therefore the position of each carpal bone will be preserved after being imported into the finite element software.

## **4.1. Cartilage modelling**

82 Finite Element Analysis – New Trends and Developments

element programs such as Ansys, Abaqus and others.

were used, resulting in an element density of about 10 elements/mm3.

The quality of the mesh of the finite element model will determine the quality of the solution. The process of meshing the three-dimensional objects using an automated meshing tool, which many of the image processing softwares packages discussed in previous chapter have incorporated, has significantly decreased the time and effort to create high quality meshes. The software packages then give the option of importing the meshes into finite

The versatility of the tetrahedral elements have made them popular candidates for the automatic meshing tools in the software packages. The tetrahedral elements are capable of capturing a high degree of geometric non-linearity and are the most popular elements used in biomechanical modelling research today. The problem with the tetrahedral elements is the stiffness of the 4 node tetrahedral element which can give too high stress values compared to the 10 node tetrahedral element. If using a 4 node tetrahedral element, the user must be confident that a sufficient number of elements is being used to capture the nonlinear geometry. For the presented models an average of roughly 430 thousand elements

Hexahedral elements can also be used in biomechanical finite element models. In 2005 Ramos and Simões compared the performance of first and second order hexahedral elements and tetrahedral elements on a femur model and reported that there was little difference in the accuracy of the two types of tetrahedral elements. The tetrahedral elements were closer to a theoretical result, also calculated than the hexahedral elements. The hexahedral elements though showed a higher degree of stability and were less influenced by

As with other finite element models, the mesh quality will play a significant role in the overall solution quality. In a multibody analysis needing contact formulation, obtaining high element quality at the articular surfaces is important, as cartilage elements are soft and tend to deform to a greater extent than the bone elements. Therefore an ill shaped cartilage element, undergoing large deformations, is likely to be excessively distorted and cause divergence of the solution.

With increased computing power, the automatic meshing tool have become extremely powerful and have made it possible that the user will not need to spend much time on producing a high quality mesh, making it possible to model larger numbers of models and

During the creation of the finite element model, the best practice is to import each carpal bone individually allowing the user to keep control over whole assembly. Most of the image processing software packages will take into account the coordinates of individual pixels from the MRI or CT scans. Therefore the position of each carpal bone will be preserved after

**3. Meshing** 

the number of elements.

incorporating subject specific models.

**4. Creation of the finite element model** 

being imported into the finite element software.

Modelling the cartilage is one of the greatest challenges faced by researchers working on joint modelling. Cartilage is not visible from CT scans, but can be identified using MRI scans. In clinical 3 Tesla scans it can be difficult to determine exactly where the cartilage boundary layer is located in three-dimensional space, making it difficult to create the cartilage layer via masking of the scans. In doing so, the researcher will need to interpolate the shape of the cartilage layer often resulting in an irregular shape causing meshing problems. Another aspect regarding incorporating the cartilage layer into the bone model is the scattering of stiff cortical bone elements and soft cartilage elements. That could cause numerical instabilities in the solution phase. A more practical approach is to extrude the external surfaces of the bones at the articulation and creating a solid volume layer representing the cartilage. Using this method will give a distinct boundary between the bone and the cartilage layer. Another possibility would be to extrude the elements directly creating a layer of wedge elements.

## **4.2. Material modelling**

## *4.2.1. Bone*

Most finite element models of joints have used elastic material properties for both the cortical shell and for the cancellous bone. Bone is a viscoelastic material and its properties will depend on the strain rate. All published multibone joint finite element models have focussed on a quasi static analysis of the joint and therefore applying the loads slowly. The material properties used for bone material can be seen in Table 2 and are obtained from Rho et al (1997).


### **Table 2.** Bone material properties

The simplified material values presented in Table 2 will give an idea about the parameters that can be applied to a macroscopical finite element model of a bone. A more refined material model incorporating bone mineral density, the orthotropical behaviour and viscoelastic properties would add a substantial amount of complexity to the model.

## *4.2.2. Cartilage*

Many finite element studies have simulated the mechanical properties of the articular cartilage as elastic material which can be subjected to large errors. Articular cartilage is a complex material that has the properties of a fluid and a solid and has been researched extensively in the literature. Much of that research hasn't been applied into the finite element modelling of multibody joints, although many finite element models exist of cartilage only focussing on the material behaviour. Attempts have been made (Gislason 10 and Bajuri 12) to incorporate the non-linearities of the articular cartilage behaviour into the

finite element models, by using Mooney-Rivlin hyper-elastic material properties using the material data obtained from Li et al (2007).

Finite Element Modelling of a Multi-Bone Joint: The Human Wrist 85

Once the bones have all been incorporated into the finite element software and assembled together bone by bone and cartilage constructed, the contact formulation between the bones needs to be formulated. A surface-to-surface contact is most common method defining the contact between the bones, but node-to-surface configuration can also be implemented. Most finite element models will allow the user different contact models, such as the Lagrange method, the penalty method etc. The availability of these different contact models can be limited to the type of solution algorithm used. Additionally the user can determine the stiffness of the contact, but usually as "hard contact" is applied which is defined by

 ൌ Ͳǡ ݄ ൏ Ͳ ݄ ൌ Ͳǡ Ͳ Where p is the contact pressure, and h is the over closure between the two surfaces. Using kinematic contact method is generally preferred over the penalty contact as it introduces an additional stiffness to the system. Frictionless contact properties or friction using a low friction coefficient should also implemented on the articulating surfaces. By using

It has been reported in the literature (Kauer 1986) that there is little or no movement between certain articulations, such as the articulations between the bones in the distal row of the wrist and the metacarpals (in the carpometacarpal joint). For those joints, it is possible to use a tie constraint so that no relative motion occurs between the two bones. That will

frictionless contact, it is ensured that no shear stresses occur at the articulations.

help to simplify the model. The model can be seen in Figure 4

**Figure 4.** Finite element model

*4.2.4. Contact setup* 

## *4.2.3. Ligaments*

Evaluating the material properties of ligaments pose a great challenge to researchers in multibone joint modelling as they operate only in tension and show viscoelastic material properties. In tension the ligaments show a non-linear characteristic at the initial stages of the load application (usually referred to as the toe region) but once a given reference strain or extension has been exceeded, the ligaments respond in a linear manner to loading. The reason for the non linearities in the toe region is due to the fiber orientation within the ligaments. The collagen fibers are placed in a "wavy" type of fashion and the initial load applied to the ligament goes to straighten the fibers and then they can be stretched in a linear fashion. Another reason is that the fiber lengths within the ligament differ and the initial loading goes to pull the fibers to the same length (Amis 1985). After the linear region then the ligaments follow another period of nonlinear behaviour where the stiffness decreases due to fibre failure until it reaches complete failure

The extrinsic ligaments are generally stiffer but weaker than the intrinsic ligaments which are elastic and strong. In 1991, Logan and Nowak carried out a study where two extrinsic ligaments (the radiocapitate (RC) and the radiolunate (RL)) and two intrinsic ligaments (the scapholunate (SL) and the lunotriquetrum (LT)) were tested to demonstrate the biomechanical difference between the two types of ligaments. Table 3 shows the findings from the study from Logan and Nowak.


**Table 3.** Results from Logan and Nowak on ligament material properties

From the table it can be seen that the loading rate primarily affects the extrinsic ligaments, making them stiffer and stronger under a rapid loading. This mechanism helps preventing ligament injury during fall, as the extrinsic ligaments anchor the mobile carpal bones to the radius and the ulna.

Tensile experiments on ligaments are difficult to carry out in practice. Wrist ligaments in particular are too short to be tested on their own, so the attaching bones are dissected along with the ligament and are held rigid in the tensile machine. It can be difficult to compare ligament tensile studies because they can be performed under different conditions which can have profound effects on the experimental results on which modellers of the joint rely. Other material studies have been carried out and published in the literature on wrist ligament properties (Berger 1997, Bettinger 1999).

#### *4.2.4. Contact setup*

84 Finite Element Analysis – New Trends and Developments

material data obtained from Li et al (2007).

decreases due to fibre failure until it reaches complete failure

**Table 3.** Results from Logan and Nowak on ligament material properties

ligament properties (Berger 1997, Bettinger 1999).

from the study from Logan and Nowak.

radius and the ulna.

*4.2.3. Ligaments* 

finite element models, by using Mooney-Rivlin hyper-elastic material properties using the

Evaluating the material properties of ligaments pose a great challenge to researchers in multibone joint modelling as they operate only in tension and show viscoelastic material properties. In tension the ligaments show a non-linear characteristic at the initial stages of the load application (usually referred to as the toe region) but once a given reference strain or extension has been exceeded, the ligaments respond in a linear manner to loading. The reason for the non linearities in the toe region is due to the fiber orientation within the ligaments. The collagen fibers are placed in a "wavy" type of fashion and the initial load applied to the ligament goes to straighten the fibers and then they can be stretched in a linear fashion. Another reason is that the fiber lengths within the ligament differ and the initial loading goes to pull the fibers to the same length (Amis 1985). After the linear region then the ligaments follow another period of nonlinear behaviour where the stiffness

The extrinsic ligaments are generally stiffer but weaker than the intrinsic ligaments which are elastic and strong. In 1991, Logan and Nowak carried out a study where two extrinsic ligaments (the radiocapitate (RC) and the radiolunate (RL)) and two intrinsic ligaments (the scapholunate (SL) and the lunotriquetrum (LT)) were tested to demonstrate the biomechanical difference between the two types of ligaments. Table 3 shows the findings

**Rate SL [N] LT [N] RL [N] RC [N]**  1 mm/min 197.1 ± 35.5 241.1 ± 41.8 50.8 ± 14.8 84.3 ± 16.0 100 mm/min 232.6 ± 10.9 353.7 ± 69.2 107.2 ± 14.5 151.6 ± 23.0

From the table it can be seen that the loading rate primarily affects the extrinsic ligaments, making them stiffer and stronger under a rapid loading. This mechanism helps preventing ligament injury during fall, as the extrinsic ligaments anchor the mobile carpal bones to the

Tensile experiments on ligaments are difficult to carry out in practice. Wrist ligaments in particular are too short to be tested on their own, so the attaching bones are dissected along with the ligament and are held rigid in the tensile machine. It can be difficult to compare ligament tensile studies because they can be performed under different conditions which can have profound effects on the experimental results on which modellers of the joint rely. Other material studies have been carried out and published in the literature on wrist Once the bones have all been incorporated into the finite element software and assembled together bone by bone and cartilage constructed, the contact formulation between the bones needs to be formulated. A surface-to-surface contact is most common method defining the contact between the bones, but node-to-surface configuration can also be implemented. Most finite element models will allow the user different contact models, such as the Lagrange method, the penalty method etc. The availability of these different contact models can be limited to the type of solution algorithm used. Additionally the user can determine the stiffness of the contact, but usually as "hard contact" is applied which is defined by

$$\begin{array}{cc} p = 0, & h < 0 \\ h = 0, & p > 0 \end{array}$$

Where p is the contact pressure, and h is the over closure between the two surfaces. Using kinematic contact method is generally preferred over the penalty contact as it introduces an additional stiffness to the system. Frictionless contact properties or friction using a low friction coefficient should also implemented on the articulating surfaces. By using frictionless contact, it is ensured that no shear stresses occur at the articulations.

It has been reported in the literature (Kauer 1986) that there is little or no movement between certain articulations, such as the articulations between the bones in the distal row of the wrist and the metacarpals (in the carpometacarpal joint). For those joints, it is possible to use a tie constraint so that no relative motion occurs between the two bones. That will help to simplify the model. The model can be seen in Figure 4

**Figure 4.** Finite element model

## **5. Soft tissue modelling**

Due to the high mobility of joints such as the wrist and the ankle, they need to be constrained through a large and complicated set of ligaments to ensure structural integrity of the joint. Without any structural contribution from the ligaments, any finite model of the wrist or the ankle would diverge. As previously discussed then the material properties of each ligament will vary depending on its function and location.

Finite Element Modelling of a Multi-Bone Joint: The Human Wrist 87

Modelling the ligaments as hyperelastic resulted in larger motion of the ligaments than allowed by using the elastic properties, but less using the non-linear springs. The springs are most probably under- constraining the whole system, but using three dimensional ligaments with elastic material properties are probably over-constraining the system. More research needs to be carried out on the soft tissue properties of multibone joints and the constraining

With a computational model of the wrist in place, analysis of surgical procedures such as arthrodesis and arthroplasty can be carried out. Arthrodesis is a procedure that fuses together joints to reduce mobility. In the wrist and the ankle there are many individual joints and should just a single joint be fused, the procedure is called partial arthrodesis and a total arthrodesis if the whole joint is fused. This is a recognised surgical procedure to reduce pain and increase stability in the arthritic wrist. Simulating such procedures can be done using a finite element model, where instead of applying contact formulation a tie constraint is applied at the articulating joints. That will treat the two articulating bones as a single unit, not allowing any relative movement between them. After such a procedure it can be seen that the overall load transfer will be altered as additional constraints have been introduced to the system. This can be seen in particular on radiolunate fusion where high joint contact forces were seen on the capitolunate joint. Figure 6 shows the changes in load transfer in the midcarpal joint following radiolunate (RL), radioscaphoid (RS) and radioscapholunate (RSL)

effects various modelling techniques will have on the overall system.

**Figure 5.** Ligaments modelled as three dimensional surfaces.

**6. Modelling of surgical procedures** 

fusion compared to the untreated wrist

The geometry of the wrist ligaments is complex and difficult to incorporate into a finite element model. Some ligaments wrap around the carpal bones without attaching to them, thus providing additional dorsal/volar constraints on the carpus. This can be seen for the dorsal radiotriquetral ligament which originates at the distal end of the radius and attaches to the proximal pole of the triquetrum, overlapping the lunate and adding to the transverse stability of the carpus.

Previous models have incorporated the ligaments as one dimensional spring elements (Carrigan, Gislason, Bajuri), which is the simplest approach of creating the geometry. Although this method will give a relatively good representation regarding the overall constraints of the carpus, the problem will persist that the spring elements will only constrain the carpal bones in the direction of the springs. Using non-linear springs, the user must make sure that the springs do not take any tensile forces. The literature gives a range of ultimate strength and strain values (Berger 1999, Nowak 1991) for various ligaments which can be used to recreate a non-linear stress-strain or force-displacement curve in the form of

$$F = \begin{cases} 0, & x < 0\\ \frac{\alpha x^2}{2\varepsilon\_{ref}}, & 0 \le x < \varepsilon\_{ref} \\\ a x + b \,, & x \ge \varepsilon\_{ref} \end{cases}$$

Where F is the ligament force, x is the strain and α, a and b are constants. The force values can be converted into stress, by using measurements of the cross sectional areas of the ligaments as presented by by Feipel et al (1998).

Another possibility is to model the ligament as three dimensional surfaces using two dimensional elements, by identifying the insertion node points and creating the external lines of the ligament using splines, finally an area is defined from the lines and meshed using shell elements. Modelling the material behaviour can be modelled by implementing stress-strain curves for each ligament using hyperelastic material properties. The challenge in soft tissue modelling, beside the geometrical representation of the ligaments, is not over or under constraining the model. A figure of the model where ligaments are represented as three dimensional surfaces can be seen in Figure 5.

In a pilot study carried out on ligament modelling, it was seen that by using the elastic springs, there was a significant translation of the carpal bones, which decreased drastically by assuming linear elastic material properties of the ligaments. That over-constrained the system to a great extent and allowed extremely little bone movement under loading. Modelling the ligaments as hyperelastic resulted in larger motion of the ligaments than allowed by using the elastic properties, but less using the non-linear springs. The springs are most probably under- constraining the whole system, but using three dimensional ligaments with elastic material properties are probably over-constraining the system. More research needs to be carried out on the soft tissue properties of multibone joints and the constraining effects various modelling techniques will have on the overall system.

**Figure 5.** Ligaments modelled as three dimensional surfaces.

## **6. Modelling of surgical procedures**

86 Finite Element Analysis – New Trends and Developments

each ligament will vary depending on its function and location.

Due to the high mobility of joints such as the wrist and the ankle, they need to be constrained through a large and complicated set of ligaments to ensure structural integrity of the joint. Without any structural contribution from the ligaments, any finite model of the wrist or the ankle would diverge. As previously discussed then the material properties of

The geometry of the wrist ligaments is complex and difficult to incorporate into a finite element model. Some ligaments wrap around the carpal bones without attaching to them, thus providing additional dorsal/volar constraints on the carpus. This can be seen for the dorsal radiotriquetral ligament which originates at the distal end of the radius and attaches to the proximal pole of the triquetrum, overlapping the lunate and adding to the transverse

Previous models have incorporated the ligaments as one dimensional spring elements (Carrigan, Gislason, Bajuri), which is the simplest approach of creating the geometry. Although this method will give a relatively good representation regarding the overall constraints of the carpus, the problem will persist that the spring elements will only constrain the carpal bones in the direction of the springs. Using non-linear springs, the user must make sure that the springs do not take any tensile forces. The literature gives a range of ultimate strength and strain values (Berger 1999, Nowak 1991) for various ligaments which can be used

Ͳ ൏ ݔͲǡ

ߝ ݔ ǡܾ ݔܽ

Where F is the ligament force, x is the strain and α, a and b are constants. The force values can be converted into stress, by using measurements of the cross sectional areas of the

Another possibility is to model the ligament as three dimensional surfaces using two dimensional elements, by identifying the insertion node points and creating the external lines of the ligament using splines, finally an area is defined from the lines and meshed using shell elements. Modelling the material behaviour can be modelled by implementing stress-strain curves for each ligament using hyperelastic material properties. The challenge in soft tissue modelling, beside the geometrical representation of the ligaments, is not over or under constraining the model. A figure of the model where ligaments are represented as

In a pilot study carried out on ligament modelling, it was seen that by using the elastic springs, there was a significant translation of the carpal bones, which decreased drastically by assuming linear elastic material properties of the ligaments. That over-constrained the system to a great extent and allowed extremely little bone movement under loading.

ߝ൏ݔͲ ǡ

to recreate a non-linear stress-strain or force-displacement curve in the form of

ൌ ܨ

ligaments as presented by by Feipel et al (1998).

three dimensional surfaces can be seen in Figure 5.

ە ۖ ۔ ۖ

 ۓ ଶݔߙ ߝʹ

**5. Soft tissue modelling** 

stability of the carpus.

With a computational model of the wrist in place, analysis of surgical procedures such as arthrodesis and arthroplasty can be carried out. Arthrodesis is a procedure that fuses together joints to reduce mobility. In the wrist and the ankle there are many individual joints and should just a single joint be fused, the procedure is called partial arthrodesis and a total arthrodesis if the whole joint is fused. This is a recognised surgical procedure to reduce pain and increase stability in the arthritic wrist. Simulating such procedures can be done using a finite element model, where instead of applying contact formulation a tie constraint is applied at the articulating joints. That will treat the two articulating bones as a single unit, not allowing any relative movement between them. After such a procedure it can be seen that the overall load transfer will be altered as additional constraints have been introduced to the system. This can be seen in particular on radiolunate fusion where high joint contact forces were seen on the capitolunate joint. Figure 6 shows the changes in load transfer in the midcarpal joint following radiolunate (RL), radioscaphoid (RS) and radioscapholunate (RSL) fusion compared to the untreated wrist

Finite Element Modelling of a Multi-Bone Joint: The Human Wrist 89

**Figure 7.** Finite element model of a total wrist arthroplasty

**Figure 8.** Load transmission through the implanted wrist

The stresses on the carpal bones and the implant can be seen in Figure 8

**Figure 6.** Changes in joint contact forces following a partial wrist arthrodesis.

Using the finite element method can be a useful tool to predict a possible surgical outcome, as can be seen with the radiolunate fusion, an extremely high force can be seen acting on the capitolunate joint. This can be explained by the fact that during gripping (and most other tasks) the thumb will be angled in such a way that the joint contact forces acting on the first carpometacarpal joint will tend to push the carpus ulnarly. This can be seen in Figure 4 how the thumb forces tend to ulnarly translate. With the lunate anchored to the radius and the capitate free to translate, it can be seen that under such ulnarly directed forces the capitate will be excessively constrained by the lunate thus causing such high joint contact forces. It can be seen that by fusing both the radius and the lunate, the model predicts more evenly distributed load through the midcarpal joints, however at the expense of a smaller range of motion.

Finite element models on total hip and knee arthroplasty have been prominent in the literature and extensive research has been carried out on the stress distribution in the femur following a total hip arthroplasty and has contributed to the clinical success of the joint replacements. Little has been written about total wrist arthroplasty and the effects it has on the distribution of load within the wrist. Grosland et al have reported on wrist implants in terms of design and carried out ex-vivo analysis, but a model is missing that captures a full three dimensional features of the implanted wrist. A preliminary model was created of the implanted wrist under physiological loading. It showed how the majority of the load was transmitted through the implant and onto the radius. The finite element model can be seen in Figure 7.

**Figure 7.** Finite element model of a total wrist arthroplasty

88 Finite Element Analysis – New Trends and Developments

**Figure 6.** Changes in joint contact forces following a partial wrist arthrodesis.

motion.

in Figure 7.

Using the finite element method can be a useful tool to predict a possible surgical outcome, as can be seen with the radiolunate fusion, an extremely high force can be seen acting on the capitolunate joint. This can be explained by the fact that during gripping (and most other tasks) the thumb will be angled in such a way that the joint contact forces acting on the first carpometacarpal joint will tend to push the carpus ulnarly. This can be seen in Figure 4 how the thumb forces tend to ulnarly translate. With the lunate anchored to the radius and the capitate free to translate, it can be seen that under such ulnarly directed forces the capitate will be excessively constrained by the lunate thus causing such high joint contact forces. It can be seen that by fusing both the radius and the lunate, the model predicts more evenly distributed load through the midcarpal joints, however at the expense of a smaller range of

Finite element models on total hip and knee arthroplasty have been prominent in the literature and extensive research has been carried out on the stress distribution in the femur following a total hip arthroplasty and has contributed to the clinical success of the joint replacements. Little has been written about total wrist arthroplasty and the effects it has on the distribution of load within the wrist. Grosland et al have reported on wrist implants in terms of design and carried out ex-vivo analysis, but a model is missing that captures a full three dimensional features of the implanted wrist. A preliminary model was created of the implanted wrist under physiological loading. It showed how the majority of the load was transmitted through the implant and onto the radius. The finite element model can be seen The stresses on the carpal bones and the implant can be seen in Figure 8

**Figure 8.** Load transmission through the implanted wrist

From a finite element perspective, modelling a total wrist arthroplasty is a simpler task than modelling the healthy wrist as a few of the carpal bones will be removed during the procedure which will decrease the number of contact surfaces. However problems regarding the fixation of the implant into the radius and the distal row will arise as well as contact between the proximal and distal part. In the pilot study, it was assumed that the implant was fully fixed in the radius as well as the distal component fully tied to the carpal bones in the distal row. There are many different types of wrist implants commercially available and the personal preference of the surgeon will in many cases determine which implant will be used. A finite element model will allow to virtually implant a prosthesis into the carpus and calculate the stresses under static loading. The main problem with carrying out such experiments is that the size and the location of the implant could be erroneous which will have a large impact on the overall solution.

Finite Element Modelling of a Multi-Bone Joint: The Human Wrist 91

Subject 1 Subject 2 Subject 3

 Fx [N] Fy [N] Fz [N] Fx [N] Fy [N] Fz [N] Fx [N] Fy [N] Fz [N] Digit 1 144.1 -545.1 -44.6 80.8 -536.1 -8.4 139.7 -452.2 -12.0 Digit 2 253.2 -270.7 141.8 84.1 -294.2 10.5 110.7 -156.8 87.4 Digit 3 348.5 -274.4 172.8 135.1 -126.2 72.8 125.6 -237.7 98.9 Digit 4 117.3 -236.1 29.2 67.0 -94.0 54.7 113.7 -198.0 78.5 Digit 5 111.1 -200.0 -3.8 42.5 -103.0 10.6 53.5 -160.5 19.3

As can be seen from Table 4, the contact forces were primarily directed, ulnarly, proximally and dorsally. The joint contact forces were applied to the model as nodal forces where a subset of nodes was chosen and the total force acting on each metacarpal was divided

The proximal ends of the radius and ulna were kept fixed and compressive forces applied to

**Figure 9.** Grip pattern used for the analysis

**Table 4.** Internal loading on the digits

the distal end of the metacarpals.

 Positive x-direction denotes ulnar direction Positive y direction denotes distal direction Positive z-direction denotes dorsal direction

Where

between the nodes.

The finite element method can be used as a tool to evaluate the different implant designs available on the market. Given the high failure rate of the implants, there is a demand to investigate closer the effects that a total wrist arthroplasty has on the overall load transfer through the wrist and what can be done to design for longevity and functionality of the implant.

## **7. Loading conditions**

Applying in vivo loading conditions on the finite element model, is an extremely challenging aspect of the modelling, especially since there has been very little written about the biomechanical modelling of the wrist. Most studies have applied arbitrary loading conditions, 15 N compressive force acting on the distal end of the capitate (Carrigan et al), a combined compressive load of 100 N applied to the 2nd and 3rd metacarpal (Guo et al) and a combined 1000 N load acting on the scaphoid and lunate (Ulrich et al). The load cases are better defined when dealing with joints in the lower limb and the fundamental question, researchers must ask themselves is "what activity is characteristic for loading on the upper limb?". The answer to that is not clear cut and can range from compressive forces acting on the proximal part of the palm with subject trying to push an object to forces action on the fingers via gripping. There are many grip patterns defined in the literature (chuck grip, power grip, pinch grip etc.) which all contribute in a unique manner to the loading distribution through the fingers.

For the analysis presented in this chapter a grip pattern, seen in Figure 9 was used.

The gripping forces were obtained through a biomechanical study where the gripping strength of 50 subjects were measured using five 6-degrees of freedom force transducer (Nano 25-E and Nano 17, ATI Industrial Automation Inc, USA). Simultaneous collection of position data using an 8 camera motion capture system (Vicon, Oxford Metrics Ltd) was carried out to capture both the kinetic and the kinematic data. The external forces were converted in to joint contact forces acting on the metacarpals using a biomechanical model as described by Fowler and Nicol (2000). More detailed analysis on execution of the biomechanical trials can be found in Gislason et al (2009). The wrist models created were subject specific and the joint contact forces applied can be seen in Table 4

**Figure 9.** Grip pattern used for the analysis


**Table 4.** Internal loading on the digits

Where

90 Finite Element Analysis – New Trends and Developments

which will have a large impact on the overall solution.

implant.

**7. Loading conditions** 

distribution through the fingers.

From a finite element perspective, modelling a total wrist arthroplasty is a simpler task than modelling the healthy wrist as a few of the carpal bones will be removed during the procedure which will decrease the number of contact surfaces. However problems regarding the fixation of the implant into the radius and the distal row will arise as well as contact between the proximal and distal part. In the pilot study, it was assumed that the implant was fully fixed in the radius as well as the distal component fully tied to the carpal bones in the distal row. There are many different types of wrist implants commercially available and the personal preference of the surgeon will in many cases determine which implant will be used. A finite element model will allow to virtually implant a prosthesis into the carpus and calculate the stresses under static loading. The main problem with carrying out such experiments is that the size and the location of the implant could be erroneous

The finite element method can be used as a tool to evaluate the different implant designs available on the market. Given the high failure rate of the implants, there is a demand to investigate closer the effects that a total wrist arthroplasty has on the overall load transfer through the wrist and what can be done to design for longevity and functionality of the

Applying in vivo loading conditions on the finite element model, is an extremely challenging aspect of the modelling, especially since there has been very little written about the biomechanical modelling of the wrist. Most studies have applied arbitrary loading conditions, 15 N compressive force acting on the distal end of the capitate (Carrigan et al), a combined compressive load of 100 N applied to the 2nd and 3rd metacarpal (Guo et al) and a combined 1000 N load acting on the scaphoid and lunate (Ulrich et al). The load cases are better defined when dealing with joints in the lower limb and the fundamental question, researchers must ask themselves is "what activity is characteristic for loading on the upper limb?". The answer to that is not clear cut and can range from compressive forces acting on the proximal part of the palm with subject trying to push an object to forces action on the fingers via gripping. There are many grip patterns defined in the literature (chuck grip, power grip, pinch grip etc.) which all contribute in a unique manner to the loading

For the analysis presented in this chapter a grip pattern, seen in Figure 9 was used.

subject specific and the joint contact forces applied can be seen in Table 4

The gripping forces were obtained through a biomechanical study where the gripping strength of 50 subjects were measured using five 6-degrees of freedom force transducer (Nano 25-E and Nano 17, ATI Industrial Automation Inc, USA). Simultaneous collection of position data using an 8 camera motion capture system (Vicon, Oxford Metrics Ltd) was carried out to capture both the kinetic and the kinematic data. The external forces were converted in to joint contact forces acting on the metacarpals using a biomechanical model as described by Fowler and Nicol (2000). More detailed analysis on execution of the biomechanical trials can be found in Gislason et al (2009). The wrist models created were


As can be seen from Table 4, the contact forces were primarily directed, ulnarly, proximally and dorsally. The joint contact forces were applied to the model as nodal forces where a subset of nodes was chosen and the total force acting on each metacarpal was divided between the nodes.

The proximal ends of the radius and ulna were kept fixed and compressive forces applied to the distal end of the metacarpals.

Many studies have applied arbitrary boundary conditions onto the wrist, which will not give information about the possible in-vivo behaviour of the carpal bones under loading. By applying physiologically relevant loading conditions, it is possible to determine in more detail the mechanical features within the wrist that control the loading. Due to the extensive research carried out on the biomechanics of the hip and knee, modellers are able to apply physiologically relevant loading conditions onto their models and predict in-vivo loading.

Finite Element Modelling of a Multi-Bone Joint: The Human Wrist 93

In the finite element models, the largest stress was seen at the in the cortical shell and were on average a magnitude higher than the stresses in the cancellous bone. On average the stresses in the cortical shell were around 18.6 MPa, and in the cancellous bone they were

Ligaments opposing ulnar translation were more active than others in the model, in particular the dorsal radiotriquetral ligament which showed high degree of force going through it. That result is in agreement with the theoretical findings of Garcia-Elias (1995) who stated that in order to maintain stability, the dorsal radiotriquetral ligament would

The force through the radius and ulna was distributed so that majority of the load was taken by the radius, ranging from 79-93% which is in agreement with the findings of Palmer and Werner (1984) who measured the load distribution between the two forearm bones using a

Validation is an important procedure to verify that the assumptions used for the computational model are correct. In 2005, guidelines were written by Viceconti regarding the methodology of producing a clinically relevant finite element model. There two

load cell and reported that 80% of the loading was transmitted through the radius.

play an integral part in stabilisation of the carpus during gripping.

around 1.1 MPa. The stress distribution for one of the model can be seen in Figure 10

**Figure 10.** von Mises stresses in a single model

**9.2. Validation** 

## **8. Solution algorithms**

For a multibody computational models, it is virtually impossible to solve an implicit model where convergence needs to be obtained for each contact surface for each loadstep. High residual forces at the boundaries of the contact surfaces are primarily seen that cause the solution to diverge. Damping can be introduced between the bones, which can be released gradually as the load step progresses and will be fully released when all of the loading has been applied. Experiments showed that the load step progressed well at the initial stages of the load step, but once the effects of the damping became less, cutbacks were seen in the solution process which increased as the solution reached towards the end of the load step. The solution never will reach the end of the loadstep. This is a classical behaviour of the proper contact not being established between the bones. It has been previously demonstrated in the literature how nonlinearities can cause divergence using the implicit code (Harewood 2007).

Most multibody analyses use the explicit algorithm to solve the model. The explicit algorithm assumes dynamic behaviour of the model and no convergence checks are carried out on the contact surfaces, which makes the explicit algorithm extremely robust in solving such a multi body system. The solution for time step t +Δt is based on the status of the model at the previous time step, t. In contrast for the implicit code the solution is based on the same time step. The time step in the explicit analysis is determined from the characteristic element length and material properties and is given by

$$
\Delta t \le \frac{2}{\omega\_{\max}}
$$

where ωmax is the maximum eigenvalue in the system. Generally the time steps, Δt, are very small, resulting in long run times. The criteria for assuming a quasi static solution, is that the kinetic energy of the system does not exceed 5% of the strain energy.

## **9. Results**

## **9.1. Finite element results**

The results from the finite element model have shown that anatomical features play an integral role in the stress distribution through the wrist and therefore it is difficult to generalise about the results of a single standard model. However due to the complexity and time commitment creating the finite element models, it is not possible to generate a large cohort of models.

In the finite element models, the largest stress was seen at the in the cortical shell and were on average a magnitude higher than the stresses in the cancellous bone. On average the stresses in the cortical shell were around 18.6 MPa, and in the cancellous bone they were around 1.1 MPa. The stress distribution for one of the model can be seen in Figure 10

**Figure 10.** von Mises stresses in a single model

Ligaments opposing ulnar translation were more active than others in the model, in particular the dorsal radiotriquetral ligament which showed high degree of force going through it. That result is in agreement with the theoretical findings of Garcia-Elias (1995) who stated that in order to maintain stability, the dorsal radiotriquetral ligament would play an integral part in stabilisation of the carpus during gripping.

The force through the radius and ulna was distributed so that majority of the load was taken by the radius, ranging from 79-93% which is in agreement with the findings of Palmer and Werner (1984) who measured the load distribution between the two forearm bones using a load cell and reported that 80% of the loading was transmitted through the radius.

### **9.2. Validation**

92 Finite Element Analysis – New Trends and Developments

**8. Solution algorithms** 

**9. Results** 

cohort of models.

**9.1. Finite element results** 

Many studies have applied arbitrary boundary conditions onto the wrist, which will not give information about the possible in-vivo behaviour of the carpal bones under loading. By applying physiologically relevant loading conditions, it is possible to determine in more detail the mechanical features within the wrist that control the loading. Due to the extensive research carried out on the biomechanics of the hip and knee, modellers are able to apply physiologically relevant loading conditions onto their models and predict in-vivo loading.

For a multibody computational models, it is virtually impossible to solve an implicit model where convergence needs to be obtained for each contact surface for each loadstep. High residual forces at the boundaries of the contact surfaces are primarily seen that cause the solution to diverge. Damping can be introduced between the bones, which can be released gradually as the load step progresses and will be fully released when all of the loading has been applied. Experiments showed that the load step progressed well at the initial stages of the load step, but once the effects of the damping became less, cutbacks were seen in the solution process which increased as the solution reached towards the end of the load step. The solution never will reach the end of the loadstep. This is a classical behaviour of the proper contact not being established between the bones. It has been previously demonstrated in the literature

how nonlinearities can cause divergence using the implicit code (Harewood 2007).

characteristic element length and material properties and is given by

kinetic energy of the system does not exceed 5% of the strain energy.

Most multibody analyses use the explicit algorithm to solve the model. The explicit algorithm assumes dynamic behaviour of the model and no convergence checks are carried out on the contact surfaces, which makes the explicit algorithm extremely robust in solving such a multi body system. The solution for time step t +Δt is based on the status of the model at the previous time step, t. In contrast for the implicit code the solution is based on the same time step. The time step in the explicit analysis is determined from the

�� � �

2 ���� where ωmax is the maximum eigenvalue in the system. Generally the time steps, Δt, are very small, resulting in long run times. The criteria for assuming a quasi static solution, is that the

The results from the finite element model have shown that anatomical features play an integral role in the stress distribution through the wrist and therefore it is difficult to generalise about the results of a single standard model. However due to the complexity and time commitment creating the finite element models, it is not possible to generate a large

Validation is an important procedure to verify that the assumptions used for the computational model are correct. In 2005, guidelines were written by Viceconti regarding the methodology of producing a clinically relevant finite element model. There two

important assessment tools for finite element models were introduced, *verification* and *validation*. The term verification is used to check numerical accuracy, that is how well the underlying equations are solved. To verify the model, the user can check that forces at all reactions sum up to give the input forces. Another example of verification can be seen when energy values are compared to check whether the solution is portraying quasi-static behaviour. The term validation is used to assess how well the underlying equations describe the physical phenomena. Validation must be carried out in the lab to test a specimen under the same conditions used in the computational model. Computational models are capable of creating complex load cases, so through validation some simplification generally must be done, which then can then be re-created through the computational model.

Validation of the computational model was carried out through two separate experiments. One measured the strain on the radius and ulnar with the carpus loaded through pull of the tendons (MacLeod 2007). The second measured the joint contact pressure of the radioscaphoid joint using a pressure sensitive film. Setup of the two experiments can be seen in Figure 11.

It was measured using the strain gauges on the radius and ulna that the load through the radius is around 70% and the remaining 30% through the ulna. These values are slightly lower than what the finite element model was predicting, but both recognise the radius as the main load bearing structure of the forearm.

The measurements of the contact pressure on the radioscaphoid joint showed that the joint contact pressure ranged between 4-5 MPa under a 600 N compressive load which is in agreement with the findings of the finite element model which predicted 6.5 MPa contact pressure on the joint under the same loading conditions.

Finite Element Modelling of a Multi-Bone Joint: The Human Wrist 95

Creating a finite element model of the wrist and other multibody joints is a complex task where many different aspects of the modelling need to be addressed. The most important aspect contributing to a high quality finite element model is the construction of high integrity geometrical model and the soft tissue modelling. High integrity geometrical model of the articulating surfaces will aid the contact analysis, as a high degree of incongruence of the articulating surfaces can lead to element distortion, especially on soft cartilage elements. The external soft tissue constraints are important in order to maintain mechanical equilibrium as well as allowing the bones to translate and rotate under loading. These two

Finite element models of such complex joints such as the wrist and the ankle are likely to become more prominent in the future as computational power and modelling software quality increases. That will make modellers able to create models incorporation a higher

It is inevitable that errors are introduced in such complex models. The errors can either be within the control of the modeller or without. This chapter has discussed the procedures that the modeller can carry out to minimise the sources of errors in the model. However the modeller will have little control over errors that can be generated through using previously published material properties and geometrical representation of the ligaments and soft

Using the finite element method predicting the load transfer through the healthy and the pathological wrist can give clinicians important information regarding the choice of treatment which can lead to higher procedure success rates and improve the quality of life

*Department of Mechanical and Aerospace Engineering, University of Strathclyde, Glasgow, UK* 

[1] Maragoni RD and Glaser AA: Viscoelastic properties of soft tissue model

[2] Rybicki EF, Simonen FA, Weis Jr EB: On the mathematical analysis of stress in human

[3] Bergmann G, Graichen F, Rohlmann A: Hip joint loading during walking and running,

[4] Taylor M, Barrett, DS: Explicit finite element simulation of eccentric loading in total knee replacement. Clinical Orthopaedics and Related Research, 2003, 414:162-171.

factors will play an integral role in the success of the finite element model.

degree of detail than previously has been published.

Magnús Kjartan Gíslason and David H. Nash

femur, J Biomech, 1972, 5(2):203-215.

characterization, J Biomech, 1968, 1(1): 33-36.

measured in two patients. J Biomech, 1993, 26(8):969–90.

**10. Conclusions** 

tissue.

for many patients.

**Author details** 

**11. References** 

**Figure 11.** Validation of the finite element model

## **10. Conclusions**

94 Finite Element Analysis – New Trends and Developments

the main load bearing structure of the forearm.

**Figure 11.** Validation of the finite element model

pressure on the joint under the same loading conditions.

seen in Figure 11.

important assessment tools for finite element models were introduced, *verification* and *validation*. The term verification is used to check numerical accuracy, that is how well the underlying equations are solved. To verify the model, the user can check that forces at all reactions sum up to give the input forces. Another example of verification can be seen when energy values are compared to check whether the solution is portraying quasi-static behaviour. The term validation is used to assess how well the underlying equations describe the physical phenomena. Validation must be carried out in the lab to test a specimen under the same conditions used in the computational model. Computational models are capable of creating complex load cases, so through validation some simplification generally must be

Validation of the computational model was carried out through two separate experiments. One measured the strain on the radius and ulnar with the carpus loaded through pull of the tendons (MacLeod 2007). The second measured the joint contact pressure of the radioscaphoid joint using a pressure sensitive film. Setup of the two experiments can be

It was measured using the strain gauges on the radius and ulna that the load through the radius is around 70% and the remaining 30% through the ulna. These values are slightly lower than what the finite element model was predicting, but both recognise the radius as

The measurements of the contact pressure on the radioscaphoid joint showed that the joint contact pressure ranged between 4-5 MPa under a 600 N compressive load which is in agreement with the findings of the finite element model which predicted 6.5 MPa contact

done, which then can then be re-created through the computational model.

Creating a finite element model of the wrist and other multibody joints is a complex task where many different aspects of the modelling need to be addressed. The most important aspect contributing to a high quality finite element model is the construction of high integrity geometrical model and the soft tissue modelling. High integrity geometrical model of the articulating surfaces will aid the contact analysis, as a high degree of incongruence of the articulating surfaces can lead to element distortion, especially on soft cartilage elements. The external soft tissue constraints are important in order to maintain mechanical equilibrium as well as allowing the bones to translate and rotate under loading. These two factors will play an integral role in the success of the finite element model.

Finite element models of such complex joints such as the wrist and the ankle are likely to become more prominent in the future as computational power and modelling software quality increases. That will make modellers able to create models incorporation a higher degree of detail than previously has been published.

It is inevitable that errors are introduced in such complex models. The errors can either be within the control of the modeller or without. This chapter has discussed the procedures that the modeller can carry out to minimise the sources of errors in the model. However the modeller will have little control over errors that can be generated through using previously published material properties and geometrical representation of the ligaments and soft tissue.

Using the finite element method predicting the load transfer through the healthy and the pathological wrist can give clinicians important information regarding the choice of treatment which can lead to higher procedure success rates and improve the quality of life for many patients.

## **Author details**

Magnús Kjartan Gíslason and David H. Nash *Department of Mechanical and Aerospace Engineering, University of Strathclyde, Glasgow, UK* 

## **11. References**

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[19] Cheung JTM, Zhang M, Leung AKL and Fan YB: Three Dimensional Finite Element Analysis of the Foot During Standing: A Material Sensitive Study, Journal of

[20] Louis O, Willnecker J, Soykens S, Van den Winkel P and Osteaux M: Cortical Thickness Assessed by Peripheral Quantitative Computed Tomography: Accuracy Evaluated on

[21] Ramos A and Simões J. Tetrahedral versus hexahedral finite elements in numerical modelling of the proximal femur. Medical Engineering & Physics, 2006, 28(9):916-924. [22] Rho, JY, Tsui TY and Pharr GM: Elastic properties of human cortical and trabecular lamellar bone measured by nanoindentation, Biomaterials, 1997, 18(20), 1325- 1330. [23] Li Z, Kim JE, Davidson JS, Etheridge BS, Alonso JE and Eberhardt AW: Biomechanical Response of the Pubic Symphysis in Lateral Pelvic Impacts: A Finite Element Study,

[24] A. Amis. Ligament Injuries and Their Treatment, chapter Biomechanics of Ligaments,

[25] Logan S and Nowak M: Distinguishing biomechanical properties and intrinsic and extrinsic human wrist ligaments. Journal of Biomechanical Engineering, 1991, 113(1):85–

[27] Bettinger PC, Linscheid, RL, Berger RA,Cooney WP and An KN: An anatomic study of the stabilizing ligaments of the trapezium and trapeziometacarpal joint. J. Hand Surg.

[28] Berger R, Imeada T, Berglund L and An K: Constraint and material properties of the subregions of the scapholunate interosseous ligament. J Hand Surg Am, 24(5):953–62,

[29] Nowak M: Biomechanics of the Wrist Joint, chapter Material Properties of Ligaments.

[30] Kauer JM: The mechanism of the carpal joint, Clinical Orthopaedics and Related

[31] Feipel V, Salvia P and Rooze M: A new method for measuring wrist joint ligament length changes during sagittal and frontal motion, Clinical Biomechanics, 1998, 13(2):

[32] Grosland N, Rogge RD and Adams BD: Influence of articular geometry on prosthetic wrist stability, Clinical Orthopaedics and Related Research, 2004, 421:134-142. [33] Fowler NK and Nicol AC: Interphalangeal Joint and Tendon Forces: Normal Model and Biomechanical Consequences of Surgical Reconstruction, Journal of Biomechanics, 2000,

[34] Harewood FJ and McHugh PE: Comparison of the Implicit and Explicit Finite Element Methods Using Crystal Plasticity, Computational Materials Science, 2007, 39,

Radius Specimens, Osteoporosis International, 1995, 5, 446-449.

[26] R. Berger. The ligaments of the wrist. Hand Clinics, 1997, 13(1):63–82.

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[5] Garcia-Elias M, Ribe M, Rodriguez J, Cost J, and Casas J: Influence of joint laxity on

[6] Anderson DD and Daniel TE: A Contact-Coupled Finite Element Analysis of the

[7] Miyake T, H. Hashizumea H , Inouea H, Shia Q, Nagayama N: Malunited Colles' fracture Analysis of stress distribution, Journal of Hand Surgery (European volume),

[8] Anderson DD, Deshpande BR, Daniel TE and Baratz ME: A Three-Dimensional Finite Element Model of the Radiocarpal Joint: Distal Radius Fracture Step-off and Stress

[9] Schuind F, Cooney WP, Linscheid RL, An KN and Chao EYS: Force and Pressure Transmission Through the Normal Wrist: A Theoretical Two-Dimensional Study in the

[10] Ulrich D, van Rietbergen B, Laib A and Ruegsegger P: Load transfer analysis of the distal radius from in-vivo high resolution ct-imaging. Journal of Biomechanics, 1999,

[11] Oda M, Hashizume H, Miyake T, Inoue H and Nagayama N. A stress distribution analysis of a ceramic lunate replacement for kienbok's disease. Journal of Hand Surgery

[12] Carrigan SD, Whiteside RA, Pichora DR and Small CF: Developement of a Three Dimensional Finite Element Model for Carpal Load Transmission in a Static Neutral

[13] Nedoma J, Klézl Z, Fousek J, Kestřánek Z, Stehlík J: Numerical Simulation of Some Biomechanical Problems, Mathematics and Computers in Simulation, 2003, 61, 283-295. [14] Gislason M, Nash DH, Nicol A, Kanellopoulos A, Bransby-Zachary M, Hems TEJ, Condon B and Stansfield B.: A Three Dimensional Finite element Model of Maximal Grip Loading in the Human Wrist, Proc. IMechE Part H, Engineering in Medicine, 2009,

[15] Gislason M, Stansfield B and Nash D: Finite element creation and stability considerations of complex biological articulations: The human wrist joint, Medical

[16] Guo X, Fan Y and Li ZM: Effects of Dividing the Transverse Carpal Ligament on the Mechanical Behaviour of the Carpal Bones under Axial Compressive Load: A Finite

[17] Bajuria MN,Mohammed Rafiq Abdul Kadira, Murali Malliga Ramanb, Kamarul T: Mechanical and functional assessment of the wrist affected by rheumatoid arthritis: A

[18] Chen WP, Ju CW and Tang FT: Effect of Total Contact Insoles on the Plantar Stress Redistribution: A Finite Element Analysis, Clinical Biomechanics,2003, 18, 17-24.

scaphoid kinematics. Journal of Hand Surgery, 1995, 20B(3):379–382.

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**Section 2** 

**Recent Advances of Finite Element Analysis** 

**in "Electrical Engineering"** 


**Recent Advances of Finite Element Analysis in "Electrical Engineering"** 

98 Finite Element Analysis – New Trends and Developments

13(1):151–158.

Taiwan 2007.

Res, 1984, 187:26–35.

[35] Garcia-Elias M: Kinetic analysis of carpal stability during grip. Hand Clinics, 1997,

[36] Palmer A and Werner F. Biomechanics of the distal radioulnar joint. Clin. Orthop. Rel.

[37] Viceconti M, Olsen S, Nolte L and Burton K: Extracting clinically relevant data from finite element simulations (editorial). Clinical Biomechanics, 2005, 20:451–454. [38] Macleod NA, Nash DH, Stansfield BW, Bransby-Zachary M and Hems T: Cadaveric Analysis of the Wrist and Forearm Load Distribution for Finite Element Validation, Proceedings of the 6th International Hand and Wrist Biomechanics Symposium, Tainan,

**Chapter 5** 

© 2012 Virjoghe et al., licensee InTech. This is an open access chapter distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

© 2012 Virjoghe et al., licensee InTech. This is a paper distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

**Finite Element Analysis of** 

**Stationary Magnetic Field** 

Additional information is available at the end of the chapter

infinite points, the function values are in infinite number.

the analytical way for solving the problems.

http://dx.doi.org/10.5772/50846

(Morozionkov et al., 2008).

obtained (Gârbea, 1990).

**1. Introduction** 

Elena Otilia Virjoghe, Diana Enescu, Mihail-Florin Stan and Marcel Ionel

Computer-aided analysis of field distribution for evaluating electromagnetic device or component performance has become the most advantageous way of design. Analytical methods have limited uses and experimental methods are time intensive and expensive

The problems of magnetic fields calculation are aimed at determining the value of one or more unknown functions for the field considered, such as magnetic field intensity, magnetic flux density, magnetic scalar potential and magnetic vector potential. As the field has

Physical phenomena of electromagnetic nature are described by Maxwell's equations from the mathematical point of view. These are differential equations with the given boundary conditions. By means of them, the exact solution of the problem is obtained. In this way, the value of function or functions in any point of the studied range is calculated. This represents

Analytical methods (conformable representation method, method of separation of the variables, Green function method) are applied to solve relatively simple problems. Problems which occur in practice are often complex concerning the geometric construction, material heterogeneity, loading conditions, boundary conditions, so that the integration of differential equations is difficult or sometimes impossible. In this case, the analytical solution can be carried out only by creating a simplified model so that the integration of differential equations is possible. Therefore, an exact solution for a simplified model can be

It is sometimes preferable to obtain, instead of the exact solution of the simplified model, an approximate solution of the real problem. Approximate solutions which are obtained by numerical methods reflect better the reality than exact solutions of a simplified model.

## **Finite Element Analysis of Stationary Magnetic Field**

Elena Otilia Virjoghe, Diana Enescu, Mihail-Florin Stan and Marcel Ionel

Additional information is available at the end of the chapter

http://dx.doi.org/10.5772/50846

## **1. Introduction**

Computer-aided analysis of field distribution for evaluating electromagnetic device or component performance has become the most advantageous way of design. Analytical methods have limited uses and experimental methods are time intensive and expensive (Morozionkov et al., 2008).

The problems of magnetic fields calculation are aimed at determining the value of one or more unknown functions for the field considered, such as magnetic field intensity, magnetic flux density, magnetic scalar potential and magnetic vector potential. As the field has infinite points, the function values are in infinite number.

Physical phenomena of electromagnetic nature are described by Maxwell's equations from the mathematical point of view. These are differential equations with the given boundary conditions. By means of them, the exact solution of the problem is obtained. In this way, the value of function or functions in any point of the studied range is calculated. This represents the analytical way for solving the problems.

Analytical methods (conformable representation method, method of separation of the variables, Green function method) are applied to solve relatively simple problems. Problems which occur in practice are often complex concerning the geometric construction, material heterogeneity, loading conditions, boundary conditions, so that the integration of differential equations is difficult or sometimes impossible. In this case, the analytical solution can be carried out only by creating a simplified model so that the integration of differential equations is possible. Therefore, an exact solution for a simplified model can be obtained (Gârbea, 1990).

It is sometimes preferable to obtain, instead of the exact solution of the simplified model, an approximate solution of the real problem. Approximate solutions which are obtained by numerical methods reflect better the reality than exact solutions of a simplified model.

© 2012 Virjoghe et al., licensee InTech. This is an open access chapter distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. © 2012 Virjoghe et al., licensee InTech. This is a paper distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

The software package ANSYS can be used for investigation of the magnetic field distribution (the magnetic flux density, the magnetic field intensity and the magnetic vector potential) and basic electromagnetic characteristics (inductance and electromagnetic force). A typical magnetic field problem is described by defining the geometry, material properties, currents, boundary conditions, and the field system equations. The computer requires the input dates, the numerical solution of the field equation and output of desired parameters. If the values are found unsatisfactory, the design modified and parameters are recalculated. The process is repeated until optimum values for the design parameters are obtained.

Finite Element Analysis of Stationary Magnetic Field 103

(1)

*div B* <sup>0</sup> (2)

<sup>0</sup> (3)

*M MH t t* . (4)

(5)

is the magnetic

and the magnetic field

is the magnetic polarization. This

(6)

is the total current density, *B*

is called the temporary magnetization of the material and

*t* 

. A

moving 0 *<sup>v</sup>* and the electric and magnetic quantities are invariable in time, <sup>0</sup> .

equations:

Here, *H*

 

intensity *H*

where *Mp*

1975) :

0 

induction, quantity *Mt*

permanent magnets ( 0 *J*





7 1

**2.2. Models for the** *B***-***H* **relation** 

*2.2.1. Linear and isotropic materials* 

is the magnetic field intensity, *<sup>J</sup>*

4 10 *H m* is the vacuum permeability.

; 0 *<sup>v</sup>* ).

Depending on the relation between the magnetic induction *B*

is called the permanent magnetization and *pI*

stationary magnetic field in a conducting domain satisfies the following system of

*rot H J* 

*B HM* 

, a few types of materials are distinguished (Andrei et al. 2012).

The most important type of materials consists of the linear and isotropic materials, in which:

<sup>0</sup> *<sup>p</sup> <sup>p</sup> B H M HI*

category includes the materials for which the temporary magnetization law is (Răduleţ,

*Mt m <sup>H</sup>*

 

Magnetostatics is the branch in electromagnetism that studies the stationary magnetic states that do not accompany the conduction electric currents. This magnetic field is produced by

The ANSYS program is based on the finite element method (FEM) for solving Maxwell's equations and can be used for electromagnetic field modeling, where the field is electrostatics, magnetostatics, eddy currents, time-invariant or time-harmonic and permanent magnets (ANSYS Documentation).

The finite elements method assures sufficient accuracy of electromagnetic field computation and very good flexibility when geometry is modeled and field sources are loaded.

## **2. The fundamental relations of the stationary magnetic field**

In this section, we discuss the particular forms of the electromagnetic field theory laws for the magnetic stationary field. We consider the models of the magnetic induction versus magnetic field intensity (*B*-*H*) relation, passing conditions through discontinuity surfaces, the enunciation of stationary magnetic field (the sources of the field, boundary conditions), the enunciation of scalar magnetic potential - magnetostatic field problems (Dirichlet conditions, Neumann conditions) and the enunciations using the magnetic vector potential (stationary magnetic field problems). The general formulation of the uniqueness conditions gets particular forms, adapted to some geometrical configurations (plane-parallel fields, with rotation symmetry, etc.).

Depending on the relation between the magnetic induction and the intensity of the magnetic field, a few types of materials are distinguished, the most important being linear and isotropic materials, linear and non-isotropic materials, linear and non-isotropic materials, non-linear and isotropic materials, without permanent magnetization, non-linear and nonisotropic materials, materials with hysteresis.

Non-linear and isotropic materials, without permanent magnetization, are ferromagnetic materials, which are frequently used in the production of electric equipment.

## **2.1. Particular forms of the electromagnetic field theory laws for the stationary magnetic field**

The stationary magnetic field is established by non-moving, permanently magnetized bodies and by non-moving connecting wires crossed by direct current (Mocanu, 1981). Fundamental magnetic field relationships result by customizing the general laws and material laws of the electromagnetic field in the following conditions: bodies are nonmoving 0 *<sup>v</sup>* and the electric and magnetic quantities are invariable in time, <sup>0</sup> . *t* . A stationary magnetic field in a conducting domain satisfies the following system of equations:


$$
\vec{I} \text{rot}\,\vec{H} = \vec{J} \tag{1}
$$


102 Finite Element Analysis – New Trends and Developments

permanent magnets (ANSYS Documentation).

with rotation symmetry, etc.).

**magnetic field** 

isotropic materials, materials with hysteresis.

The software package ANSYS can be used for investigation of the magnetic field distribution (the magnetic flux density, the magnetic field intensity and the magnetic vector potential) and basic electromagnetic characteristics (inductance and electromagnetic force). A typical magnetic field problem is described by defining the geometry, material properties, currents, boundary conditions, and the field system equations. The computer requires the input dates, the numerical solution of the field equation and output of desired parameters. If the values are found unsatisfactory, the design modified and parameters are recalculated.

The process is repeated until optimum values for the design parameters are obtained.

The ANSYS program is based on the finite element method (FEM) for solving Maxwell's equations and can be used for electromagnetic field modeling, where the field is electrostatics, magnetostatics, eddy currents, time-invariant or time-harmonic and

The finite elements method assures sufficient accuracy of electromagnetic field computation

In this section, we discuss the particular forms of the electromagnetic field theory laws for the magnetic stationary field. We consider the models of the magnetic induction versus magnetic field intensity (*B*-*H*) relation, passing conditions through discontinuity surfaces, the enunciation of stationary magnetic field (the sources of the field, boundary conditions), the enunciation of scalar magnetic potential - magnetostatic field problems (Dirichlet conditions, Neumann conditions) and the enunciations using the magnetic vector potential (stationary magnetic field problems). The general formulation of the uniqueness conditions gets particular forms, adapted to some geometrical configurations (plane-parallel fields,

Depending on the relation between the magnetic induction and the intensity of the magnetic field, a few types of materials are distinguished, the most important being linear and isotropic materials, linear and non-isotropic materials, linear and non-isotropic materials, non-linear and isotropic materials, without permanent magnetization, non-linear and non-

Non-linear and isotropic materials, without permanent magnetization, are ferromagnetic

**2.1. Particular forms of the electromagnetic field theory laws for the stationary** 

The stationary magnetic field is established by non-moving, permanently magnetized bodies and by non-moving connecting wires crossed by direct current (Mocanu, 1981). Fundamental magnetic field relationships result by customizing the general laws and material laws of the electromagnetic field in the following conditions: bodies are non-

materials, which are frequently used in the production of electric equipment.

and very good flexibility when geometry is modeled and field sources are loaded.

**2. The fundamental relations of the stationary magnetic field** 

$$d\dot{v}\,\vec{B} = 0\tag{2}$$


$$
\vec{B} = \mu\_0 \left(\vec{H} + \vec{M}\right) \tag{3}
$$


$$
\vec{M}\_t = \vec{M}\_t \left(\vec{H}\right). \tag{4}
$$

Here, *H* is the magnetic field intensity, *<sup>J</sup>* is the total current density, *B* is the magnetic induction, quantity *Mt* is called the temporary magnetization of the material and 7 1 0 4 10 *H m* is the vacuum permeability.

Magnetostatics is the branch in electromagnetism that studies the stationary magnetic states that do not accompany the conduction electric currents. This magnetic field is produced by permanent magnets ( 0 *J* ; 0 *<sup>v</sup>* ).

#### **2.2. Models for the** *B***-***H* **relation**

Depending on the relation between the magnetic induction *B* and the magnetic field intensity *H* , a few types of materials are distinguished (Andrei et al. 2012).

#### *2.2.1. Linear and isotropic materials*

The most important type of materials consists of the linear and isotropic materials, in which:

$$
\vec{B} = \mu \vec{H} + \mu\_0 \vec{M}\_p = \mu \vec{H} + \vec{I}\_p \tag{5}
$$

where *Mp* is called the permanent magnetization and *pI* is the magnetic polarization. This category includes the materials for which the temporary magnetization law is (Răduleţ, 1975) :

$$
\vec{M}\_t = \chi\_m \vec{H} \tag{6}
$$

where *m* is called the magnetic susceptibility, representing a dimensionless and constant scalar quantity.

In the absence of permanent magnetization ( *Mp* =0, *pI* =0) the relation becomes:

$$
\vec{B} = \mu \vec{H} \tag{7}
$$

Finite Element Analysis of Stationary Magnetic Field 105

This is, usually, the behavior of ferromagnetic materials, which are frequently used in the

In hysteresis materials, the instantaneous value of the magnetic induction depends not only on the value of the intensity of the magnetic field, but also on the previous evolution of

Assume that the magnetic field intensity is gradually reduced after following the first

The curve obtained during the magnetic field intensity reduction differs from the first magnetization curve. When *H* is null, the magnetic induction has a value different from zero

> *r r* <sup>0</sup> *B M*

For further reduction of the magnetic induction, the sense of magnetic field intensity is changed (as well as the sense of magnetization current), with respect to the initial one.

The magnetic field intensity necessary to compensate the magnetic induction is called the coercitive field *Hc*. Increasing the field in the contrary sense to *–Hmax* and then returning to the values of *H* up to *Hmax*, the hysteresis cycle is obtained. By repeating several times the magnetization cycle between the limits *+Hmax* and *-Hmax*, a closed curve and a stabilized cycle are obtained, with the reversal points *A* and *A'* symmetrical with respect to the origin of the

It is important to mention that in the case of a periodic magnetization, the existence of the hysteresis cycle leads to energy losses that occur in the ferromagnetic core as heat. These

energy losses are called the hysteresis iron losses (Şora, 1982).

(13)

magnetization curve *OA,* corresponding to a value *+Hmax* (Figure 1).

production of electric equipment.

**Figure 1.** The *B*-*H* relation for a hysteresis material

called the residual magnetic induction:

where *Mr* is the residual magnetization.

coordinate system.

*2.2.4. Hysteresis materials* 

these quantities.

$$
\vec{H} = \nu \vec{B} \,. \tag{8}
$$

The quantity is the magnetic permeability and <sup>1</sup> is called the reluctivity. The *<sup>B</sup>* and *H* vectors are collinear.

#### *2.2.2. Linear and non-isotropic materials*

In these materials, the *B* and *H* vectors are not, generally, collinear, but the connection between them remains linear.

For some crystalline materials, the dependence between *Mt* and *<sup>H</sup>* is linear, but each component of the temporary magnetization depends on all components of the magnetic field.

The relation between them can be written, in the absence of permanent magnetization, under the form:

$$
\vec{B} = \mu \vec{H} \tag{9}
$$

where is a tensor. In Cartesian coordinates, the relation becomes:

$$
\begin{bmatrix} B\_x \\ B\_y \\ B\_z \end{bmatrix} = \begin{bmatrix} \mu\_{xx} & \mu\_{xy} & \mu\_{xz} \\ \mu\_{yx} & \mu\_{yy} & \mu\_{yz} \\ \mu\_{zx} & \mu\_{zy} & \mu\_{zz} \end{bmatrix} \cdot \begin{bmatrix} H\_x \\ H\_y \\ H\_z \end{bmatrix} \tag{10}
$$

The permittivity matrix is symmetrical ( *ij ji* ) and positively defined. In these conditions, there are three orthogonal directions, called main directions, with respect to which the relation between *B* and *H* becomes (Hănţilă, 2004):

$$
\begin{bmatrix} B\_1 \\ B\_2 \\ B\_3 \end{bmatrix} = \begin{bmatrix} \mu\_1 & 0 & 0 \\ 0 & \mu\_2 & 0 \\ 0 & 0 & \mu\_3 \end{bmatrix} \cdot \begin{bmatrix} H\_1 \\ H\_2 \\ H\_3 \end{bmatrix} \tag{11}
$$

#### *2.2.3. Non-linear and isotropic materials, without permanent magnetization*

In these materials, the *B* and *H* vectors are collinear, but the relation between them is nonlinear:

$$B = f\begin{pmatrix} H \end{pmatrix} \quad f: R^3 \to R^3. \tag{12}$$

This is, usually, the behavior of ferromagnetic materials, which are frequently used in the production of electric equipment.

#### *2.2.4. Hysteresis materials*

104 Finite Element Analysis – New Trends and Developments

In the absence of permanent magnetization ( *Mp*

is called the magnetic susceptibility, representing a dimensionless and constant

=0) the relation becomes:

(7)

. (8)

 and

is linear, but each

is called the reluctivity. The *<sup>B</sup>*

vectors are not, generally, collinear, but the connection

and *<sup>H</sup>*

(9)

) and positively defined. In these

(10)

(11)

=0, *pI*

*B H* 

*H B* 

component of the temporary magnetization depends on all components of the magnetic field. The relation between them can be written, in the absence of permanent magnetization,

> *B H*

*xx xy xz x x y yx yy yz y z zx zy zz z*

*B H B H B H*

 

conditions, there are three orthogonal directions, called main directions, with respect to

1 1 1 2 22 3 3 3

*B H B H B H*

*2.2.3. Non-linear and isotropic materials, without permanent magnetization* 

0 0 0 0 0 0

vectors are collinear, but the relation between them is non-

3 3 *B fH f R R* : . (12)

is a tensor. In Cartesian coordinates, the relation becomes:

is the magnetic permeability and <sup>1</sup>

where *m*

scalar quantity.

The quantity

under the form:

In these materials, the *B*

linear:

where 

*H*

*2.2.2. Linear and non-isotropic materials* 

The permittivity matrix is symmetrical ( *ij ji*

and *H*

which the relation between *B* and *H* becomes (Hănţilă, 2004):

and *H*

For some crystalline materials, the dependence between *Mt*

vectors are collinear.

In these materials, the *B*

between them remains linear.

In hysteresis materials, the instantaneous value of the magnetic induction depends not only on the value of the intensity of the magnetic field, but also on the previous evolution of these quantities.

Assume that the magnetic field intensity is gradually reduced after following the first magnetization curve *OA,* corresponding to a value *+Hmax* (Figure 1).

**Figure 1.** The *B*-*H* relation for a hysteresis material

The curve obtained during the magnetic field intensity reduction differs from the first magnetization curve. When *H* is null, the magnetic induction has a value different from zero called the residual magnetic induction:

$$B\_r = \mu\_0 \cdot M\_r \tag{13}$$

where *Mr* is the residual magnetization.

For further reduction of the magnetic induction, the sense of magnetic field intensity is changed (as well as the sense of magnetization current), with respect to the initial one.

The magnetic field intensity necessary to compensate the magnetic induction is called the coercitive field *Hc*. Increasing the field in the contrary sense to *–Hmax* and then returning to the values of *H* up to *Hmax*, the hysteresis cycle is obtained. By repeating several times the magnetization cycle between the limits *+Hmax* and *-Hmax*, a closed curve and a stabilized cycle are obtained, with the reversal points *A* and *A'* symmetrical with respect to the origin of the coordinate system.

It is important to mention that in the case of a periodic magnetization, the existence of the hysteresis cycle leads to energy losses that occur in the ferromagnetic core as heat. These energy losses are called the hysteresis iron losses (Şora, 1982).

#### *2.2.5. Non-linear and non-isotropic materials*

In these materials, the *B* and *H* vectors are not, generally, collinear and the relation between them is non-linear.

#### **2.3. The magnetic vector potential**

The magnetic vector potential is a vector field, which does not have a specific physical meaning. Its utilization allows simplification of the mathematical approach of many physical problems.

The condition *div B* <sup>0</sup> , which expresses the continuity of the magnetic flux, is identically satisfied if the *B* vector is expressed under the form of an auxiliary vector *A* , called the magnetic potential vector:

$$
\vec{B} = \text{rot}\,\vec{A}\tag{14}
$$

Finite Element Analysis of Stationary Magnetic Field 107

(19)

(21)

*<sup>A</sup>* <sup>0</sup> (20)

(23)

(24)

(25)

 

0 00 *A AA x yz* . (22)

(18)

*rotrot A J graddivA A* 

> *A <sup>J</sup>*

Solving Equations (19) and (20) requires the boundary condition to be known. The vector equations are divided after the Cartesian coordinates in scalar equations of Poisson type

*Axx yy zz*

The integral of Equation (19) in all space is determined by using the scalar forms (21).

 **<sup>A</sup>** 

The magnetic vector potential of the filiform circuit with current *i* is expressed as:

 *J AJ AJ* 

> <sup>d</sup> , <sup>4</sup> *i l R*

> > 3

rot d . <sup>4</sup> *i l*

 *R*

<sup>3</sup> <sup>4</sup> *<sup>D</sup> J R <sup>B</sup>*

 

*R*

 **A R <sup>H</sup>** 

the magnetic vector potential verifies the *Poisson's* vector equation:

, it verifies the *Laplace's* equation:

respectively, scalar equations of Laplace type

**Figure 2.** Biot–Savart–Laplace relation for filiform conductors

The magnetic field intensity is (Figure 2):

The Biot-Savart-Laplace relation becomes:

and if 0 *J*

*dl*

is line unit vector.

The potential vector *A* is univocally determined only after 0 *div A* is chosen. The choice of the value for *B* is called the calibration of the vector potential and the respective condition is called calibration condition. Depending on the context, one can adopt convenient *calibration conditions*. One of the most used calibration conditions is the Coulomb condition:

$$\operatorname{div}\vec{A} = 0\tag{15}$$

If the calculation of the magnetic flux through an open surface is expressed by means of magnetic induction, then the magnetic vector potential must be taken into account by the Stokes' theorem (Moraru, 2002).

The *magnetic flux* through a surface *S<sup>Γ</sup>* bounded by a contour Γ can be computed as a contour integral of the vector potential:

$$\Phi\_{S\_{\Gamma}} = \int\_{S\_{\Gamma}} \vec{B} \, d\vec{A} = \int\_{S\_{\Gamma}} \operatorname{rot} \vec{A} \, d\vec{A} = \oint\_{\Gamma} \vec{A} \, d\vec{I} \tag{16}$$

The magnetic flux through the surface *SΓ* is equal to the line integral of the magnetic vector potential along the contour *Γ* on which this surface is supported. Equation (16) relieves the fact that the value of a magnetic flux does not depend on the surface shape, as it is computed only by considering the contour on which that surface is supported. Let us consider a material with linear magnetic properties and without permanent magnetization, for which *B H* . From the magnetic circuit law, it results that (Hănţilă, 2004):

$$\operatorname{rot}\vec{H} = \operatorname{rot}\left[\frac{1}{\mu}\operatorname{rot}\vec{A}\right] = \vec{J} \tag{17}$$

In linear and homogeneous mediums, where µ is constant:

Finite Element Analysis of Stationary Magnetic Field 107

$$
\vec{A} \,\text{rot} \,\text{rot} \,\vec{A} = \mu \vec{\J} = \text{grad} \,\text{div} \,\vec{A} - \Delta \vec{A} \tag{18}
$$

the magnetic vector potential verifies the *Poisson's* vector equation:

$$
\Delta \vec{A} = -\mu \,\vec{\J} \tag{19}
$$

and if 0 *J* , it verifies the *Laplace's* equation:

$$
\vec{A} = 0\tag{20}
$$

Solving Equations (19) and (20) requires the boundary condition to be known. The vector equations are divided after the Cartesian coordinates in scalar equations of Poisson type

$$
\Delta A\_x = -\mu \, I\_x \qquad \Delta A\_y = -\mu \, I\_y \qquad \Delta A\_z = -\mu \, I\_z \tag{21}
$$

respectively, scalar equations of Laplace type

$$
\Delta A\_x = 0 \qquad \Delta A\_y = 0 \qquad \Delta A\_z = 0 \tag{22}
$$

The integral of Equation (19) in all space is determined by using the scalar forms (21).

The magnetic vector potential of the filiform circuit with current *i* is expressed as:

$$\vec{\mathbf{A}} = \frac{\mu \cdot i}{4\pi} \oint\_{\Gamma} \frac{\mathbf{d}\vec{l}}{R} \,' \,\tag{23}$$

*dl* is line unit vector.

106 Finite Element Analysis – New Trends and Developments

**2.3. The magnetic vector potential** 

0 *div A*

*S*

In linear and homogeneous mediums, where µ is constant:

Stokes' theorem (Moraru, 2002).

integral of the vector potential:

for which *B H*

magnetic potential vector:

The potential vector *A*

of the value for *B*

condition:

In these materials, the *B*

physical problems.

satisfied if the *B*

between them is non-linear.

*2.2.5. Non-linear and non-isotropic materials* 

and *H*

The magnetic vector potential is a vector field, which does not have a specific physical meaning. Its utilization allows simplification of the mathematical approach of many

The condition *div B* <sup>0</sup> , which expresses the continuity of the magnetic flux, is identically

*B rot A* 

is univocally determined only after 0 *div A*

condition is called calibration condition. Depending on the context, one can adopt convenient *calibration conditions*. One of the most used calibration conditions is the Coulomb

If the calculation of the magnetic flux through an open surface is expressed by means of magnetic induction, then the magnetic vector potential must be taken into account by the

The *magnetic flux* through a surface *S<sup>Γ</sup>* bounded by a contour Γ can be computed as a contour

*BdA rot AdA Adl* 

The magnetic flux through the surface *SΓ* is equal to the line integral of the magnetic vector potential along the contour *Γ* on which this surface is supported. Equation (16) relieves the fact that the value of a magnetic flux does not depend on the surface shape, as it is computed only by considering the contour on which that surface is supported. Let us consider a material with linear magnetic properties and without permanent magnetization,

. From the magnetic circuit law, it results that (Hănţilă, 2004):

<sup>1</sup> *rot H rot rot A J* 

 

*S S*

is called the calibration of the vector potential and the respective

vector is expressed under the form of an auxiliary vector *A*

vectors are not, generally, collinear and the relation

is chosen. The choice

(14)

(15)

(17)

(16)

, called the

**Figure 2.** Biot–Savart–Laplace relation for filiform conductors

The magnetic field intensity is (Figure 2):

$$\mathbf{H} = \frac{\operatorname{rot}\vec{\mathbf{A}}}{\mu} = \frac{i}{4\pi} \oint\_{\Gamma} \frac{\mathbf{d}\vec{I} \times \vec{\mathbf{R}}}{R^3}. \tag{24}$$

The Biot-Savart-Laplace relation becomes:

$$
\vec{B} = \frac{\mu}{4\pi} \int\_{D\_\pi} \frac{\vec{J} \times \vec{R}}{R^3} \tag{25}
$$

#### **2.4. The scalar magnetic potential**

The magnetic field is not irrotational for a circuit with current flow, therefore this can be deduced fom a scalar potential. But the rotor of magnetic field intensity is equal to zero if there is no current which flows, 0 *J* . If there is no conductor in the considered space, the following equation is available (Mocanu, 1981):

$$
tau \vec{H} = 0\tag{26}$$

Finite Element Analysis of Stationary Magnetic Field 109

(31)

(32)

(33)

(34)

(35)

(36)

(37)

*div A H HrotA ArotH* (38)

*W HrotAdv div A H dv AJdv* (39)

4 *<sup>m</sup> <sup>i</sup> <sup>V</sup>* 

is:

is calculated:

is (Şora, 1982):

In this case, the magnetic field intensity *H*

If the medium is linear (µ=constant), then:

In this case, the following expressions are obtained:

medium, the magnetic energy can be written as:

taking into account the vector operation. In other words, the magnetic energy is:

**2.5. Energy of the stationary magnetic field** 

where is the solid angle under which the contour is seen from the point where the field

3 *S*

1 4 *H i* 

Magnetic energy is located in a magnetic field with a volume density *wm* whose expression

0 *B w HdB <sup>m</sup>*

2 2 22 2 *<sup>m</sup> H B HB*

1 <sup>2</sup> *<sup>m</sup> <sup>v</sup> W HBdv*

Assuming that inside the field limited by a closed surface *Σ* and considering an isotropic

 11 1 22 2 *<sup>m</sup> vv v*

*<sup>H</sup> HdB Hd H d*

*w*

2 2

*<sup>R</sup> dS R*

Therefore, *H* can be deduced by a scalar potential:

$$
\vec{H} = -\text{grad}\,V\_m\tag{27}
$$

where is *Vm* is the scalar magnetic potential.

In the presence of some conductors crossed by electric current, the scalar magnetic potential is not uniform:

$$\int\_{1}^{2} \vec{H}d\vec{l} \cdot \mathbf{i} = -\int\_{1}^{2} \nabla V\_{m} dl = -\oint\_{1}^{2} dV = V\_{m1} - V\_{m2} \neq 0\tag{28}$$

For uniformity, a cut can be introduced an arbitrary surface bounded by the contour crossed by current (Figure 3).

**Figure 3.** Cut in order to uniform the scalar magnetic potential

In the absence of permanently magnetized bodies, the partial derivative equation of the scalar magnetic potential is deducted from the magnetic flux law:

$$
\vec{B} = -\mu \nabla V\_m \qquad \text{div}\,\vec{B} = 0 \quad \rightarrow -\text{div}\left(\mu \nabla V\_m\right) = 0 \tag{29}
$$

In homogeneous materials (where =cost), the Laplace equation is obtained:

$$
\nabla V\_m = 0 \tag{30}
$$

This expression is used to determine the scalar magnetic potential of the magnetic field produced by a filiform circuit crossed by electric current.

Applying the magnetic circuit law for a closed curve which surrounds the conductor, the scalar magnetic potential is written as:

Finite Element Analysis of Stationary Magnetic Field 109

$$V\_m = \frac{\dot{l}}{4\pi} \Omega$$

where is the solid angle under which the contour is seen from the point where the field is calculated:

$$
\Omega = \int\_{S\_{\Gamma}} \frac{R}{R^3} dS \tag{32}
$$

In this case, the magnetic field intensity *H* is:

108 Finite Element Analysis – New Trends and Developments

The magnetic field is not irrotational for a circuit with current flow, therefore this can be deduced fom a scalar potential. But the rotor of magnetic field intensity is equal to zero if

*H gradVm*

In the presence of some conductors crossed by electric current, the scalar magnetic potential

<sup>0</sup> *Hdl i V dl dV V V <sup>m</sup> m m*

For uniformity, a cut can be introduced an arbitrary surface bounded by the contour crossed

In the absence of permanently magnetized bodies, the partial derivative equation of the

This expression is used to determine the scalar magnetic potential of the magnetic field

Applying the magnetic circuit law for a closed curve which surrounds the conductor, the

 *m m* 0 0 (29)

0 *Vm* (30)

*B V divB div V*

In homogeneous materials (where =cost), the Laplace equation is obtained:

. If there is no conductor in the considered space, the

(26)

(27)

1 2

(28)

**2.4. The scalar magnetic potential** 

there is no current which flows, 0 *J*

Therefore, *H*

is not uniform:

by current (Figure 3).

following equation is available (Mocanu, 1981):

where is *Vm* is the scalar magnetic potential.

0 *rot H*

can be deduced by a scalar potential:

2 22

1 11

**Figure 3.** Cut in order to uniform the scalar magnetic potential

scalar magnetic potential is deducted from the magnetic flux law:

produced by a filiform circuit crossed by electric current.

scalar magnetic potential is written as:

$$
\vec{H} = -\frac{1}{4\pi} i \nabla \Omega \tag{33}
$$

#### **2.5. Energy of the stationary magnetic field**

Magnetic energy is located in a magnetic field with a volume density *wm* whose expression is (Şora, 1982):

$$
\omega v\_m = \int\_0^B \vec{H} d\vec{B} \tag{34}
$$

If the medium is linear (µ=constant), then:

$$dHdB = Hd\left(\mu H\right) = d\left(\frac{\mu H^2}{2}\right) \tag{35}$$

In this case, the following expressions are obtained:

$$
\omega w\_m = \frac{\mu H^2}{2} = \frac{B^2}{2\mu} = \frac{\vec{H}\vec{B}}{2} \tag{36}
$$

$$\mathcal{W}\_m = \frac{1}{2} \int\_{\upsilon} \vec{H} \vec{B} d\upsilon \tag{37}$$

Assuming that inside the field limited by a closed surface *Σ* and considering an isotropic medium, the magnetic energy can be written as:

$$\operatorname{div}\left(\vec{A}\cdot\vec{H}\right) = \vec{H}\operatorname{rot}\vec{A} - \vec{A}\operatorname{rot}\vec{H} \tag{38}$$

taking into account the vector operation.

In other words, the magnetic energy is:

$$\mathcal{W}\_m = \frac{1}{2} \int\_v \vec{H}rot \vec{A} dv = \frac{1}{2} \int\_v div \left( \vec{A} \cdot \vec{H} \right) dv + \frac{1}{2} \int\_v \vec{A} \vec{J} dv\tag{39}$$

Applying the Gauss-Ostrogradski's theorem to the first term on the right-hand side, the following expression is obtained:

$$\mathcal{W}\_m = \frac{1}{2} \int\_{\Sigma} (\vec{A} \cdot \vec{H}) d\vec{s} + \frac{1}{2} \int\_{v} \vec{A} \vec{J} dv \tag{40}$$

Finite Element Analysis of Stationary Magnetic Field 111

; *H P fp P S <sup>t</sup>* (45)

; *<sup>n</sup> B P gp P S* (46)

Boundary conditions, that can be of the following types (Figure 4):



**Figure 4.** The uniqueness theorem for the stationary magnetic field (Andrei et al. 2012)

*2.7.2. The enouncing by scalar magnetic potential (magnetostatic field problems)* 

permanent magnetization of the bodies (Andrei et al., 2012).

at the points on the surface, denoted by SD:

potentials.

types:

The theorem stands for linear materials, or for non-linear materials, but having *B-H* monotone magnetization characteristics. The case of materials which have hysteresis is not included. Particular forms can be deducted from this general formulation, expressed by field

In magnetostatic regime problems, the sources of the magnetic field are represented by the

The boundary conditions, expressed by the scalar magnetic potential, are of the following

a. Dirichlet Conditions, which consist of imposing values for the scalar magnetic potential

These conditions imply knowing the value of the tangent component of the intensity of the magnetic field in the respective points, which is equal to the derivative by the

tangent direction of the scalar magnetic potential (Flueraşu & Flueraşu, 2007).

; *VP fP P SD* (47)

#### **2.6. Generalized forces in the stationary magnetic field**

In the case of the stationary magnetic field, the general expressions of the generalized forces *Xk* associated to a generalized coordinate *xk* are given by one of the two generalized forces theorems in a stationary magnetic field (Timotin, 1970):

$$X\_k = -\left[\frac{\partial \mathcal{W}\_{em}}{\partial \mathbf{x}\_k}\right]\_{\Phi = ct} \tag{41}$$

$$X\_k = \left[\frac{\partial \mathcal{W}\_{em}}{\partial \mathbf{x}\_k}\right]\_{i=ct} \tag{42}$$

in which the transformations are supposed to be done at constant fluxes on any surface, respectively to constant currents through any conducting contour.

## **2.7. Uniqueness theorems of the solutions of the equations of stationary and magnetostatic magnetic fields**

#### *2.7.1. The enunciation of stationary and magnetostatic magnetic field*

In stationary magnetic field problems, the electric currents distribution (the *J* field) is supposed to be known (for example, by solving a stationary electrokinetic stationary regime, in the case of massive conductors, or by indicating the value of the current through the coils in the domain that is being studied).

In magnetostatic field problems, the sources of the field are represented by the distribution of the permanent magnetization ( *Mp* or the permanent magnetic polarization), which are supposed to be known (Andrei et al, 2012).

In conformity to the general uniqueness theorem of the solutions of the stationary magnetic fields equations, the solution of the electromagnetic field equations in a domain *D* bounded by closed surface  *=SHSB* is uniquely determined by the following uniqueness conditions:

The electric currents distribution in the domain:

$$J(P); \quad \forall P \in D\_{\Sigma} \tag{43}$$

The distribution of the permanent magnetization:

$$M\_p\left(P\right); \quad \forall P \in D\_\Sigma\tag{44}$$

Boundary conditions, that can be of the following types (Figure 4):

110 Finite Element Analysis – New Trends and Developments

following expression is obtained:

**magnetostatic magnetic fields** 

in the domain that is being studied).

of the permanent magnetization ( *Mp*

 *=SH*

by closed surface

supposed to be known (Andrei et al, 2012).

The electric currents distribution in the domain:

The distribution of the permanent magnetization:

Applying the Gauss-Ostrogradski's theorem to the first term on the right-hand side, the

 1 1 2 2 *<sup>m</sup>*

*k*

*k*

*X*

respectively to constant currents through any conducting contour.

*2.7.1. The enunciation of stationary and magnetostatic magnetic field* 

*X*

**2.6. Generalized forces in the stationary magnetic field** 

theorems in a stationary magnetic field (Timotin, 1970):

*W A H ds AJdv*

In the case of the stationary magnetic field, the general expressions of the generalized forces *Xk* associated to a generalized coordinate *xk* are given by one of the two generalized forces

*em*

*em*

*W*

in which the transformations are supposed to be done at constant fluxes on any surface,

supposed to be known (for example, by solving a stationary electrokinetic stationary regime, in the case of massive conductors, or by indicating the value of the current through the coils

In magnetostatic field problems, the sources of the field are represented by the distribution

In conformity to the general uniqueness theorem of the solutions of the stationary magnetic

fields equations, the solution of the electromagnetic field equations in a domain *D*

**2.7. Uniqueness theorems of the solutions of the equations of stationary and** 

In stationary magnetic field problems, the electric currents distribution (the *J*

*W*

 

*x*

*k ct*

*k i ct*

*x* 

*v*

(40)

or the permanent magnetic polarization), which are

*JP P D* ; (43)

; *Mp P PD* (44)

*SB* is uniquely determined by the following uniqueness conditions:

(41)

(42)

bounded

field) is


$$H\_t\left(P\right) = f\left(p\right); \quad \forall P \in \mathcal{S}\_\Sigma \tag{45}$$


$$B\_n\left(P\right) = \mathcal{g}\left(p\right); \quad \forall P \in \mathcal{S}\_\Sigma \tag{46}$$

**Figure 4.** The uniqueness theorem for the stationary magnetic field (Andrei et al. 2012)

The theorem stands for linear materials, or for non-linear materials, but having *B-H* monotone magnetization characteristics. The case of materials which have hysteresis is not included. Particular forms can be deducted from this general formulation, expressed by field potentials.

#### *2.7.2. The enouncing by scalar magnetic potential (magnetostatic field problems)*

In magnetostatic regime problems, the sources of the magnetic field are represented by the permanent magnetization of the bodies (Andrei et al., 2012).

The boundary conditions, expressed by the scalar magnetic potential, are of the following types:

a. Dirichlet Conditions, which consist of imposing values for the scalar magnetic potential at the points on the surface, denoted by SD:

$$V\left(P\right) = f\left(P\right); \quad \forall P \subset S\_D\tag{47}$$

These conditions imply knowing the value of the tangent component of the intensity of the magnetic field in the respective points, which is equal to the derivative by the tangent direction of the scalar magnetic potential (Flueraşu & Flueraşu, 2007).

b. Neumann Conditions, which consist of imposing the values of the derivative of the scalar magnetic potential iny the direction of the normal to the surface, denoted by SN. Practically, this type of conditions imposes the normal component of the magnetic induction in the respective points on the surface:

$$B\_n = -\mu \frac{dV}{dn} = g\left(P\right); \quad \forall P \subset S\_N\tag{48}$$

Finite Element Analysis of Stationary Magnetic Field 113

. (52)

(51)

(53)

The shape function coefficients *ak*, *bk*, and *ck* are called the generalized coordinates. These coefficients are constant because they depend on the constant coordinates of the nodes only. The values of the shape functions vary between 0 and 1. They are equal to 1 in node *k* and liniarly decrease in the elements adjacent to this node, being null in the rest of the nodes and

, 1 1,2,3

<sup>1</sup> for node 1 results from the formula (Stammberg, 1995):

12 21

*D xy xy xy xy xy xy* 23 32 13 12 31 21 (54)

(55)

*ii ii*

11 1 22 1 33 1

*xy a xy b xy c*

1 1 1 0 1 0

1 2

*xy xy <sup>a</sup> D*

*i i*

*y y <sup>b</sup> D x x*

*i i*

*D*

*k*

*k*

*k*

*c*

clockwise order. Writing with *Se*, the area of element "*e*":

2 1

and the index *i* takes the values by circular permutations in nodal set of an element "*e*", in

*S xy*

*e*

*x y*

*x y*

 

*kkk kii*

**Figure 6.** Graphical representation of the shape function

The shape function coefficients are:

, 0

*xxyy k x xy y i k*

 

elements (Figure 6). Thus:

The shape function

where:

Considering:

c. mixed conditions, that consist of imposing a condition in the form of a linear combination between the two above condition types, on a portion SM of the surface.

$$
\alpha V + \beta \frac{dV}{dn} = h\left(P\right); \quad \forall P \subset S\_M\tag{49}
$$

#### **3. Finite element analysis**

#### **3.1. Triangular finite elements**

The first step in solving the problems using the Finite Element Method (FEM) begins by dividing the analysis area in finite elements, as well as the choice of the finite element type. Currently, a wide range of finite elements is used, but their classification, their description, as well as their criteria presentation for choosing adequate finite element types does not represent the subject of this chapter. In the presented application, the triangular finite element with three nodes is used. At the same time with the choice of finite element type, the shape functions are chosen, so that the description of finite elements is followed by the associated shape function presentation. Concerning the shape functions, the interpolating polynomials are mainly used due to the facility in their derivation and their integration. The interpolation on a triangle supposes a shape or interpolating function which links the nodal values (triangle vertices). An approximation of the solution of the magnetic vector potential *A* is allowed at the level of each triangular element "*e*" (Figure 5), according to the following interpolating polynomial (Stammberg, 1995):

$$
\alpha\_k = a\_k + b\_k \cdot \mathbf{x} + c\_k \cdot \mathbf{y} \tag{50}
$$

**Figure 5.** The triangular element "*e*"

The shape function coefficients *ak*, *bk*, and *ck* are called the generalized coordinates. These coefficients are constant because they depend on the constant coordinates of the nodes only.

The values of the shape functions vary between 0 and 1. They are equal to 1 in node *k* and liniarly decrease in the elements adjacent to this node, being null in the rest of the nodes and elements (Figure 6). Thus:

$$\begin{aligned} \alpha\_k \left( \mathbf{x} = \mathbf{x}\_{k'}, y = y\_k \right) &= 1 \qquad k = 1, 2, 3\\ \alpha\_k \left( \mathbf{x} = \mathbf{x}\_{i'}, y = y\_i \right) &= 0 \qquad i \neq k \end{aligned} \tag{51}$$

$$\mathbf{x} = \underbrace{\begin{bmatrix} \mathbf{x} & \mathbf{x} \\ \mathbf{x} & \mathbf{x} \end{bmatrix} \alpha\_{k-1}}\_{\text{i}} \alpha\_{k-1}$$

**Figure 6.** Graphical representation of the shape function

The shape function <sup>1</sup> for node 1 results from the formula (Stammberg, 1995):

$$
\begin{pmatrix} 1 & x\_1 & y\_1 \\ 1 & x\_2 & y\_2 \\ 1 & x\_3 & y\_3 \end{pmatrix} \cdot \begin{pmatrix} a\_1 \\ b\_1 \\ c\_1 \end{pmatrix} = \begin{pmatrix} 1 \\ 0 \\ 0 \end{pmatrix}. \tag{52}
$$

The shape function coefficients are:

$$\begin{aligned} a\_k &= \frac{\boldsymbol{\chi}\_{i+1} \cdot \boldsymbol{y}\_{i+2} - \boldsymbol{\chi}\_{i+2} \cdot \boldsymbol{y}\_{i+1}}{D} \\ b\_k &= \frac{\boldsymbol{y}\_{i+1} - \boldsymbol{y}\_{i+2}}{D} \\ c\_k &= \frac{\boldsymbol{\chi}\_{i+2} - \boldsymbol{\chi}\_{i+1}}{D} \end{aligned} \tag{53}$$

where:

112 Finite Element Analysis – New Trends and Developments

**3. Finite element analysis** 

*A* 

**3.1. Triangular finite elements** 

**Figure 5.** The triangular element "*e*"

induction in the respective points on the surface:

b. Neumann Conditions, which consist of imposing the values of the derivative of the scalar magnetic potential iny the direction of the normal to the surface, denoted by SN. Practically, this type of conditions imposes the normal component of the magnetic

> ; *n N dV B gP P S dn*

c. mixed conditions, that consist of imposing a condition in the form of a linear combination between the two above condition types, on a portion SM of the surface.

*dV V hP P S*

The first step in solving the problems using the Finite Element Method (FEM) begins by dividing the analysis area in finite elements, as well as the choice of the finite element type. Currently, a wide range of finite elements is used, but their classification, their description, as well as their criteria presentation for choosing adequate finite element types does not represent the subject of this chapter. In the presented application, the triangular finite element with three nodes is used. At the same time with the choice of finite element type, the shape functions are chosen, so that the description of finite elements is followed by the associated shape function presentation. Concerning the shape functions, the interpolating polynomials are mainly used due to the facility in their derivation and their integration. The interpolation on a triangle supposes a shape or interpolating function which links the nodal values (triangle vertices). An approximation of the solution of the magnetic vector potential

is allowed at the level of each triangular element "*e*" (Figure 5), according to the

*kkk k*

*dn*

 

following interpolating polynomial (Stammberg, 1995):

; *<sup>M</sup>*

(49)

*a bxcy* (50)

(48)

$$D = \mathbf{x}\_2 y\_3 - \mathbf{x}\_3 y\_2 - \mathbf{x}\_1 y\_3 + \mathbf{x}\_1 y\_2 + \mathbf{x}\_3 y\_1 - \mathbf{x}\_2 y\_1 \tag{54}$$

and the index *i* takes the values by circular permutations in nodal set of an element "*e*", in clockwise order. Writing with *Se*, the area of element "*e*":

$$S\_e = \frac{1}{2} \begin{vmatrix} 1 & \chi\_1 & \chi\_1 \\ 1 & \chi\_2 & \chi\_2 \\ 1 & \chi\_3 & \chi\_3 \end{vmatrix} \tag{55}$$

Considering:

$$D = 2S\_e \tag{56}$$

Finite Element Analysis of Stationary Magnetic Field 115

(60)

, is orientated as one axis of the coordinate

*<sup>A</sup>* <sup>0</sup> (64)

(65)

(61)

and

(59)

*E grad V*

*B rot A* 

The energy functional associated with the stationary magnetic field produced by the direct

D

(62)

2 2

is obtained by minimizing the functional:

(63)

0

Two-dimensional problems of stationary magnetic field are by definition problems in which

2D problems in Cartesian coordinates (x,y,z) are called parallel-plane. The current density *J*

*A A B rotA i j k gradA xy z y x Axy*

<sup>2</sup> <sup>2</sup> <sup>2</sup> .

In the case of stationary magnetic field, the functional is related to the physical size of a

1 2 *<sup>m</sup> V W H Bdv*

*AA A B rot A grad A xy x*

known issue, such as the total energy of the magnetic field inside the domain *D*:

is oriented by the axis Oz and the magnetic vector potential has the structure *A Ak*

D

system and depends on the other two coordinates of the system (Stammberg, 1995).

 

*B A HdB JA d* 

currents and in case by permanent magnets is expressed as:

the unknown, the vector magnetic potential *A*

The square of magnetic induction is:

The magnetic vector potential *A*

its orientation is also by the axis Oz. The magnetic induction is written as:

*ij k*

00 ,

0

 

*A y*

and

The magnetic vector potential in an arbitrary point (*x,y,z*) is obtained with the following equation:

$$A\left(\mathbf{x},\mathbf{y},\mathbf{z}\right) = \sum\_{k=1}^{n(\epsilon)} a\_k \left(\mathbf{x},\mathbf{y},\mathbf{z}\right) A\_k \tag{57}$$

where:

 *<sup>k</sup>* - shape function *n e* - number of nodes on element *Ak* - magnetic vector potential of node *k* 

### **3.2. FEM application for two-dimensional problems of stationary magnetic field**

Finite element methods (FEM) use most of the times a variation principle. According to the variation computation, solving a differential equation in a field and under certain boundary conditions is equivalent with minimizing, in that field, a functional corresponding to the differential equation with its boundary conditions. A functional integral is an integral expression, a function that depends on the unknown functions. The functional integral has a finite value.

The problem concerning solving the system of differential equations of the electromagnetic field with some boundary conditions is equivalent with the problem of finding a function which gives the integral minimum by which the energy system is expressed.

Let's consider the energy functional associated with the arbitrary three-dimensional field D:

$$\mathbf{C} = \int\_{\mathcal{D}} \left\{ \left( \overset{\vec{E}}{\int} \overset{\vec{D}}{D} d\vec{E} - \overset{\vec{B}}{\int} \overset{\vec{B}}{H} d\vec{B} \right) + \left( \overrightarrow{f}\vec{A} - \rho\_V V \right) \right\} d\mathcal{D} \tag{58}$$

where *D EB H* , , , are the vectors associated with electric and magnetic fields, *<sup>A</sup>* is the magnetic vector potential, *V* is the scalar electric potential, *J* is the density vector of conduction electric current and is the volume density of electric charge (Silvester & Ferrari, 1996).

The first parenthesis of the integrand represents the difference between the volume density of the electric and magnetic energy. The second parenthesis represents the difference between the volume density of interaction energies between the conduction current and magnetic field, as well as between the electric charge and the electric field. The interaction energies are equal to the work done by the field forces in order to bring the current density, respectively electric charge, from infinity, where the potentials are considered *A V* 0, 0 , , to the states characterized by the values *<sup>A</sup>* and *V* .

The electromagnetic potentials *A* and *V* define the vectors *E* and *B* of the electromagnetic field:

Finite Element Analysis of Stationary Magnetic Field 115

$$
\vec{E} = -\text{grad } V \tag{59}
$$

and

114 Finite Element Analysis – New Trends and Developments

equation:

where: 

finite value.

*<sup>k</sup>* - shape function

*n e* - number of nodes on element *Ak* - magnetic vector potential of node *k* 

2 *D Se* (56)

The magnetic vector potential in an arbitrary point (*x,y,z*) is obtained with the following

 

**3.2. FEM application for two-dimensional problems of stationary magnetic field** 

Finite element methods (FEM) use most of the times a variation principle. According to the variation computation, solving a differential equation in a field and under certain boundary conditions is equivalent with minimizing, in that field, a functional corresponding to the differential equation with its boundary conditions. A functional integral is an integral expression, a function that depends on the unknown functions. The functional integral has a

The problem concerning solving the system of differential equations of the electromagnetic field with some boundary conditions is equivalent with the problem of finding a function

Let's consider the energy functional associated with the arbitrary three-dimensional field D:

 

*DdE HdB JA V d <sup>V</sup>*

The first parenthesis of the integrand represents the difference between the volume density of the electric and magnetic energy. The second parenthesis represents the difference between the volume density of interaction energies between the conduction current and magnetic field, as well as between the electric charge and the electric field. The interaction energies are equal to the work done by the field forces in order to bring the current density, respectively electric charge, from infinity, where the potentials are considered

and *V* define the vectors *E*

and *V* .

and *B*

D

(58)

of the electromagnetic

is the magnetic

is the density vector of conduction electric

which gives the integral minimum by which the energy system is expressed.

0 0

where *D EB H* , , , are the vectors associated with electric and magnetic fields, *<sup>A</sup>*

current and is the volume density of electric charge (Silvester & Ferrari, 1996).

D

*A V* 0, 0 , , to the states characterized by the values *<sup>A</sup>*

The electromagnetic potentials *A*

field:

vector potential, *V* is the scalar electric potential, *J*

*E B*

*k A xyz xyzA* 

1 ,, ,, *n e*

*k k*

(57)

$$
\vec{B} = \text{rot } \vec{A} \tag{60}
$$

The energy functional associated with the stationary magnetic field produced by the direct currents and in case by permanent magnets is expressed as:

$$\mathfrak{T}\left(\vec{A}\right) = \int\_{\mathcal{D}} \left(\int\_{0}^{\vec{B}} \vec{H}d\vec{B} - \vec{J}\vec{A}\right)d\mathcal{D} \tag{61}$$

Two-dimensional problems of stationary magnetic field are by definition problems in which the unknown, the vector magnetic potential *A* , is orientated as one axis of the coordinate system and depends on the other two coordinates of the system (Stammberg, 1995).

2D problems in Cartesian coordinates (x,y,z) are called parallel-plane. The current density *J* is oriented by the axis Oz and the magnetic vector potential has the structure *A Ak* and its orientation is also by the axis Oz. The magnetic induction is written as:

$$
\vec{B} = \operatorname{rot} \vec{A} = \begin{vmatrix}
\vec{i} & \vec{j} & \vec{k} \\
\vec{\mathcal{O}} & \vec{\mathcal{O}} & \vec{\mathcal{O}} \\
\vec{\mathcal{O}\mathbf{x}} & \vec{\mathcal{O}y} & \vec{\mathcal{O}z} \\
\mathbf{0} & \mathbf{0} & A \begin{pmatrix} \mathbf{x}, \mathbf{y} \end{pmatrix}
\end{pmatrix} = \vec{i}\,\frac{\vec{\mathcal{O}A}}{\vec{\mathcal{O}y}} - \vec{j}\,\frac{\vec{\mathcal{O}A}}{\vec{\mathcal{O}x}} = \vec{k} \times \left( -\operatorname{grad} A \right) \tag{62}
$$

The square of magnetic induction is:

$$
\bar{B}^2 = \left(\operatorname{rot}\bar{A}\right)^2 = \left(-\frac{\hat{\mathcal{O}A}}{\hat{\mathcal{O}\mathcal{X}}}\right) = \left(\frac{\hat{\mathcal{O}A}}{\hat{\mathcal{O}\mathcal{Y}}}\right)^2 + \left(\frac{\hat{\mathcal{O}A}}{\hat{\mathcal{O}\mathcal{X}}}\right)^2 = \left(\operatorname{grad}\bar{A}\right)^2. \tag{63}
$$

The magnetic vector potential *A* is obtained by minimizing the functional:

$$\mathcal{L}\mathfrak{T}(\vec{A}) = 0\tag{64}$$

In the case of stationary magnetic field, the functional is related to the physical size of a known issue, such as the total energy of the magnetic field inside the domain *D*:

$$\mathcal{W}\_m = \frac{1}{2} \int\_V \vec{H} \cdot \vec{B} dv \tag{65}$$

The magnetic energy is located in the magnetic field with the volume density *wm* :

$$
\hbar w\_{\text{av}} = \frac{\vec{H} \cdot \vec{B}}{2} = \frac{\vec{B}^2}{2\mu} \tag{66}
$$

Finite Element Analysis of Stationary Magnetic Field 117

where the function represents the magnetic energy computed on the surface *SΓ* bounded by the curve Γ. The unknown function *Ax y*, for node *i* is determined by minimizing the

> *i i ii S A AA A A J dxdy A xA x yA y A*

*q i*

*k*

*f*

1

*z i*

The differentials involved in Eq. (71) are written according to the shape functions and have

*n e n e k k k k*

*k k <sup>A</sup> <sup>A</sup> A b*

*x x*

*y y*

*i*

*i*

*n e*

*i i k*

*i S k k*

finite element, being independent of the *x* and *y* coordinates:

where D is the determinant computed according to Eq.(56).

*e*

*e*

*i*

Eq. (79) can be written under matrix form as:

1

*n e n e*

1 1

1 1

*n e n e k k k k*

*k k <sup>A</sup> <sup>A</sup> A c*

> *Ax x*

> > *A*

*Ay y* 

*<sup>A</sup> <sup>A</sup> a bxcy A A*

*b A b c A c J dxdy <sup>A</sup>*

A homogeneous medium is considered, thus the magnetic permittivity *µ* is constant for each

2 6

*D D A bb cc A bb cc A bb cc J <sup>A</sup>*

*ii ii ii*

 

1 1

*i i*

*i i*

*kk i i i i*

<sup>1</sup> <sup>0</sup>

11 1 22 2 33 3 0

(79)

(78)

*kk i kk i i*

*c*

*<sup>A</sup> <sup>b</sup>*

*A*

Therefore, the following equations system is obtained:

*Ai* is the magnetic vector potential of the node *i* of the element *z*.

the following expressions (Stammberg, 1995; Eckhardt, 1978):

<sup>1</sup> <sup>0</sup>

0

2

(72)

(73)

(74)

(75)

(77)

(76)

(71)

functional:

Eckhardt describes in detail the solution to this problem. The magnetic field density is reduced to the following scalar equation (Stammberg, 1995; Eckhardt, 1978):

$$\delta w\_m = \frac{1}{2\mu} B^2 - J \cdot A = \frac{1}{2\mu} \left[ \left( \frac{\partial A}{\partial \mathbf{x}} \right)^2 + \left( \frac{\partial A}{\partial y} \right)^2 \right] - J \cdot A \tag{67}$$

In the case of the parallel-plane fileds, the boundary conditions are:

$$A\begin{pmatrix} x, y \end{pmatrix} = f\begin{pmatrix} P \\ \end{pmatrix}, \quad P \in \mathbb{C}\_D \tag{68}$$

$$-\frac{1}{\mu} \frac{dA}{dm}\bigg|\_{\mathbb{C}\_N} = \lg\left(P\right)\_\prime \quad P \in \mathbb{C}\_N \tag{69}$$

where Eq. (68) represents the Dirichlet boundary conditions on the boundary CD and Eq. (69) represents the Neumann boundary conditions on the boundary CN. The unknown function *Ax y*, is the solution of the Poisson's equation in a two-dimensional domain, the boundary *Γ* being composed of two disjoint parts CD and CN where the Neumann and Dirichlet conditions are described (Figure 7):

**Figure 7.** The domain of computation for a two-dimensional problem (Andrei et al. 2012)

$$\mathcal{L} = \int\_{S} \left[ \frac{1}{2\mu} \left[ \left( \frac{\partial A}{\partial \mathbf{x}} \right)^{2} + \left( \frac{\partial A}{\partial y} \right)^{2} \right] - J \cdot A \right] d\mathbf{x} dy \tag{70}$$

where the function represents the magnetic energy computed on the surface *SΓ* bounded by the curve Γ. The unknown function *Ax y*, for node *i* is determined by minimizing the functional:

$$\frac{\partial \ell}{\partial A\_i} = \oint\_S \left[ \frac{1}{\mu} \left[ \frac{\partial A}{\partial \mathbf{x}} \cdot \frac{\partial}{\partial A\_i} \left( \frac{\partial A}{\partial \mathbf{x}} \right) + \frac{\partial A}{\partial y} \cdot \frac{\partial}{\partial A\_i} \left( \frac{\partial A}{\partial y} \right)^2 \right] - f \frac{\partial A}{\partial A\_i} \right] dx dy = 0 \tag{71}$$

Therefore, the following equations system is obtained:

116 Finite Element Analysis – New Trends and Developments

Dirichlet conditions are described (Figure 7):

The magnetic energy is located in the magnetic field with the volume density *wm* :

*w*

reduced to the following scalar equation (Stammberg, 1995; Eckhardt, 1978):

1 1 <sup>2</sup> 2 2 *<sup>m</sup>*

In the case of the parallel-plane fileds, the boundary conditions are:

2 2 *<sup>m</sup> HB B*

Eckhardt describes in detail the solution to this problem. The magnetic field density is

*A A w B JA J A*

 

 <sup>1</sup> , *N*

where Eq. (68) represents the Dirichlet boundary conditions on the boundary CD and Eq. (69) represents the Neumann boundary conditions on the boundary CN. The unknown function *Ax y*, is the solution of the Poisson's equation in a two-dimensional domain, the boundary *Γ* being composed of two disjoint parts CD and CN where the Neumann and

*C*

**Figure 7.** The domain of computation for a two-dimensional problem (Andrei et al. 2012)

 *x y* 

1 2 *<sup>S</sup>*

2 2

*A A J A dxdy*

(70)

*dA*

 

2

*x y*

*g P PC*

2 2

*N*

*A* , , *<sup>D</sup> xy f P P C* (68)

*dn* (69)

(66)

(67)

$$f\_k = \sum\_{z=1}^{q(i)} \frac{\partial \ell}{\partial A\_i} = 0 \tag{72}$$

*Ai* is the magnetic vector potential of the node *i* of the element *z*.

The differentials involved in Eq. (71) are written according to the shape functions and have the following expressions (Stammberg, 1995; Eckhardt, 1978):

$$\frac{\partial A}{\partial \mathbf{x}} = \sum\_{k=1}^{n(\epsilon)} A\_k \frac{\partial \alpha\_k}{\partial \mathbf{x}} = \sum\_{k=1}^{n(\epsilon)} A\_k \cdot b\_k \tag{73}$$

$$\frac{\partial A}{\partial y} = \sum\_{k=1}^{n(\epsilon)} A\_k \frac{\partial \alpha\_k}{\partial y} = \sum\_{k=1}^{n(\epsilon)} A\_k \cdot c\_k \tag{74}$$

$$\frac{\partial}{\partial A\_i} \left( \frac{\partial A}{\partial \mathbf{x}} \right) = \frac{\partial a\_i}{\partial \mathbf{x}} = b\_i \tag{75}$$

$$\frac{\partial}{\partial A\_i} \left( \frac{\partial A}{\partial y} \right) = \frac{\partial a\_i}{\partial y} = c\_i \tag{76}$$

$$\frac{\partial A}{\partial A\_i} = \frac{\partial}{\partial A\_i} \left( \sum\_{k=1}^{n(\epsilon)} \alpha\_k A\_k \right) = \alpha\_i = a\_i + b\_i \cdot \mathbf{x} + c\_i \cdot \mathbf{y} \tag{77}$$

$$\frac{\partial \ell\_e}{\partial A\_i} = \int\_S \left[ \frac{1}{\mu} \left[ \left( \sum\_{k=1}^{n(e)} b\_k \cdot A\_k \right) \cdot b\_i + \left( \sum\_{k=1}^{n(e)} c\_k \cdot A\_k \right) \cdot c\_i \right] - J \alpha\_i \right] dxdy = 0 \tag{78}$$

A homogeneous medium is considered, thus the magnetic permittivity *µ* is constant for each finite element, being independent of the *x* and *y* coordinates:

$$\frac{\partial \ell\_e}{\partial A\_i} = \frac{D}{2\mu} \left[ A\_1 \left( b\_1 b\_i + c\_1 c\_i \right) + A\_2 \left( b\_2 b\_i + c\_2 c\_i \right) + A\_3 \left( b\_3 b\_i + c\_3 c\_i \right) \right] - J \frac{D}{6} = 0 \tag{79}$$

where D is the determinant computed according to Eq.(56).

Eq. (79) can be written under matrix form as:

$$\frac{\partial \ell\_e}{\partial A\_i} = \frac{D}{2\mu} \begin{pmatrix} b\_1 b\_i + c\_1 c\_i & b\_2 b\_i + c\_2 c\_i & b\_3 b\_i + c\_3 c\_i \end{pmatrix} \cdot \begin{pmatrix} A\_1 \\ A\_2 \\ A\_3 \end{pmatrix} - J \frac{D}{6} = 0 \tag{80}$$

The functional is sum of contributions other than the "*ne*" finite elements. Stationarization of the functional requires (Stammberg, 1995; Silvester & Ferrari, 1996):

$$\sum\_{e=1}^{ne} \frac{\partial \ell\_e}{\partial A\_i} = 0 \tag{81}$$

Finite Element Analysis of Stationary Magnetic Field 119

A typical magnetic field problem is described by defining its geometry, material properties, currents, boundary conditions, and the field system equations. The computer requires the input data and provides the numerical solution of the field equation and the output of desired parameters. If the values are found unsatisfactory, the design is modified and parameters are recalculated. The process is repeated until optimum values for the design

In order to define the physics environment for an analysis, it is necessary to enter in the ANSYS preprocessor (PREP7) and to establish a mathematical simulation model of the physical problem. In order to this, the following steps are presented below: set GUI Preferences, define the analysis title, define the element types and options, define the element coordinate systems, set real constants and define a system of units, and define the

The Global Cartesian coordinate system is the default. A different coordinate system can be specified by the user by indicating its origin location and orientation angles. The coordinate

Some materials with magnetic properties are defined in the ANSYS material library. The materials can be modified to correspond more closely to the analysed problem and to be loaded in the ANSYS database. The copper property shows temperature which depends on resistivity and relative permeability. All other properties are described in terms of *B-H* curves. Most of the materials included in ANSYS are used for modeling the electromagnetic phenomenon. The element types are used to establish the physics of the problem domain. Some element types and options are defined to represent the different regions in the model. If some laminated materials are aligned in an arbitrary form, the element coordinate system or systems have to be identified and used. The applications presented in this chapter use the PLANE53 element in the two-dimensional problem and the SOLID97 element for the three-

In order to obtain the magnetic field values, the Maxwell's equations are solved by using the imput data. The nodal values of the magnetic vector potential are considered as main or primary unknows. Their derivatives (e.g., flux density) are the secondary unknows. After this, it is possible to choose the type of solver to be used. The available options include Sparse solver (default), Frontal solver, Jacobi Conjugate Gradient (JCG) solver, JCG out-ofmemory solver, Incomplete Cholesky Conjugate Gradient (ICCG) solver, Preconditioned Conjugate Gradient solver (PCG), and PCG out-of-memory solver (ANSYS Documentation). The results of the calculations are shown in the postprocessing phase, which is a graphical program. Here, it can be observed if the applied loads affect the design, if the finite element mesh is good, and so on. The resulting fields in the form of contour and density plots are displayed by this praphical program. The analysis of the field at arbitrary points, the evaluation of a number of different integrals, and the plot of some quantities along predefined contours are also made with this program. The plotted results are saved in the

system types are Cartesian, cylindrical (circular or elliptical), spherical, and toroidal.

parameters are obtained.

dimensional problem.

Extended Metafile (EMF) format.

material properties (ANSYS Documentation).

where "*ne*" is the total number of finite elements.

$$\begin{Bmatrix} F \end{Bmatrix} = \begin{bmatrix} K \\ \end{bmatrix} \cdot \begin{Bmatrix} A \end{Bmatrix} - \begin{bmatrix} P \\ \end{bmatrix} + \begin{bmatrix} Q \\ \end{bmatrix} \tag{82}$$

The term *P* is used for the case of the "source" type elements where the current density *J* is non-null, and the term *Q* is used for the case of the elements which have in one of the sizes a non-homogeneous Neumann boundary. By assembling the"*ne*" equations in Eq. (81), a linear system of equations of magnetic vector potential values in the mesh nodes is obtained (Ioan, 1993).

## **4. Applications**

Direct current (DC), which was one of the main means of distributing electric power, is still widespread today in the electrical plants supplying particular industrial applications. The advantages in terms of settings, offered by the applicants of DC motors and by supply through a single line, make direct current supply a good solution for railway and underground systems, trams, lifts and other transport means. Current-limiting circuit breakers play an important role in electrical low-voltage circuits. Due to the high short-circuit currents it is necessary a very short time to switch off the faulted branch. For this reason the current limiting circuit breakers are conceived as elaborated solutions especially for the arc quenching system, meaning the path of current and the arcing chamber (Vîrjoghe, 2010).

This section presents the calculation of the magnetic field in the arcing chamber of a currentlimiting d.c. circuit breaker of 1250 A, 750 V, and in a DC circuit breaker-separator of 3150 A, 1000 V. The authors present few optimization solutions of some quenching systems which will lead to more performing constructive choices. Two-dimensional (2D) and threedimensional (3D) problems of stationary magnetic field are addressed.

The finite element software package ANSYS is used for calculation of the magnetic field components. This tool includes three stages: preprocessor, solver and postprocessor. The procedure for carrying out a static magnetic analysis consists of following main steps: create the physics environment, build and mesh the model and assign physics attributes to each region within the model, apply boundary conditions and loads (excitation), obtain the solution, and review the results (ANSYS Documentation).

A typical magnetic field problem is described by defining its geometry, material properties, currents, boundary conditions, and the field system equations. The computer requires the input data and provides the numerical solution of the field equation and the output of desired parameters. If the values are found unsatisfactory, the design is modified and parameters are recalculated. The process is repeated until optimum values for the design parameters are obtained.

118 Finite Element Analysis – New Trends and Developments

*e*

*i*

where "*ne*" is the total number of finite elements.

obtained (Ioan, 1993).

**4. Applications** 

the functional requires (Stammberg, 1995; Silvester & Ferrari, 1996):

meaning the path of current and the arcing chamber (Vîrjoghe, 2010).

dimensional (3D) problems of stationary magnetic field are addressed.

solution, and review the results (ANSYS Documentation).

1

*ne <sup>e</sup> <sup>e</sup> Ai*

*F KA P Q* 

The term *P* is used for the case of the "source" type elements where the current density *J* is non-null, and the term *Q* is used for the case of the elements which have in one of the sizes a non-homogeneous Neumann boundary. By assembling the"*ne*" equations in Eq. (81), a linear system of equations of magnetic vector potential values in the mesh nodes is

Direct current (DC), which was one of the main means of distributing electric power, is still widespread today in the electrical plants supplying particular industrial applications. The advantages in terms of settings, offered by the applicants of DC motors and by supply through a single line, make direct current supply a good solution for railway and underground systems, trams, lifts and other transport means. Current-limiting circuit breakers play an important role in electrical low-voltage circuits. Due to the high short-circuit currents it is necessary a very short time to switch off the faulted branch. For this reason the current limiting circuit breakers are conceived as elaborated solutions especially for the arc quenching system,

This section presents the calculation of the magnetic field in the arcing chamber of a currentlimiting d.c. circuit breaker of 1250 A, 750 V, and in a DC circuit breaker-separator of 3150 A, 1000 V. The authors present few optimization solutions of some quenching systems which will lead to more performing constructive choices. Two-dimensional (2D) and three-

The finite element software package ANSYS is used for calculation of the magnetic field components. This tool includes three stages: preprocessor, solver and postprocessor. The procedure for carrying out a static magnetic analysis consists of following main steps: create the physics environment, build and mesh the model and assign physics attributes to each region within the model, apply boundary conditions and loads (excitation), obtain the

*iiiiii*

*D D bb cc bb cc bb cc A J <sup>A</sup> <sup>A</sup>*

112233 2

The functional is sum of contributions other than the "*ne*" finite elements. Stationarization of

0

2 6

1

*A*

(80)

0

3

(81)

(82)

In order to define the physics environment for an analysis, it is necessary to enter in the ANSYS preprocessor (PREP7) and to establish a mathematical simulation model of the physical problem. In order to this, the following steps are presented below: set GUI Preferences, define the analysis title, define the element types and options, define the element coordinate systems, set real constants and define a system of units, and define the material properties (ANSYS Documentation).

The Global Cartesian coordinate system is the default. A different coordinate system can be specified by the user by indicating its origin location and orientation angles. The coordinate system types are Cartesian, cylindrical (circular or elliptical), spherical, and toroidal.

Some materials with magnetic properties are defined in the ANSYS material library. The materials can be modified to correspond more closely to the analysed problem and to be loaded in the ANSYS database. The copper property shows temperature which depends on resistivity and relative permeability. All other properties are described in terms of *B-H* curves. Most of the materials included in ANSYS are used for modeling the electromagnetic phenomenon. The element types are used to establish the physics of the problem domain. Some element types and options are defined to represent the different regions in the model. If some laminated materials are aligned in an arbitrary form, the element coordinate system or systems have to be identified and used. The applications presented in this chapter use the PLANE53 element in the two-dimensional problem and the SOLID97 element for the threedimensional problem.

In order to obtain the magnetic field values, the Maxwell's equations are solved by using the imput data. The nodal values of the magnetic vector potential are considered as main or primary unknows. Their derivatives (e.g., flux density) are the secondary unknows. After this, it is possible to choose the type of solver to be used. The available options include Sparse solver (default), Frontal solver, Jacobi Conjugate Gradient (JCG) solver, JCG out-ofmemory solver, Incomplete Cholesky Conjugate Gradient (ICCG) solver, Preconditioned Conjugate Gradient solver (PCG), and PCG out-of-memory solver (ANSYS Documentation).

The results of the calculations are shown in the postprocessing phase, which is a graphical program. Here, it can be observed if the applied loads affect the design, if the finite element mesh is good, and so on. The resulting fields in the form of contour and density plots are displayed by this praphical program. The analysis of the field at arbitrary points, the evaluation of a number of different integrals, and the plot of some quantities along predefined contours are also made with this program. The plotted results are saved in the Extended Metafile (EMF) format.

## **4.1. Numerical modelling of stationary magnetic field in area slope-sliderferromagnetic profile of arc chamber in case of a current-limiting DC circuit breaker – 2D application**

Finite Element Analysis of Stationary Magnetic Field 121

To observe the influence of the ferromagnetic slider and of the ferromagnetic plate on the magnetic fied distribution, the magnetic induction is computed and the magnetic flux lines are drawn. The slider and the ferromagnetic plate case were studied independently of each

Figure 10 and Figure 11 respectively show the magnetic induction spectrum and the

**Figure 10.** The magnetic induction spectrum in the presence of the ferromagnetic slider

using the ferromagnetic profile and the ferromagnetic slider (Vîrjoghe, 2004).

Figure 12 and Figure 13 respectively show the magnetic induction spectrum and the magnetic field lines only, for the case of *I* shaped ferromagnetic profile. Figure 14 and Figure 15 respectively show the magnetic induction spectrum and the magnetic field lines when

Analyzing these simulations, a strong influence of the ferromagnetic slider on the orientation of the magnetic field was observed. When using only the ferromagnetic slider, a shielding of the field lines is observed, and the maximum values of the magnetic induction is 0.907 T. The *I* shaped ferromagnetic profile makes a good shielding of the field lines

**Figure 9.** The magnetization characteristic for the M3 steel

magnetic field lines only, for the case of ferromagnetic slider.

other.

The problem of magnetic field distribution in the arc chamber of DC a circuit breaker with rated current 1250A was numerically solved. The conductor where a current of 1250 A flows is located in the immediate vicinity of a ferromagnetic profile. This has the role of enhancing and orienting the magnetic field in the arc-quenching chamber for obtaining a strong force that moves the arc up inside the extinction chamber (Vîrjoghe, 2004).

It is considered the plane parallel model, whose cross section is shown in Figure 8. This model is an *I* shaped ferromagnetic profile, with cross section in the vertical plane and the dimmensions of 60x3 mm2. The cross section of copper conductor is 5x15 mm2. The conductor is surrounded by a slider with *U* shaped cross section and a thickness of 1mm.

**Figure 8.** The physical model in the area with ramp, slider and ferromagnetic profile

For numerical computation the *PLANE53* element was chosen, which allows twodimensional modeling of the magnetic field in plane parallel and axisymmetric problems. This element is based on the magnetic vector potential formulation with Coulomb calibration. This element is also applicable to the stationary magnetic field with the possibility of modeling the magnetic nonlinearities. The material used for other two ferromagnetic profiles is a steel chosen from the ANSYS library and having the properties in the *emagM3.SI\_MPL* folder. The material is M3 steel and its magnetization curve is shown in Figure 9. This domain was discretized in a number of 2436 triangular finite elements uniformly distributed (Vîrjoghe, 2004).

The boundary conditions and loads are applied to a 2-D static magnetic analysis either on the solid model (key points, lines, and areas) or on the finite element model (nodes and elements). The loads applied to the solid model to the mesh during solution are automatically transferred by ANSYS (ANSYS Documentation).

To observe the influence of the ferromagnetic slider and of the ferromagnetic plate on the magnetic fied distribution, the magnetic induction is computed and the magnetic flux lines are drawn. The slider and the ferromagnetic plate case were studied independently of each other.

**Figure 9.** The magnetization characteristic for the M3 steel

120 Finite Element Analysis – New Trends and Developments

**breaker – 2D application** 

**4.1. Numerical modelling of stationary magnetic field in area slope-sliderferromagnetic profile of arc chamber in case of a current-limiting DC circuit** 

that moves the arc up inside the extinction chamber (Vîrjoghe, 2004).

**Figure 8.** The physical model in the area with ramp, slider and ferromagnetic profile

uniformly distributed (Vîrjoghe, 2004).

automatically transferred by ANSYS (ANSYS Documentation).

For numerical computation the *PLANE53* element was chosen, which allows twodimensional modeling of the magnetic field in plane parallel and axisymmetric problems. This element is based on the magnetic vector potential formulation with Coulomb calibration. This element is also applicable to the stationary magnetic field with the possibility of modeling the magnetic nonlinearities. The material used for other two ferromagnetic profiles is a steel chosen from the ANSYS library and having the properties in the *emagM3.SI\_MPL* folder. The material is M3 steel and its magnetization curve is shown in Figure 9. This domain was discretized in a number of 2436 triangular finite elements

The boundary conditions and loads are applied to a 2-D static magnetic analysis either on the solid model (key points, lines, and areas) or on the finite element model (nodes and elements). The loads applied to the solid model to the mesh during solution are

The problem of magnetic field distribution in the arc chamber of DC a circuit breaker with rated current 1250A was numerically solved. The conductor where a current of 1250 A flows is located in the immediate vicinity of a ferromagnetic profile. This has the role of enhancing and orienting the magnetic field in the arc-quenching chamber for obtaining a strong force

It is considered the plane parallel model, whose cross section is shown in Figure 8. This model is an *I* shaped ferromagnetic profile, with cross section in the vertical plane and the dimmensions of 60x3 mm2. The cross section of copper conductor is 5x15 mm2. The conductor is surrounded by a slider with *U* shaped cross section and a thickness of 1mm.

> Figure 10 and Figure 11 respectively show the magnetic induction spectrum and the magnetic field lines only, for the case of ferromagnetic slider.

**Figure 10.** The magnetic induction spectrum in the presence of the ferromagnetic slider

Figure 12 and Figure 13 respectively show the magnetic induction spectrum and the magnetic field lines only, for the case of *I* shaped ferromagnetic profile. Figure 14 and Figure 15 respectively show the magnetic induction spectrum and the magnetic field lines when using the ferromagnetic profile and the ferromagnetic slider (Vîrjoghe, 2004).

Analyzing these simulations, a strong influence of the ferromagnetic slider on the orientation of the magnetic field was observed. When using only the ferromagnetic slider, a shielding of the field lines is observed, and the maximum values of the magnetic induction is 0.907 T. The *I* shaped ferromagnetic profile makes a good shielding of the field lines

obtaining the maximum values of magnetic induction of 0.153 T. If both methods of magnetic field orientation are used together then a maximum value of magnetic induction of 0.947 T is obtained. To obtain the system optimization in this area, the simulations for a thickness of 2 mm have been repeated.

Finite Element Analysis of Stationary Magnetic Field 123

**Figure 13.** The magnetic equipotential lines in the presence of the ferromagnetic profile

**Figure 14.** The magnetic induction spectrum in the area with ramp, slider and ferromagnetic profile.

**Figure 15.** The magnetic equipotential lines in the area with ramp, slider and ferromagnetic profile.

**Figure 11.** The magnetic equipotential lines in the presence of the ferromagnetic slider

**Figure 12.** The magnetic induction spectrum in the case with the ferromagnetic profile

Thus, the obtained results are plotted in comparison with those presented for the slider of 1 mm (Figure 16). For the slider with the thickness of 2 mm, the values of magnetic induction are lower (up to 0.5 T). Using the slider of 1 mm thickness a better orientation of the field lines, as well as a better arc transmission toward arc-quenching chamber are observed. The path for the displayed charts is chosen between two points placed symmetrically one from another in the middle of a figure which contains the conductor, slider and ferromagnetic profile (Vîrjoghe, 2004).

**Figure 13.** The magnetic equipotential lines in the presence of the ferromagnetic profile

thickness of 2 mm have been repeated.

obtaining the maximum values of magnetic induction of 0.153 T. If both methods of magnetic field orientation are used together then a maximum value of magnetic induction of 0.947 T is obtained. To obtain the system optimization in this area, the simulations for a

**Figure 11.** The magnetic equipotential lines in the presence of the ferromagnetic slider

**Figure 12.** The magnetic induction spectrum in the case with the ferromagnetic profile

profile (Vîrjoghe, 2004).

Thus, the obtained results are plotted in comparison with those presented for the slider of 1 mm (Figure 16). For the slider with the thickness of 2 mm, the values of magnetic induction are lower (up to 0.5 T). Using the slider of 1 mm thickness a better orientation of the field lines, as well as a better arc transmission toward arc-quenching chamber are observed. The path for the displayed charts is chosen between two points placed symmetrically one from another in the middle of a figure which contains the conductor, slider and ferromagnetic

**Figure 14.** The magnetic induction spectrum in the area with ramp, slider and ferromagnetic profile.

**Figure 15.** The magnetic equipotential lines in the area with ramp, slider and ferromagnetic profile.

Finite Element Analysis of Stationary Magnetic Field 125

106 A/m2. As boudary conditions

These profiles form a rectangular prism with length of 150 mm, height of 100 mm and thickness of 5 mm. The electric arc ramps 1 and 2 are made of copper and have width of 10mm and thickness of 2.5 mm. The left ramp is inclined to the vertical with an angle of 45º. The arc chamber model together with ramps 1 and 2 are incorporated in a boundary volume, where the air is defined as material. In order to achieve the circuit continuity, two ramps have been unified with a bar 3 having the same dimensions of the ramps (Figure 18). In the preprocessing phase, the materials are defined and chosen. For the current path, consisting of two ramps and the connecting bar, the copper was chosen. For the two ferromagnetic plates, from ANSYS library a *M3* steel is chosen, having the properties contained in the *emagM3.SI\_MPL* file (ANSYS Documentation). The next step in the preprocessor phase is the mesh generation and load application upon the elements (Figure 19). In this application, for modeling the three-dimensional stationary magnetic field a *SOLID97* element is chosen. For the numerical computation of the stationary magnetic field, the model of the DC circuit-breaker together with the boundary volume is discretized in a

In the postprocessing phase is also applied the load on elements and boundary conditions.

The load on elements is represented by the conduction current density. For *3D* analysis, a positive value indicates current flowing in the *+Z* direction in the plan case and the *-Z* (loop) direction in the asymmetrical case. The current density is directly applied on the finite

number of 1268 nodes and 3623 triangular elements (Vîrjoghe, 2004).

elements which form the conductors and its value is 125.

**Figure 18.** The current path for DC circuit breaker – separator of 3125 A

the Dirichlet condition, *A*=0, is applied.

**Figure 16.** Magnetic induction variation depending on the thickness of the slider.

## **4.2. Numerical modelling of stationary magnetic field of arc-quenching chamber in case of a DC circuit breaker-separator – 3D application**

The physical model of the arc-quenching chamber in case of a DC circuit breaker-separator of 3125 A having the ramps-ferromagnetic profiles is shown in Figure 17. In this model, two profiles composed of a ferromagnetic material are presented. The magnetization curve of the two profiles is shown in Figure 9.

**Figure 17.** The DC circuit breaker-separator model with a current of 3125A.

These profiles form a rectangular prism with length of 150 mm, height of 100 mm and thickness of 5 mm. The electric arc ramps 1 and 2 are made of copper and have width of 10mm and thickness of 2.5 mm. The left ramp is inclined to the vertical with an angle of 45º.

124 Finite Element Analysis – New Trends and Developments

**Figure 16.** Magnetic induction variation depending on the thickness of the slider.

**in case of a DC circuit breaker-separator – 3D application** 

**Figure 17.** The DC circuit breaker-separator model with a current of 3125A.

two profiles is shown in Figure 9.

**4.2. Numerical modelling of stationary magnetic field of arc-quenching chamber** 

The physical model of the arc-quenching chamber in case of a DC circuit breaker-separator of 3125 A having the ramps-ferromagnetic profiles is shown in Figure 17. In this model, two profiles composed of a ferromagnetic material are presented. The magnetization curve of the The arc chamber model together with ramps 1 and 2 are incorporated in a boundary volume, where the air is defined as material. In order to achieve the circuit continuity, two ramps have been unified with a bar 3 having the same dimensions of the ramps (Figure 18).

In the preprocessing phase, the materials are defined and chosen. For the current path, consisting of two ramps and the connecting bar, the copper was chosen. For the two ferromagnetic plates, from ANSYS library a *M3* steel is chosen, having the properties contained in the *emagM3.SI\_MPL* file (ANSYS Documentation). The next step in the preprocessor phase is the mesh generation and load application upon the elements (Figure 19). In this application, for modeling the three-dimensional stationary magnetic field a *SOLID97* element is chosen. For the numerical computation of the stationary magnetic field, the model of the DC circuit-breaker together with the boundary volume is discretized in a number of 1268 nodes and 3623 triangular elements (Vîrjoghe, 2004).

In the postprocessing phase is also applied the load on elements and boundary conditions.

The load on elements is represented by the conduction current density. For *3D* analysis, a positive value indicates current flowing in the *+Z* direction in the plan case and the *-Z* (loop) direction in the asymmetrical case. The current density is directly applied on the finite elements which form the conductors and its value is 125. 106 A/m2. As boudary conditions the Dirichlet condition, *A*=0, is applied.

**Figure 18.** The current path for DC circuit breaker – separator of 3125 A

Finite Element Analysis of Stationary Magnetic Field 127

induction is 1.44 T. Hence, the optimal material for construction of these profiles is EmagVanad. Although the steel with vanadium is an expensive material, it assures an optimal value of magnetic induction. The high price is compensated by improving the arcquenching chamber performance and thus increase the breaking capacity of the device

**Figure 20.** The magnetic induction spectrum in arc-quenching chamber of DC circuit breaker-separator

**Figure 21.** Magnetic induction distribution for the ferromagnetic material EmagSilicon

(Vîrjoghe, 2004).

of 3125 A

**Figure 19.** The discretized model of the DC circuit breaker-separator of 3125 A

The Maxwell's equations solver is based on the Finite Element Method (FEM). The results are the nodal values of the primary unknowns (magnetic vector potential) and derivatives of these values for obtaining the secondary unknowns (magnetic induction).

In the postprocessing phase, the tool allows visualization of magnetic induction spectrum, determination of magnetic sizes in arbitrarily chosen points, as well as the evaluation of the different charts. Figure 20 shows the magnetic induction spectrum in the arc-quenching chamber of DC circuit breaker-separator of 3125 A. A maximum value of magnetic induction 2.149 T is obtained.

The DC circuit breaker-separator is designed as a particularly elaborated solution for the current path and the arc-quenching chamber. It is widely known that in electromechanic design of a switching device, the arc-quenching chamber together with current paths and contacts represent the essential element due to their switching performances in normal operating conditions and in abnormal conditions. An optimization criterion of this arc-quenching chamber concerns the ferromagnetic material used in the construction of the ferromagnetic profiles. Simulation was performed for three different steels. For the two ferromagnetic plates, three different steels from ANSYS library are chosen, namely (ANSYS Documentation):


It was established that in the case of steel EmagSilicon utilization, the maximum value of magnetic induction is 1.883 T, in the case of EmagVanad the maximum value of magnetic induction is 1.975 T and in the case of EmagSa1010 the maximum value of magnetic induction is 1.44 T. Hence, the optimal material for construction of these profiles is EmagVanad. Although the steel with vanadium is an expensive material, it assures an optimal value of magnetic induction. The high price is compensated by improving the arcquenching chamber performance and thus increase the breaking capacity of the device (Vîrjoghe, 2004).

126 Finite Element Analysis – New Trends and Developments

induction 2.149 T is obtained.

**Figure 19.** The discretized model of the DC circuit breaker-separator of 3125 A

these values for obtaining the secondary unknowns (magnetic induction).

The Maxwell's equations solver is based on the Finite Element Method (FEM). The results are the nodal values of the primary unknowns (magnetic vector potential) and derivatives of

In the postprocessing phase, the tool allows visualization of magnetic induction spectrum, determination of magnetic sizes in arbitrarily chosen points, as well as the evaluation of the different charts. Figure 20 shows the magnetic induction spectrum in the arc-quenching chamber of DC circuit breaker-separator of 3125 A. A maximum value of magnetic

The DC circuit breaker-separator is designed as a particularly elaborated solution for the current path and the arc-quenching chamber. It is widely known that in electromechanic design of a switching device, the arc-quenching chamber together with current paths and contacts represent the essential element due to their switching performances in normal operating conditions and in abnormal conditions. An optimization criterion of this arc-quenching chamber concerns the ferromagnetic material used in the construction of the ferromagnetic profiles. Simulation was performed for three different steels. For the two ferromagnetic plates,

three different steels from ANSYS library are chosen, namely (ANSYS Documentation):

SA1010 steel with material properties contained in the *emagSa1010.SI\_MPL* file.

 carpenter (silicon) steel with material properties contained in *emagSilicon.SI\_MPL* file; iron cobalt vanadium steel with material properties contained in the *emagVanad.SI\_MPL file*;

It was established that in the case of steel EmagSilicon utilization, the maximum value of magnetic induction is 1.883 T, in the case of EmagVanad the maximum value of magnetic induction is 1.975 T and in the case of EmagSa1010 the maximum value of magnetic

**Figure 20.** The magnetic induction spectrum in arc-quenching chamber of DC circuit breaker-separator of 3125 A

**Figure 21.** Magnetic induction distribution for the ferromagnetic material EmagSilicon

Finite Element Analysis of Stationary Magnetic Field 129

Comparing the magnetic flux density spectrums in the three cases it can be observed that the maximum arc-quenching effect is obtained by using EmagVanad for the ferromagnetic shapes. For this material an optimal distribution for the magnetic field in the circuit breaker arcing chamber is obtained, which leads to a rapid movement of the electric arc towards the ferromagnetic plates. Arc quenching and arc voltage limiting occur in base of the niche

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Morozionkov, J., Virbalis, J.A. (2008). Investigation of Electric Reactor Magnetic Field using

Răduleţ, R. (1975). Bazele teoretice ale electrotehnicii, Litografia Învăţământului, Bucureşti. Silvester P.P., Ferrari, R.L. (1996). Finite Elements for Electrical Engineers, 3rd Edition,

Stammberger, H. (1995). Magnetfeld-und Kraftberech-nungen für strombegrenzende

Timotin, A., Hortopan, V., Ifrim, A., Preda, V., (1970). Lecţii de bazele electrotehnicii,

Finite Element Method, *Electronics and Electrical Enginnering*, No.5(85).

Şora, C., (1982). Bazele electrotehnicii, *Didactică şi Pedagogică*, Bucureşti, 258-260.

Niederspannungs-Schaltgeräte, *VDI Verlag*, Elektrotechnik, Bonn, pp.12-20.

Truşcă, V., Truşcă, B. (2001). Elektrische anlagen – vorlesungen- , *Printech*, Bukarest. Vîrjoghe, E.O., (2004). Aparate electrice de comutaţie, *Electra*, Bucureşti, 93-99, 122-129.

effect principle along with the electrode effect (Hortopan, 1996).

Elena Otilia Virjoghe, Diana Enescu, Mihail-Florin Stan and Marcel Ionel

Flueraşu, C., Flueraşu, C. (2007). Electromagnetism, in *Printech*, Bucureşti. Gârbea, D. (1990). Analiză cu elemente finite, in Tehnică, Bucureşti, 7-8.

Hănţilă, I.F. (2004). Electrotehnica teoretică, in *Electra*, Bucureşti.

*University Press*, Cambridge, 28-29, 118-120, 183-187.

*Didactică şi Pedagogică*, Bucureşti.

**Author details** 

**6. References** 

*Valahia University of Targoviste, Romania* 

*Politehnica*, Bucureşti.

460.

Bucureşti.

Engineering, in *Electra*, Bucureşti, 427-436.

**Figure 22.** Magnetic induction distribution for the ferromagnetic material EmagVanad

**Figure 23.** Magnetic induction distribution for the ferromagnetic material EmagSa1010

## **5. Conclusion**

It is well known that in electromechanical construction of a switching device, the arcing chamber along with current paths and contacts represents the all-important elements concerning switching performances of these in normal operating conditions as well as in operation under faults (Truşcă & Truşcă, 2001).

Comparing the magnetic flux density spectrums in the three cases it can be observed that the maximum arc-quenching effect is obtained by using EmagVanad for the ferromagnetic shapes. For this material an optimal distribution for the magnetic field in the circuit breaker arcing chamber is obtained, which leads to a rapid movement of the electric arc towards the ferromagnetic plates. Arc quenching and arc voltage limiting occur in base of the niche effect principle along with the electrode effect (Hortopan, 1996).

## **Author details**

128 Finite Element Analysis – New Trends and Developments

**Figure 22.** Magnetic induction distribution for the ferromagnetic material EmagVanad

**Figure 23.** Magnetic induction distribution for the ferromagnetic material EmagSa1010

It is well known that in electromechanical construction of a switching device, the arcing chamber along with current paths and contacts represents the all-important elements concerning switching performances of these in normal operating conditions as well as in

**5. Conclusion** 

operation under faults (Truşcă & Truşcă, 2001).

Elena Otilia Virjoghe, Diana Enescu, Mihail-Florin Stan and Marcel Ionel *Valahia University of Targoviste, Romania* 

#### **6. References**


Vîrjoghe, E.O., Enescu, D., Ionel, M., Stan, M-F. (2010). 3D Finite Element Analysis for arcing chamber optimization of the current-limiting circuit breaker, *WSEAS TRANSACTIONS on POWER SYSTEMS*, ISSN: 1790-5060, Issue 1, Volume 5. ANSYS Release 11.0 Documentation.

**Chapter 6** 

© 2012 Thangavel, licensee InTech. This is an open access chapter distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

© 2012 Thangavel, licensee InTech. This is a paper distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

The finite difference method (FDM) is an alternative way of approximating solutions of

 The most attractive feature of the FEM is its ability to handle complicated geometries (and boundaries) with relative ease. While FDM in its basic form is restricted to handle

**Finite Element Analysis of the Direct** 

The **finite element method (FEM)** (its practical application often known as **finite element analysis** (FEA)) is a numerical technique for finding approximate solutions of partial differential equations (PDE) as well as integral equations. The solution approach is based either on eliminating the differential equation completely (steady state problems), or rendering the PDE into an approximating system of ordinary differential equations, which are then numerically integrated using standard techniques such as Euler's method, Runge-Kutta, etc. In solving partial differential equations, the primary challenge is to create an equation that approximates the equation to be studied, but is numerically stable, meaning that errors in the input and intermediate calculations do not accumulate and cause the resulting output to be meaningless. There are many ways of doing this, all with advantages and disadvantages. The finite element method is a good choice for solving partial differential equations over complicated domains (like cars and oil pipelines), when the domain changes (as during a solid state reaction with a moving boundary), when the desired precision varies over the entire domain, or when the solution lacks smoothness. For instance, in a frontal crash simulation it is possible to increase prediction accuracy in "important" areas like the front of the car and reduce it in its rear (thus reducing cost of the simulation). Another example would be in Numerical weather prediction, where it is more important to have accurate predictions over developing highly-nonlinear phenomena (such as tropical cyclones in the atmosphere, or eddies in the ocean) rather

Additional information is available at the end of the chapter

**Drive PMLOM** 

Govindaraj Thangavel

http://dx.doi.org/10.5772/46169

than relatively calm areas.

PDEs. The differences between FEM and FDM are:

**1. Introduction** 

## **Finite Element Analysis of the Direct Drive PMLOM**

Govindaraj Thangavel

130 Finite Element Analysis – New Trends and Developments

ANSYS Release 11.0 Documentation.

Vîrjoghe, E.O., Enescu, D., Ionel, M., Stan, M-F. (2010). 3D Finite Element Analysis for arcing chamber optimization of the current-limiting circuit breaker, *WSEAS TRANSACTIONS* 

*on POWER SYSTEMS*, ISSN: 1790-5060, Issue 1, Volume 5.

Additional information is available at the end of the chapter

http://dx.doi.org/10.5772/46169

## **1. Introduction**

The **finite element method (FEM)** (its practical application often known as **finite element analysis** (FEA)) is a numerical technique for finding approximate solutions of partial differential equations (PDE) as well as integral equations. The solution approach is based either on eliminating the differential equation completely (steady state problems), or rendering the PDE into an approximating system of ordinary differential equations, which are then numerically integrated using standard techniques such as Euler's method, Runge-Kutta, etc. In solving partial differential equations, the primary challenge is to create an equation that approximates the equation to be studied, but is numerically stable, meaning that errors in the input and intermediate calculations do not accumulate and cause the resulting output to be meaningless. There are many ways of doing this, all with advantages and disadvantages. The finite element method is a good choice for solving partial differential equations over complicated domains (like cars and oil pipelines), when the domain changes (as during a solid state reaction with a moving boundary), when the desired precision varies over the entire domain, or when the solution lacks smoothness. For instance, in a frontal crash simulation it is possible to increase prediction accuracy in "important" areas like the front of the car and reduce it in its rear (thus reducing cost of the simulation). Another example would be in Numerical weather prediction, where it is more important to have accurate predictions over developing highly-nonlinear phenomena (such as tropical cyclones in the atmosphere, or eddies in the ocean) rather than relatively calm areas.

The finite difference method (FDM) is an alternative way of approximating solutions of PDEs. The differences between FEM and FDM are:

 The most attractive feature of the FEM is its ability to handle complicated geometries (and boundaries) with relative ease. While FDM in its basic form is restricted to handle

© 2012 Thangavel, licensee InTech. This is an open access chapter distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. © 2012 Thangavel, licensee InTech. This is a paper distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

rectangular shapes and simple alterations thereof, the handling of geometries in FEM is theoretically straightforward.

Finite Element Analysis of the Direct Drive PMLOM 133

Tradeoffs between accuracy, robustness, and speed are central issues in numerical analysis, and here they receive careful consideration. The principal purpose of the work is to evaluate the performances of the PMLOM models when implemented in the FEM analysis of electrical machines. The developed methods are applied in an in-house FEM code, specialized for the design and analysis of electrical machines. The FEM simulations and the analysis on axial flux PMLOM, and the numerical results are validated experimentally. The techniques developed for the calculation of integral parameters involve particular

LINEAR motors are finding increasing applications in different specific areas like highspeed transport, electric hammers, looms, reciprocating pumps, heart pumps etc. [1]-[7]. They are also well suited for manufacturing automation applications. Therefore, design of energy efficient and high force to weight ratio motors and its performance assessment has become a research topic for quite a few years. The Permanent Magnet Linear Oscillating Motors (PMLOMs) are one of the derivatives of the linear motors in the low power applications having the advantages of higher efficiency. They can be supplied with dc or ac voltages [4]-[7] of which, the dc motors are having better efficiency due to the absence of the

The motor designed and analyzed in this paper finds the suitability of application in the loads having low frequency and short stroke requirements. One such application is the heart pump, where frequency of oscillation is to be adjusted between 0.5 to 1.5 Hz, with the requirement of variable thrust depending on the condition of the heart under treatment. For analysis of such motors the main task is to determine the essential equivalent circuit parameters, which are its resistances and inductances. The resistances, for the machine, though vary with operating conditions due to temperature, do not affect much on its performance assessment. However, the inductances for these machines are mover position dependent and mostly affect the machine performance. Therefore, determination of these parameters is essentially required for analyzing the machine model. There are several works [6], [9] executed which assumes the machine inductance to be constant for simplicity of the model although different other works [4], [7]and [8] dynamically estimate the inductance through FEM and field analysis and control[10-15] for getting correct results. In this paper, the machine under consideration is an axial flux machine and the mover is having a non-magnetic structure, which is aluminium. Also the rare earth permanent magnets used in the mover are having a relative permeability nearly equal to unity and therefore the magnetic circuit under consideration will be unsaturated due to major presence of air in the flux path. Hence, consideration of constant inductance is quite errorless for such kind of machines, which also conforms to the experimental data shown later. Finally the machine is analyzed with the help of the field equations and solved for forces and resultant flux densities through FEMLAB6.2 WITH MATHWORKS backed by suitable experimental results. A controller using PIC16F877A microcontroller has been developed for its speed and thrust control for successful implementation in the

assumptions and simplifications and present specific advantages.

core losses.

proposed application.


Generally, FEM is the method of choice in all types of analysis in structural mechanics (i.e. solving for deformation and stresses in solid bodies or dynamics of structures) while computational fluid dynamics (CFD) tends to use FDM or other methods like finite volume method (FVM). CFD problems usually require discretization of the problem into a large number of cells/gridpoints (millions and more), therefore cost of the solution favors simpler, lower order approximation within each cell. This is especially true for 'external flow' problems, like air flow around the car or airplane, or weather simulation.A variety of specializations under the umbrella of the mechanical engineering discipline (such as aeronautical, biomechanical, and automotive industries) commonly use integrated FEM in design and development of their products. Several modern FEM packages include specific components such as thermal, electromagnetic, fluid, and structural working environments. In a structural simulation, FEM helps tremendously in producing stiffness and strength visualizations and also in minimizing weight, materials, and costs.FEM allows detailed visualization of where structures bend or twist, and indicates the distribution of stresses and displacements. FEM software provides a wide range of simulation options for controlling the complexity of both modeling and analysis of a system. Similarly, the desired level of accuracy required and associated computational time requirements can be managed simultaneously to address most engineering applications. FEM allows entire designs to be constructed, refined, and optimized before the design is manufactured.

The 3-D finite element method (FEM) involves important computational methods. Many efforts have been undertaken in order to use 3-D FEM (FEMLAB6.2 WITH MATHWORKS). Analytical and Numerical Analysis have been developed for the analysis of the end zones of electrical machine.This paper presents different methodologies based on 3-D geometries using analytical solutions, This method has been implemented in conjunction with various geometry optimization techniques as it provides very fast solutions and has exhibited very good convergence with gradient free algorithms. Interior permanent magnet motors are widely applied to the industry because of many advantages. Also the characteristics of magnetic materials are important to the performance and efficiency of electrical devices. Tradeoffs between accuracy, robustness, and speed are central issues in numerical analysis, and here they receive careful consideration. The principal purpose of the work is to evaluate the performances of the PMLOM models when implemented in the FEM analysis of electrical machines. The developed methods are applied in an in-house FEM code, specialized for the design and analysis of electrical machines. The FEM simulations and the analysis on axial flux PMLOM, and the numerical results are validated experimentally. The techniques developed for the calculation of integral parameters involve particular assumptions and simplifications and present specific advantages.

132 Finite Element Analysis – New Trends and Developments

between grid points is poor in FDM.

contrary can be provided.

theoretically straightforward.

triangles.

rectangular shapes and simple alterations thereof, the handling of geometries in FEM is

 The most attractive feature of finite differences is that it can be very easy to implement. There are several ways one could consider the FDM a special case of the FEM approach. E.g., first order FEM is identical to FDM for Poisson's equation, if the problem is discretized by a regular rectangular mesh with each rectangle divided into two

 There are reasons to consider the mathematical foundation of the finite element approximation more sound, for instance, because the quality of the approximation

 The quality of a FEM approximation is often higher than in the corresponding FDM approach, but this is extremely problem-dependent and several examples to the

Generally, FEM is the method of choice in all types of analysis in structural mechanics (i.e. solving for deformation and stresses in solid bodies or dynamics of structures) while computational fluid dynamics (CFD) tends to use FDM or other methods like finite volume method (FVM). CFD problems usually require discretization of the problem into a large number of cells/gridpoints (millions and more), therefore cost of the solution favors simpler, lower order approximation within each cell. This is especially true for 'external flow' problems, like air flow around the car or airplane, or weather simulation.A variety of specializations under the umbrella of the mechanical engineering discipline (such as aeronautical, biomechanical, and automotive industries) commonly use integrated FEM in design and development of their products. Several modern FEM packages include specific components such as thermal, electromagnetic, fluid, and structural working environments. In a structural simulation, FEM helps tremendously in producing stiffness and strength visualizations and also in minimizing weight, materials, and costs.FEM allows detailed visualization of where structures bend or twist, and indicates the distribution of stresses and displacements. FEM software provides a wide range of simulation options for controlling the complexity of both modeling and analysis of a system. Similarly, the desired level of accuracy required and associated computational time requirements can be managed simultaneously to address most engineering applications. FEM allows entire designs to be

constructed, refined, and optimized before the design is manufactured.

The 3-D finite element method (FEM) involves important computational methods. Many efforts have been undertaken in order to use 3-D FEM (FEMLAB6.2 WITH MATHWORKS). Analytical and Numerical Analysis have been developed for the analysis of the end zones of electrical machine.This paper presents different methodologies based on 3-D geometries using analytical solutions, This method has been implemented in conjunction with various geometry optimization techniques as it provides very fast solutions and has exhibited very good convergence with gradient free algorithms. Interior permanent magnet motors are widely applied to the industry because of many advantages. Also the characteristics of magnetic materials are important to the performance and efficiency of electrical devices. LINEAR motors are finding increasing applications in different specific areas like highspeed transport, electric hammers, looms, reciprocating pumps, heart pumps etc. [1]-[7]. They are also well suited for manufacturing automation applications. Therefore, design of energy efficient and high force to weight ratio motors and its performance assessment has become a research topic for quite a few years. The Permanent Magnet Linear Oscillating Motors (PMLOMs) are one of the derivatives of the linear motors in the low power applications having the advantages of higher efficiency. They can be supplied with dc or ac voltages [4]-[7] of which, the dc motors are having better efficiency due to the absence of the core losses.

The motor designed and analyzed in this paper finds the suitability of application in the loads having low frequency and short stroke requirements. One such application is the heart pump, where frequency of oscillation is to be adjusted between 0.5 to 1.5 Hz, with the requirement of variable thrust depending on the condition of the heart under treatment. For analysis of such motors the main task is to determine the essential equivalent circuit parameters, which are its resistances and inductances. The resistances, for the machine, though vary with operating conditions due to temperature, do not affect much on its performance assessment. However, the inductances for these machines are mover position dependent and mostly affect the machine performance. Therefore, determination of these parameters is essentially required for analyzing the machine model. There are several works [6], [9] executed which assumes the machine inductance to be constant for simplicity of the model although different other works [4], [7]and [8] dynamically estimate the inductance through FEM and field analysis and control[10-15] for getting correct results. In this paper, the machine under consideration is an axial flux machine and the mover is having a non-magnetic structure, which is aluminium. Also the rare earth permanent magnets used in the mover are having a relative permeability nearly equal to unity and therefore the magnetic circuit under consideration will be unsaturated due to major presence of air in the flux path. Hence, consideration of constant inductance is quite errorless for such kind of machines, which also conforms to the experimental data shown later. Finally the machine is analyzed with the help of the field equations and solved for forces and resultant flux densities through FEMLAB6.2 WITH MATHWORKS backed by suitable experimental results. A controller using PIC16F877A microcontroller has been developed for its speed and thrust control for successful implementation in the proposed application.

## **2. Machine construction**

The construction of the prototype PMLOM is shown in Fig.1 below. Also the dimensional details of the motor are shown in Fig.2. There are two concentric coils on the surface of the stators connected in such polarities that the fluxes for both the coils aid each other to form the poles in the iron parts. The formation of the N and the S poles of the electromagnet of the stator are shown in the Fig.2.

Finite Element Analysis of the Direct Drive PMLOM 135

*H J* (1)

*B* 0 (2)

1 2 *nH H* ˆ 0 (4)

1 2 *nB B* ˆ 0 (5)

*B A* (6)

<sup>1</sup> *<sup>A</sup> <sup>J</sup>* (7)

*J Jz* ˆ (8)

*A Az* ˆ (9)

<sup>1</sup> *A J* (10)

(3)

**3. Simulation and experimental results** 

and experiment is given in Table-1.

of the magnetic field.

Where 

If

Then,

Thus, (7) reduces to,

The proposed scheme is simulated under FEMLAB6.2 WITH MATHWORKS environment, which provides a finite element analysis. The machine specification used for both simulation

Where *H* is magnetic field intensity, *B* is magnetic flux density and *J* is the current density

*B H* 

Since the divergence of the curl of any vector must always be zero, it follows from (2) that

denotes material permeability. Boundary conditions that must be satisfied at the

Subject to a constitutive relationship between B and H for each material:

interface between two materials having finite conductivities are,

there exists a so-called magnetic vector potential A such that,

Substituting (3) and (6) into (1) and taking a curl on both sides yields

The classical description of static magnetic fields are provided by Maxwell's equations

**Figure 1.** Construction details of the developed PMLOM (i) Stators to be mounted on both sides of the mover and (ii) the mover (iii) the PMLOM machine

Al – Aluminium material PM-N42 Permanent Magnet Attraction Force *AF* and Repulsion Force *RF* Coil 1 – aa' and bb' Coil 2 – cc' and dd'

**Figure 2.** Dimensional details of the developed PMLOM

### **3. Simulation and experimental results**

The proposed scheme is simulated under FEMLAB6.2 WITH MATHWORKS environment, which provides a finite element analysis. The machine specification used for both simulation and experiment is given in Table-1.

The classical description of static magnetic fields are provided by Maxwell's equations

$$
\nabla \times \mathbf{H} = \mathbf{J} \tag{1}
$$

$$
\nabla \cdot B = 0 \tag{2}
$$

Where *H* is magnetic field intensity, *B* is magnetic flux density and *J* is the current density of the magnetic field.

Subject to a constitutive relationship between B and H for each material:

$$B = \mu H \tag{3}$$

Where denotes material permeability. Boundary conditions that must be satisfied at the interface between two materials having finite conductivities are,

$$
\hat{m} \times \left( H\_1 - H\_2 \right) = 0 \tag{4}
$$

$$
\hat{n} \cdot \left( B\_1 - B\_2 \right) = 0 \tag{5}
$$

Since the divergence of the curl of any vector must always be zero, it follows from (2) that there exists a so-called magnetic vector potential A such that,

$$B = \nabla \times A \tag{6}$$

Substituting (3) and (6) into (1) and taking a curl on both sides yields

$$\nabla \times \left(\frac{1}{\mu} \nabla \times A\right) = J \tag{7}$$

If

134 Finite Element Analysis – New Trends and Developments

mover and (ii) the mover (iii) the PMLOM machine

Al – Aluminium material PM-N42 Permanent Magnet

**Figure 2.** Dimensional details of the developed PMLOM

Attraction Force *AF* and Repulsion Force *RF* Coil 1 – aa' and bb' Coil 2 – cc' and dd'

The construction of the prototype PMLOM is shown in Fig.1 below. Also the dimensional details of the motor are shown in Fig.2. There are two concentric coils on the surface of the stators connected in such polarities that the fluxes for both the coils aid each other to form the poles in the iron parts. The formation of the N and the S poles of the electromagnet of

**Figure 1.** Construction details of the developed PMLOM (i) Stators to be mounted on both sides of the

**2. Machine construction** 

the stator are shown in the Fig.2.

*J Jz* ˆ (8)

Then,

$$A = A\hat{z}\tag{9}$$

Thus, (7) reduces to,

$$-\nabla \cdot \left(\frac{1}{\mu} \nabla A\right) = J \tag{10}$$

The above equation (10) may be written in the expanded form as,

$$\frac{\partial}{\partial \mathbf{x}} \left( \frac{1}{\mu} \frac{\partial A}{\partial \mathbf{x}} \right) + \frac{\partial}{\partial y} \left( \frac{1}{\mu} \frac{\partial A}{\partial y} \right) = -J \tag{11}$$

Finite Element Analysis of the Direct Drive PMLOM 137

**Figure 3.** (a) Finite element mesh of PMLOM while mover is oscillating with in Stator 1. (b) Magnetic flux plotting of PMLOM while mover is oscillating with in Stator 1, at 1 Hz, 4Amps, (c) Magnetic flux plotting of PMLOM while mover is oscillating with in Stator 1, at 0 Hz. (d) Finite element Magnetic flux plotting at upper and lower part of the airgap while mover oscillates within stator 1. Now Mover is

(d)

(c)

attracted to the Stator 1

This equation (11) represents the *scalar Poisson equation*.

The mover consists of aluminium structure embedded with rare earth permanent magnets with the polarities as shown. The force developed will be attractive on one side and simultaneously repulsive on the other side. These two forces act in the same direction to enhance the total force on the mover, assisting the linear oscillation of the mover cyclically.

(b)

The above equation (10) may be written in the expanded form as,

This equation (11) represents the *scalar Poisson equation*.

 

*xxyy* (11)

 1 1 *A A <sup>J</sup>*

The mover consists of aluminium structure embedded with rare earth permanent magnets with the polarities as shown. The force developed will be attractive on one side and simultaneously repulsive on the other side. These two forces act in the same direction to enhance the total force on the mover, assisting the linear oscillation of the mover cyclically.

(a)

(b)

(d)

**Figure 3.** (a) Finite element mesh of PMLOM while mover is oscillating with in Stator 1. (b) Magnetic flux plotting of PMLOM while mover is oscillating with in Stator 1, at 1 Hz, 4Amps, (c) Magnetic flux plotting of PMLOM while mover is oscillating with in Stator 1, at 0 Hz. (d) Finite element Magnetic flux plotting at upper and lower part of the airgap while mover oscillates within stator 1. Now Mover is attracted to the Stator 1

138 Finite Element Analysis – New Trends and Developments


Finite Element Analysis of the Direct Drive PMLOM 139

*<sup>r</sup>* =1.048) was supposed. Its

accurately discretized with fine meshes. Symmetry was exploited to reduce the problem

The halved longitudinal cross section of the motor has created the calculation area, with Dirichlet boundary conditions (Fig. 3*(d)*). Thus, the magnetic field has been analyzed. For the

coercive force was assumed to be *Hc* = 925 KA/m and the magnetization vector direction were adopted for the calculations. Very small air gaps compared with the main motor dimensions between permanent magnets and ferromagnetic rings were neglected due to very small

The finite-element mesh (Fig. 3*(d)*) is dense in the air gap between stator cylinder sleeve and the mover. The fine mesh is also used near the edges of ferromagnetic parts where the magnetic field is expected to vary rapidly (Fig. 3*(d)*)*.* In order to predict the integral parameters of the PMLOM, it is necessary to analyze the magnetic field distribution in the stator and mover. Obviously, it is possible to optimize the construction by making changes in the stator and mover geometries. The improvements of the structure result from knowledge of the magnetic field distribution. The presented results have been obtained for

The control block diagram along with the experimental set-up power electronic control circuit is shown in Fig.4. Here the thrust control is provided with the help of phase controlled ac supply which can vary the input voltage. The frequency control is provided

The set up is reliable and provides a scope for portability to any remote place. Fig. 5 shows the plot of the input voltage and current of the machine at 5 Hz. From which the assumption of constant inductance for the machine can be well validated. Fig. 6 shows the characteristics plot of input power, voltage and force as a function of current for the machine taken at a frequency of 1Hz. Figure 7 shows Force at different axial airgap length. Force observed by

with the help of a low cost and commercially available microcontroller PIC16F877A.

measurement is compared with the theoretical Force value and shown in fig. 8.

**Figure 5.** Current waveform of PMLOM taken from Tektronix make Storage Oscilloscope

domain to half of the axial cross section of the overall motor.

calculations, material linearity of the NdFeB permanent magnet *(*

magnetic permeability of the permanent magnets, it is acceptable.

one variant of the motor construction.

**Table 1.** PMLOM Design Parameters

Figure 3(a), shows the FEM mesh configuration for the PMLOM Prototype. Figure 3(b) shows the Magnetic flux plotting of PMLOM while mover is oscillating within Stator 1, at 1 Hz, 4Amps.Figure 3(c) is the corresponding flux plotting of the machine while mover is oscillating within Stator 1, at 0 Hz. Figure 3(d) illustrates the finite element analysis of the PMLOM at the axial airgap. Thus, the geometries of the mover and stator have been accurately discretized with fine meshes. Symmetry was exploited to reduce the problem domain to half of the axial cross section of the overall motor.

138 Finite Element Analysis – New Trends and Developments

**Table 1.** PMLOM Design Parameters

**Figure 4.** Power Circuit of PMLOM

Rated Input Voltage 70V Rated input power 200 watts Stroke length 10 mm Outer Diameter (Stator) 85 mm

Thickness of lamination 0.27 mm Stator length 60 mm

Number of turns in Coil aa',cc' 800 Number of turns in Coil bb',dd' 400

Permanent Magnet Length 2 mm

Stator core type CRGO Silicon Steel

Coil resistance 17.8 ohms Slot depth 45 mm Permanent Magnet Type Rare Earth N42, Nd-Fe-B

Coercivity 925000 A/m Remanence 1.3 T Outer diameter (Mover) 65 mm Shaft diameter 8 mm Coil Inductance 0.18 Henry

Figure 3(a), shows the FEM mesh configuration for the PMLOM Prototype. Figure 3(b) shows the Magnetic flux plotting of PMLOM while mover is oscillating within Stator 1, at 1 Hz, 4Amps.Figure 3(c) is the corresponding flux plotting of the machine while mover is oscillating within Stator 1, at 0 Hz. Figure 3(d) illustrates the finite element analysis of the PMLOM at the axial airgap. Thus, the geometries of the mover and stator have been The halved longitudinal cross section of the motor has created the calculation area, with Dirichlet boundary conditions (Fig. 3*(d)*). Thus, the magnetic field has been analyzed. For the calculations, material linearity of the NdFeB permanent magnet *( <sup>r</sup>* =1.048) was supposed. Its coercive force was assumed to be *Hc* = 925 KA/m and the magnetization vector direction were adopted for the calculations. Very small air gaps compared with the main motor dimensions between permanent magnets and ferromagnetic rings were neglected due to very small magnetic permeability of the permanent magnets, it is acceptable.

The finite-element mesh (Fig. 3*(d)*) is dense in the air gap between stator cylinder sleeve and the mover. The fine mesh is also used near the edges of ferromagnetic parts where the magnetic field is expected to vary rapidly (Fig. 3*(d)*)*.* In order to predict the integral parameters of the PMLOM, it is necessary to analyze the magnetic field distribution in the stator and mover. Obviously, it is possible to optimize the construction by making changes in the stator and mover geometries. The improvements of the structure result from knowledge of the magnetic field distribution. The presented results have been obtained for one variant of the motor construction.

The control block diagram along with the experimental set-up power electronic control circuit is shown in Fig.4. Here the thrust control is provided with the help of phase controlled ac supply which can vary the input voltage. The frequency control is provided with the help of a low cost and commercially available microcontroller PIC16F877A.

The set up is reliable and provides a scope for portability to any remote place. Fig. 5 shows the plot of the input voltage and current of the machine at 5 Hz. From which the assumption of constant inductance for the machine can be well validated. Fig. 6 shows the characteristics plot of input power, voltage and force as a function of current for the machine taken at a frequency of 1Hz. Figure 7 shows Force at different axial airgap length. Force observed by measurement is compared with the theoretical Force value and shown in fig. 8.

**Figure 5.** Current waveform of PMLOM taken from Tektronix make Storage Oscilloscope

Finite Element Analysis of the Direct Drive PMLOM 141

A simple control method along with the development of an axial flux PMLOM suitable for low frequency and short stroke application is presented. Analytical solution to the forces and determination method of the integral parameters of a PMLOM are shown. Finite element method with FEMLAB6.2 WITH MATHWORKS is used for the field analysis of the different values of the exciting current and for variable mover position. Computer simulations for the magnetic field distribution, forces are given. To obtain experimentally the field distribution and its integral parameters, a physical model of the motor together with its electronic controller system has been developed and tested. The Prototype has been operated in the oscillatory mode with small loads at low frequency up to 5 Hz. The theoretically calculated results are compared with the measured ones and found a good

*Department of Electrical and Electronics Engineering, Muthayammal Engineering College, India* 

[1] Kou Baoquan, Li Liyi, and Zhang Chengming, "Analysis and Optimization of Thrust Characteristics of Tubular Linear Electromagnetic Launcher for Space-Use,"*IEEE* 

[2] Ge Baoming, Aníbal T. de Almeida, and Fernando J. T. E.Ferreira, "Design of Transverse Flux Linear Switched Reluctance Motor,"*IEEE Trans. Magn., vol 45, no.1,*

[3] H. D. Chai*, Electromechanical Motion Devices*. Upper Saddle River, NJ: Prentice Hall,

[4] G. Kang, J. Hong, and G. Kim, "Design and analysis of slotless-type permanent- magnet linear brushless motor by using equivalent magnetizing current," *IEEE Trans. Ind. Appl.*,

[5] S. A. Nasar and I. Boldea, *Linear Electric Motors*..Englewood Cliffs, NJ: Prentice-Hall,

[6] B. Tomczuk and M. Sobol, "Influence of the supply voltage on the dynamics of the onephase tubular motor with reversal motion," in *Proc. 39th Int. Symp. Electrical Machines—*

[7] N. Sadowski, R. Carlson, A. M. Beckert, and J. P. A.Bastos, "Dynamic modeling of a newly designed linear Actuator using 3D edge elements analysis," *IEEE Trans. Magn.*,

[8] D. G. Taylor and N. Chayopitak, "Time-optimal position control of electric motors with steady-state temperature constraints," in *Proc.IEEE Int. Symp. Industrial Electronics*,

*SME'2003*, Gdansk/Jurata, Poland, Jun. 9–11, 2003, pp. 417–426

*Trans.Magn., vol.. 45, no. 1*, pp. 250- 255,Jan.2009

**4. Conclusions** 

conformity.

**Author details** 

**5. References** 

1998.

1987.

Govindaraj Thangavel

pp.113-119 Jan 2009

vol. 37, no. 5, pp. 1241–1247,2001.

vol. 32, no. 3, pp. 1633–1636, May 1996.

Montreal, QC, Canada, Jul 2006, pp. 3142– 3146.

**Figure 6.** Measured Coil current versus Power( (W).Voltage(V),Force(N) Characteristics of PMLOM

**Figure 7.** Axial Airgap Length versus Force

**Figure 8.** Comparison of measured Force Versus Theoretical Force

## **4. Conclusions**

140 Finite Element Analysis – New Trends and Developments

**Force(N)**

**Figure 7.** Axial Airgap Length versus Force

**3**

**3.5**

**4**

**4.5**

**5**

**Figure 8.** Comparison of measured Force Versus Theoretical Force

**5.5**

**6**

**6.5**

**7**

**7.5**

**8**

**8.5**

**9**

**9.5**

**9.5**

**Airgap (mm)**

**9**

**8.5**

**8**

**7.5**

**7**

**6.5**

**6**

**5.5**

**5**

**4.5**

**4**

**3.5**

**3**

**Figure 6.** Measured Coil current versus Power( (W).Voltage(V),Force(N) Characteristics of PMLOM

**Axial Airgap versus Force**

A simple control method along with the development of an axial flux PMLOM suitable for low frequency and short stroke application is presented. Analytical solution to the forces and determination method of the integral parameters of a PMLOM are shown. Finite element method with FEMLAB6.2 WITH MATHWORKS is used for the field analysis of the different values of the exciting current and for variable mover position. Computer simulations for the magnetic field distribution, forces are given. To obtain experimentally the field distribution and its integral parameters, a physical model of the motor together with its electronic controller system has been developed and tested. The Prototype has been operated in the oscillatory mode with small loads at low frequency up to 5 Hz. The theoretically calculated results are compared with the measured ones and found a good conformity.

## **Author details**

Govindaraj Thangavel *Department of Electrical and Electronics Engineering, Muthayammal Engineering College, India* 

## **5. References**

	- [9] S. Vaez-Zadeh and A. Isfahani, "Multi-objective design optimization of air-core linear permanent-magnet synchronous motors for improved thrust and low magnet consumption," *IEEE Trans. Magn.*, vol.42, no. 3, pp. 446–452, 2006.

**Chapter 7** 

© 2012 Calado et al., licensee InTech. This is an open access chapter distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

© 2012 Calado et al., licensee InTech. This is a paper distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

This chapter presents the Finite Element Method for the calculation of a rod resistance in a two layer soil. Theoretical basis of the method are presented. The FEM mesh was tested in

**The IEEE Model for a Ground Rod in** 

**a Two Layer Soil – A FEM Approach** 

Additional information is available at the end of the chapter

example. Comparison with experimental result is also made.

http://dx.doi.org/10.5772/48252

**1. Introduction** 

António Martins, Sílvio Mariano and Maria do Rosário Calado

The calculation of a ground electrode resistance, using a two layer soil model, has been widely presented in literature. Several methods had been used. Formulas for grid in two layers soil using the synthetic-asymptote approach have been developed in (Salama et al., 1995). Berberovic explored the Method of Moments in the calculation of ground resistance, using higher order polynomials approximation in the unknown current distribution in (Berberovic et al., 2003), and the Galerkin's Moment Method with a variation was used in (Sharma & De Four, 2006). Another theoretical tool commonly used is the Boundary Element Method, as in (Colominas et al., 1998, 2002a, 2002b; Adriano et al., 2003). These authors transformed the differential equation that governs the physical phenomenon into an equivalent boundary integral equation. The Matrix/Integration Method for calculating the mutual resistance segment in one and two layered soil was adopted by (Coa, 2006) and an optimised method of images for multilayer soils was used in (Ma et al., 1996). Even in the study of ionization phenomena, the two layer ground model was used in (Liu et al., 2004). In general these works used the theory of images, which implies infinite series for the expanded Green function as in (Berberovic et al., 2003). Recently, a work presented the effect of low resistance materials filling in a pit surrounding a rod, working with two different soil resistivity's (Al-Arayny et al., 2011). This type of research was also presented in (Zhenghua et al., 2011), that even considered the use of electrolytic materials. A Finite Element Method (FEM) application to grounding can also be found in (Manikandan et al., 2011) to the analysis of wind turbines grounding. In this chapter the FEM is presented in a theoretical basis for cylindrical symmetry problems, using a ground rod resistance calculation as an


## **The IEEE Model for a Ground Rod in a Two Layer Soil – A FEM Approach**

António Martins, Sílvio Mariano and Maria do Rosário Calado

Additional information is available at the end of the chapter

http://dx.doi.org/10.5772/48252

## **1. Introduction**

142 Finite Element Analysis – New Trends and Developments

18, 2009, pp. 351- 355

[9] S. Vaez-Zadeh and A. Isfahani, "Multi-objective design optimization of air-core linear permanent-magnet synchronous motors for improved thrust and low magnet

[10] Govindaraj T, Debashis Chatterjee, and Ashoke K. Ganguli, "A Permanent Magnet Linear Oscillating Motor for Short Strokes,"in *Proc.Int .Conf. Electrical Energy Systems & Power Electronics in Emerging Economies- ICEESPEEE 09, SRM University, India,* Apl. 16-

[11] Govindaraj T, Debashis Chatterjee, and Ashoke K. Ganguli,"Development, Finite Element Analysis and Electronic Controlled Axial Flux Permanent Magnet Linear Oscillating Motor direct drive suitable for short Strokes" in *Proc. Int. Conf. Control,Automation, Communication and Energy Conservation -INCACEC 2009, Kongu* 

[12] Govindaraj T, Debashis Chatterjee, and Ashoke K. Ganguli,"FE Magnetic Field Analysis Simulation Models based Design, Development, Control and Testing of An Axial Flux Permanent Magnet Linear Oscillating Motor" in *Proc. Int. Conf. Electrical andElectronics Engineering,ICEEE2009, Int. Association of Engineers,World Congress on Engineering* 

[13] Govindaraj T, Debashis Chatterjee, and Ashoke K. Ganguli, "Development, Analysis and Control of an Axial Flux Permanent Magnet Linear Oscillating Motor suitable for Short Strokes" in *Proc. 2009 IEEE Int. Sym. on Industrial Electronics,IEEE ISIE 2009*, Seoul

[14] Govindaraj T, Debashis Chatterjee, and Ashoke K. Ganguli,"Development, Control and Testing of a New Axial Flux Permanent Magnet Linear Oscillating Motor using FE Magnetic Field Analysis Simulation Models", *Proc. 2009 Int. Conf. on Mechanical and Electronics Engineering, ICMEE2009,Int. Association of Computer Science and Information* 

[15] D.G. Holmes, T. A. Lipo, B. P. McGrath, and W.Y. Kong," Optimized Design of Stationary Frame Three phase AC Current Regulators"*IEEE Trans. Power Electronics, vol.* 

consumption," *IEEE Trans. Magn.*, vol.42, no. 3, pp. 446–452, 2006.

*Engineering College, India* Jun.4–6, 2009, pp. 479–483.

*2009vol 1,* London, United Kingdom, July 1-3, 2009

Olympic Parktel, Seoul, Korea, July 5-8, 2009, 29-34

*24, no. 11*, pp 2417-2426, Nov 2009.

*Technology, IACSIT*, Chennai, India, July 24-26, 2009, pp 191 - 195

The calculation of a ground electrode resistance, using a two layer soil model, has been widely presented in literature. Several methods had been used. Formulas for grid in two layers soil using the synthetic-asymptote approach have been developed in (Salama et al., 1995). Berberovic explored the Method of Moments in the calculation of ground resistance, using higher order polynomials approximation in the unknown current distribution in (Berberovic et al., 2003), and the Galerkin's Moment Method with a variation was used in (Sharma & De Four, 2006). Another theoretical tool commonly used is the Boundary Element Method, as in (Colominas et al., 1998, 2002a, 2002b; Adriano et al., 2003). These authors transformed the differential equation that governs the physical phenomenon into an equivalent boundary integral equation. The Matrix/Integration Method for calculating the mutual resistance segment in one and two layered soil was adopted by (Coa, 2006) and an optimised method of images for multilayer soils was used in (Ma et al., 1996). Even in the study of ionization phenomena, the two layer ground model was used in (Liu et al., 2004). In general these works used the theory of images, which implies infinite series for the expanded Green function as in (Berberovic et al., 2003). Recently, a work presented the effect of low resistance materials filling in a pit surrounding a rod, working with two different soil resistivity's (Al-Arayny et al., 2011). This type of research was also presented in (Zhenghua et al., 2011), that even considered the use of electrolytic materials. A Finite Element Method (FEM) application to grounding can also be found in (Manikandan et al., 2011) to the analysis of wind turbines grounding. In this chapter the FEM is presented in a theoretical basis for cylindrical symmetry problems, using a ground rod resistance calculation as an example. Comparison with experimental result is also made.

This chapter presents the Finite Element Method for the calculation of a rod resistance in a two layer soil. Theoretical basis of the method are presented. The FEM mesh was tested in

© 2012 Calado et al., licensee InTech. This is an open access chapter distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. © 2012 Calado et al., licensee InTech. This is a paper distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

homogeneous soil, first a cylindrical approach to IEEE model was considered, in order to guarantee that mesh is voltage and energy adapted, and then the whole model was discretized. Resistance was calculated using Joule's law in a FEM energetic approach. The zero volt Dirichlet boundary was located within a distance of 3 cm, 15 cm and 7.6 m from the ground electrode, to analyze if the resistance relations indicated by IEEE in homogeneous soil could be also used in a two layer soil. The simulated results were compared with those obtained from Tagg formula. In homogeneous soil the errors are less than 1%. In two layer soil, resistance error to Tagg formula decreased from values of 28% with zero volt boundary at 3 cm to 22% with zero volt boundary at 15 cm; for the whole model discretized the error is near -18%. The results were unsatisfactory, thus the percentage resistances at these distances cannot be generalized in a two layer soil. However the error between simulated and field measured values is of 4.6%, turning the FEM analysis a valuable simulation tool.

## **2. Finite element method**

The first order finite elements method using triangular elements may be regarded as twodimensional generalizations of piecewise-linear approximation techniques as in reference (Sylvester & Ferrari, 1990), widely used in Electrical Engineering. The method allows several choices for mesh types and an easy treatment for boundary shapes.

Several problems in electrical engineering require the solution of Laplace equation in two or three dimensions with two kinds of boundary conditions, such as prescribed potential values along the referred boundaries, Dirichlet conditions, and vanishing normal derivative along the symmetry planes, Neumann condition. As an example, in power system grounding, where capacitive and inductive effects are not considered, since industrial frequency is too low, the soil potential satisfies Laplace equation. On the other hand grounding systems dimensions are much smaller than power line wavelength, so that propagation phenomenon is not considered. Laplace equation solution is equivalent, according to the minimum potential energy principle, to the following energy ( *W u* ) functional minimization, that stores field energy per unit volume, as in (Sylvester & Ferrari, 1990):

$$\mathcal{W}(u) = \frac{1}{2} \iiint\_{V} \left| \nabla u \right|^{2} dV \tag{1}$$

The IEEE Model for a Ground Rod in a Two Layer Soil – A FEM Approach 145

**Figure 1.** IEEE model as in (IEEE Std 142, 2007)

**Figure 2.** Model discretization near rod top end

**2.2. First order triangular elements** 

�2.025

�2.02

�2.015

�2.01

�2.005

Depth (m)

�1.995

�1.99

�1.985

�2

Due to isotropy, the problem was discretized in the '*rz*' plane, since cylindrical coordinates were used. To obtain an approximate solution by FEM the problem region is subdivided into triangular elements. The solution mesh is presented in Fig. 2, near the rod top end.

r 3.0 m

7.6 m

The essence of the method lies in first approximating the potential *u* within each element in a standardized fashion, and thereafter interrelating the potential distribution in the various elements so as to constrain the potential to be continuous across interelement boundaries, as

�0.01�0.005 0 0.005 0.01 0.015 0.02 0.025 0.03 0.035 0.04

Radius (m)

where *u* is the potential and *V* the volume.

The integral is evaluated over all the volume defined by the problem boundary.

#### **2.1. Discretization**

In Fig. 1 the IEEE model for a ground rod is presented. The boundary conditions within a distance of 7.6 m from the electrode have zero potential. At surface, normal derivative potential is also zero, since there is no current flowing in the air.

**Figure 1.** IEEE model as in (IEEE Std 142, 2007)

a valuable simulation tool.

**2. Finite element method** 

homogeneous soil, first a cylindrical approach to IEEE model was considered, in order to guarantee that mesh is voltage and energy adapted, and then the whole model was discretized. Resistance was calculated using Joule's law in a FEM energetic approach. The zero volt Dirichlet boundary was located within a distance of 3 cm, 15 cm and 7.6 m from the ground electrode, to analyze if the resistance relations indicated by IEEE in homogeneous soil could be also used in a two layer soil. The simulated results were compared with those obtained from Tagg formula. In homogeneous soil the errors are less than 1%. In two layer soil, resistance error to Tagg formula decreased from values of 28% with zero volt boundary at 3 cm to 22% with zero volt boundary at 15 cm; for the whole model discretized the error is near -18%. The results were unsatisfactory, thus the percentage resistances at these distances cannot be generalized in a two layer soil. However the error between simulated and field measured values is of 4.6%, turning the FEM analysis

The first order finite elements method using triangular elements may be regarded as twodimensional generalizations of piecewise-linear approximation techniques as in reference (Sylvester & Ferrari, 1990), widely used in Electrical Engineering. The method allows several

Several problems in electrical engineering require the solution of Laplace equation in two or three dimensions with two kinds of boundary conditions, such as prescribed potential values along the referred boundaries, Dirichlet conditions, and vanishing normal derivative along the symmetry planes, Neumann condition. As an example, in power system grounding, where capacitive and inductive effects are not considered, since industrial frequency is too low, the soil potential satisfies Laplace equation. On the other hand grounding systems dimensions are much smaller than power line wavelength, so that propagation phenomenon is not considered. Laplace equation solution is equivalent, according to the minimum potential energy principle, to the following energy ( *W u* ) functional minimization, that

<sup>1</sup> <sup>2</sup> ( ) <sup>2</sup> *<sup>V</sup>*

In Fig. 1 the IEEE model for a ground rod is presented. The boundary conditions within a distance of 7.6 m from the electrode have zero potential. At surface, normal derivative

The integral is evaluated over all the volume defined by the problem boundary.

*W u u dV* (1)

choices for mesh types and an easy treatment for boundary shapes.

stores field energy per unit volume, as in (Sylvester & Ferrari, 1990):

potential is also zero, since there is no current flowing in the air.

where *u* is the potential and *V* the volume.

**2.1. Discretization** 

Due to isotropy, the problem was discretized in the '*rz*' plane, since cylindrical coordinates were used. To obtain an approximate solution by FEM the problem region is subdivided into triangular elements. The solution mesh is presented in Fig. 2, near the rod top end.

**Figure 2.** Model discretization near rod top end

#### **2.2. First order triangular elements**

The essence of the method lies in first approximating the potential *u* within each element in a standardized fashion, and thereafter interrelating the potential distribution in the various elements so as to constrain the potential to be continuous across interelement boundaries, as in reference (Sylvester & Ferrari, 1990). This standardized fashion is done using a first order polynomial for potential within each element, which is the approximate solution to the actual one.

$$
\ln(r, z) = a + br + cz \tag{2}
$$

The IEEE Model for a Ground Rod in a Two Layer Soil – A FEM Approach 147

2

(7)

(8)

(9)

(10)

(11)

(5)

(6)

potentials are already known. In order to calculate these potentials is useful to represent potential within the element as a function of nodes potentials, so equation (2) became:

> 23 32 2 3 3 2 1 31 13 3 1 1 3 2 12 21 1 2 2 1 3

*rz rz z z r r r z U u rz rz z z r r r z U A rz rz z z r r r z U*

The factors multiplying nodes potentials, after being divided by 2A, are known as shape

1 12 23 3 *urz rzU rzU rzU* ,, , ,

 3

1 , , *i i i urz U rz* 

The finite element energy can now be calculated using equation (1). Node potentials,

*u U rz* 

The energy can now be evaluated, using equation (8) and (1). Considering that the integrand has nine sums, due to the square of potential gradient, energy functional for a single

3 3

1 1 <sup>1</sup> . <sup>2</sup>

. *ij i j <sup>V</sup> S d V*

3 3

1 1

*W u U U dV*

*ij i j <sup>V</sup> i j*

<sup>1</sup> . . <sup>2</sup>

*i ij j <sup>V</sup> i j*

 *dV U*

 

3

1

*i*

, *i i*

 

 

functions, so we can rewrite previous equation using these functions.

1. Their value is one in the associated node and zero in the two others.

Or in a more elegant manner:

element is:

Integrating by decomposition:

Defining the *S* variable as:

Shape functions have two important properties;

2. In any point inside the finite element their sum is one.

although unknown, are constants so the potential gradient is

*Wu U*

*e*

*e*

The true potential distribution is thus replaced by a piecewise planar function and within each triangle side, potential is obtained by a linear interpolation of node potentials.

Considering a generic triangular finite element, as shown in Fig. 3, equation (2) must satisfy node potentials. Using the equation in the three nodes a system is obtained, allowing the constants '*a*', '*b*' and '*c*' to be calculated as functions of node potentials.

$$\begin{aligned} LU\_1 &= a + br\_1 + cz\_1 \\ LU\_2 &= a + br\_2 + cz\_2 \\ LU\_3 &= a + br\_3 + cz\_3 \end{aligned} \tag{3}$$

where *U1*, *U2* and *U3* are node potentials.

**Figure 3.** Finite element with nodes numbered

Using Cramer´s rule the referred constants are easily obtained. The formula for '*a*' is:

$$a = \frac{\begin{vmatrix} \mathcal{U}\_1 & r\_1 & z\_1 \\ \mathcal{U}\_2 & r\_2 & z\_2 \\ \mathcal{U}\_3 & r\_3 & z\_3 \end{vmatrix}}{\begin{vmatrix} 1 & r\_1 & z\_1 \\ 1 & r\_2 & z\_2 \\ 1 & r\_3 & z\_3 \end{vmatrix}} \tag{4}$$

The denominator is recognized as twice the triangle area (2A) as in reference (Sylvester & Ferrari, 1990). It must be pointed that this formula was found by integration so that area become negative if the nodes are clockwise numbered. Equation (2) is useful when node potentials are already known. In order to calculate these potentials is useful to represent potential within the element as a function of nodes potentials, so equation (2) became:

$$u = \begin{cases} \left[ \left( r\_2 z\_3 - r\_3 z\_2 \right) + \left( z\_2 - z\_3 \right) r + \left( r\_3 - r\_2 \right) z \right] U\_1 + \\ \left[ \left( r\_3 z\_1 - r\_1 z\_3 \right) + \left( z\_3 - z\_1 \right) r + \left( r\_1 - r\_3 \right) z \right] U\_2 + \\ \left[ \left( r\_1 z\_2 - r\_2 z\_1 \right) + \left( z\_1 - z\_2 \right) r + \left( r\_2 - r\_1 \right) z \right] U\_3 \end{cases} \right] \tag{5}$$

The factors multiplying nodes potentials, after being divided by 2A, are known as shape functions, so we can rewrite previous equation using these functions.

$$u(r,z) = \alpha\_1(r,z)\mathcal{U}\_1 + \alpha\_2(r,z)\mathcal{U}\_2 + \alpha\_3(r,z)\mathcal{U}\_3\tag{6}$$

Or in a more elegant manner:

146 Finite Element Analysis – New Trends and Developments

where *U1*, *U2* and *U3* are node potentials.

**Figure 3.** Finite element with nodes numbered

actual one.

in reference (Sylvester & Ferrari, 1990). This standardized fashion is done using a first order polynomial for potential within each element, which is the approximate solution to the

The true potential distribution is thus replaced by a piecewise planar function and within

Considering a generic triangular finite element, as shown in Fig. 3, equation (2) must satisfy node potentials. Using the equation in the three nodes a system is obtained, allowing the

> 1 11 2 22 3 33

*U a br cz U a br cz U a br cz*

Using Cramer´s rule the referred constants are easily obtained. The formula for '*a*' is:

1 1 1

*a*

*r z r z r z*

The denominator is recognized as twice the triangle area (2A) as in reference (Sylvester & Ferrari, 1990). It must be pointed that this formula was found by integration so that area become negative if the nodes are clockwise numbered. Equation (2) is useful when node

(4)

*Urz Urz Urz*

 

each triangle side, potential is obtained by a linear interpolation of node potentials.

constants '*a*', '*b*' and '*c*' to be calculated as functions of node potentials.

*u r z a br cz* (,) (2)

(3)

$$
\mu\left(r, z\right) = \sum\_{i=1}^{3} \mathcal{U}\_i \; \alpha\_i\left(r, z\right) \tag{7}
$$

Shape functions have two important properties;


The finite element energy can now be calculated using equation (1). Node potentials, although unknown, are constants so the potential gradient is

$$\stackrel{\rightarrow}{\nabla} \stackrel{\rightarrow}{\mu} = \stackrel{\rightarrow}{\sum} \mathcal{U}\_i \stackrel{\rightarrow}{\nabla} \mathcal{a}\_i \left( r, z \right) \tag{8}$$

The energy can now be evaluated, using equation (8) and (1). Considering that the integrand has nine sums, due to the square of potential gradient, energy functional for a single element is:

$$\mathcal{W}^{\varepsilon}\left(\boldsymbol{\mu}\right) = \frac{1}{2} \iiint\_{V} \sum\_{i=1}^{3} \sum\_{j=1}^{3} \mathcal{U}\_{i} \mathcal{U}\_{j} \stackrel{\rightarrow}{\nabla} \boldsymbol{\alpha}\_{i} \stackrel{\rightarrow}{\nabla} \boldsymbol{\alpha}\_{j} dV \tag{9}$$

Integrating by decomposition:

$$\mathcal{W}^{\varepsilon}\left(\boldsymbol{u}\right) = \frac{1}{2} \sum\_{i=1}^{3} \sum\_{j=1}^{3} \mathcal{U}\_{i} \coprod\_{V} \bigtimes^{\rightarrow} \alpha\_{i} \bigtimes^{\rightarrow} \alpha\_{j} dV \,\mathcal{U}\_{j} \tag{10}$$

Defining the *S* variable as:

$$S\_{ij} = \iiint\_{V} \stackrel{\rightarrow}{\nabla} \alpha\_i \stackrel{\rightarrow}{\nabla} \alpha\_j dV \tag{11}$$

As shape functions are first degrees polynomials, gradients are constants, such as their inner product, so the integral is obtained by the calculus of the revolution volume of the finite element, which is 2 *<sup>c</sup> r* where *<sup>c</sup> r* is the centroid radius. The *S* variable became:

1. For equal index

$$S\_{11}^{\epsilon} = \frac{\left(z\_2 - z\_3\right)^2 + \left(r\_2 - r\_3\right)^2}{2A} \pi \, r\_c \tag{12}$$

The IEEE Model for a Ground Rod in a Two Layer Soil – A FEM Approach 149

*<sup>T</sup> Wu U S U dis dis dis* (17)

**0 S** (16)

The *dis S* matrix is formed by the elementary elements matrices.

Ferrari, 1990). The energy of the two elements is:

**Figure 5.** Global node numbering

would be:

In all cases would be:

nodal numbering for element assembly illustrated in Fig. 5.

*dis <sup>S</sup>* **1**

This matrix is tridiagonal and is called the Dirichlet matrix, as in reference (Sylvester &

Potentials continuity implies that the values for the same node are equal. So potential at node three is equal to potential at node six, in Fig. 4, and so on. It can be defined a global

The potential continuity for corresponding nodes will be guaranteed by a linear transformation that relates disjoint nodes numbering with global numbering. In this case it

> 2 1 3 2 4 3 5 4

Introducing this relation in equation (15) it can be obtained an energy formulation in

*U U U U U U U U*

*conj*

*U CU dis conj* (19)

(18)

1

*U*

6

*U*

function of global numbering nodal potentials.

*dis*

 <sup>1</sup> 2

**S 0**

**2**

2. For different index

$$S\_{12}^{\varepsilon} = \frac{\left(z\_2 - z\_3\right)\left(z\_3 - z\_1\right) + \left(r\_3 - r\_2\right)\left(r\_1 - r\_3\right)}{2A}\pi \, r\_c \tag{13}$$

The remaining terms are obtained by index cyclic rotation. The finite element energy is finally represented as:

$$W^{\varepsilon} \left( \mu \right) = \frac{1}{2} \mathcal{U}^{T} \mathcal{S} \mathcal{U} \tag{14}$$

where *S* is a 3x3 matrix and *U* the nodes potential vector.

#### **2.3. Elements assembly**

The total energy model is the sum of all finite elements energy. Consider the following two elements with disjoint nodal numbering:

**Figure 4.** Disjoint numbering of finite element nodes

The nodes potential vector is:

$$\mathbf{U}\_{\rm dis}^{T} = \begin{bmatrix} \mathbf{U}\_1 & \mathbf{U}\_2 & \mathbf{U}\_3 & \mathbf{U}\_4 & \mathbf{U}\_5 & \mathbf{U}\_6 \end{bmatrix} \tag{15}$$

The *dis S* matrix is formed by the elementary elements matrices.

$$S\_{dis} = \begin{bmatrix} \mathbf{S\_1} & \mathbf{0} \\ \mathbf{0} & \mathbf{S\_2} \end{bmatrix} \tag{16}$$

This matrix is tridiagonal and is called the Dirichlet matrix, as in reference (Sylvester & Ferrari, 1990). The energy of the two elements is:

$$\mathcal{W}\{\boldsymbol{u}\} = \frac{1}{2}\mathcal{U}\_{dis}^T \mathcal{S}\_{dis} \mathcal{U}\_{dis} \tag{17}$$

Potentials continuity implies that the values for the same node are equal. So potential at node three is equal to potential at node six, in Fig. 4, and so on. It can be defined a global nodal numbering for element assembly illustrated in Fig. 5.

**Figure 5.** Global node numbering

148 Finite Element Analysis – New Trends and Developments

*r* where *<sup>c</sup>*

*e*

where *S* is a 3x3 matrix and *U* the nodes potential vector.

element, which is 2 *<sup>c</sup>*

1. For equal index

2. For different index

finally represented as:

**2.3. Elements assembly** 

elements with disjoint nodal numbering:

**Figure 4.** Disjoint numbering of finite element nodes

The nodes potential vector is:

As shape functions are first degrees polynomials, gradients are constants, such as their inner product, so the integral is obtained by the calculus of the revolution volume of the finite

> 2 2 2 3 23

2 3 3 1 3213

*z z z z rrrr S r A*

The remaining terms are obtained by index cyclic rotation. The finite element energy is

 <sup>1</sup> 2

The total energy model is the sum of all finite elements energy. Consider the following two

123456

*<sup>T</sup> U UUUUUU dis* 

*z z rr S r A*

<sup>11</sup> 2

*e*

<sup>12</sup> 2

*r* is the centroid radius. The *S* variable became:

*c*

(13)

(12)

*c*

*W u U SU e T* (14)

(15)

The potential continuity for corresponding nodes will be guaranteed by a linear transformation that relates disjoint nodes numbering with global numbering. In this case it would be:

$$
\begin{bmatrix}
\mathcal{U}\_1\\ \mathcal{U}\_2\\ \mathcal{U}\_3\\ \mathcal{U}\_4\\ \mathcal{U}\_5\\ \mathcal{U}\_6
\end{bmatrix}\_{\text{dis}} = \begin{bmatrix}
1 & 0 & 0 & 0\\ 0 & 1 & 0 & 0\\ 0 & 0 & 0 & 1\\ 0 & 1 & 0 & 0\\ 0 & 0 & 1 & 0\\ 0 & 0 & 0 & 1
\end{bmatrix} \begin{bmatrix}
\mathcal{U}\_1\\ \mathcal{U}\_2\\ \mathcal{U}\_3\\ \mathcal{U}\_4
\end{bmatrix}\_{\text{conj}}\tag{18}
$$

In all cases would be:

$$\mathcal{L}\mathcal{L}\_{dis} = \mathcal{C}\mathcal{L}\_{\alpha\alpha\dot{\jmath}}\tag{19}$$

Introducing this relation in equation (15) it can be obtained an energy formulation in function of global numbering nodal potentials.

$$\mathcal{W}\{\boldsymbol{u}\} = \frac{1}{2}\boldsymbol{\mathcal{U}}\_{\text{conj}}^T \boldsymbol{S} \boldsymbol{U}\_{\text{conj}} \tag{20}$$

The IEEE Model for a Ground Rod in a Two Layer Soil – A FEM Approach 151

*v r* (26)

0.03 ( ) 166.4 ln *<sup>r</sup> v r* (27)

in the same axis. The inner cylinder has the rod radius and the outer cylinder has a 3 cm radius, which is the zero volt boundary condition. The finite elements are shown in Fig. 6.

<sup>0</sup> ( ) ln ln

with '*b*' the outer cylinder radius, '*a*' the inner cylinder radius, and *V*0 the potential

With '*b*'=3 cm, '*a*'=8 mm and a voltage difference of 220 V between the two cylinders,

The numerical values for voltage were calculated by FEM. For 2 m length cylinders, the

The potential is almost constant along a parallel line 1 cm away from rod axis, as expected. In the first point its potential is 183.2 V and the last has the value of 182.1 V. The potential

0 0.005 0.01 0.015 0.02 0.025 0.03 0.035

Radius (m)

potential at 1 cm away from the common axis is presented in Fig. 7.

*V*

*b b r a*

The voltage at any point between the cylindrical surfaces is easily obtained by:

difference applied between the cylindrical surfaces.

where '*r*' is the distance to common axis.

**Figure 6.** Cylindrical region Discretization

�0.03

�0.025

�0.02

�0.015

Depth (m)

�0.01

�0.005

0

variation with the distance to axis '*r*' is presented in Table 1.

equation (27) becomes:

with

$$\mathbf{S} = \mathbf{C}^T \mathbf{S}\_{dis} \mathbf{C} \tag{21}$$

In order to minimize energy expression *W u* , it is necessary to calculate the derivative, in equation (21), with respect to potential vector and solve the equation

$$S\mathcal{L}I = 0\tag{22}$$

Also, in order to avoid trivial solution *U* 0 the *S* matrix should be partitioned in blocks, as well as potential vector:

$$\left[\begin{array}{cc} \mathcal{S}\_n & \mathcal{S}\_k \\ \end{array}\right] \left[\begin{smallmatrix} \mathcal{U}\_n \\ \mathcal{U}\_k \end{smallmatrix}\right] = 0 \tag{23}$$

Where *Un* is a vector of unknown node potentials and *Uk* is the known potential vector. Unknown potential nodes must be the first to be numbered and after the known potentials that satisfy boundary conditions. The *<sup>n</sup> S* and *KS* dimensions must allow matrix multiplication. Final solution for the unknown potential is given by:

$$
\delta \mathcal{U}\_n = -\mathcal{S}\_n^{-1} \mathcal{S}\_K \mathcal{U}\_K \tag{24}
$$

This FEM solution is called stored energy approach.

#### **2.4. Electric field**

Knowing the node potentials, the constants in equation (8) are easy to find as well as the electric field strength, which is given by:

$$\stackrel{\frown}{E} = -b\stackrel{\frown}{\mathbf{j}} - c\stackrel{\frown}{\mathbf{j}}\tag{25}$$

#### **3. IEEE model in homogeneous soil**

The analysis of a ground rod was carried out with first order triangular finite element only in the 3 cm near the rod, since 25 % of the rod resistance is within this region as in reference (IEEE Std 142, 2007). This avoids the discretization of the IEEE entire model with a zero volt boundary condition at 7.6 m away from the rod, being the rod resistance four times the calculated value.

#### **3.1. The IEEE model for cylindrical region**

In order to test the FEM mesh, it was considered only the cylindrical region in homogeneous soil. This problem has a theoretical solution, since it is considered as two cylinders centered in the same axis. The inner cylinder has the rod radius and the outer cylinder has a 3 cm radius, which is the zero volt boundary condition. The finite elements are shown in Fig. 6.

The voltage at any point between the cylindrical surfaces is easily obtained by:

$$w(r) = \frac{V\_0}{\ln \frac{h}{a}} \ln \frac{h}{r} \tag{26}$$

with '*b*' the outer cylinder radius, '*a*' the inner cylinder radius, and *V*0 the potential difference applied between the cylindrical surfaces.

With '*b*'=3 cm, '*a*'=8 mm and a voltage difference of 220 V between the two cylinders, equation (27) becomes:

$$v(r) = 166.4 \text{ ln} \frac{0.03}{r} \tag{27}$$

where '*r*' is the distance to common axis.

150 Finite Element Analysis – New Trends and Developments

with

well as potential vector:

**2.4. Electric field** 

<sup>1</sup> . <sup>2</sup>

*T*

In order to minimize energy expression *W u* , it is necessary to calculate the derivative, in

Also, in order to avoid trivial solution *U* 0 the *S* matrix should be partitioned in blocks, as

*n k*

*S S*

0 *<sup>n</sup>*

*k U*

*U* 

Where *Un* is a vector of unknown node potentials and *Uk* is the known potential vector. Unknown potential nodes must be the first to be numbered and after the known potentials that satisfy boundary conditions. The *<sup>n</sup> S* and *KS* dimensions must allow matrix

<sup>1</sup> *U S SU n nKK*

Knowing the node potentials, the constants in equation (8) are easy to find as well as the

*E bc <sup>i</sup> <sup>j</sup>* 

The analysis of a ground rod was carried out with first order triangular finite element only in the 3 cm near the rod, since 25 % of the rod resistance is within this region as in reference (IEEE Std 142, 2007). This avoids the discretization of the IEEE entire model with a zero volt boundary condition at 7.6 m away from the rod, being the rod resistance four times the calculated value.

In order to test the FEM mesh, it was considered only the cylindrical region in homogeneous soil. This problem has a theoretical solution, since it is considered as two cylinders centered

equation (21), with respect to potential vector and solve the equation

multiplication. Final solution for the unknown potential is given by:

This FEM solution is called stored energy approach.

electric field strength, which is given by:

**3. IEEE model in homogeneous soil** 

**3.1. The IEEE model for cylindrical region** 

*<sup>T</sup> W u U SU conj conj* (20)

*dis S CS C* (21)

*SU* 0 (22)

(24)

(25)

(23)

The numerical values for voltage were calculated by FEM. For 2 m length cylinders, the potential at 1 cm away from the common axis is presented in Fig. 7.

**Figure 6.** Cylindrical region Discretization

The potential is almost constant along a parallel line 1 cm away from rod axis, as expected. In the first point its potential is 183.2 V and the last has the value of 182.1 V. The potential variation with the distance to axis '*r*' is presented in Table 1.



The IEEE Model for a Ground Rod in a Two Layer Soil – A FEM Approach 153

(29)

ln

2 *<sup>b</sup> <sup>R</sup> L a* 

cylindrical radius.

**3.2. The complete IEEE model** 

**Figure 8.** Bottom rod FEM discretization

�2.025

�2.02

�2.015

�2.01

�2.005

Depth (m)

�1.995

�1.99

�1.985

�1.98

�2

theoretical one.

with *P* the electric resistivity, *L* the length, *b* the outer surface radius and *a* the inner

The value given by equation (30) is 21.0 Ω. The simulated value of 20.8 Ω is 1% less, which

To achieve the discretization of the entire IEEE model the mesh was altered in the rod bottom as shown in Fig. 8. Triangular finite elements have inner angles greater than 5º

For a rod with these dimensions the Dwight formula (Dwight, 1936) gives a value for resistance of 94.0 Ω. According to the standard (IEEE Std 142, 2007), this formula has 13% excess, so the corrected theoretical value is 83.2 Ω. The FEM simulated value for resistance, with the zero volt boundary at 0.03 m, is 20.8 Ω, representing 25 % of total resistance, so the numerical simulated value is four times 20.8 which results in 83.2 Ω. This value is equal to the

�0.01 0 0.01 0.02 0.03 0.04

Radius (m)

avoiding triangle areas close to zero, allowing stiffness matrix to be well defined.

allows the conclusion that the mesh is also adapted in energy.

**Table 1.** Potential variation between cylindrical surfaces

**Figure 7.** Potential along a parallel line 1 cm away from rod axis

The obtained figures for potential at distances of 1.4 cm and 2.1 cm are similar to Fig. 7. It was concluded that this mesh is voltage adapted. For the resistance between the two cylinders was used the FEM energetic approach given by (Martins & Antunes, 1997),

$$R = \frac{\upsilon^2}{P\_{\text{JOLLE}}} = \frac{\upsilon^2}{\sigma \int\_V \left| E \right|^2 dV} \tag{28}$$

where *v* is the voltage between the cylinders, the conductivity of the material between the two cylinders, *E* the electric field intensity and *V* the revolution volume due to axial symmetry, generated by each finite element. The obtained value for the electric resistance was 20.8 Ω considering a medium with 200 Ωm for the resistivity. That value should be compared with the theoretical value for the resistance between two cylinders given by (Purcell, 1998):

The IEEE Model for a Ground Rod in a Two Layer Soil – A FEM Approach 153

$$R = \frac{\rho}{2\pi} \ln \frac{b}{a} \tag{29}$$

with *P* the electric resistivity, *L* the length, *b* the outer surface radius and *a* the inner cylindrical radius.

The value given by equation (30) is 21.0 Ω. The simulated value of 20.8 Ω is 1% less, which allows the conclusion that the mesh is also adapted in energy.

#### **3.2. The complete IEEE model**

152 Finite Element Analysis – New Trends and Developments

Distance (m) Theoretical value

(Volt)

**Figure 7.** Potential along a parallel line 1 cm away from rod axis

where *v* is the voltage between the cylinders,

(Purcell, 1998):

182.5

Potential (V)

183

183.5

**Table 1.** Potential variation between cylindrical surfaces

Mean value (Volt)

0.01 182.9 182.6 -0.2 -0.4 0.014 126.9 126.6 -0.2 -0.6 0.021 59.4 59.3 -0.2 -0.2

The obtained figures for potential at distances of 1.4 cm and 2.1 cm are similar to Fig. 7. It was concluded that this mesh is voltage adapted. For the resistance between the two

<sup>0</sup> 0.2 0.4 0.6 0.8 <sup>1</sup> 1.2 1.4 1.6 1.8 <sup>2</sup> <sup>182</sup>

Depth (m)

2 2

*E dV*

*V*

the two cylinders, *E* the electric field intensity and *V* the revolution volume due to axial symmetry, generated by each finite element. The obtained value for the electric resistance was 20.8 Ω considering a medium with 200 Ωm for the resistivity. That value should be compared with the theoretical value for the resistance between two cylinders given by

2

(28)

the conductivity of the material between

cylinders was used the FEM energetic approach given by (Martins & Antunes, 1997),

*JOULE*

*P*

*v v <sup>R</sup>*

Mean error (%)

Maximum error (%)

> To achieve the discretization of the entire IEEE model the mesh was altered in the rod bottom as shown in Fig. 8. Triangular finite elements have inner angles greater than 5º avoiding triangle areas close to zero, allowing stiffness matrix to be well defined.

**Figure 8.** Bottom rod FEM discretization

For a rod with these dimensions the Dwight formula (Dwight, 1936) gives a value for resistance of 94.0 Ω. According to the standard (IEEE Std 142, 2007), this formula has 13% excess, so the corrected theoretical value is 83.2 Ω. The FEM simulated value for resistance, with the zero volt boundary at 0.03 m, is 20.8 Ω, representing 25 % of total resistance, so the numerical simulated value is four times 20.8 which results in 83.2 Ω. This value is equal to the theoretical one.

## **4. IEEE model in a two layer soil**

## **4.1. Zero volt equipotential at 3 cm**

As in previous examples, it was supposed that the Dirichlet border was 3 cm away from the rod axis, accounting for 25% of rod resistance, a supposition that needs validation in a two layer soil. The results, for a 2 m length rod, 8 mm radius, buried at ground level, with an upper soil layer of 100 Ωm resistivity, and a 500 Ωm resistivity in the lower layer, are summarized in Table 2. Theoretical resistance was obtained using Tagg formula, as in reference (Tagg, 1964). The results are unacceptable. The assumption that 25% of resistance in a two layer soil is also in the first 3 cm is probably wrong. Changing the values of the layers resistivity for the same rod, the results are presented in Table 3.

The IEEE Model for a Ground Rod in a Two Layer Soil – A FEM Approach 155

value x2 (Ω) error (%)

value x2 (Ω) error (%)

FEM simulated

FEM simulated

FEM simulated

value (Ω) error (%)

Upper layer thickness (m)

Upper layer thickness (m)

presented in Fig. 9.

sufficiently accurate.

Upper layer thickness (m)

Table 6.

problem.

Theoretical resistance (Ω)

Theoretical resistance (Ω)

**Table 4.** Resistance variation for positive voltage reflexion coefficient

**Table 5.** Resistance variation for negative voltage reflexion coefficient

**4.3. IEEE entire model discretization** 

0.6 124 103 -17 1.2 83.4 67.1 -20 1.8 63.4 49.6 -22

0.6 62.8 59.0 -6 1.2 89.1 85.3 -4 1.8 162 154 -5

In the last simulation the IEEE whole model was discretized. The complete solution mesh is

In order to validate the mesh, it was used the same conductance for all finite elements. It was chosen the value of 0.005 S/m. With this value the used rod, 2 m length and 8 mm radius, has a resistance, using Dwight formula, of 94 Ω. According to IEEE, this formula has a 13% excess to real value that should be of 83 Ω. The computer program developed returned a value of 87 Ω, which is 4.4 % higher than IEEE value, making the mesh

The next step was to simulate a two layer soil model, with an upper soil layer of 100 Ωm resistivity, and a 500 Ωm resistivity in the lower layer. The results are presented in the

> 0.5 134 118 -11.9 1.0 93.4 81.4 -12.8 1.5 72.0 62.0 -13.9

Interchanging the values of resistivity layers, new results were obtained and are shown in Table 7. The results are acceptable, unfortunately by default. The last one is surprisingly high, but within 20% of theoretical value. It can be concluded than FEM meshes can provide useful results, considering the whole IEEE model in discretizing the

Theoretical resistance (Ω)

**Table 6.** Resistance variation for positive voltage reflexion coefficient

The results are acceptable but would be better if they were greater than the theoretical ones, as a safe margin.




**Table 3.** Resistance variation for negative voltage reflexion coefficient

## **4.2. Zero volt equipotential at 15 cm**

The zero volt Dirichlet border was moved to 15 cm, where in homogeneous soil remains 50 % of the rod resistance. The mesh was changed improving the rod bottom discretization as shown in Fig. 2. The results in Table 4 were obtained for a 2 m length rod, 8 mm radius, buried at ground level, with an upper soil layer of 100 Ωm resistivity, and a 500 Ωm resistivity in the lower layer. The analysis of the results shows that the errors are quite high. The assumption that the first 15 cm contain 50 % of rod resistance seems to be incorrect. Interchanging the values of the layers resistivity, new results were obtained, and are shown in Table 5. In this case the errors are acceptable bur not conservative.


**Table 4.** Resistance variation for positive voltage reflexion coefficient


**Table 5.** Resistance variation for negative voltage reflexion coefficient

## **4.3. IEEE entire model discretization**

154 Finite Element Analysis – New Trends and Developments

**4. IEEE model in a two layer soil** 

**4.1. Zero volt equipotential at 3 cm** 

in Table 3.

as a safe margin.

Upper layer thickness (m)

Upper layer thickness (m)

conservative.

As in previous examples, it was supposed that the Dirichlet border was 3 cm away from the rod axis, accounting for 25% of rod resistance, a supposition that needs validation in a two layer soil. The results, for a 2 m length rod, 8 mm radius, buried at ground level, with an upper soil layer of 100 Ωm resistivity, and a 500 Ωm resistivity in the lower layer, are summarized in Table 2. Theoretical resistance was obtained using Tagg formula, as in reference (Tagg, 1964). The results are unacceptable. The assumption that 25% of resistance in a two layer soil is also in the first 3 cm is probably wrong. Changing the values of the layers resistivity for the same rod, the results are presented

The results are acceptable but would be better if they were greater than the theoretical ones,

0.5 134 104 -22.4 1.0 93.4 69.4 -25.7 1.5 72.0 52.0 -27.8

0.5 60 51.9 -13.5 1.0 77 69.1 -10.3 1.5 114 103 -9.6

The zero volt Dirichlet border was moved to 15 cm, where in homogeneous soil remains 50 % of the rod resistance. The mesh was changed improving the rod bottom discretization as shown in Fig. 2. The results in Table 4 were obtained for a 2 m length rod, 8 mm radius, buried at ground level, with an upper soil layer of 100 Ωm resistivity, and a 500 Ωm resistivity in the lower layer. The analysis of the results shows that the errors are quite high. The assumption that the first 15 cm contain 50 % of rod resistance seems to be incorrect. Interchanging the values of the layers resistivity, new results were obtained, and are shown in Table 5. In this case the errors are acceptable bur not

FEM simulated

FEM simulated

value x4 (Ω) error (%)

valuex4 (Ω) error (%)

Theoretical resistance (Ω)

Theoretical resistance (Ω)

**Table 2.** Resistance variation for positive voltage reflexion coefficient

**Table 3.** Resistance variation for negative voltage reflexion coefficient

**4.2. Zero volt equipotential at 15 cm** 

In the last simulation the IEEE whole model was discretized. The complete solution mesh is presented in Fig. 9.

In order to validate the mesh, it was used the same conductance for all finite elements. It was chosen the value of 0.005 S/m. With this value the used rod, 2 m length and 8 mm radius, has a resistance, using Dwight formula, of 94 Ω. According to IEEE, this formula has a 13% excess to real value that should be of 83 Ω. The computer program developed returned a value of 87 Ω, which is 4.4 % higher than IEEE value, making the mesh sufficiently accurate.

The next step was to simulate a two layer soil model, with an upper soil layer of 100 Ωm resistivity, and a 500 Ωm resistivity in the lower layer. The results are presented in the Table 6.


**Table 6.** Resistance variation for positive voltage reflexion coefficient

Interchanging the values of resistivity layers, new results were obtained and are shown in Table 7. The results are acceptable, unfortunately by default. The last one is surprisingly high, but within 20% of theoretical value. It can be concluded than FEM meshes can provide useful results, considering the whole IEEE model in discretizing the problem.

The IEEE Model for a Ground Rod in a Two Layer Soil – A FEM Approach 157

*<sup>h</sup>* (30)

was considered the average resistivity value of the last four points values, which are almost in a horizontal line. The average value is 94.5 Ωm. The upper layer thickness is obtained considering the point where concavity changes. The three first points are almost in a straight line, and concavity is detected only after the third point. It was considered the point where resistivity curve crosses the 400 Ωm ordinate where abscissae seem to be 2.3 m. Using this value in Lancaster-Jones rule one obtains

> <sup>3</sup> 2.3 2

The ground rod was discretized as indicated in Fig. 9. The measured value was 108 Ω and the simulated value using FEM is 113 Ω, which is 4.6 % higher. The equipotential lines were

Equipotential lines in bottom layer are closer to rod, since this layer has a smaller resistivity.

<sup>0</sup> <sup>1</sup> <sup>2</sup> <sup>3</sup> <sup>4</sup> <sup>5</sup> <sup>6</sup> <sup>7</sup> <sup>8</sup> <sup>0</sup>

Depth (m)

The upper layer thickness '*h*' is 1.53 m. This value was rounded to 1.5 m.

(Lancaster-Jones, 1930):

calculated and presented in Fig. 11.

**Figure 10.** Resistivity measurements with depth

200

400

600

800

1000

Resistivity (Ohmxm)

1200

1400

1600

1800

**Figure 9.** Entire IEEE model discretization



## **5. Field measurements**

In order to experimentally validate the model, resistivity measurements were done in a sandy soil. The obtained values are presented in Table 8.


**Table 8.** Resistivity variation with depth

The distance referred in the first row is the distance between test rods in Wenner method, as in (Telford et al., 1990). The resistivity curve is presented in Fig. 10. To choose a value for top layer resistivity, it was considered only the first resistivity measurement, since it wasn't found an upper asymptote. For bottom layer resistivity it was considered the average resistivity value of the last four points values, which are almost in a horizontal line. The average value is 94.5 Ωm. The upper layer thickness is obtained considering the point where concavity changes. The three first points are almost in a straight line, and concavity is detected only after the third point. It was considered the point where resistivity curve crosses the 400 Ωm ordinate where abscissae seem to be 2.3 m. Using this value in Lancaster-Jones rule one obtains (Lancaster-Jones, 1930):

$$\frac{3h}{2} = 2.3\tag{30}$$

The upper layer thickness '*h*' is 1.53 m. This value was rounded to 1.5 m.

156 Finite Element Analysis – New Trends and Developments

**Figure 9.** Entire IEEE model discretization

Theoretical resistance (Ω)

**Table 7.** Resistance variation for negative voltage reflexion coefficient

sandy soil. The obtained values are presented in Table 8.

FEM simulated

0.5 60 52.6 -12.5 1.0 77 66.7 -13.4 1.5 114 93.2 -18.3

�2 0 2 4 6 8

In order to experimentally validate the model, resistivity measurements were done in a

Distance (m) 0.5 1 2 3 4 5 6 8 Resistance (Ω) 552 207 41 7 4 2 3 2 Resistivity (Ωm) 1734 1301 515 132 101 63 113 108

The distance referred in the first row is the distance between test rods in Wenner method, as in (Telford et al., 1990). The resistivity curve is presented in Fig. 10. To choose a value for top layer resistivity, it was considered only the first resistivity measurement, since it wasn't found an upper asymptote. For bottom layer resistivity it

value (Ω) error (%)

Upper layer thickness (m)

�9

�8

�7

�6

�5

�4

�3

�2

�1

0

**5. Field measurements** 

**Table 8.** Resistivity variation with depth

The ground rod was discretized as indicated in Fig. 9. The measured value was 108 Ω and the simulated value using FEM is 113 Ω, which is 4.6 % higher. The equipotential lines were calculated and presented in Fig. 11.

Equipotential lines in bottom layer are closer to rod, since this layer has a smaller resistivity.

**Figure 10.** Resistivity measurements with depth

The IEEE Model for a Ground Rod in a Two Layer Soil – A FEM Approach 159

Adriano, U.; Bottauscio, O. & Zucca, M. (2003). Boundary Element Approach for the analysis and design of grounding s*y*stems in presence of non-homogeneousness, *IEE Proceedings* 

Al-Arayny, A.; Khan, Y.; Qureshi, M. & Malik Puzheri, F. (2011). Optimized Pit Configuration for Efficient Grounding oh the Power System in High Resistivity Soils using Low Resistivity Materials, *4th International Conference on Modeling, Simulation and* 

Berberovic, S.; Haznadar, Z. & Stih, Z. (2003). Method of moments in analysis of grounding systems. *Engineering Analysis with Boundary Elements,* Vol. 27, No. 4, (April 2003), pp.

Coa, L. (2006). Comparative Study between IEEE Std.80-2000 and Finite Elements Method application for Grounding System Analysis, *Transmission & Distribution Conference and* 

Colominas, I.; Aneiros, J.; Navarrina, F. & Casteleiro, M. (1998). A BEM Formulation for Computational Design of Grounding Systems in Stratified Soils, *Proc. Computational* 

Colominas, I.; Gomez-Calvino, J.; Navarrina, F. & Casteleiro, M. (2002). A general numeric model for grounding analysis in layered soils. *Advances in Engineering Software,* Vol. 33,

Colominas, I.; Navarrina, F. & Casteleiro, M. (2002). A Numerical Formulation for Grounding Analysis in Stratified Soils. *IEEE Transactions on Power Delivery*, Vol. 17, No.

Dwight, H. (1936). Calculation of Resistances to Ground. *Transactions of the American Institute* 

Lancaster-Jones, E. (1930). The Earth-Resistivity Method of Electrical Prospect. *Mining* 

Liu, Y.; Theethayi, N.; Thottappillil, R.; Gonzalez, R. & Zitnik, M. (2004). An improved model for soil ionization around grounding systems and its application to stratified soil.

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**Author details** 

António Martins,

**7. References** 

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*University of Beira Interior, Portugal* 

Sílvio Mariano and Maria do Rosário Calado

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2, (April 2002), pp. 587-595.

*Magazine*, Vol. 42, pp. 352-355.

**Figure 11.** Equipotential lines with depth for a 2 m rod

## **6. Conclusions**

In this chapter the Finite Element Method for the calculation of a rod resistance in a two layer soil was presented. The developed meshes were tested in homogeneous soil and in a cylindrical problem, which has theoretical solutions, and they are well adapted considering potential distribution or energy dissipation.

In homogeneous soil, 25% of rod resistance is in the 3 cm rod closest soil, according with IEEE and as validated in this work.

For an upper layer soil, with resistivity smaller than the resistivity of the lower layer soil, the assumption that 25% of rod resistance is in the nearest 3 cm and 50% in the nearest 15 cm is wrong and cannot be generalized.

For an upper layer soil, with resistivity bigger than the resistivity of the lower layer soil, the assumption that 25% of rod resistance is in the nearest 3 cm and 50% in the nearest 15 cm is acceptable, but the results are not conservative.

Discretizing the whole IEEE model allowed obtaining results with less than 20% error, but these results are not conservative. The whole mesh was tested considering equal resistivity, obtaining results similar to the ones gotten from the homogeneous soil simulation.

The comparison with the field measurement is good, since simulated value for resistance is only 4.6% higher.

## **Author details**

158 Finite Element Analysis – New Trends and Developments

**Figure 11.** Equipotential lines with depth for a 2 m rod

potential distribution or energy dissipation.

acceptable, but the results are not conservative.

IEEE and as validated in this work.

wrong and cannot be generalized.

In this chapter the Finite Element Method for the calculation of a rod resistance in a two layer soil was presented. The developed meshes were tested in homogeneous soil and in a cylindrical problem, which has theoretical solutions, and they are well adapted considering

<sup>0</sup> <sup>1</sup> <sup>2</sup> <sup>3</sup> <sup>4</sup> <sup>5</sup> <sup>6</sup> <sup>7</sup> �7

In homogeneous soil, 25% of rod resistance is in the 3 cm rod closest soil, according with

For an upper layer soil, with resistivity smaller than the resistivity of the lower layer soil, the assumption that 25% of rod resistance is in the nearest 3 cm and 50% in the nearest 15 cm is

For an upper layer soil, with resistivity bigger than the resistivity of the lower layer soil, the assumption that 25% of rod resistance is in the nearest 3 cm and 50% in the nearest 15 cm is

Discretizing the whole IEEE model allowed obtaining results with less than 20% error, but these results are not conservative. The whole mesh was tested considering equal resistivity,

The comparison with the field measurement is good, since simulated value for resistance is

obtaining results similar to the ones gotten from the homogeneous soil simulation.

**6. Conclusions** 

�6

�5

�4

�3

�2

�1

0

only 4.6% higher.

António Martins, *Polytechnic Institute of Guarda, Portugal* 

Sílvio Mariano and Maria do Rosário Calado *University of Beira Interior, Portugal* 

### **7. References**


Manikandan, P.; Rajamani, M.; Subburaj, P. & Venkatkumar, D. (2011). Design and Analysis of grounding systems for Wind turbines using Finite Element Method, *International Conference on Emerging Trends in Electrical and Computer Technology*, pp. 148-153, Tamil Nadu, India, 23-24 March 2011.

**Chapter 8** 

© 2012 Ren and Yao, licensee InTech. This is an open access chapter distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

© 2012 Ren and Yao, licensee InTech. This is a paper distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

**FEA in Micro-Electro-Mechanical Systems** 

Since the Nobel Prize winner, Richard Feynman gave the presentation "there is plenty of room at the bottom" [1], a variety of micromachined sensors, actuators, and systems have emerged and made encouraging progress in the past 50 years, based on technological innovations and increased market demand [2]. To date, Micro-Electro-Mechanical Systems (MEMS) have been developed into an interdisciplinary subject which involves electrical, mechanical, thermal, optical, and biological knowledge. Due to its significant potential, which has partially been demonstrated by the success of inertial MEMS devices (accelerometers, gyroscopes, *etc* [3, 4]) radio frequency (RF) MEMS devices (switches, filters, resonators, *etc* [5- 7]) and optical MEMS devices (Digital Light Processing, DLP [8, 9]), the research in MEMS has attracted worldwide interest. Figure 1 shows a typical process of a MEMS device from design goal to system integration. We can see that the structure and fabrication process of MEMS device are designed according to the design goal. Then before fabrication, we need to perform modeling to the structure. By modeling, we can estimate the performance to see if it satisfies the design goal and then optimize it to achieve the best performance. By performing modeling, substantial time and money can be saved, which increases the throughput and reduces the

Modeling applied in MEMS applications can mainly be divided into two categories, theoretical modeling and numerical modeling. The theoretical modeling is to apply exact equations to obtain exact solutions. It is a direct approach which is easy to interpret intuitively [10]. However, it has limitations that solutions can only be obtained for few standard cases, and it is incapable or difficult in the following situations: (1) shape, boundary conditions, and loadings are complex; (2) material properties are anisotropic; (3) structure has more than one material; (4) problems with material and geometric non-

**(MEMS) Applications: A Micromachined** 

**Spatial Light Modulator (μSLM)** 

Additional information is available at the end of the chapter

cost. As a result, modeling is critical for MEMS research.

Hao Ren and Jun Yao

http://dx.doi.org/10.5772/48532

**1. Introduction** 

