Preface

This book aims to address finite element analysis, sensing and inspection techniques, and applications for bridge condition monitoring and damage identification. It mainly focuses on finite element analysis, sensory systems, and signal-processing methodologies, enabling the determination of defects from response analysis and also the assessment of bridge condition.

Bridge condition monitoring is a complex task, involving multidisciplinary contributions, various techniques, and methodologies that have been developed over the past few decades. These include sensing techniques such as X-ray testing, ultrasonic testing, optical fiber sensing, piezoelectric transduction, and magnetic particle testing.

Each chapter has been written by specialists in that field, with rich experience in practical engineering. This book will provide a useful reference to beginners in this field, and the diverse introduction of techniques and approaches will also be invaluable for experienced researchers and engineers.

> **Dr. Yun Lai Zhou** Department of Civil and Environmental Engineering at Hong Kong Polytechnic University, China

#### **Magd Abdel Wahab**

Professor, Ghent University, Ghent, Belgium

Section 1

Introduction

1

Section 1 Introduction

Chapter 1

1. Introduction

Figure 1.

3

Diagram for (a) suspension bridge; (b) cable-stayed bridge.

Introductory Chapter: Some

Condition Monitoring

Yun-Lai Zhou and Linya Liu

area application in civil engineering.

Insights into Bridge Structural

Bridge structural condition monitoring has become a hot spot in both research and engineering fields. Bridges, serving as a connection between cliffs, shallow rivers, or special environmental conditions, have a lot of forms in functionality, economy, and art consideration. For instance, concrete bridge, steel bridges, cable stayed bridge, suspension bridges shown in Figure 1, and so on have been served in various cities [1]. The initial use of bridge is for functionality like footbridge [2], and then other considerations are included. As shown in Figure 1, both the suspension bridge and cable-stayed bridge are extending for long span and large

Since the bridges provide the convenient transportation for passengers and vehicles, the in-service safety shall be the most essential issue in the lifecycle service of bridges, providing timely early stage warning and suggestion for the possible maintenance [3–7]. For instance, fiber optic sensors are applied for full-scale destructive bridge condition monitoring [3]. A comprehensive discussion on bridge instrumentation and monitoring for structural diagnosis is conducted in [4], expressing the general steps for bridge management system for condition monitoring including experimental tests, nondestructive tests, performance evaluation, and so on; in [5], the temperature effect is studied between the temperature and the frequency ratio for model plate-girder bridges under uncertain temperature condi-

tions; and also the modal strain energy is extended to predict the location

and severity of the damages; in [6], the acoustic emission is applied for monitoring the prestressed concrete bridges health condition after constructing a referencesignals database; in [7], Poisson process is applied to simulate the arrival of vehicles traversing a bridge, and a stochastic model of traffic excitation on bridges is constructed to be incorporated in a Bayesian framework, to assess the properties and update the uncertainty for condition evaluation of the bridge superstructure.

#### Chapter 1

### Introductory Chapter: Some Insights into Bridge Structural Condition Monitoring

Yun-Lai Zhou and Linya Liu

#### 1. Introduction

Bridge structural condition monitoring has become a hot spot in both research and engineering fields. Bridges, serving as a connection between cliffs, shallow rivers, or special environmental conditions, have a lot of forms in functionality, economy, and art consideration. For instance, concrete bridge, steel bridges, cable stayed bridge, suspension bridges shown in Figure 1, and so on have been served in various cities [1]. The initial use of bridge is for functionality like footbridge [2], and then other considerations are included. As shown in Figure 1, both the suspension bridge and cable-stayed bridge are extending for long span and large area application in civil engineering.

Since the bridges provide the convenient transportation for passengers and vehicles, the in-service safety shall be the most essential issue in the lifecycle service of bridges, providing timely early stage warning and suggestion for the possible maintenance [3–7]. For instance, fiber optic sensors are applied for full-scale destructive bridge condition monitoring [3]. A comprehensive discussion on bridge instrumentation and monitoring for structural diagnosis is conducted in [4], expressing the general steps for bridge management system for condition monitoring including experimental tests, nondestructive tests, performance evaluation, and so on; in [5], the temperature effect is studied between the temperature and the frequency ratio for model plate-girder bridges under uncertain temperature conditions; and also the modal strain energy is extended to predict the location and severity of the damages; in [6], the acoustic emission is applied for monitoring the prestressed concrete bridges health condition after constructing a referencesignals database; in [7], Poisson process is applied to simulate the arrival of vehicles traversing a bridge, and a stochastic model of traffic excitation on bridges is constructed to be incorporated in a Bayesian framework, to assess the properties and update the uncertainty for condition evaluation of the bridge superstructure.

Figure 1. Diagram for (a) suspension bridge; (b) cable-stayed bridge.

#### 1.1 Sensing techniques

Sensing techniques include a lot of conventional technologies and advanced technologies developed in recent decades. The conventional technologies include impact echo testing, dye penetration, strain gage, electrical magnetic testing, piezoelectric gages, acoustic emission, leakage testing, magnetic testing, ultrasonic testing, radiographic testing, eddy current testing, infrared thermography testing, microwave testing, and so on. Advanced technologies include phased array ultrasonic testing, eddy current array testing, microelectromechanical sensors, air-coupled sensors, vision sensors with cameras, radar sensors, and so on. In [8], the condition monitoring system for Tsing Ma Bridge is thoroughly introduced: the wind and structural health monitoring (WASHMS) in the Tsing Ma Bridge used about 300 sensors: anemometers, temperature sensors, accelerometers, strain gauges, displacement transducers, weigh-in-motion sensors, and so on. The Global Positioning System-On-Structure Instrumentation System (GPS-OSIS) was installed to improve the bridge displacement response monitoring. In [9], the Global Navigation Satellite System (GNSS): BeiDou Navigation Satellite System (BDS) and Global Positioning System (GPS) are applied to monitoring the bridge displacement responses. In [10], image processing is applied to construct a visionbased monitoring system for cable tension estimation under various weather conditions in the cable-stayed bridge, proving that the natural frequencies can be obtained up to the third and fifth modes. In [11], the radar sensor techniques were employed to predict the changes in the natural frequencies of bridge girders with certain characteristics that control the structural performance with being incorporated with computational modeling. In [12], commercially available remote sensors for Highway Bridge condition evaluation such as ground penetrating radar (GPR), optical interferometry, digital image processing (DIC), and so on are summarized.

detection [16]; transmissibility is extended to apply in the responses analysis of ultrasonic testing [19]; the transmissibility still encounters difficulty in both theory development and engineering application. This study tries to extend the transmissibility theory for estimating and reconstructing mode shape from structural responses solely. Remaining work can be summarized as follows: Section 2 gives the theoretical development of transmissibility mode shape (TMS) and comparison between transmissibility-based OMA and frequency response functions (FRFs) based EMA, Section 3 gives the possible damage indicators, and Section 4 gives the

Introductory Chapter: Some Insights into Bridge Structural Condition Monitoring

Transmissibility has several kinds of definitions with existing reviews [14], while the fundamental concept is the ratio between two structural responses, which

> <sup>T</sup>ð Þ <sup>i</sup>; <sup>s</sup> <sup>¼</sup> Xi Xs

where i, s mean the response locations, while X<sup>i</sup> and X<sup>j</sup> represent the frequency

Transmissibility can be assessed with several ways, for instance, to use FRFs if

where r denotes the excitation location (assuming single load). H represents the

 

For a single load linear elastic structural system, the FRF can also be expressed as

where p denotes the pth mode, n means the number of modes considered. kp, mp, and cp mean modal stiffness, mass, and damping, respectively, ϕ means the mode

ϕi pϕr p kp � ω<sup>2</sup>mp þ jωcp

n p¼1

Hð Þ <sup>i</sup>;<sup>r</sup> ð Þ¼ ω ∑

Similar to the application of coherence in FRFs analysis, TC is also raised and

TCð Þ <sup>i</sup>; <sup>s</sup> <sup>¼</sup> <sup>G</sup><sup>2</sup>

where G means the auto- and cross spectrum. TC is initially developed for damage/small nonlinearity detection and quantification, and later it is advanced for

<sup>¼</sup> Hir Hsr

is GiiGss  

<sup>T</sup>ð Þ <sup>i</sup>; <sup>s</sup> <sup>¼</sup> Xi Xs (1)

(2)

(3)

(4)

numerical case study; conclusions are finally summarized.

2.1 Transmissibility and transmissibility coherence

spectrum of dynamic response xi and xs in time domain.

2. Structural condition monitoring

DOI: http://dx.doi.org/10.5772/intechopen.85742

can be expressed as

available,

FRFs.

5

defined as

natural frequency extraction.

2.2 Transmissibility mode shape (TMS)

shape, and ω represents the frequency.

#### 1.2 Damage identification

Damage identification, part of structural health monitoring (SHM), has appealed lots of attention since the occurrence of defect/damage demands repairing and maintenance, simultaneously leading to economic loss. Damage identification techniques can be summarized into two categories: model based and data based, and this has been discussed in [1].

Modal testing serves as the fundamental and most essential technique in SHM, and along with the development of technology, modal testing underwent experimental modal analysis (EMA) and operational modal analysis (OMA). The key difference between them is EMA needs, while OMA does not demand the measurement of excitation. During the last decades, new measurement techniques also arise in engineering application. For instance, ultrasonic testing, eddy current, magnetic particle testing, acoustic emission, and so on are all imposed in SHM.

Comparing with the model-based techniques, data based, especially the output only based damage identification suggests a wide applicable potential since its merit relying on structural responses solely. Transmissibility is a typical output only based technique [13], which has been developed in the past decades for system identification, damage detection, localization, quantification, and assessment [14]. Review can refer to [14]. Even a lot of investigations about transmissibility can be accessed to [15–19], for instance, transmissibility coherence (TC) is raised [14, 17]; cosine-based indicator is constructed from the modal assurance criterion (MAC) and incorporated with transmissibility for damage detection and quantification relatively [15]; Mahalanobis distance is also applied to transmissibility for damage

Introductory Chapter: Some Insights into Bridge Structural Condition Monitoring DOI: http://dx.doi.org/10.5772/intechopen.85742

detection [16]; transmissibility is extended to apply in the responses analysis of ultrasonic testing [19]; the transmissibility still encounters difficulty in both theory development and engineering application. This study tries to extend the transmissibility theory for estimating and reconstructing mode shape from structural responses solely. Remaining work can be summarized as follows: Section 2 gives the theoretical development of transmissibility mode shape (TMS) and comparison between transmissibility-based OMA and frequency response functions (FRFs) based EMA, Section 3 gives the possible damage indicators, and Section 4 gives the numerical case study; conclusions are finally summarized.

#### 2. Structural condition monitoring

1.1 Sensing techniques

Bridge Optimization - Inspection and Condition Monitoring

are summarized.

4

1.2 Damage identification

this has been discussed in [1].

Sensing techniques include a lot of conventional technologies and advanced technologies developed in recent decades. The conventional technologies include impact echo testing, dye penetration, strain gage, electrical magnetic testing, piezoelectric gages, acoustic emission, leakage testing, magnetic testing, ultrasonic testing, radiographic testing, eddy current testing, infrared thermography testing, microwave testing, and so on. Advanced technologies include phased array ultrasonic testing, eddy current array testing, microelectromechanical sensors, air-coupled sensors, vision sensors with cameras, radar sensors, and so on. In [8], the condition monitoring system for Tsing Ma Bridge is thoroughly introduced: the wind and structural health monitoring (WASHMS) in the Tsing Ma Bridge used about 300 sensors: anemometers, temperature sensors, accelerometers, strain gauges, displacement transducers, weigh-in-motion sensors, and so on. The Global

Positioning System-On-Structure Instrumentation System (GPS-OSIS) was installed to improve the bridge displacement response monitoring. In [9], the Global Navigation Satellite System (GNSS): BeiDou Navigation Satellite System (BDS) and Global Positioning System (GPS) are applied to monitoring the bridge displacement responses. In [10], image processing is applied to construct a visionbased monitoring system for cable tension estimation under various weather conditions in the cable-stayed bridge, proving that the natural frequencies can be obtained up to the third and fifth modes. In [11], the radar sensor techniques were employed to predict the changes in the natural frequencies of bridge girders with certain characteristics that control the structural performance with being incorporated with computational modeling. In [12], commercially available remote sensors for Highway Bridge condition evaluation such as ground penetrating radar (GPR), optical interferometry, digital image processing (DIC), and so on

Damage identification, part of structural health monitoring (SHM), has appealed lots of attention since the occurrence of defect/damage demands repairing and maintenance, simultaneously leading to economic loss. Damage identification techniques can be summarized into two categories: model based and data based, and

Modal testing serves as the fundamental and most essential technique in SHM, and along with the development of technology, modal testing underwent experimental modal analysis (EMA) and operational modal analysis (OMA). The key difference between them is EMA needs, while OMA does not demand the

measurement of excitation. During the last decades, new measurement techniques also arise in engineering application. For instance, ultrasonic testing, eddy current, magnetic particle testing, acoustic emission, and so on are all imposed in SHM. Comparing with the model-based techniques, data based, especially the output only based damage identification suggests a wide applicable potential since its merit relying on structural responses solely. Transmissibility is a typical output only based technique [13], which has been developed in the past decades for system identification, damage detection, localization, quantification, and assessment [14]. Review can refer to [14]. Even a lot of investigations about transmissibility can be accessed

to [15–19], for instance, transmissibility coherence (TC) is raised [14, 17]; cosine-based indicator is constructed from the modal assurance criterion (MAC) and incorporated with transmissibility for damage detection and quantification relatively [15]; Mahalanobis distance is also applied to transmissibility for damage

#### 2.1 Transmissibility and transmissibility coherence

Transmissibility has several kinds of definitions with existing reviews [14], while the fundamental concept is the ratio between two structural responses, which can be expressed as

$$T\_{(i,s)} = \frac{\mathbf{X}\_i}{\mathbf{X}\_s} \tag{1}$$

where i, s mean the response locations, while X<sup>i</sup> and X<sup>j</sup> represent the frequency spectrum of dynamic response xi and xs in time domain.

Transmissibility can be assessed with several ways, for instance, to use FRFs if available,

$$T\_{(i,s)} = \frac{X\_i}{X\_s} = \frac{H\_{ir}}{H\_{sr}} \tag{2}$$

where r denotes the excitation location (assuming single load). H represents the FRFs.

Similar to the application of coherence in FRFs analysis, TC is also raised and defined as

$$TC\_{(i,s)} = \left| \frac{G\_{\text{is}}^2}{G\_{\text{ii}} G\_{\text{ss}}} \right| \tag{3}$$

where G means the auto- and cross spectrum. TC is initially developed for damage/small nonlinearity detection and quantification, and later it is advanced for natural frequency extraction.

#### 2.2 Transmissibility mode shape (TMS)

For a single load linear elastic structural system, the FRF can also be expressed as

$$H\_{(i,r)}(\boldsymbol{\alpha}) = \sum\_{p=1}^{n} \frac{\phi\_p^i \phi\_p^r}{k\_p - \alpha^2 m\_p + j\alpha c\_p} \tag{4}$$

where p denotes the pth mode, n means the number of modes considered. kp, mp, and cp mean modal stiffness, mass, and damping, respectively, ϕ means the mode shape, and ω represents the frequency.

Then, the transmissibility illustrated above can be further expressed as

$$T\_{(i,s)} = \frac{X\_i}{X\_s} = \frac{H\_{ir}}{H\_{sr}} = \frac{\sum\_{p=1}^n \frac{\phi\_p^i \phi\_p^r}{k\_p - \alpha^2 m\_p + j\alpha c\_p}}{\sum\_{p=1}^n \frac{\phi\_p^r \phi\_p^r}{k\_p - \alpha^2 m\_p + j\alpha c\_p}} = \frac{\phi^r}{\phi^r} \tag{5}$$

internal change. For instance, the cross section reduction will result in stiffness reduction, which later changes the structural dynamic responses. Then, features can be constructed from structural dynamic responses to assess the stiffness reduction

Introductory Chapter: Some Insights into Bridge Structural Condition Monitoring

Certainly, damage has more kinds, like spalling in concrete structures, corrosion induced defects, and so on. These kinds of defects at initial stage may not cause a clear change in stiffness reduction; thus, special techniques like acoustic emission

TMS<sup>u</sup> ð Þ<sup>T</sup> � TMS<sup>u</sup> ð Þ � TMS<sup>d</sup> <sup>T</sup>

Of course, herein, more indicators can be constructed, since TMS and natural frequencies are assessed by transmissibility in OMA, curvature, higher order derivative, and so on, and all these modal parameters based indicators can further be

In order to illustrate the feasibility of the proposed methodology, a pined-pined

in the node 7 with 10 elements discretized on average in the whole beam. Dynamic responses are considered in the further OMA. The schematic diagram in Figure 2 shows the beam. Different damage levels are simulated with reducing the stiffness in element 3, and for damage level D1, D2, D3, and D4, the stiffness reduced from 5,

Results for the aforementioned methodology is computed and discussed in this section. Figure 3 illustrates the mode shapes for first four modes, while Figure 4 demonstrates the TMSs for first four modes, where one can find that both mode shapes and TMSs share the similar shapes, and their well agreement implies the

beam is numerically analyzed. Young's modulus is 185.2 GPa, dimension is

TMS<sup>u</sup> ð Þ<sup>T</sup> � TMS<sup>d</sup>

<sup>d</sup> denote the value under undamaged and damaged

2

� TMS<sup>d</sup>

, and a vertical impulse is excited

(7)

For the damage illustrated in this study-stiffness reduction related damage, vibration-based techniques are taken into consideration. To construct damage indicator, the change of feature is the commonest, and one may use MAC for achieving a comparable indicator without needing normalization, which can be denoted as

(kind of damage).

should be adopted in further investigation.

DOI: http://dx.doi.org/10.5772/intechopen.85742

<sup>u</sup> and (TMS)

0.005 � 0.006 � 1.000 m, density is 7800 Kg/m<sup>3</sup>

DI ¼ 1 � MAC ¼ 1 �

applied in damage detection [23].

10, 15, to 20% accordingly.

4. Results and discussion

Schematic diagram of the pined-pined beam.

Figure 2.

7

potential use of TMS in damage identification.

where (TMS)

states, respectively.

3. Case study

Note, this relation shall be better if obtained by using Laplace transform. As to Eq. (5), if fixing s, transmissibility will allow assessing the mode shape, or scalar mode shape. For each mode, transmissibility will express the scalar mode shape at each location, if further obtaining the direction of the scalar mode shape in each measurement location, and then the transmissibility-based mode shape TMS (fullunscaled mode shape ϕ1s, ϕ 2s, ϕ 3s, …, ϕNs, N is the number of measured responses) will be obtained. A general definition can be denoted as

$$\text{CTMS}\_{(i,p)}^{\vert\_{B}} = \int\_{f\_{B1}}^{f\_{B2}} T\_{(i,s)} df \tag{6}$$

where B denotes the frequency boundary [B1, B2] around the natural frequency, and f indicates the frequency domain. TMSp means pth TMS. All the TMSs will later contribute for further OMA [14].

#### 2.3 Comparison between transmissibility and FRF

Table 1 illustrates the comparison between EMA and transmissibility-based OMA, and it can be found that transmissibility has been developed by analog of FRF, where transmissibility can also perform the same function as FRF, like in damage detection, system identification, and so on. Note that transmissibility has not been thoroughly investigated; and further study is still needed to unveil new features.

Since transmissibility can assess TMS and natural frequencies, then, the extended parameters based on modal parameters can later similarly be applied in transmissibility-based OMA.

#### 2.4 Transmissibility application for outlier identification

Damage identification includes several stages: detection, locating, quantification, and remaining life assessment. All damage identifications follow the same procedure, (1) operational evaluation; (2) data acquisition, fusion, and cleansing; (3) feature extraction and information condensation; and (4) statistical mode development for feature discrimination [20]. The most essential step is feature extraction. Generally, feature means the property associated with the structural


Table 1.

Comparison between EMA and transmissibility-based OMA.

Introductory Chapter: Some Insights into Bridge Structural Condition Monitoring DOI: http://dx.doi.org/10.5772/intechopen.85742

internal change. For instance, the cross section reduction will result in stiffness reduction, which later changes the structural dynamic responses. Then, features can be constructed from structural dynamic responses to assess the stiffness reduction (kind of damage).

Certainly, damage has more kinds, like spalling in concrete structures, corrosion induced defects, and so on. These kinds of defects at initial stage may not cause a clear change in stiffness reduction; thus, special techniques like acoustic emission should be adopted in further investigation.

For the damage illustrated in this study-stiffness reduction related damage, vibration-based techniques are taken into consideration. To construct damage indicator, the change of feature is the commonest, and one may use MAC for achieving a comparable indicator without needing normalization, which can be denoted as

$$DI = 1 - \text{MAC} = 1 - \frac{\left(\left(\text{TMS}^{u}\right)^{T} \times \left(\text{TMS}^{d}\right)\right)^{2}}{\left(\left(\text{TMS}^{u}\right)^{T} \times \left(\text{TMS}^{u}\right)\right) \times \left(\left(\text{TMS}^{d}\right)^{T} \times \left(\text{TMS}^{d}\right)\right)}\tag{7}$$

where (TMS) <sup>u</sup> and (TMS) <sup>d</sup> denote the value under undamaged and damaged states, respectively.

Of course, herein, more indicators can be constructed, since TMS and natural frequencies are assessed by transmissibility in OMA, curvature, higher order derivative, and so on, and all these modal parameters based indicators can further be applied in damage detection [23].

#### 3. Case study

Then, the transmissibility illustrated above can be further expressed as

¼ ∑n p¼1

∑<sup>n</sup> p¼1

Note, this relation shall be better if obtained by using Laplace transform. As to Eq. (5), if fixing s, transmissibility will allow assessing the mode shape, or scalar mode shape. For each mode, transmissibility will express the scalar mode shape at each location, if further obtaining the direction of the scalar mode shape in each measurement location, and then the transmissibility-based mode shape TMS (fullunscaled mode shape ϕ1s, ϕ 2s, ϕ 3s, …, ϕNs, N is the number of measured responses)

> ð f B2

f B1

where B denotes the frequency boundary [B1, B2] around the natural frequency, and f indicates the frequency domain. TMSp means pth TMS. All the TMSs will later

Table 1 illustrates the comparison between EMA and transmissibility-based OMA, and it can be found that transmissibility has been developed by analog of FRF, where transmissibility can also perform the same function as FRF, like in damage detection, system identification, and so on. Note that transmissibility has not been thoroughly investigated; and further study is still needed to unveil new features. Since transmissibility can assess TMS and natural frequencies, then, the extended parameters based on modal parameters can later similarly be applied in

Damage identification includes several stages: detection, locating, quantification, and remaining life assessment. All damage identifications follow the same procedure, (1) operational evaluation; (2) data acquisition, fusion, and cleansing; (3) feature extraction and information condensation; and (4) statistical mode development for feature discrimination [20]. The most essential step is feature extraction. Generally, feature means the property associated with the structural

Modal analysis EMA Transmissibility-based OMA

ϕi pϕr p kp�ω<sup>2</sup>mpþjωcp

ϕs pϕr p kp�ω<sup>2</sup>mpþjωcp ¼ ϕr

Tð Þ <sup>i</sup>; <sup>s</sup> df (6)

Frequency extraction techniques like TC

based [14]

<sup>ϕ</sup><sup>s</sup> (5)

<sup>¼</sup> Hir Hsr

<sup>T</sup>ð Þ <sup>i</sup>; <sup>s</sup> <sup>¼</sup> Xi Xs

Bridge Optimization - Inspection and Condition Monitoring

will be obtained. A general definition can be denoted as

2.3 Comparison between transmissibility and FRF

2.4 Transmissibility application for outlier identification

Kernel FRF Transmissibility

Mode shape TMS

Coherence FRF coherence TC [17]

like SSI

Comparison between EMA and transmissibility-based OMA.

Frequency extraction techniques

contribute for further OMA [14].

transmissibility-based OMA.

Modal parameters

Table 1.

6

TMS � � � B ð Þ <sup>i</sup>; <sup>p</sup> ¼

> In order to illustrate the feasibility of the proposed methodology, a pined-pined beam is numerically analyzed. Young's modulus is 185.2 GPa, dimension is 0.005 � 0.006 � 1.000 m, density is 7800 Kg/m<sup>3</sup> , and a vertical impulse is excited in the node 7 with 10 elements discretized on average in the whole beam. Dynamic responses are considered in the further OMA. The schematic diagram in Figure 2 shows the beam. Different damage levels are simulated with reducing the stiffness in element 3, and for damage level D1, D2, D3, and D4, the stiffness reduced from 5, 10, 15, to 20% accordingly.

Figure 2. Schematic diagram of the pined-pined beam.

#### 4. Results and discussion

Results for the aforementioned methodology is computed and discussed in this section. Figure 3 illustrates the mode shapes for first four modes, while Figure 4 demonstrates the TMSs for first four modes, where one can find that both mode shapes and TMSs share the similar shapes, and their well agreement implies the potential use of TMS in damage identification.

Figure 5 gives the structural detection results from the constructed damage indicator DI. From this figure, it can be found that all damage levels are detected. It should also be noted that the change of DI for all the four modes are not very much, this suggests that TMSs vary small before and after the occurrence of damage, and further enhancement should be conducted in order to achieve a better damage

Introductory Chapter: Some Insights into Bridge Structural Condition Monitoring

This study tries to discuss some insights for bridge condition monitoring, also extends the mode shape into transmissibility-based OMA, and by using transmissibility, TMS is assessed by analog of mode shape in EMA, which paves the way for further investigation of extending the mode shape-based indicators to TMS-based analysis [21, 22]. MAC is used to construct a damage indicator with being verified by a pined-pined beam. The damage detection performance implies further neces-

sary investigation for obtaining a better and deeper understanding.

detection performance.

Author details

Yun-Lai Zhou1

9

Singapore, Singapore

\* and Linya Liu<sup>2</sup>

provided the original work is properly cited.

1 Department of Civil and Environmental Engineering, National University of

2 Engineering Research Center of Railway Environment Vibration and Noise, Ministry of Education, East China Jiaotong University, Nanchang, China

© 2020 The Author(s). Licensee IntechOpen. This chapter is 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,

\*Address all correspondence to: zhouyunlai168168@gmail.com

5. Concluding remarks

DOI: http://dx.doi.org/10.5772/intechopen.85742

Figure 3. Mode shapes of the beam for first four modes.

Figure 4. TMSs of the beam for first four modes.

Figure 5. DI for first four modes.

Introductory Chapter: Some Insights into Bridge Structural Condition Monitoring DOI: http://dx.doi.org/10.5772/intechopen.85742

Figure 5 gives the structural detection results from the constructed damage indicator DI. From this figure, it can be found that all damage levels are detected. It should also be noted that the change of DI for all the four modes are not very much, this suggests that TMSs vary small before and after the occurrence of damage, and further enhancement should be conducted in order to achieve a better damage detection performance.

#### 5. Concluding remarks

Figure 3.

Figure 4.

Figure 5.

8

DI for first four modes.

TMSs of the beam for first four modes.

Mode shapes of the beam for first four modes.

Bridge Optimization - Inspection and Condition Monitoring

This study tries to discuss some insights for bridge condition monitoring, also extends the mode shape into transmissibility-based OMA, and by using transmissibility, TMS is assessed by analog of mode shape in EMA, which paves the way for further investigation of extending the mode shape-based indicators to TMS-based analysis [21, 22]. MAC is used to construct a damage indicator with being verified by a pined-pined beam. The damage detection performance implies further necessary investigation for obtaining a better and deeper understanding.

#### Author details

Yun-Lai Zhou1 \* and Linya Liu<sup>2</sup>

1 Department of Civil and Environmental Engineering, National University of Singapore, Singapore

2 Engineering Research Center of Railway Environment Vibration and Noise, Ministry of Education, East China Jiaotong University, Nanchang, China

\*Address all correspondence to: zhouyunlai168168@gmail.com

© 2020 The Author(s). Licensee IntechOpen. This chapter is 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.

### References

[1] Cao H, Qian X, Zhou Y, Chen Z, Zhu H. Feasible range for midtower lateral stiffness in three-tower suspension bridges. Journal of Bridge Engineering. 2018;23(3):06017009

[2] Qin S, Zhou Y, Kang J. Footbridge serviceability analysis: From system identification to tuned mass damper implementation. KSCE Journal of Civil Engineering. 2019;23(2):754-762

[3] Storoy H, Sether J, Johannessen K. Fiber optical condition monitoring during a full scale destructive bridge test. Journal of Intelligent Material Systems and Structures. 1997;8:633-643

[4] Farhey DN. Bridge instrumentation and monitoring for structural diagnostics. Structural Health Monitoring. 2005;4(4):0301-0318

[5] Kim J-T, Park J-H, Lee B-J. Vibrationbased damage monitoring in model plate-girder bridges under uncertain temperature conditions. Engineering Structures. 2007;29:1354-1365

[6] Kalicka M. Acoustic emission as a monitoring method in prestressed concrete bridges health condition evaluation. Journal of Acoustic Emission. 2009;27:18-26

[7] Chen Y, Feng MQ, Tan C-A. Bridge structural condition assessment based on vibration and traffic monitoring. Journal of Engineering Mechanics. 2009;135(8):747-758

[8] Xu Y-L. Making good use of structural health monitoring systems of long-span cable-supported bridges. Journal of Civil Structural Health Monitoring. 2018;8:477-497

[9] Xi R, Jiang W, Meng X, Chen H, Chen Q. Bridge monitoring using BDS-RTK and GPS-RTK techniques. Measurement. 2018;120:128-139

[10] Kim S-W, Jeon B-G, Cheung J-H, Kim S-D, Park J-B. Stay cable tension estimation using a vision-based monitoring system under various weather conditions. Journal of Civil Structural Health Monitoring. 2017;7: 343-357

quantification using transmissibility coherence analysis. Shock and

DOI: http://dx.doi.org/10.5772/intechopen.85742

Introductory Chapter: Some Insights into Bridge Structural Condition Monitoring

Vibration. 2015. Article ID 290714, 16

[18] Zhou Y, Cao H, Liu Q, Wahab MA. Output-based structural damage detection by using correlation analysis

[19] Zhou Y, Qian X, Birnie A, Zhao X. A reference free ultrasonic phased array to identify surface cracks in welded steel pipes based on transmissibility.

International Journal of Pressure Vessels

[20] Sohn H, Farrar CR, Hemez FM, Shunk DD, Stinemates DW, Nadler BR, Czarnecki JJ. (2004). A Review of Structural Health Monitoring Literature:

1996–2001, los Alamos National

[21] Kerrouche A, Boyle WJO,

Laboratory Report, LA-13976-MS, 2004

Gebremichael Y, Sun T, Grattan KTV, Täljsten B, et al. Field tests of fibre bragg grating sensors incorporated into CFRP for railway bridge strengthening condition monitoring. Sensors and Actuators A. 2008;148:68-74

[22] Dissanayake PBR, Karunananda PAK. Reliability index for structural health monitoring of aging bridges. Structural Health Monitoring. 2008;

[23] Zhou Y, Maia NMM, Wahab MA. Damage detection using transmissibility compressed by principal component analysis enhanced with distance measure. Journal of Vibration and Control. 2018;24(10):2001-2019

7(2):0175-0179

11

together with transmissibility.

and Piping. 2018;168:66-78

Materials. 2017;10:866

pages

[11] Maizuar M, Zhang L, Miramini S, Mendis P, Thompson RG. Detecting structural damage to bridge girders using radar interferometry and computational modeling. Journal of Structural Control and Health Monitoring. 2017;10(24):1-6

[12] Vaghefi K, Oats RC, harris DK, Ahlborn TM, Broooks CN, Endsley KA, et al. Evaluation of commercially available remote sensors for highway bridge condition assessment. Journal of Bridge Engineering. 2012;17(6):886-895

[13] Zhou Y, Maia NMM, Sampaio RPC, Wahab MA. Structural damage detection using transmissibility together hierarchical clustering analysis and similarity measure. Structural Health Monitoring-An International Journal. 2017;16(6):711-731

[14] Zhou Y. Structural Health Monitoring by Using Transmissibility [PhD thesis]. Spain: Technical University of Madrid; 2015

[15] Zhou Y, Wahab MA. Cosine based and extended transmissibility damage indicators for structural damage detection. Engineering Structures. 2017; 141:175-183

[16] Zhou Y, Figueiredo E, Maia NMM, Sampaio R, Perera R. Damage detection in structures using a transmissibilitybased Mahalanobis distance. Structural Control and Health Monitoring. 2015;22: 1209-1222

[17] Zhou Y, Figueiredo E, Maia NMM, Perera R. Damage detection and

Introductory Chapter: Some Insights into Bridge Structural Condition Monitoring DOI: http://dx.doi.org/10.5772/intechopen.85742

quantification using transmissibility coherence analysis. Shock and Vibration. 2015. Article ID 290714, 16 pages

References

2018;23(3):06017009

[1] Cao H, Qian X, Zhou Y, Chen Z, Zhu H. Feasible range for midtower lateral stiffness in three-tower suspension bridges. Journal of Bridge Engineering.

Bridge Optimization - Inspection and Condition Monitoring

[10] Kim S-W, Jeon B-G, Cheung J-H, Kim S-D, Park J-B. Stay cable tension estimation using a vision-based monitoring system under various weather conditions. Journal of Civil Structural Health Monitoring. 2017;7:

[11] Maizuar M, Zhang L, Miramini S, Mendis P, Thompson RG. Detecting structural damage to bridge girders using radar interferometry and computational modeling. Journal of Structural Control and Health Monitoring. 2017;10(24):1-6

[12] Vaghefi K, Oats RC, harris DK, Ahlborn TM, Broooks CN, Endsley KA, et al. Evaluation of commercially available remote sensors for highway bridge condition assessment. Journal of Bridge Engineering. 2012;17(6):886-895

[13] Zhou Y, Maia NMM, Sampaio RPC,

detection using transmissibility together hierarchical clustering analysis and similarity measure. Structural Health Monitoring-An International Journal.

Wahab MA. Structural damage

[14] Zhou Y. Structural Health

[PhD thesis]. Spain: Technical University of Madrid; 2015

Monitoring by Using Transmissibility

[15] Zhou Y, Wahab MA. Cosine based and extended transmissibility damage indicators for structural damage

detection. Engineering Structures. 2017;

[16] Zhou Y, Figueiredo E, Maia NMM, Sampaio R, Perera R. Damage detection in structures using a transmissibilitybased Mahalanobis distance. Structural Control and Health Monitoring. 2015;22:

[17] Zhou Y, Figueiredo E, Maia NMM, Perera R. Damage detection and

2017;16(6):711-731

141:175-183

1209-1222

343-357

[2] Qin S, Zhou Y, Kang J. Footbridge serviceability analysis: From system identification to tuned mass damper implementation. KSCE Journal of Civil Engineering. 2019;23(2):754-762

[3] Storoy H, Sether J, Johannessen K. Fiber optical condition monitoring during a full scale destructive bridge test. Journal of Intelligent Material Systems and Structures. 1997;8:633-643

[4] Farhey DN. Bridge instrumentation

[5] Kim J-T, Park J-H, Lee B-J. Vibrationbased damage monitoring in model plate-girder bridges under uncertain temperature conditions. Engineering Structures. 2007;29:1354-1365

[6] Kalicka M. Acoustic emission as a monitoring method in prestressed concrete bridges health condition evaluation. Journal of Acoustic Emission. 2009;27:18-26

[7] Chen Y, Feng MQ, Tan C-A. Bridge structural condition assessment based on vibration and traffic monitoring. Journal of Engineering Mechanics.

structural health monitoring systems of long-span cable-supported bridges. Journal of Civil Structural Health Monitoring. 2018;8:477-497

[9] Xi R, Jiang W, Meng X, Chen H, Chen Q. Bridge monitoring using BDS-

RTK and GPS-RTK techniques. Measurement. 2018;120:128-139

10

2009;135(8):747-758

[8] Xu Y-L. Making good use of

and monitoring for structural diagnostics. Structural Health Monitoring. 2005;4(4):0301-0318 [18] Zhou Y, Cao H, Liu Q, Wahab MA. Output-based structural damage detection by using correlation analysis together with transmissibility. Materials. 2017;10:866

[19] Zhou Y, Qian X, Birnie A, Zhao X. A reference free ultrasonic phased array to identify surface cracks in welded steel pipes based on transmissibility. International Journal of Pressure Vessels and Piping. 2018;168:66-78

[20] Sohn H, Farrar CR, Hemez FM, Shunk DD, Stinemates DW, Nadler BR, Czarnecki JJ. (2004). A Review of Structural Health Monitoring Literature: 1996–2001, los Alamos National Laboratory Report, LA-13976-MS, 2004

[21] Kerrouche A, Boyle WJO, Gebremichael Y, Sun T, Grattan KTV, Täljsten B, et al. Field tests of fibre bragg grating sensors incorporated into CFRP for railway bridge strengthening condition monitoring. Sensors and Actuators A. 2008;148:68-74

[22] Dissanayake PBR, Karunananda PAK. Reliability index for structural health monitoring of aging bridges. Structural Health Monitoring. 2008; 7(2):0175-0179

[23] Zhou Y, Maia NMM, Wahab MA. Damage detection using transmissibility compressed by principal component analysis enhanced with distance measure. Journal of Vibration and Control. 2018;24(10):2001-2019

Section 2

Application of Artificial

Intelligence

13

Section 2

## Application of Artificial Intelligence

Chapter 2

Abstract

Computational Intelligence and Its

Applications in Uncertainty-Based

The large computational cost, the curse of dimensionality and the multidisciplinary nature are known as the main challenges in dealing with real-world engineering optimization problems. The consideration of inevitable uncertainties in such problems will exacerbate mentioned difficulties as much as possible. Therefore, the computational intelligence methods (also known as surrogate-models or metamodels, which are computationally cheaper approximations of the true expensive function) have been considered as powerful paradigms to overcome or at least to alleviate the mentioned issues over the last three decades. This chapter presents an extensive survey on surrogate-assisted optimization (SAO) methods. The main focus areas are the working styles of surrogate-models and the management of the metamodels during the optimization process. In addition, challenges and future trends of this field of study are introduced. Then, a comparison study will be carried out by employing a novel evolution control strategies (ECS) and recently developed efficient global optimization (EGO) method in the framework of uncertainty-based design optimization (UDO). To conclude, some open research

Keywords: computational intelligence, metamodeling, surrogate-assisted

Motivated by industrial demands and development of more powerful optimization techniques, the engineering design community has undergone a major transformation. They are continually seeking new optimization challenges and to solve increasingly more complicated real-world engineering problems in the shortest feasible time. In order to achieve the best solution in dealing with complex realworld engineering optimization, the classical optimization methods are weak in convergence. In solving these design problems, an evolutionary algorithm may require thousands of function evaluations in order to provide a satisfactory solution, whereas each evaluation requires hours of computer run-time. To overcome such difficulties, researchers have applied sampling-based learning methods such as artificial neural networks, radial basis functions, and polynomial model. These methods can 'learn' the problem behaviors and approximate the function value. These approximation models can speed up the function evaluation as well as

Design Optimization

Ali Asghar Bataleblu

questions in this area are discussed.

1. Introduction

15

optimization, uncertainty-based design optimization

#### Chapter 2

## Computational Intelligence and Its Applications in Uncertainty-Based Design Optimization

Ali Asghar Bataleblu

#### Abstract

The large computational cost, the curse of dimensionality and the multidisciplinary nature are known as the main challenges in dealing with real-world engineering optimization problems. The consideration of inevitable uncertainties in such problems will exacerbate mentioned difficulties as much as possible. Therefore, the computational intelligence methods (also known as surrogate-models or metamodels, which are computationally cheaper approximations of the true expensive function) have been considered as powerful paradigms to overcome or at least to alleviate the mentioned issues over the last three decades. This chapter presents an extensive survey on surrogate-assisted optimization (SAO) methods. The main focus areas are the working styles of surrogate-models and the management of the metamodels during the optimization process. In addition, challenges and future trends of this field of study are introduced. Then, a comparison study will be carried out by employing a novel evolution control strategies (ECS) and recently developed efficient global optimization (EGO) method in the framework of uncertainty-based design optimization (UDO). To conclude, some open research questions in this area are discussed.

Keywords: computational intelligence, metamodeling, surrogate-assisted optimization, uncertainty-based design optimization

#### 1. Introduction

Motivated by industrial demands and development of more powerful optimization techniques, the engineering design community has undergone a major transformation. They are continually seeking new optimization challenges and to solve increasingly more complicated real-world engineering problems in the shortest feasible time. In order to achieve the best solution in dealing with complex realworld engineering optimization, the classical optimization methods are weak in convergence. In solving these design problems, an evolutionary algorithm may require thousands of function evaluations in order to provide a satisfactory solution, whereas each evaluation requires hours of computer run-time. To overcome such difficulties, researchers have applied sampling-based learning methods such as artificial neural networks, radial basis functions, and polynomial model. These methods can 'learn' the problem behaviors and approximate the function value. These approximation models can speed up the function evaluation as well as

estimation the function value with an acceptable accuracy. Also, they can improve optimization performance and provide a better final solution. However, the application of computational intelligence methods to expensive optimization problems is not straightforward. It is important to note that accuracy is the most important criterion for evaluating a metamodel, since metamodels with a low accuracy may lead to local optima, or may even fail to obtain a satisfactory solution. Nevertheless, the choice of the surrogate model is depending on the design problem [1].

optimization (EGO) that operates based on the approximation of responses using a

In order to build globally accurate metamodels, offline SAO methods may require more sample points and they may be computationally expensive. Instead, the online SAO methods could train with fewer sample points. One of the weaknesses of online SAO methods is that the few numbers of sample points in the first iterations can lead to a poor predictive capability of the metamodel. Therefore, this can entice the optimizer into a local optimum or infeasible regions in the design space [1]. To surmount them all, researchers have presented some techniques to call

Computational Intelligence and Its Applications in Uncertainty-Based Design Optimization

real models and metamodels beside each other during the optimization. For

instance, the real model can be used to correct the fitness value of some/all individuals in some generations of evolutionary optimization algorithms. This is known as the management of the metamodels or the evolution control and has been applied in many literatures [12]. However, the best time for calling the real model or the metamodel is still a major challenge for the metamodels' management.

This chapter is introduced for the wide field of research and can be applied by readerships who are interested in the development of computational intelligence techniques for nowadays' expensive optimization problems. Moreover, a novel ECS that benefits from managing the use of metamodels for increasing the optimization accuracy is proposed in this chapter. The performance benefits of this proposed strategy include decreasing the computational cost as well as providing a global or near-global optimum solution. It is important to note that this strategy can be applied for both deterministic and non-deterministic optimization problems, with

This chapter is organized as follows. Section 2 introduces the metamodeling approximation. Section 3 presents the proposed ECS strategy. Section 4 presents applications of the proposed strategy to some mathematical benchmark problems, and numerical results are discussed in detail. Section 5 presents the research con-

Extensive research on design and optimization of engineering problems using metamodeling techniques has been done. These research fields are including sampling, metamodeling, validation, management and application, and so on. Over the years it has become prove that metamodeling provides a decision criteria role for

Metamodeling involves (a) choosing an experimental design for generating design points, (b) function evaluation of generated design points, and then (c) choosing a model to represent the data and fitting the model to the observed data

After building the metamodels from the available dataset, the accuracy of the models should carefully be checked. When the metamodel is found to have acceptable accuracy, it can be employed for considered design and optimization studies. The metamodel type that is suitable for the approximation could vary depending on the intended use or the underlying problem's physic and design space. Different datasets could be appropriate for building metamodels. The process of where pick out the design points in the design space, i.e. how to spread the design points within

There are several options for each of metamodeling steps as shown in Figure 2, and three predominant ones are highlighted. For example, response surface methodology usually employs central composite designs, second order polynomials, and

the complete design space, is called the design of experiments (DOE) [21].

clusions as well as some directions for the future research.

Gaussian process (e.g. Kriging or SVR) [1].

DOI: http://dx.doi.org/10.5772/intechopen.81689

any optimization algorithms [1].

2. Meta-modeling

designers [7].

(see Figure 1).

17

Viana et al. [2] and Jin [3] have presented a survey study about metamodeling techniques and their application in the design and analysis of computer experiments. Moreover, they proposed the future work directions to handle more complex simulations. The metamodeling/surrogate-modeling techniques approximate the real model in the entire design space bound that could help to reduce the running time of a complex problem considerably [4]. The simulation-based design problems using metamodels is reviewed extensively in [5–8]. Furthermore, researchers at Boeing and Rice University have proposed a number of mathematical techniques for application of metamodels in optimization problems [2, 9]. They have introduced some software packages to expedite the design and optimization process by using metamodels. These packages include "Optimus" developed by Tzannetakis et al. [10] and "DAKOTA" developed by Adams et al. [11]. Some of the common metamodeling techniques are including response surface method (RSM), artificial neural networks (ANN), Kriging, radial basis functions (RBF), and support vector regression (SVR) [12].

In the recent years, the large number of research and literature indicates the importance of using metamodeling techniques in the optimization. Horng and Lin [13] have proposed an evolutionary algorithm optimizer using metamodels in the ordinal optimization framework. They have used their algorithm to solve a stochastic optimization problem with a huge discrete design space. Sóbester et al. [14] have presented a research on improving the accuracy of metamodels in engineering design problems. Gong et al. [15] have proposed a metamodel with the small computational cost for design by evolutionary optimization algorithms. In order to consider the low and high fidelity model's information and make a trade-off between accuracy and computational cost, Zhou et al. [16] have introduced an active learning strategy for application of metamodeling. Belyaev et al. [17] have presented a new tool namely GTApprox to generate medium-scale metamodels for industrial design. Sun et al. [18] have introduced a swarm optimization algorithm based on the surrogate models. Recently, a strategy for reducing the running time has presented by Sayyafzadeh [19] that is based on a self-adaptive metamodeling approach. Also, a number of metamodeling strategies that could be used for the uncertainty-based design optimization have reviewed by Chatterjee et al. [20].

Despite the recent advances in the design optimization tools, researchers are still trying to surmount some other issues such as curse of dimensionality, the numerical noise, and handling mixed discrete/continuous variables. Surrogate-assisted optimization (SAO) and the evolution control strategies (ECS) are two newly developed methods in this field of area [1]. Both of these strategies can be applied offline or online. The main difference between these two methods is the management of using metamodels instead of real models during the optimization process. In the SAO strategy, metamodels are substituted for real models directly, but in the ECS strategy, metamodels are substituted for real models in some of the optimizer design points. Furthermore, metamodels that are used offline are not updated while the optimization is ongoing, whereas online metamodels are adaptively updated during the optimization process and can progressively improve the accuracy of the metamodels [12]. One of the known SAO strategies is known as the efficient global

#### Computational Intelligence and Its Applications in Uncertainty-Based Design Optimization DOI: http://dx.doi.org/10.5772/intechopen.81689

optimization (EGO) that operates based on the approximation of responses using a Gaussian process (e.g. Kriging or SVR) [1].

In order to build globally accurate metamodels, offline SAO methods may require more sample points and they may be computationally expensive. Instead, the online SAO methods could train with fewer sample points. One of the weaknesses of online SAO methods is that the few numbers of sample points in the first iterations can lead to a poor predictive capability of the metamodel. Therefore, this can entice the optimizer into a local optimum or infeasible regions in the design space [1]. To surmount them all, researchers have presented some techniques to call real models and metamodels beside each other during the optimization. For instance, the real model can be used to correct the fitness value of some/all individuals in some generations of evolutionary optimization algorithms. This is known as the management of the metamodels or the evolution control and has been applied in many literatures [12]. However, the best time for calling the real model or the metamodel is still a major challenge for the metamodels' management.

This chapter is introduced for the wide field of research and can be applied by readerships who are interested in the development of computational intelligence techniques for nowadays' expensive optimization problems. Moreover, a novel ECS that benefits from managing the use of metamodels for increasing the optimization accuracy is proposed in this chapter. The performance benefits of this proposed strategy include decreasing the computational cost as well as providing a global or near-global optimum solution. It is important to note that this strategy can be applied for both deterministic and non-deterministic optimization problems, with any optimization algorithms [1].

This chapter is organized as follows. Section 2 introduces the metamodeling approximation. Section 3 presents the proposed ECS strategy. Section 4 presents applications of the proposed strategy to some mathematical benchmark problems, and numerical results are discussed in detail. Section 5 presents the research conclusions as well as some directions for the future research.

#### 2. Meta-modeling

estimation the function value with an acceptable accuracy. Also, they can improve optimization performance and provide a better final solution. However, the application of computational intelligence methods to expensive optimization problems is not straightforward. It is important to note that accuracy is the most important criterion for evaluating a metamodel, since metamodels with a low accuracy may lead to local optima, or may even fail to obtain a satisfactory solution. Nevertheless,

Viana et al. [2] and Jin [3] have presented a survey study about metamodeling techniques and their application in the design and analysis of computer experiments. Moreover, they proposed the future work directions to handle more complex simulations. The metamodeling/surrogate-modeling techniques approximate the real model in the entire design space bound that could help to reduce the running time of a complex problem considerably [4]. The simulation-based design problems using metamodels is reviewed extensively in [5–8]. Furthermore,

researchers at Boeing and Rice University have proposed a number of mathematical techniques for application of metamodels in optimization problems [2, 9]. They have introduced some software packages to expedite the design and optimization process by using metamodels. These packages include "Optimus" developed by Tzannetakis et al. [10] and "DAKOTA" developed by Adams et al. [11]. Some of the common metamodeling techniques are including response surface method (RSM), artificial neural networks (ANN), Kriging, radial basis functions (RBF), and sup-

In the recent years, the large number of research and literature indicates the importance of using metamodeling techniques in the optimization. Horng and Lin [13] have proposed an evolutionary algorithm optimizer using metamodels in the ordinal optimization framework. They have used their algorithm to solve a stochastic optimization problem with a huge discrete design space. Sóbester et al. [14] have presented a research on improving the accuracy of metamodels in engineering design problems. Gong et al. [15] have proposed a metamodel with the small computational cost for design by evolutionary optimization algorithms. In order to consider the low and high fidelity model's information and make a trade-off between accuracy and computational cost, Zhou et al. [16] have introduced an active learning strategy for application of metamodeling. Belyaev et al. [17] have presented a new tool namely GTApprox to generate medium-scale metamodels for industrial design. Sun et al. [18] have introduced a swarm optimization algorithm based on the surrogate models. Recently, a strategy for reducing the running time has presented by Sayyafzadeh [19] that is based on a self-adaptive

metamodeling approach. Also, a number of metamodeling strategies that could be used for the uncertainty-based design optimization have reviewed by Chatterjee

the optimization process and can progressively improve the accuracy of the metamodels [12]. One of the known SAO strategies is known as the efficient global

Despite the recent advances in the design optimization tools, researchers are still trying to surmount some other issues such as curse of dimensionality, the numerical noise, and handling mixed discrete/continuous variables. Surrogate-assisted optimization (SAO) and the evolution control strategies (ECS) are two newly developed methods in this field of area [1]. Both of these strategies can be applied offline or online. The main difference between these two methods is the management of using metamodels instead of real models during the optimization process. In the SAO strategy, metamodels are substituted for real models directly, but in the ECS strategy, metamodels are substituted for real models in some of the optimizer design points. Furthermore, metamodels that are used offline are not updated while the optimization is ongoing, whereas online metamodels are adaptively updated during

the choice of the surrogate model is depending on the design problem [1].

Bridge Optimization - Inspection and Condition Monitoring

port vector regression (SVR) [12].

et al. [20].

16

Extensive research on design and optimization of engineering problems using metamodeling techniques has been done. These research fields are including sampling, metamodeling, validation, management and application, and so on. Over the years it has become prove that metamodeling provides a decision criteria role for designers [7].

Metamodeling involves (a) choosing an experimental design for generating design points, (b) function evaluation of generated design points, and then (c) choosing a model to represent the data and fitting the model to the observed data (see Figure 1).

After building the metamodels from the available dataset, the accuracy of the models should carefully be checked. When the metamodel is found to have acceptable accuracy, it can be employed for considered design and optimization studies. The metamodel type that is suitable for the approximation could vary depending on the intended use or the underlying problem's physic and design space. Different datasets could be appropriate for building metamodels. The process of where pick out the design points in the design space, i.e. how to spread the design points within the complete design space, is called the design of experiments (DOE) [21].

There are several options for each of metamodeling steps as shown in Figure 2, and three predominant ones are highlighted. For example, response surface methodology usually employs central composite designs, second order polynomials, and

Figure 1.

The concept of metamodel creation [21].

2.1.1 Classical experimental designs

(CCD), box-behnken design (BBD) and Koshal) [21].

DOI: http://dx.doi.org/10.5772/intechopen.81689

Figure 3.

these methods, one may refer to [21].

presented in Ref. [21].

19

2.1.2 Experimental designs for complex metamodels

2.2 Metamodel choice and metamodel fitting

inputs and outputs, and can thus be used as a metamodel.

The idea of the classical DOE is to reach as much information as possible from a

Experimental designs in three variables for fitting second order models (Full factorial, central composite design

Computational Intelligence and Its Applications in Uncertainty-Based Design Optimization

As computer experiments involve mostly systematic error rather than random error as in physical experiments, researchers stated that in the presence of such sources of error, a good experimental design has to be space filling and noncollapsing rather than to concentrate on the boundary [5]. Also, in dealing with a complex design space, the metamodel's training samples should be spread the design points within the complete design space so that no prediction be too far from training points. Four types of space filling sampling methods that relatively more often used in the literature are orthogonal arrays, various Latin hypercube designs,

After selecting an appropriate DOE strategy and performing the necessary computer runs, the next step is to choose a metamodel and fitting method. As alluded to earlier in the introduction, many machine learning methods such as ANN, Kriging, RBF and SVR have been used to approximate complex relations between a set of

Despite the various metamodel types that have been introduced so far, which model is suitable for use? Different metamodels have their unique properties and consequently, there is no universal model that always is the best choice. Instead, the

Hammersley sequences, and uniform designs. Details of these methods are

limited number of experiments. The focus of these methods is on planning the experiments so that the random error from the physical experiments has minimum influence in the approval or disapproval of a hypothesis. Therefore, a classical experimental design represents a sequence of experiments to be performed, expressed in terms of factors (design variables) set at specified levels (predefined values) [5]. Widely used "classical DOE" include factorial or fractional factorial designs (FFD), central composite designs (CCD), Box-Behnken designs (BBD), and Koshal designs (KD). Schematic illustration of these methods is presented in Figure 3. These classic methods tend to pick out the sample points around boundaries of the design space and leave a few at the center of the design space. To view the details of

Figure 2. Metamodeling techniques [5].

least squares regression analysis while building a neural network involves fitting a network of neurons by means of back-propagation to data which is typically hand selected [5].

Several strategies exist for finding the optimal solution using metamodel-based design optimization (MBDO). In what follows, a brief overview of several DOEs, metamodel choice and metamodel fitting, and some strategies of MBDO will be explained, respectively.

#### 2.1 Design of experiments

The gathering a set of input-output date is the first step to build a metamodel and this dataset is known as the training set. The DOE is also the theory that helps to select best samples from the design space to cover everywhere. Based on the DOE theory, it is better that the training sets be space-filling and non-collapsing [1]. This signifies the importance of sampling efficiency in the generation of the training set for building an appropriate metamodel. This field has been a challenging research area among metamodeling researchers.

Computational Intelligence and Its Applications in Uncertainty-Based Design Optimization DOI: http://dx.doi.org/10.5772/intechopen.81689

Figure 3.

Experimental designs in three variables for fitting second order models (Full factorial, central composite design (CCD), box-behnken design (BBD) and Koshal) [21].

#### 2.1.1 Classical experimental designs

The idea of the classical DOE is to reach as much information as possible from a limited number of experiments. The focus of these methods is on planning the experiments so that the random error from the physical experiments has minimum influence in the approval or disapproval of a hypothesis. Therefore, a classical experimental design represents a sequence of experiments to be performed, expressed in terms of factors (design variables) set at specified levels (predefined values) [5].

Widely used "classical DOE" include factorial or fractional factorial designs (FFD), central composite designs (CCD), Box-Behnken designs (BBD), and Koshal designs (KD). Schematic illustration of these methods is presented in Figure 3. These classic methods tend to pick out the sample points around boundaries of the design space and leave a few at the center of the design space. To view the details of these methods, one may refer to [21].

#### 2.1.2 Experimental designs for complex metamodels

As computer experiments involve mostly systematic error rather than random error as in physical experiments, researchers stated that in the presence of such sources of error, a good experimental design has to be space filling and noncollapsing rather than to concentrate on the boundary [5]. Also, in dealing with a complex design space, the metamodel's training samples should be spread the design points within the complete design space so that no prediction be too far from training points. Four types of space filling sampling methods that relatively more often used in the literature are orthogonal arrays, various Latin hypercube designs, Hammersley sequences, and uniform designs. Details of these methods are presented in Ref. [21].

#### 2.2 Metamodel choice and metamodel fitting

After selecting an appropriate DOE strategy and performing the necessary computer runs, the next step is to choose a metamodel and fitting method. As alluded to earlier in the introduction, many machine learning methods such as ANN, Kriging, RBF and SVR have been used to approximate complex relations between a set of inputs and outputs, and can thus be used as a metamodel.

Despite the various metamodel types that have been introduced so far, which model is suitable for use? Different metamodels have their unique properties and consequently, there is no universal model that always is the best choice. Instead, the

least squares regression analysis while building a neural network involves fitting a network of neurons by means of back-propagation to data which is typically hand

Several strategies exist for finding the optimal solution using metamodel-based design optimization (MBDO). In what follows, a brief overview of several DOEs, metamodel choice and metamodel fitting, and some strategies of MBDO will be

The gathering a set of input-output date is the first step to build a metamodel and this dataset is known as the training set. The DOE is also the theory that helps to select best samples from the design space to cover everywhere. Based on the DOE theory, it is better that the training sets be space-filling and non-collapsing [1]. This signifies the importance of sampling efficiency in the generation of the training set for building an appropriate metamodel. This field has been a challenging research

selected [5].

18

Figure 2.

Figure 1.

The concept of metamodel creation [21].

Bridge Optimization - Inspection and Condition Monitoring

explained, respectively.

Metamodeling techniques [5].

2.1 Design of experiments

area among metamodeling researchers.

suitable metamodel depends on the problem at hand [21]. They are bound to their special domains, and thus no comparative studies have been conducted on them.

In complex real-world design problems, achieving a flawless metamodel is almost impossible. Therefore, in order to take advantage of metamodels in MBDO, it is better to manage using metamodels based on their accuracy in design points/ spaces. As shown in Figure 5, evolution control and migration are two major classes of management strategies for utilizing metamodels [12]. In the evolution control class, metamodels are called beside the real-models during the optimization process where the real-models are used in some/all individuals and in some/all generations. In model Migration, the entire population is divided into several sub-populations with its local metamodel. Also, the individuals in various sub-populations can migrate into other sub-populations [1]. To study the details of evolution control and migration, readers may refer to Tenne and Goh [12]. To improve the applicability of MBDO for complex real-world design problems, a novel ECS is developed that is

Computational Intelligence and Its Applications in Uncertainty-Based Design Optimization

In this section, a novel management strategy for application of the metamodels is introduced. This strategy relies on the Mean-Squared-Displacement (MSD) concept and is based on the evolution control class of metamodel management strategies. The MSD means the deviation of a particle's position relative to a reference

During the optimization process, the value of MSD for each design point must be

∑ Ntrain n¼1

where Ntrain and Xn indicate the number and the vector of design variables of metamodel training data, respectively. Xind is the vector of optimizer design vari-

In order to use the proposed strategy in the optimization process, the MSD value of each sample that is used as metamodel's test point based on the all of training data set has to be computed. Then, using these MSD values, two MSD values for the first and last iteration of the optimization process have to be selected. These two values that are named initial MSD (IMSD) and final MSD (FMSD) respectively indicate

ð Þ xn � xind

<sup>2</sup> (1)

introduced in the next section.

DOI: http://dx.doi.org/10.5772/intechopen.81689

position that is a statistical concept.

ables, iteratively.

Figure 5.

21

Metamodel management strategies in MBDO [12].

3. Proposed evolution control strategy (ECS)

computed that is named as calculated MSD (CMSD).

CMSD <sup>¼</sup> <sup>1</sup>

Ntrain

On the other hand, the performance of metamodels is depending on the problem to be addressed, and multiple criteria need to be considered. Model accuracy is probably the most important criterion, since approximate models with a low accuracy may lead the optimization process to local optima or even diverge from the optimal solution. Model accuracy also should be evaluated based on the new random sample points instead of the training data set points. The reason for this is that for some models overfitting is a common difficulty. In the case of overfitting, the model yields good accuracy on training data but may have poor performance on new sample points. The optimization process could easily go in the wrong direction if it is assisted by models with low accuracy [12].

There are some accuracy measures that may be used to evaluate the metamodels. The coefficient of determination R2 is a measure of how well the metamodel is able to capture the variability in the dataset. Other common ways for accuracy measures include: the maximum absolute error (MAE), the average absolute error (AAE), the mean squared error (MSE) and the root mean squared error (RMSE) [21].

#### 2.3 Metamodel-based design optimization

Metamodel-based design optimization can be applied using different strategies. The main issue with MBDO is the error that is introduced when approximating the real simulations with metamodels. The optimization process can be performed using the detailed simulation model, using its surrogate model, or both of them. Most common types of MBDO strategies are illustrated in Figure 4. In the first strategy (Figure 4a), a global metamodel will be built and then will be used during optimization. This approach uses a relatively large number of sample points at the outset and is commonly seen in the literature. The second strategy (Figure 4b) is based on the online metamodeling and involves the validation and/or optimization in the loop in deciding the resampling and remodeling strategy. In this strategy, samples will be generated iteratively to update the train data and related metamodel to maintain the model accuracy. In the third strategy (Figure 4c), the optimization is performed by adaptive sampling alone and no formal optimization process is used. This strategy directly generates new sample points toward the optimum with the guidance of a metamodel [7].

Figure 4. MBDO strategies: (a) sequential approach; (b) adaptive MBDO; and (c) direct sampling approach [7].

Computational Intelligence and Its Applications in Uncertainty-Based Design Optimization DOI: http://dx.doi.org/10.5772/intechopen.81689

In complex real-world design problems, achieving a flawless metamodel is almost impossible. Therefore, in order to take advantage of metamodels in MBDO, it is better to manage using metamodels based on their accuracy in design points/ spaces. As shown in Figure 5, evolution control and migration are two major classes of management strategies for utilizing metamodels [12]. In the evolution control class, metamodels are called beside the real-models during the optimization process where the real-models are used in some/all individuals and in some/all generations. In model Migration, the entire population is divided into several sub-populations with its local metamodel. Also, the individuals in various sub-populations can migrate into other sub-populations [1]. To study the details of evolution control and migration, readers may refer to Tenne and Goh [12]. To improve the applicability of MBDO for complex real-world design problems, a novel ECS is developed that is introduced in the next section.

#### 3. Proposed evolution control strategy (ECS)

In this section, a novel management strategy for application of the metamodels is introduced. This strategy relies on the Mean-Squared-Displacement (MSD) concept and is based on the evolution control class of metamodel management strategies. The MSD means the deviation of a particle's position relative to a reference position that is a statistical concept.

During the optimization process, the value of MSD for each design point must be computed that is named as calculated MSD (CMSD).

$$\text{CMSD} = \frac{1}{N\_{train}} \sum\_{n=1}^{N\_{train}} (\boldsymbol{\kappa}\_n - \boldsymbol{\kappa}\_{ind})^2 \tag{1}$$

where Ntrain and Xn indicate the number and the vector of design variables of metamodel training data, respectively. Xind is the vector of optimizer design variables, iteratively.

In order to use the proposed strategy in the optimization process, the MSD value of each sample that is used as metamodel's test point based on the all of training data set has to be computed. Then, using these MSD values, two MSD values for the first and last iteration of the optimization process have to be selected. These two values that are named initial MSD (IMSD) and final MSD (FMSD) respectively indicate

Figure 5. Metamodel management strategies in MBDO [12].

suitable metamodel depends on the problem at hand [21]. They are bound to their special domains, and thus no comparative studies have been conducted on them. On the other hand, the performance of metamodels is depending on the problem

There are some accuracy measures that may be used to evaluate the metamodels. The coefficient of determination R2 is a measure of how well the metamodel is able to capture the variability in the dataset. Other common ways for accuracy measures include: the maximum absolute error (MAE), the average absolute error (AAE), the

Metamodel-based design optimization can be applied using different strategies. The main issue with MBDO is the error that is introduced when approximating the real simulations with metamodels. The optimization process can be performed using the detailed simulation model, using its surrogate model, or both of them. Most common types of MBDO strategies are illustrated in Figure 4. In the first strategy (Figure 4a), a global metamodel will be built and then will be used during optimization. This approach uses a relatively large number of sample points at the outset and is commonly seen in the literature. The second strategy (Figure 4b) is based on the online metamodeling and involves the validation and/or optimization in the loop in deciding the resampling and remodeling strategy. In this strategy, samples will be generated iteratively to update the train data and related metamodel to maintain the model accuracy. In the third strategy (Figure 4c), the optimization is performed by adaptive sampling alone and no formal optimization process is used. This strategy directly generates new sample points toward the optimum with

MBDO strategies: (a) sequential approach; (b) adaptive MBDO; and (c) direct sampling approach [7].

mean squared error (MSE) and the root mean squared error (RMSE) [21].

to be addressed, and multiple criteria need to be considered. Model accuracy is probably the most important criterion, since approximate models with a low accuracy may lead the optimization process to local optima or even diverge from the optimal solution. Model accuracy also should be evaluated based on the new random sample points instead of the training data set points. The reason for this is that for some models overfitting is a common difficulty. In the case of overfitting, the model yields good accuracy on training data but may have poor performance on new sample points. The optimization process could easily go in the wrong direction

if it is assisted by models with low accuracy [12].

Bridge Optimization - Inspection and Condition Monitoring

2.3 Metamodel-based design optimization

the guidance of a metamodel [7].

Figure 4.

20

the acceptable accuracy of metamodels from the first until the last iteration of the optimization process. An adaptive threshold namely predetermined MSD (PMSD) is proposed for the management of the decreasing PMSD value from IMSD to FMSD. The adaptive threshold of the PMSD enables the optimizer to call more metamodels vs. real models in the first iterations of optimization. Also, it enables that while the optimization is ongoing, the number of metamodel's call functions decrease slowly and the real model's call functions will increase. Here, the proposed adaptive PMSD threshold relies on the inverse hyperbolic cosecant concept, as follows:

$$\text{PMSD} = \text{IMSD} + (\text{IMSD} - \text{FMSD}) \times a \csc h(k) \tag{2}$$

The design variable x follows a normal distribution x � N(x, σ<sup>x</sup>

solution and its cost function is defined as:

DOI: http://dx.doi.org/10.5772/intechopen.81689

prediction capability of metamodels.

Figure 6.

23

Real robust function and its metamodels.

metamodel of Eq. (4) is constructed (Figure 6).

σ<sup>x</sup> = 0.08 and x ∈ [0, 1]. The objective of this example is to find the robust optimum

Computational Intelligence and Its Applications in Uncertainty-Based Design Optimization

Based on four initial samples at x = [0, 0.33, 0.67, 1], a Kriging and ANN

In Figure 6, the cross-mark and square mark represent the test and train sample points of metamodels, respectively. As illustrated in Figure 6, the robust optimum points resulted from Kriging and ANN metamodels are different from the real model one. The robust solution of Kriging and ANN are located at point x = 0.38, which are far away from the real solution x = 0.28. Therefore, due to the relatively large error of these metamodels, the obtained robust solution cannot be accepted. In order to resolve this issue, it is essential to add more samples to improve the

Another way to overcome this problem is by using proposed ECS in this work. To do this, MSD value of all sample points that are used for metamodels test should be calculated. Then, based on these MSD values, a setting of the PMSD parameters (Eq. (2)) including IMSD and FMSD should be done. Figure 7 illustrates the MSD value related to testing sample points. According to MSD values in Figure 7, the value of IMSD and FMSD are considered as 1.3 and 0.5, respectively. Now, it is time to select the adaptive threshold variable (k in Eq. (2)). This variable should increase iteratively while optimization is ongoing and its bound has a direct impact on how the PMSD threshold decreases adaptively. For example, by considering the interval [�12.5, �1] for the variable k and assuming the maximum iteration of the optimization process be 10, the PMSD adaptive threshold variation is illustrated in Figure 8. Now, the optimization problem defined in this example is solved through proposed strategy along with simulated annealing optimizer and x = 1 as a start point. Convergence process in comparing with using only metamodels and real model is illustrated in Figure 9. Also, switching between real model and ANN metamodel

Minimize F xð Þ¼ μfð Þþ x 3σfð Þ x (4)

2

), where

The variable k has to iteratively increase within an interval (e.g. from �12.5 to �1) while the optimization is ongoing. The introduced strategy is summarized iteratively, as follows [1]:


The introduced strategy could be applied to all class of the optimization algorithms and both deterministic and non-deterministic optimization problems. In the next section, a number of benchmark problems are solved to present the ability of the proposed strategy.

#### 4. MBDO of benchmark problems

In this section, the performance of the proposed strategy in achieving the global or at least the near-global optimum is investigated through solving some benchmark problems.

#### 4.1 Analytical problem: one dimensional

Here, a one-dimensional nonlinear analytical example from Ref. [8] is used to illustrate the implementation of the proposed strategy. The mathematical formulation is shown as:

$$f(\mathbf{x}) = \left(\mathbf{6x} - \mathbf{2}\right)^2 \sin(\mathbf{12x} - \mathbf{4})\tag{3}$$

Computational Intelligence and Its Applications in Uncertainty-Based Design Optimization DOI: http://dx.doi.org/10.5772/intechopen.81689

The design variable x follows a normal distribution x � N(x, σ<sup>x</sup> 2 ), where σ<sup>x</sup> = 0.08 and x ∈ [0, 1]. The objective of this example is to find the robust optimum solution and its cost function is defined as:

$$Minimize \ F(\mathbf{x}) = \mu\_f(\mathbf{x}) + \mathbf{3}\,\sigma\_f(\mathbf{x}) \tag{4}$$

Based on four initial samples at x = [0, 0.33, 0.67, 1], a Kriging and ANN metamodel of Eq. (4) is constructed (Figure 6).

In Figure 6, the cross-mark and square mark represent the test and train sample points of metamodels, respectively. As illustrated in Figure 6, the robust optimum points resulted from Kriging and ANN metamodels are different from the real model one. The robust solution of Kriging and ANN are located at point x = 0.38, which are far away from the real solution x = 0.28. Therefore, due to the relatively large error of these metamodels, the obtained robust solution cannot be accepted. In order to resolve this issue, it is essential to add more samples to improve the prediction capability of metamodels.

Another way to overcome this problem is by using proposed ECS in this work. To do this, MSD value of all sample points that are used for metamodels test should be calculated. Then, based on these MSD values, a setting of the PMSD parameters (Eq. (2)) including IMSD and FMSD should be done. Figure 7 illustrates the MSD value related to testing sample points. According to MSD values in Figure 7, the value of IMSD and FMSD are considered as 1.3 and 0.5, respectively. Now, it is time to select the adaptive threshold variable (k in Eq. (2)). This variable should increase iteratively while optimization is ongoing and its bound has a direct impact on how the PMSD threshold decreases adaptively. For example, by considering the interval [�12.5, �1] for the variable k and assuming the maximum iteration of the optimization process be 10, the PMSD adaptive threshold variation is illustrated in Figure 8.

Now, the optimization problem defined in this example is solved through proposed strategy along with simulated annealing optimizer and x = 1 as a start point. Convergence process in comparing with using only metamodels and real model is illustrated in Figure 9. Also, switching between real model and ANN metamodel

Figure 6. Real robust function and its metamodels.

the acceptable accuracy of metamodels from the first until the last iteration of the optimization process. An adaptive threshold namely predetermined MSD (PMSD) is proposed for the management of the decreasing PMSD value from IMSD to FMSD. The adaptive threshold of the PMSD enables the optimizer to call more metamodels vs. real models in the first iterations of optimization. Also, it enables that while the optimization is ongoing, the number of metamodel's call functions decrease slowly and the real model's call functions will increase. Here, the proposed adaptive PMSD threshold relies on the inverse hyperbolic cosecant concept, as

Bridge Optimization - Inspection and Condition Monitoring

PMSD ¼ IMSD þ ðIMSD � FMSDÞ � a csc h kð Þ (2)

The variable k has to iteratively increase within an interval (e.g. from �12.5 to �1) while the optimization is ongoing. The introduced strategy is summarized iteratively,

1. Calculate MSD value of metamodel's test points and initialize the IMSD and

2. Determine the value of the k for PMSD estimation during the optimization

4.Calculate the CMSD for each optimizer design point, compare CMSD value with PMSD and take a decision on using metamodels or real models.

5. Evaluate objective functions and constraints based on the decision in step 4.

The introduced strategy could be applied to all class of the optimization algorithms and both deterministic and non-deterministic optimization problems. In the next section, a number of benchmark problems are solved to present the ability of

In this section, the performance of the proposed strategy in achieving the global or at least the near-global optimum is investigated through solving some benchmark

Here, a one-dimensional nonlinear analytical example from Ref. [8] is used to illustrate the implementation of the proposed strategy. The mathematical formula-

sinð Þ 12x � 4 (3)

f xð Þ¼ ð Þ <sup>6</sup><sup>x</sup> � <sup>2</sup> <sup>2</sup>

6.Go to the next iteration of the optimizer and update the PMSD value.

7. Check the optimization convergence criterion and go to step 4.

3. Start the optimization process using an initial design point.

follows:

as follows [1]:

FMSD.

process.

the proposed strategy.

problems.

tion is shown as:

22

4. MBDO of benchmark problems

4.1 Analytical problem: one dimensional

Figure 7. MSD values related to the metamodels test points.

Figure 8.

PMSD variation vs. optimization iteration.

based on the CMSD value of each design point and PMSD value in the related iteration is shown in Figure 10. Table 1 illustrates that proposed strategy with 3 real model call functions is achieved to near global robust optimum point compared to other methods.

re-test the metamodel. Since the metamodel is poor in the first iterations, this can mislead the optimization process into local optimum or infeasible regions in the

Computational Intelligence and Its Applications in Uncertainty-Based Design Optimization

As illustrated in Figure 10, the proposed strategy allows the optimizer to call metamodel in the first iterations. As optimization ongoing, metamodel accuracy in each design point will be checked and the real model will be called if necessary to prevent the optimizer from going to the wrong direction. Therefore, with proper

design space.

25

Figure 10.

Figure 9.

Optimization convergence—one dimensional example.

DOI: http://dx.doi.org/10.5772/intechopen.81689

Switching between ANN metamodel and real model.

In Table 1, the methods developed by Zhang et al. [22] that are based on the Kriging metamodel have been reached the near global optimal point through adding a new sample to the training set iteratively. For every new design point metamodel has been re-trained. Every time that a new point is added, you need to re-train and

Computational Intelligence and Its Applications in Uncertainty-Based Design Optimization DOI: http://dx.doi.org/10.5772/intechopen.81689

Figure 9. Optimization convergence—one dimensional example.

Figure 10. Switching between ANN metamodel and real model.

re-test the metamodel. Since the metamodel is poor in the first iterations, this can mislead the optimization process into local optimum or infeasible regions in the design space.

As illustrated in Figure 10, the proposed strategy allows the optimizer to call metamodel in the first iterations. As optimization ongoing, metamodel accuracy in each design point will be checked and the real model will be called if necessary to prevent the optimizer from going to the wrong direction. Therefore, with proper

based on the CMSD value of each design point and PMSD value in the related iteration is shown in Figure 10. Table 1 illustrates that proposed strategy with 3 real model call functions is achieved to near global robust optimum point compared to

In Table 1, the methods developed by Zhang et al. [22] that are based on the Kriging metamodel have been reached the near global optimal point through adding a new sample to the training set iteratively. For every new design point metamodel has been re-trained. Every time that a new point is added, you need to re-train and

other methods.

PMSD variation vs. optimization iteration.

Figure 8.

24

Figure 7.

MSD values related to the metamodels test points.

Bridge Optimization - Inspection and Condition Monitoring

#### Bridge Optimization - Inspection and Condition Monitoring


iterations of optimization, there is not any guarantee to achieve global optimization,

Computational Intelligence and Its Applications in Uncertainty-Based Design Optimization

In order to the implementation of the proposed strategy to moderate this issue, PMSD equation parameters (Eq. (2)) including IMSD and FMSD should be determined based on the MSD amount of metamodels test points. Therefore, in this example, IMSD and FMSD values are considered as 2.4 and 0.8, respectively. As alluded to in the previous analytical example, to reduce the amount of PMSD threshold slowly, the interval of the variable k is assumed as [�12.5, �1]. According to the considered set of proposed strategy along with simulated annealing optimizer and x = [4, 4] as a start point, the robust design problem defined in Eq. (6) is solved using different methods. Optimization convergence process and switching between models are shown in Figures 12 and 13, respectively. Also, resulted robust opti-

As presented in Table 2, one-stage sampling and sequential sampling methods that are based on the Kriging metamodel and proposed by Zhang et al. [22] have been reached the near global optimal point through 30 and 19 training sample points, respectively. But proposed strategy with checking the accuracy of the metamodel during optimization process through 5 switching between real model

Uncertainty based design optimization of truss and frame structures is a popular

As illustrated in Figure 14, the cross-section diameter (d) and the structure height

topic in mechanical, civil, and structural engineering due to the complexity of problems and benefits to industry. In this section, the popular two-bar truss structure problem (Figure 14) is used as a benchmark problem for the multi-objective Robust Design Optimization (RDO) under epistemic uncertainties. The test case is

(H) are as the design variables. The uncertain design parameters are including

and metamodel (see Figure 13) is able to achieve near global optimum.

especially in complex real-world applications.

DOI: http://dx.doi.org/10.5772/intechopen.81689

mums of different methods are presented in Table 2.

4.3 Engineering problem—two-bar truss structure

adapted from Ref. [23].

Figure 12.

27

Optimization convergence—Haupt function robust design.

Table 1.

Robust solution resulted from different methods [1].

management of metamodels during the optimization process, the possibility of accessing the global or near global optimum will be increased.

#### 4.2 Analytical problem—two dimensional

To further investigate the benefits of the proposed strategy, the robust design of two-dimensional Haupt function is presented here [22]. The Haupt function is defined as:

$$f(\mathbf{x}) = \mathbf{x}\_1 \sin \left(4\mathbf{x}\_1\right) + \mathbf{1}.\mathbf{1}\left.\mathbf{x}\_2 \sin \left(2\mathbf{x}\_2\right)\right|\tag{5}$$

In this example, both of the design variables x1 and x2 follow a normal distribution x � N(x, σ<sup>x</sup> 2 ), where x = [x1, x2] with x ∈ [0, 4] and σ<sup>x</sup> = [σx1, σx2] = [0.2, 0.2]. Considering the effect of design variable uncertainty, the robust design formulation is defined as:

$$Minimize\ F(\mathbf{x}) = \mu\_f(\mathbf{x}\_1, \mathbf{x}\_2) + \mathbf{3}\sigma\_f(\mathbf{x}\_1, \mathbf{x}\_2) \tag{6}$$

Based on Improved LHS (ILHS), 10 points are generated as the training sample points. The real model, Kriging and ANN metamodels of Eq. (6) are shown in Figure 11. It can be seen that the constructed metamodels are not sufficiently accurate and will mislead the optimizer into local optimum or non-optimal regions. To resolve this issue, Zhang et al. [22] have been proposed methods to generate new points while optimization is ongoing and increase the metamodel accuracy, iteratively. But since the predictive capability of the metamodel is poor in the first

Figure 11. Design space and different models of Haupt function.

#### Computational Intelligence and Its Applications in Uncertainty-Based Design Optimization DOI: http://dx.doi.org/10.5772/intechopen.81689

iterations of optimization, there is not any guarantee to achieve global optimization, especially in complex real-world applications.

In order to the implementation of the proposed strategy to moderate this issue, PMSD equation parameters (Eq. (2)) including IMSD and FMSD should be determined based on the MSD amount of metamodels test points. Therefore, in this example, IMSD and FMSD values are considered as 2.4 and 0.8, respectively. As alluded to in the previous analytical example, to reduce the amount of PMSD threshold slowly, the interval of the variable k is assumed as [�12.5, �1]. According to the considered set of proposed strategy along with simulated annealing optimizer and x = [4, 4] as a start point, the robust design problem defined in Eq. (6) is solved using different methods. Optimization convergence process and switching between models are shown in Figures 12 and 13, respectively. Also, resulted robust optimums of different methods are presented in Table 2.

As presented in Table 2, one-stage sampling and sequential sampling methods that are based on the Kriging metamodel and proposed by Zhang et al. [22] have been reached the near global optimal point through 30 and 19 training sample points, respectively. But proposed strategy with checking the accuracy of the metamodel during optimization process through 5 switching between real model and metamodel (see Figure 13) is able to achieve near global optimum.

#### 4.3 Engineering problem—two-bar truss structure

Uncertainty based design optimization of truss and frame structures is a popular topic in mechanical, civil, and structural engineering due to the complexity of problems and benefits to industry. In this section, the popular two-bar truss structure problem (Figure 14) is used as a benchmark problem for the multi-objective Robust Design Optimization (RDO) under epistemic uncertainties. The test case is adapted from Ref. [23].

As illustrated in Figure 14, the cross-section diameter (d) and the structure height (H) are as the design variables. The uncertain design parameters are including

Figure 12. Optimization convergence—Haupt function robust design.

management of metamodels during the optimization process, the possibility of

Model Robust solution

Real model 0.3 0.88 Kriging 0.335 0.973 ANN 0.377 0.947 EI-based EGO [22]—(4 extra points is added to the training samples) 0.270 0.94 R-EI-based EGO [22]—(2 extra points is added to the training samples) 0.30 2.24 Proposed Strategy—(with 7 meta-model calls and 3 real model calls) 0.29 0.87

two-dimensional Haupt function is presented here [22]. The Haupt function is

To further investigate the benefits of the proposed strategy, the robust design of

In this example, both of the design variables x1 and x2 follow a normal distribu-

Based on Improved LHS (ILHS), 10 points are generated as the training sample

points. The real model, Kriging and ANN metamodels of Eq. (6) are shown in Figure 11. It can be seen that the constructed metamodels are not sufficiently accurate and will mislead the optimizer into local optimum or non-optimal regions. To resolve this issue, Zhang et al. [22] have been proposed methods to generate new points while optimization is ongoing and increase the metamodel accuracy, iteratively. But since the predictive capability of the metamodel is poor in the first

Considering the effect of design variable uncertainty, the robust design formulation

fð Þ¼ x x<sup>1</sup> sin 4ð Þþ x<sup>1</sup> 1:1 x<sup>2</sup> sin 2ð Þ x<sup>2</sup> (5)

X F

), where x = [x1, x2] with x ∈ [0, 4] and σ<sup>x</sup> = [σx1, σx2] = [0.2, 0.2].

Minimize Fð Þ¼ x μfð Þþ x1; x<sup>2</sup> 3σfð Þ x1; x<sup>2</sup> (6)

accessing the global or near global optimum will be increased.

4.2 Analytical problem—two dimensional

Robust solution resulted from different methods [1].

Bridge Optimization - Inspection and Condition Monitoring

defined as:

Table 1.

tion x � N(x, σ<sup>x</sup>

is defined as:

Figure 11.

26

Design space and different models of Haupt function.

2

#### Figure 13.

Switching between metamodel and real model of Haupt function.


FRC <sup>¼</sup> <sup>1</sup>

Two-bar truss structure [23].

Figure 14.

Table 3.

29

3 � 4

8 >>>>>>><

DOI: http://dx.doi.org/10.5772/intechopen.81689

>>>>>>>:

σvolume σt

<sup>þ</sup> <sup>σ</sup>deflection σt

<sup>þ</sup> <sup>σ</sup><sup>S</sup> σt þ σS σw þ σS σP

Approximation capability of metamodels for two-bar truss problem.

<sup>þ</sup> <sup>σ</sup>volume σw

> <sup>þ</sup> <sup>σ</sup>deflection σw

Computational Intelligence and Its Applications in Uncertainty-Based Design Optimization

Based on the procedure summarized in Section 2 and 3, four different ANN metamodels are constructed for computing normal stress, buckling stress, volume and deflection. For this purpose, the training set is provided by 100 sampling using ILHS and testing set is generated using 1000 random sampling on the design variables and uncertain parameters bounds. Metamodeling creation involves making a decision on the appropriate number of layer(s) and the number of neurons in the hidden layer(s) and selecting the best model with minimum MSE. The architecture and approximating capability of these metamodels on training and testing sets are shown in Table 3. As illustrated in Table 3, the existence of some areas without enough accuracy is inevitable, so using metamodels in optimization process requires a management strategy. In order to implement the developed management strategy, we need the value of IMSD and FMSD parameters. To make a decision on the value of these parameters, the random testing set is utilized to compute the MSD value of each design point using created metamodels. Increasing in MSD values led to an increase in

Meta-model Unit N. neurons in each layer Train Test

Normal stress MPa [5 5 5] 9.30e�6 81.22 Buckling stress MPa [5 5 5] 2.45e�5 2.28e+2 Volume mm<sup>3</sup> [5 5 5] 5.38e�6 0.0374 Deflection mm [5 5 5] 1.22e�5 4.34e+7

<sup>þ</sup> <sup>σ</sup>deflection σP

� �

<sup>þ</sup> <sup>σ</sup>deflection σE

9 >>>>>>>=

>>>>>>>;

MSE MSE

(8)

� �

� �

#### Table 2.

Robust solution resulted from different methods for Haupt function [1].

vertical force (P � N (150, 5) kN), structure width (B � N (750, 10) mm), Elastic modulus (E � N (2.1e5, 5e3) N/mm2 ), and member thickness (t � N (2.5, 0.4) mm). The RDO problem is formulated in the following equation to minimize volume, vertical displacement and robustness criteria of the structure subject to constraints of stress and buckling.

In this design problem, the robustness measure FRC given in Eq. (7) is defined as follows, with P, B, E and t as the four uncertain parameters.

$$\begin{aligned} \text{Minimize} \qquad & \left\{ \mu\_{\text{volume}}, \mu\_{\text{df}|\text{f}=0}, F\_{RC}(\sigma\_{\text{volume}}, \sigma\_{\text{df}|\text{f}=0}, \sigma\_{S}) \right\} \\ \text{Subject to} \qquad & g\_1: \mu\_S \le S\_{\text{max}} \\ \text{\$g\_2: \mu\_S \le S\_{\text{crit}}\$} \\ \text{With respect to \$20 \le d \ (mm) \le 80, 200 \le H \ (mm) \le 1000 \\ \text{volume} = 2 \pi dt \sqrt{B^2 + H^2}; \text{deflection} \qquad & \frac{P(B^2 H^2)^{\frac{2}{3}}}{(2\pi EdH)^2} \\ \text{\$S = \frac{P\sqrt{B^2 + H^2}}{2\pi dt dH}\$ \\ \text{\$S = \frac{P\sqrt{B^2 + H^2}}{2\pi dt dH}\$} \qquad & S\_{\text{crit}} = \frac{\pi^2 E \left(t^2 + d^2\right)}{8\left(B^2 + H^2\right)}, \quad S\_{\text{max}} = 400 \text{ MPa} \end{aligned} \tag{7}$$

Computational Intelligence and Its Applications in Uncertainty-Based Design Optimization DOI: http://dx.doi.org/10.5772/intechopen.81689

Figure 14. Two-bar truss structure [23].

$$F\_{RC} = \frac{1}{3 \times 4} \left\{ \begin{aligned} &\left(\frac{\sigma\_{\text{volume}}}{\sigma\_t} + \frac{\sigma\_{\text{volume}}}{\sigma\_w}\right) \\ &+ \left(\frac{\sigma\_{\text{deflection}}}{\sigma\_t} + \frac{\sigma\_{\text{deflection}}}{\sigma\_w} + \frac{\sigma\_{\text{deflection}}}{\sigma\_P} + \frac{\sigma\_{\text{deflection}}}{\sigma\_E}\right) \\ &+ \left(\frac{\sigma\_S}{\sigma\_t} + \frac{\sigma\_S}{\sigma\_w} + \frac{\sigma\_S}{\sigma\_P}\right) \end{aligned} \right\} \tag{8}$$

Based on the procedure summarized in Section 2 and 3, four different ANN metamodels are constructed for computing normal stress, buckling stress, volume and deflection. For this purpose, the training set is provided by 100 sampling using ILHS and testing set is generated using 1000 random sampling on the design variables and uncertain parameters bounds. Metamodeling creation involves making a decision on the appropriate number of layer(s) and the number of neurons in the hidden layer(s) and selecting the best model with minimum MSE. The architecture and approximating capability of these metamodels on training and testing sets are shown in Table 3.

As illustrated in Table 3, the existence of some areas without enough accuracy is inevitable, so using metamodels in optimization process requires a management strategy. In order to implement the developed management strategy, we need the value of IMSD and FMSD parameters. To make a decision on the value of these parameters, the random testing set is utilized to compute the MSD value of each design point using created metamodels. Increasing in MSD values led to an increase in


Table 3.

Approximation capability of metamodels for two-bar truss problem.

vertical force (P � N (150, 5) kN), structure width (B � N (750, 10) mm), Elastic

Model Robust solution

Real model [1.18, 2.45] �1.78 Kriging [2.72, 2.53] �2.56 ANN [1.74, 1.76] �1.99 One-stage sampling method [22]—(30 training sample points) [1.19, 2.44] �1.74 sequential sampling method [22]—(19 training sample points) [1.2, 2.47] �1.68 Proposed strategy—(with 95 meta-model calls and 5 real model calls) [1.2, 2.4] �1.77

In this design problem, the robustness measure FRC given in Eq. (7) is defined as

n o � �

; deflection <sup>¼</sup> P B<sup>2</sup>

<sup>H</sup><sup>2</sup> � �<sup>3</sup> 2 (7)

ð Þ <sup>2</sup>πEdH <sup>2</sup>

<sup>8</sup> <sup>B</sup><sup>2</sup> <sup>þ</sup> <sup>H</sup><sup>2</sup> � � , Smax <sup>¼</sup> <sup>400</sup> MPa

The RDO problem is formulated in the following equation to minimize volume, vertical displacement and robustness criteria of the structure subject to constraints of

Minimize μvolume; μdeflection; FRC σvolume; σdeflection; σ<sup>S</sup>

With respect to 20≤ d mm ð Þ≤80, 200≤ H mm ð Þ ≤1000

follows, with P, B, E and t as the four uncertain parameters.

Robust solution resulted from different methods for Haupt function [1].

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi <sup>B</sup><sup>2</sup> <sup>þ</sup> <sup>H</sup><sup>2</sup> <sup>p</sup>

<sup>2</sup>πtdH , Scrit <sup>¼</sup> <sup>π</sup><sup>2</sup>E t<sup>2</sup> <sup>þ</sup> <sup>d</sup><sup>2</sup> � �

Subject to g<sup>1</sup> : μ<sup>S</sup> ≤Smax g<sup>2</sup> : μ<sup>S</sup> ≤Scrit

Switching between metamodel and real model of Haupt function.

Bridge Optimization - Inspection and Condition Monitoring

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi <sup>B</sup><sup>2</sup> <sup>þ</sup> <sup>H</sup><sup>2</sup> <sup>p</sup>

volume ¼ 2πdt

<sup>S</sup> <sup>¼</sup> <sup>P</sup>

), and member thickness (t � N (2.5, 0.4) mm).

X F

modulus (E � N (2.1e5, 5e3) N/mm2

stress and buckling.

Figure 13.

Table 2.

28

metamodel error. So, during the optimization process, if the amount of the MSD in each individual (i.e. CMSD) be less than PMSD, the metamodel has enough accuracy and can be used to functions evaluation. Contrariwise, when the accuracy of the metamodel is not sufficient (i.e. MSD value is greater than PMSD), the real-model should be used. As stated above, here, the amount of IMSD and FMSD are considered as 2.5 and 0.5, respectively. Also, it is assumed that the adaptive threshold variable k in Eq. (2) be increase from �12.5 to �1 until final optimization generation.

According to the iterative results shown in Figure 16a, in the early iterations of the optimization process, the number of metamodels call functions are more than real-model ones and in the final iterations, real-model call functions are increased. Therefore, control of call functions in metamodel based optimization process could lead to increasing the accuracy and globality of convergence. In addition, during the

Computational Intelligence and Its Applications in Uncertainty-Based Design Optimization

optimization process, the tuned adaptive threshold (Figure 16b) is slowly

Despite recent surrogate-based design books [8, 12, 24] for engineers and extensive investigations conducted in this field, many researchers are still making efforts to push the boundaries of metamodeling. It can be noted that, despite the numerous research carried out over the last few years, the computational complexity still remains as a major challenge of this field of study. Also, today's engineering

design problems with multidisciplinary nature are extremely complex (e.g., uncertainty-based multidisciplinary design optimization). Therefore, metamodels management and their accuracy over the design space are another challenges and open fields for research. Therefore, in this chapter, a novel ECS is proposed to improve computational efficiency and make better decisions for function evalua-

effectiveness of the proposed strategy. For all case studies, ANN and Kriging metamodels are used to create metamodels based on ILHS. Also, the ILHS is utilized for uncertainty propagation and analysis. Results illustrate that the proposed strategy could lead to improving the computational efficiency, accuracy, and globality of

sional with high fidelity analysis modules and considering different sources of uncertainties. The sensitivity of the proposed strategy to other metamodeling techniques (i.e. RSM, RBF, Kriging, etc.) can be considered in the future. Also, the use of metamodels in co-simulation works to replace high fidelity analysis with inexpensive surrogate models might be an interesting research field. Determining appropriate criteria for extracting or selecting new points to update the metamodels training set during online metamodeling is another challenge of this field of study. Also, the presence of data mining approaches along with computational intelligence methods could provide the basis for the emergence of new metamodeling tech-

The assessment of the benchmark problems revealed both the efficiency and the

Future researches could be include extensions of the problem to higher dimen-

tion when facing the metamodel based design optimization problems.

decreased to increase the real-model call functions, iteratively.

5. Concluding remarks and future work

DOI: http://dx.doi.org/10.5772/intechopen.81689

the convergence process in MBDO problems.

niques, which could be very significant.

31

The explained problem is solved through NSGA-II optimizer. The LHS based on the correlation criterion is used to generate the initial population of the optimization process. The optimization setting including population size, generation, crossover, and mutation are 50, 150, 0.8 and 0.15, respectively. The ILHS method with 1000 points is employed for uncertainty propagation and analysis. Finally, the Pareto frontier with two optimality criteria and one robustness criteria using both proposed strategy and real-model simulation is illustrated in Figure 15. The number of metamodel/realmodel call function and PMSD value are shown in Figure 16, iteratively.

Figure 15. Pareto frontiers resulted from RDO for two-bar truss problem.

Figure 16. Number of call functions (a) and PMSD threshold (b) for two-bar truss problem.

Computational Intelligence and Its Applications in Uncertainty-Based Design Optimization DOI: http://dx.doi.org/10.5772/intechopen.81689

According to the iterative results shown in Figure 16a, in the early iterations of the optimization process, the number of metamodels call functions are more than real-model ones and in the final iterations, real-model call functions are increased. Therefore, control of call functions in metamodel based optimization process could lead to increasing the accuracy and globality of convergence. In addition, during the optimization process, the tuned adaptive threshold (Figure 16b) is slowly decreased to increase the real-model call functions, iteratively.

#### 5. Concluding remarks and future work

metamodel error. So, during the optimization process, if the amount of the MSD in each individual (i.e. CMSD) be less than PMSD, the metamodel has enough accuracy and can be used to functions evaluation. Contrariwise, when the accuracy of the metamodel is not sufficient (i.e. MSD value is greater than PMSD), the real-model should be used. As stated above, here, the amount of IMSD and FMSD are considered as 2.5 and 0.5, respectively. Also, it is assumed that the adaptive threshold variable k

The explained problem is solved through NSGA-II optimizer. The LHS based on the correlation criterion is used to generate the initial population of the optimization process. The optimization setting including population size, generation, crossover, and mutation are 50, 150, 0.8 and 0.15, respectively. The ILHS method with 1000 points is employed for uncertainty propagation and analysis. Finally, the Pareto frontier with two optimality criteria and one robustness criteria using both proposed strategy and real-model simulation is illustrated in Figure 15. The number of metamodel/real-

in Eq. (2) be increase from �12.5 to �1 until final optimization generation.

Bridge Optimization - Inspection and Condition Monitoring

model call function and PMSD value are shown in Figure 16, iteratively.

Figure 15.

Figure 16.

30

Pareto frontiers resulted from RDO for two-bar truss problem.

Number of call functions (a) and PMSD threshold (b) for two-bar truss problem.

Despite recent surrogate-based design books [8, 12, 24] for engineers and extensive investigations conducted in this field, many researchers are still making efforts to push the boundaries of metamodeling. It can be noted that, despite the numerous research carried out over the last few years, the computational complexity still remains as a major challenge of this field of study. Also, today's engineering design problems with multidisciplinary nature are extremely complex (e.g., uncertainty-based multidisciplinary design optimization). Therefore, metamodels management and their accuracy over the design space are another challenges and open fields for research. Therefore, in this chapter, a novel ECS is proposed to improve computational efficiency and make better decisions for function evaluation when facing the metamodel based design optimization problems.

The assessment of the benchmark problems revealed both the efficiency and the effectiveness of the proposed strategy. For all case studies, ANN and Kriging metamodels are used to create metamodels based on ILHS. Also, the ILHS is utilized for uncertainty propagation and analysis. Results illustrate that the proposed strategy could lead to improving the computational efficiency, accuracy, and globality of the convergence process in MBDO problems.

Future researches could be include extensions of the problem to higher dimensional with high fidelity analysis modules and considering different sources of uncertainties. The sensitivity of the proposed strategy to other metamodeling techniques (i.e. RSM, RBF, Kriging, etc.) can be considered in the future. Also, the use of metamodels in co-simulation works to replace high fidelity analysis with inexpensive surrogate models might be an interesting research field. Determining appropriate criteria for extracting or selecting new points to update the metamodels training set during online metamodeling is another challenge of this field of study. Also, the presence of data mining approaches along with computational intelligence methods could provide the basis for the emergence of new metamodeling techniques, which could be very significant.

Bridge Optimization - Inspection and Condition Monitoring

References

1969-4

05.001

[1] Roshanian J, Bataleblu AA, Ebrahimi M. A novel evolution control strategy

DOI: http://dx.doi.org/10.5772/intechopen.81689

Computational Intelligence and Its Applications in Uncertainty-Based Design Optimization

[8] Forrester AIJ, Keane AJ. Recent advances in surrogate-based

optimization. Progress in Aerospace Science. 2009;45(1–3):50-79. DOI: 10.1016/j.paerosci.2008.11.001

[9] Booker AJ, Dennis JE Jr, Frank PD, Serafini DB, Torczon V, Trosset MW. A rigorous framework for optimization of expensive functions by surrogates. Structural Optimization. 1999;17(1): 1-13. DOI: 10.1007/BF01197708

[10] Tzannetakis N, Van de Peer J. Design optimization through parallel-

optimization methodologies and the utility of legacy simulation software. Structural and Multidisciplinary

Optimization. 2002;23(2):170-186. DOI:

[11] Adams BM, Bohnhoff WJ, Dalbey KR, Eddy JP, Eldred MS, Gay DM, et al. DAKOTA, a Multilevel Parallel Object-Oriented Framework for Design Optimization, Parameter Estimation, Uncertainty Quantification, and Sensitivity Analysis: Version 5.0 User's Manual. Tech. Rep. SAND2010-2183. Sandia National Laboratories; 2009

[12] Tenne Y, Goh CK. Computational intelligence in expensive optimization problems. Vol. 2. Berlin: Springer Science & Business Media; 2010

[13] Horng SC, Lin SY. Evolutionary algorithm assisted by surrogate model in the framework of ordinal optimization

allocation. Information Sciences. 2013; 233:214-229. DOI: 10.1016/j.ins.

[14] Sóbester A, Forrester AI, Toal DJ, Tresidder E, Tucker S. Engineering design applications of surrogate-assisted optimization techniques. Optimization

and optimal computing budget

2013.01.024

generated surrogate models,

10.1007/s00158-002-0175-5

Multidisciplinary Optimization. 2018; 58:1255. DOI: 10.1007/s00158-018-

[2] Viana FA, Simpson TW, Balabanov V, Toropov V. Special section on multidisciplinary design optimization: Metamodeling in multidisciplinary design optimization: How far have we really come? AIAA Journal. 2014;52(4): 670-690. DOI: 10.2514/1.J052375

[3] Jin Y. Surrogate-assisted evolutionary computation: Recent advances and future challenges. Swarm and

Evolutionary Computation. 2011;1(2): 61-70. DOI: 10.1016/j.swevo.2011.

[4] Sudret B. Meta-models for structural

[5] Simpson TW, Peplinski J, Koch PN, Allen JK. Meta-models for computerbased engineering design: Survey and recommendations. Engineering Computations. 2001;17(2):129-150.

[6] Simpson TW, Booker AJ, Ghosh D,

Optimization. 2004;27(5):302-313. DOI:

quantification. In: Proc. 5th Asian-Pacific Symp. Stuctural Reliab, Its Appl. Singapore: APSSRA; 2012. pp. 53-76

reliability and uncertainty

DOI: 10.1007/PL00007198

Giunta AA, Koch PN, Yang RJ. Approximation methods in multidisciplinary analysis and optimization: A panel discussion. Structural and Multidisciplinary

10.1007/s00158-004-0389-9

33

[7] Wang GG, Shan S. Review of metamodeling techniques in support of engineering design optimization. Journal of Mechanical Design. 2007; 129(4):370-380. DOI: 10.1115/1.2429697

for surrogate-assisted design optimization. Structural and

### Author details

Ali Asghar Bataleblu Department of Aerospace Engineering, K. N. Toosi University of Technology, Tehran, Islamic Republic of Iran

\*Address all correspondence to: ali.bataleblu@gmail.com

© 2019 The Author(s). Licensee IntechOpen. This chapter is 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.

Computational Intelligence and Its Applications in Uncertainty-Based Design Optimization DOI: http://dx.doi.org/10.5772/intechopen.81689

#### References

[1] Roshanian J, Bataleblu AA, Ebrahimi M. A novel evolution control strategy for surrogate-assisted design optimization. Structural and Multidisciplinary Optimization. 2018; 58:1255. DOI: 10.1007/s00158-018- 1969-4

[2] Viana FA, Simpson TW, Balabanov V, Toropov V. Special section on multidisciplinary design optimization: Metamodeling in multidisciplinary design optimization: How far have we really come? AIAA Journal. 2014;52(4): 670-690. DOI: 10.2514/1.J052375

[3] Jin Y. Surrogate-assisted evolutionary computation: Recent advances and future challenges. Swarm and Evolutionary Computation. 2011;1(2): 61-70. DOI: 10.1016/j.swevo.2011. 05.001

[4] Sudret B. Meta-models for structural reliability and uncertainty quantification. In: Proc. 5th Asian-Pacific Symp. Stuctural Reliab, Its Appl. Singapore: APSSRA; 2012. pp. 53-76

[5] Simpson TW, Peplinski J, Koch PN, Allen JK. Meta-models for computerbased engineering design: Survey and recommendations. Engineering Computations. 2001;17(2):129-150. DOI: 10.1007/PL00007198

[6] Simpson TW, Booker AJ, Ghosh D, Giunta AA, Koch PN, Yang RJ. Approximation methods in multidisciplinary analysis and optimization: A panel discussion. Structural and Multidisciplinary Optimization. 2004;27(5):302-313. DOI: 10.1007/s00158-004-0389-9

[7] Wang GG, Shan S. Review of metamodeling techniques in support of engineering design optimization. Journal of Mechanical Design. 2007; 129(4):370-380. DOI: 10.1115/1.2429697 [8] Forrester AIJ, Keane AJ. Recent advances in surrogate-based optimization. Progress in Aerospace Science. 2009;45(1–3):50-79. DOI: 10.1016/j.paerosci.2008.11.001

[9] Booker AJ, Dennis JE Jr, Frank PD, Serafini DB, Torczon V, Trosset MW. A rigorous framework for optimization of expensive functions by surrogates. Structural Optimization. 1999;17(1): 1-13. DOI: 10.1007/BF01197708

[10] Tzannetakis N, Van de Peer J. Design optimization through parallelgenerated surrogate models, optimization methodologies and the utility of legacy simulation software. Structural and Multidisciplinary Optimization. 2002;23(2):170-186. DOI: 10.1007/s00158-002-0175-5

[11] Adams BM, Bohnhoff WJ, Dalbey KR, Eddy JP, Eldred MS, Gay DM, et al. DAKOTA, a Multilevel Parallel Object-Oriented Framework for Design Optimization, Parameter Estimation, Uncertainty Quantification, and Sensitivity Analysis: Version 5.0 User's Manual. Tech. Rep. SAND2010-2183. Sandia National Laboratories; 2009

[12] Tenne Y, Goh CK. Computational intelligence in expensive optimization problems. Vol. 2. Berlin: Springer Science & Business Media; 2010

[13] Horng SC, Lin SY. Evolutionary algorithm assisted by surrogate model in the framework of ordinal optimization and optimal computing budget allocation. Information Sciences. 2013; 233:214-229. DOI: 10.1016/j.ins. 2013.01.024

[14] Sóbester A, Forrester AI, Toal DJ, Tresidder E, Tucker S. Engineering design applications of surrogate-assisted optimization techniques. Optimization

Author details

32

Ali Asghar Bataleblu

Tehran, Islamic Republic of Iran

provided the original work is properly cited.

Department of Aerospace Engineering, K. N. Toosi University of Technology,

© 2019 The Author(s). Licensee IntechOpen. This chapter is 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,

\*Address all correspondence to: ali.bataleblu@gmail.com

Bridge Optimization - Inspection and Condition Monitoring

and Engineering. 2014;15(1):243-265. DOI: 10.1007/s11081-012-9199-x

[15] Gong W, Zhou A, Cai Z. A multioperator search strategy based on cheap surrogate models for evolutionary optimization. IEEE Transactions on Evolutionary Computation. 2015;19(5): 746-758. DOI: 10.1109/TEVC.2015. 2449293

[16] Zhou Q, Shao X, Jiang P, Gao Z, Zhou H, Shu L. An active learning variable-fidelity metamodelling approach based on ensemble of metamodels and objective-oriented sequential sampling. Journal of Engineering Design. 2016;27(4–6): 205-231. DOI: 10.1080/09544828. 2015.1135236

[17] Belyaev M, Burnaev E, Kapushev E, Panov M, Prikhodko P, Vetrov D, et al. GTApprox: Surrogate modeling for industrial design. Advances in Engineering Software. 2016;102:29-39. DOI: 10.1016/j.advengsoft.2016.09.001

[18] Sun C, Jin Y, Cheng R, Ding J, Zeng J. Surrogate-assisted cooperative swarm optimization of high-dimensional expensive problems. IEEE Transactions on Evolutionary Computation. 2017;21: 644-660. DOI: 10.1109/TEVC.2017. 2675628

[19] Sayyafzadeh M. Reducing the computation time of well placement optimisation problems using selfadaptive metamodelling. Journal of Petroleum Science and Engineering. 2017;151:143-158. DOI: 10.1016/j. petrol.2016.12.015

[20] Chatterjee T, Chakraborty S, Chowdhury R. A critical review of surrogate assisted robust design optimization. Archives of Computational Methods in Engineering. 2017:1-30. DOI: 10.1007/s11831-017- 9240-5

[21] Ryberg AB, Domeij Bäckryd R, Nilsson L. Metamodel-Based Multidisciplinary Design Optimization for Automotive Applications. Linköping: Linköping University Electronic Press; 2012

[22] Zhang SL, Zhu P, Arendt PD, Chen W. Extended objective oriented sequential sampling method for robust design of complex systems against design uncertainty. In: Proceedings of the ASME International Design Engineering Technical Conferences & Computers and Information in Engineering Conference IDETC/CIE. 2012. pp. 12-15

[23] Martinez J, Marti P. Metamodelbased multi-objective robust design optimization of structures. In: 12th International Conference on Optimum Design of Structures and Materials in Engineering; New Forest, UK. 2012

[24] Dellino G, Meloni C. Uncertainty Management in Simulation Optimization of Complex Systems: Algorithms and Applications. New York: Springer; 2015

**35**

Section 3

Inspection Techniques

Section 3
