**1. Introduction**

[18] Paulo, P. A Time-Domain Methodology For Rotor Dynamics: Analysis and Force

[19] Vakakis, A. F. Dynamic Analysis of a Unidirectional Periodic Isolator, Consisting of Identical Masses and Intermediate Distributed Resilient Blocks. Journal of Sound

[20] Vakakis, A. F, & Paipetis, S. A. Transient Response of Unidirectional Vibration Isola‐ torswith Many Degrees of Freedom, Journal of Sound and Vibration, (1985).

[21] Vakakis, A. F, & Paipetis, S. A. The Effect of a Viscously Damped Dynamic Absorber ona Linear Multi-Degree-of-Freedom System, Journal of Sound and Vibration, (1986).

[22] Liu, W. Structural Dynamic Analysis and Testing of Coupled Structures. PhD thesis.

[23] Liu, W, & Ewins, D. J. Transmissibility Properties of MDOF Systems, Proceedings of the16th International Operational Modal Analysis Conference (IMAC XVI), Santa

[24] Varoto, P. S, & Mcconnell, K. G. Single Point vs Multi Point AccelerationTransmissi‐ bility Concepts in Vibration Testing, Proceedings of the 12th International Modal Analysis Conference (IMAC XVI), Santa Barbara, California, USA,(1998). , 83-90. [25] Ribeiro, A. M. R. On the Generalization of the Transmissibility Concept, Proceedings of the NATO/ASI Conference on Modal Analysis and Testing, Sesimbra, Portugal,

[26] Fontul, M, Ribeiro, A. M. R, Silva, J. M. M, & Maia, N. M. M. Transmissibility Matrix in Harmonic and Random Processes, Shock and Vibration, (2004). , 563-571.

[27] Maia, N. M. M, Fontul, M, & Ribeiro, A. M. R. Transmissibility of Forces in Multiple-Degree-of-Freedom Systems, Proceedings of ISMA (2006). Noise and Vibration Engi‐

[28] Rao, S. S. Mechanical Vibrations. Fourth International Edition. Prentice-Hall; (2004).

Identification. MSc thesis. Instituto Superior Técnico Lisbon; (2011).

andVibration, (1985). , 25-33.

132 Advances in Vibration Engineering and Structural Dynamics

0002-2460X., 99(4), 557-562.

Imperial College London; (2000).

Barbara, California, (1998). , 847-854.

neering, Leuven, Belgium 2006.

(1), 49-60.

(1998). , 757-764.

Finite element analysis (FEA) is a well-established numerical simulation method for struc‐ tural dynamics. It serves as the main computational tool for Noise, Vibration and Harshness (NVH) analysis in the low-frequency range. Because of developments in numerical methods and advances in computer software and hardware, FEA can now handle much more com‐ plex models far more efficiently than even a few years ago. However, the demand for com‐ putational capabilities increases in step with or even beyond the pace of these improvements. For example, automotive companies are constructing more detailed models with millions of degrees of freedom (DOFs) to study vibro-acoustic problems in higher fre‐ quency ranges. Although these tasks can be performed with FEA, the computational cost can be prohibitive even for high-end workstations with the most advanced software.

For large finite element (FE) models, a modal reduction is commonly used to obtain the sys‐ tem response. An eigenanalysis is performed using the system stiffness and mass matrices and a smaller in size modal model is formed which is solved more efficiently for the re‐ sponse. The computational cost is also reduced using substructuring (superelement analy‐ sis). Modal reduction is applied to each substructure to obtain the component modes and the system level response is obtained using Component Mode Synthesis (CMS).

When design changes are involved, the FEA analysis must be repeated many times in order to obtain the optimum design. Furthermore in probabilistic analysis where parameter uncer‐ tainties are present, the FEA analysis must be repeated for a large number of sample points. In such cases, the computational cost is even higher, if not prohibitive. Reanalysis methods

are intended to analyze efficiently structures that are modified due to various changes. They estimate the structural response after such changes without solving the complete set of modified analysis equations. Several reviews have been published on reanalysis methods [1-3] which are usually based on local and global approximations. Local approximations are very efficient but they are effective only for small structural changes. Global approximations are preferable for large changes, but they are usually computationally expensive especially for cases with many design parameters. The well-known Rayleigh-Ritz reanalysis procedure [4, 5] belongs to the category of local approximation methods. The mode shapes of a nomi‐ nal design are used to form a Ritz basis for predicting the response in a small parametric zone around the nominal design point. However, it is incapable of capturing relatively large design changes.

solving a linear system is dominated by the cost of matrix decomposition way no be longer valid (see Section 3.4) and the computational savings from using the CA method may not be substantial. For this reason, we developed a modified combined approximation (MCA) and integrated it with the PROM method to improve accuracy and computational efficiency. The computational savings can be substantial for problems with a large number of design pa‐ rameters. Examples in this Chapter demonstrate the benefits of this reanalysis methodology.

Vibration and Optimization Analysis of Large-Scale Structures using Reduced-Order Models and Reanalysis Methods

http://dx.doi.org/10.5772/51402

135

**1.** for accurate and efficient vibration analysis methods of large-scale, finite-element mod‐

**2.** for efficient and yet accurate reanalysis methods for dynamic response and optimiza‐

**3.** for efficient design optimization methods to optimize structures for best vibratory re‐

The optimization is able to handle a large number of design variables and identify local and global optima. It is organized as follows. Section 2 presents an overview of reduced-order modeling and substructuring methods including modal reduction and component mode synthesis (CMS). Improvements to the CMS method are presented using interface modes and filtration of constraint modes. The section also overviews two Frequency Response Function (FRF) substructuring methods where two substructures, represented by FRFs or FE models, are assembled to form an efficient reduced-order model to calculate the dynamic re‐ sponse. Section 3 presents four reanalysis methods: the CDH/VAO method, the Parametric Reduced Order Modeling (PROM) method, the Combined Approximation (CA) method, and the Modified Combined Approximation (MCA) method. It also points out their strong and weak points in terms of efficiency and accuracy. Section 4 demonstrates how the reanal‐ ysis methods are used in vibration and optimization of large-scale structures. It also presents a new reanalysis method in Craig-Bampton substructuring with interface modes which is very useful for problems with many interface DOFs where the FRF substructuring methods cannot be used. Section 5 presents a vibration and optimization case study of a large-scale vehicle model demonstrating the value of reduced-order modeling and reanaly‐

Computational efficiency is of paramount importance in vibration analysis of large-scale, fi‐ nite-element models. Reduced-order modeling (or substructuring) is a common approach to reduce the computational effort. Substructuring methods can be classified in "mathemati‐ cal" and "physical" methods. The "mathematical" substructuring methods include the Au‐ tomatic Multi-level Substructuring (AMLS) and the Automatic Component Mode Synthesis (ACMS) in NASTRAN. The "physical" substructuring methods include the well known fixed-interface Craig-Bampton method. Both the AMLS and ACMS methods use graph theo‐

sis methods in practice. Finally, Section 6 summarizes and concludes.

**2. Reduced-Order Modeling for Dynamic Analysis**

The Chapter presents methodologies

els,

tion, and

sponse.

A parametric reduced-order modeling (PROM) method, developed by Balmes [6, 7], ex‐ pands on the Rayleigh-Ritz method by using the mode shapes from a few selected design points to predict the response throughout the design space. The PROM method belongs to the category of local approximation methods and can handle relatively larger parameter changes because it uses multiple design points. An improved component-based PROM method has been introduced by Zhang et al. [8, 9] for design changes at the component lev‐ el. The PROM method using a 'parametric' approach has been successfully applied to de‐ sign optimization and probabilistic analysis of vehicle structures. However, the 'parametric' approach is only applicable to problems where the mass and stiffness matrices can be ap‐ proximated by a polynomial function of the design parameters and their powers. A new 'parametric' approach using Kriging interpolation [10] has been recently proposed [11]. It improves efficiency by interpolating the reduced system matrices without needing to as‐ sume a polynomial relationship of the system matrices with respect to the design parame‐ ters as in [6, 7].

The Combined Approximations (CA) method [12-14] combines the strengths of both local and global approximations and can be accurate even for large design changes. It uses a com‐ bination of binomial series (local) approximations, called Neumann expansion approxima‐ tions, and reduced basis (global) approximations. The CA method is developed for linear static reanalysis and eigen-problem reanalysis [15-19]. Accurate results and significant com‐ putational savings have been reported. All reported studies on the CA method [12-19] use relatively simple frame or truss systems for static or dynamics analysis with a relatively small number of DOF and/or modes. For these problems, the computational efficiency was improved by a factor of 5 to 10. Such an improvement is beneficial for a single design change evaluation or even for gradient-based design optimization where only a limited number of reanalyses (e.g. less than 50) is performed. However, the computational efficiency of the CA method may not be adequate in simulation-based (e.g. Monte-Carlo) probabilistic dynamic analysis of large finite-element models where reanalysis must be performed hun‐ dreds or thousands of times in order to estimate accurately the reliability of a design.

A large number of modes must be calculated and used in a dynamic analysis of a large fi‐ nite-element model with a high modal density, even if a reduced-order modeling approach (Section 2) is used. In such a case, the implicit assumption of the CA method that the cost of solving a linear system is dominated by the cost of matrix decomposition way no be longer valid (see Section 3.4) and the computational savings from using the CA method may not be substantial. For this reason, we developed a modified combined approximation (MCA) and integrated it with the PROM method to improve accuracy and computational efficiency. The computational savings can be substantial for problems with a large number of design pa‐ rameters. Examples in this Chapter demonstrate the benefits of this reanalysis methodology.

The Chapter presents methodologies

are intended to analyze efficiently structures that are modified due to various changes. They estimate the structural response after such changes without solving the complete set of modified analysis equations. Several reviews have been published on reanalysis methods [1-3] which are usually based on local and global approximations. Local approximations are very efficient but they are effective only for small structural changes. Global approximations are preferable for large changes, but they are usually computationally expensive especially for cases with many design parameters. The well-known Rayleigh-Ritz reanalysis procedure [4, 5] belongs to the category of local approximation methods. The mode shapes of a nomi‐ nal design are used to form a Ritz basis for predicting the response in a small parametric zone around the nominal design point. However, it is incapable of capturing relatively large

A parametric reduced-order modeling (PROM) method, developed by Balmes [6, 7], ex‐ pands on the Rayleigh-Ritz method by using the mode shapes from a few selected design points to predict the response throughout the design space. The PROM method belongs to the category of local approximation methods and can handle relatively larger parameter changes because it uses multiple design points. An improved component-based PROM method has been introduced by Zhang et al. [8, 9] for design changes at the component lev‐ el. The PROM method using a 'parametric' approach has been successfully applied to de‐ sign optimization and probabilistic analysis of vehicle structures. However, the 'parametric' approach is only applicable to problems where the mass and stiffness matrices can be ap‐ proximated by a polynomial function of the design parameters and their powers. A new 'parametric' approach using Kriging interpolation [10] has been recently proposed [11]. It improves efficiency by interpolating the reduced system matrices without needing to as‐ sume a polynomial relationship of the system matrices with respect to the design parame‐

The Combined Approximations (CA) method [12-14] combines the strengths of both local and global approximations and can be accurate even for large design changes. It uses a com‐ bination of binomial series (local) approximations, called Neumann expansion approxima‐ tions, and reduced basis (global) approximations. The CA method is developed for linear static reanalysis and eigen-problem reanalysis [15-19]. Accurate results and significant com‐ putational savings have been reported. All reported studies on the CA method [12-19] use relatively simple frame or truss systems for static or dynamics analysis with a relatively small number of DOF and/or modes. For these problems, the computational efficiency was improved by a factor of 5 to 10. Such an improvement is beneficial for a single design change evaluation or even for gradient-based design optimization where only a limited number of reanalyses (e.g. less than 50) is performed. However, the computational efficiency of the CA method may not be adequate in simulation-based (e.g. Monte-Carlo) probabilistic dynamic analysis of large finite-element models where reanalysis must be performed hun‐

dreds or thousands of times in order to estimate accurately the reliability of a design.

A large number of modes must be calculated and used in a dynamic analysis of a large fi‐ nite-element model with a high modal density, even if a reduced-order modeling approach (Section 2) is used. In such a case, the implicit assumption of the CA method that the cost of

design changes.

134 Advances in Vibration Engineering and Structural Dynamics

ters as in [6, 7].


The optimization is able to handle a large number of design variables and identify local and global optima. It is organized as follows. Section 2 presents an overview of reduced-order modeling and substructuring methods including modal reduction and component mode synthesis (CMS). Improvements to the CMS method are presented using interface modes and filtration of constraint modes. The section also overviews two Frequency Response Function (FRF) substructuring methods where two substructures, represented by FRFs or FE models, are assembled to form an efficient reduced-order model to calculate the dynamic re‐ sponse. Section 3 presents four reanalysis methods: the CDH/VAO method, the Parametric Reduced Order Modeling (PROM) method, the Combined Approximation (CA) method, and the Modified Combined Approximation (MCA) method. It also points out their strong and weak points in terms of efficiency and accuracy. Section 4 demonstrates how the reanal‐ ysis methods are used in vibration and optimization of large-scale structures. It also presents a new reanalysis method in Craig-Bampton substructuring with interface modes which is very useful for problems with many interface DOFs where the FRF substructuring methods cannot be used. Section 5 presents a vibration and optimization case study of a large-scale vehicle model demonstrating the value of reduced-order modeling and reanaly‐ sis methods in practice. Finally, Section 6 summarizes and concludes.
