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This book combines a model reduction technique with an efficient parametrization scheme for the purpose of solving a class of complex and computationally expensive simulation-based problems involving finite element models. These problems, which have a wide range of important applications in several engineering fields, include reliability analysis, structural dynamic simulation, sensitivity analysis, reliability-based design optimization, Bayesian model validation, uncertainty quantification and propagation, etc. The solution of this type of problems requires a large number of dynamic re-analyses. To cope with this difficulty, a model reduction technique known as substructure coupling for dynamic analysis is considered. While the use of reduced order models alleviates part of the computational effort, their repetitive generation during the simulation processes can be computational expensive due to the substantial computational overhead that arises at the substructure level. In this regard, an efficient finite element model parametrization scheme is considered. When the division of the structural model is guided by such a parametrization scheme, the generation of a small number of reduced order models is sufficient to run the large number of dynamic re-analyses. Thus, a drastic reduction in computational effort is achieved without compromising the accuracy of the results. The capabilities of the developed procedures are demonstrated in a number of simulation-based problems involving uncertainty.

Inhaltsverzeichnis

Frontmatter

Reduced-Order Models

Frontmatter

Chapter 1. Model Reduction Techniques for Structural Dynamic Analyses

Abstract
This chapter presents a model reduction technique based on substructure coupling for dynamic analysis. The dynamic behavior of the substructures is described by a set of dominant fixed-interface normal modes along with a set of interface constraint modes that account for the coupling at each interface where the substructures are connected. Based on these modes, the corresponding reduced-order matrices are derived. The internal dynamic behavior of the substructures is then enhanced by consideration of the contribution of residual fixed-interface normal modes. Next, the interface degrees of freedom are reduced by consideration of a small number of characteristic constraint modes. Pseudo-codes are provided in order to illustrate how the reduced-order matrices are constructed, by including dominant and residual fixed-interface normal modes as well as interface reduction. Finally, the dynamic response of reduced-order models is discussed.
Hector Jensen, Costas Papadimitriou

Chapter 2. Parametrization of Reduced-Order Models Based on Normal Modes

Abstract
This chapter deals with the parametrization of reduced-order models based on dominant and residual fixed-interface normal modes, in terms of model parameters. The division of the original structure is guided by a parametrization scheme, which assumes that the substructure matrices for each of the introduced linear substructures depend on only one of the model parameters. Based on this assumption, a global parametrization of the reduced-order matrices is provided. Invariant issues are discussed that are related to the matrices that account for the contribution of residual normal modes. A pseudo-code is then provided in order to illustrate how the parametrization of the reduced-order matrices is constructed.
Hector Jensen, Costas Papadimitriou

Chapter 3. Parametrization of Reduced-Order Models Based on Global Interface Reduction

Abstract
An interpolation scheme for approximating the interface modes in terms of the model parameters is presented in this chapter. The approximation scheme involves a set of support points in the model parameters space and a number of interpolation coefficients that are determined by the singular value decomposition technique. The approximate interface modes are combined with the parametrization scheme introduced in Chap. 2 to derive the corresponding reduced-order matrices. Pseudo-codes are provided to illustrate how the interface modes are approximated and how the parametrization of the reduced-order matrices is constructed based on interface reduction.
Hector Jensen, Costas Papadimitriou

Application to Reliability Problems

Frontmatter

Chapter 4. Reliability Analysis of Dynamical Systems

Abstract
The use of reduced-order models in the context of reliability analysis of dynamical systems under stochastic excitation is explored in this chapter. A stochastic excitation model based on a point-source model is introduced, and it is used for the generation of ground motions. The corresponding reliability analysis represents a high-dimensional reliability problem whose solution is carried out by an advanced simulation technique. Two application problems are considered in order to evaluate the effectiveness of the proposed model reduction technique. The first example consists of a two-dimensional frame structure, while the second example considers an involved nonlinear finite element building model. The results show that an important reduction in computational effort can be achieved without compromising the accuracy of the reliability estimates.
Hector Jensen, Costas Papadimitriou

Chapter 5. Reliability Sensitivity Analysis of Dynamical Systems

Abstract
The reliability sensitivity analysis of systems subjected to stochastic loading is considered in this chapter. In particular, the change that the probability of failure undergoes due to changes in the distribution parameters of the uncertain model parameters is utilized as a sensitivity measure. A simulation-based approach that corresponds to a simple post-processing step of an advanced sampling-based reliability analysis is used to perform the sensitivity analysis. In particular, subset simulation, introduced in the previous chapter, is applied in the present formulation. The analysis does not require any additional system response evaluations. The feasibility and effectiveness of the approach is demonstrated on a finite element model of a bridge under stochastic ground excitation. The sensitivity analysis is carried out in a reduced space of generalized coordinates. The computational effort involved in the reliability sensitivity analysis of the reduced-order model is significantly decreased with respect to the corresponding analysis of the full finite element model. The reduction is accomplished without compromising the accuracy of the reliability sensitivity estimates.
Hector Jensen, Costas Papadimitriou

Chapter 6. Reliability-Based Design Optimization

Abstract
The solution of reliability-based design optimization problems by using reduced-order models is considered in this chapter. Specifically, problems involving high-dimensional stochastic dynamical systems are analyzed. The design process is formulated in terms of a constrained nonlinear optimization problem, which is solved by a class of interior point algorithms based on feasible directions. Search directions are estimated in an efficient manner as a by-product of reliability analyses. The design process generates a sequence of steadily-improved feasible designs. Three numerical examples are presented to evaluate the performance of the interior point algorithm and the effectiveness of reduced-order models in the context of complex reliability-based optimization problems. High speedup values can be obtained for the design process without changing the accuracy of the final designs.
Hector Jensen, Costas Papadimitriou

Application to Identification Problems

Frontmatter

Chapter 7. Bayesian Finite Element Model Updating

Abstract
In this chapter, the implementation of the reduced-order models within Bayesian finite element model updating is explored. The Bayesian framework for model parameter estimation, model selection, and robust predictions of output quantities of interest is first presented. Bayesian asymptotic approximations and sampling algorithms are then outlined. The framework is implemented for updating linear and nonlinear finite element models in structural dynamics using vibration measurements consisting of either identified modal frequencies or measured response time histories. For asymptotic approximations based on modal properties, the formulation for the posterior distribution is presented with respect to the modal properties of the reduced-order model. In addition, analytical expressions for the required gradients with respect to the model parameters are provided using adjoint methods. Two applications demonstrate that drastic reductions in computational demands can be achieved without compromising the accuracy of the model updating results. In the first application, a high-fidelity linear finite element model of a full-scale bridge with hundreds of thousands of degrees-of-freedom (DOFs) is updated using experimentally identified modal properties. In the second application, a nonlinear model of a base-isolated building is updated using acceleration response time histories.
Hector Jensen, Costas Papadimitriou
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