Parametrization of Reduced-Order Models Based on Normal Modes
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
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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.
2.1 Motivation The solution of complex simulation-based problems involving uncertainty such as, Bayesian finite element model updating, reliability and sensitivity analysis of dynamic systems, reliability-based design optimization, uncertainty quantification, propagation analysis, etc., is computationally very demanding, due to the large number of dynamic analyses required during the corresponding simulation processes [1, 3–5, 7, 8, 10, 12, 15–17, 21, 25–31]. In fact, the solution of these types of problems requires evaluating the system response at a large number of samples in the uncertain parameter space (of the order of hundreds or thousands) [2, 9, 13, 14, 23]. Consequently, the computational cost may become excessive when the computational time for performing a dynamic analysis is substantial. Certainly, part of the computational effort is alleviated by dividing the original model into substructures and reducing the number of physical coordinates to a much-smaller number of generalized coordinates. However, the construction of reduced-order models at each sample implies re-computing the fixed-interface normal modes and the interface constraint modes for each substructure as well as for the interface modes. This procedure can be very expensive computationally, due to the substantial computational overhead that arises at the substructure level [11, 22]. To cope with this difficulty, an efficient finite element model parametrization scheme can be considered. When © Springer Nature Switzerland AG 2019 H. Jensen and C. Papadimitriou, Sub-structure Coupling for Dynamic Analysis, Lecture Notes in Applied and Computational Mechanics 89, https://doi.org/10.1007/978-3-030-12819-7_2
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2 Parametrization of Reduced-Order Models Based on Normal Modes
dividing the structure into substructures is guided by such a parametrization scheme, dramatic computational savings can be achieved. In the framework of this chapter, it is assumed that the finite element model is parametrized by a set of parameters θ ∈ Ωθ ⊂ R n θ , which are considered uncertain. These parameters are modeled using a probability density function q(θ) that indicates the relative plausibility of the possible values of the parameters θ ∈ Ωθ . The specific characterization of the
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