Parametrization of Reduced-Order Models Based on Global Interface Reduction
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
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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.
3.1 Meta-Model for Global Interface Modes The assumption that the substructure matrices depend on only one model parameter is no longer valid for the interface matrices M I and K I , defined in Eqs. (1.45) and (1.49), respectively. In general, these matrices depend on the entire set of model parameters θ [12, 15]. Therefore, the parametrization scheme presented in the previous chapter can no longer be applied to the interface modes. In other words, a direct interface analysis must be performed for each new sample during the corresponding simulation processes. In this context, different strategies can be considered in order to avoid direct evaluation of the interface quantities for different samples of θ . For example, the interface modes can be considered constant and can be updated every few iterations during the analyses. Another possibility considered in this chapter is to use an interpolation scheme to approximate the interface modes in terms of the model parameters.
© 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_3
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3 Parametrization of Reduced-Order Models Based on Global Interface Reduction
3.1.1 Baseline Information It is assumed that the interface matrices M I and K I , defined in Eqs. (1.45) and (1.49), respectively, have been assembled at L support points in the model parameters space, or θ l , l = 1, . . . , L, and the associated eigenvalue problems K I (θ l )Υ I (θ l ) − M I (θ l )Υ I (θ l )Ω I (θ l ) = 0 , l = 1, . . . , L
(3.1)
have been solved. In addition, the nominal solution for θ 0 has also been computed. The support points θ l , l = 1, . . . , L are distributed around the nominal point θ 0 . If ˆ sbb the definition of the interface matrices and the parametrization of the matrices M s ˆ bb , given in Eq. (2.18), are considered, M I and K I evaluated at the support and K point θ l can be expressed as T ˆ 1 ˆ¯ Ns δ ]T ¯ bb δ10 , . . . , M ˜+ M I (θ l ) = T˜ [M bb Ns 0
nθ
T ˆ 1 ˆ¯ Ns δ ]Tg ¯ bb δ1 j , . . . , M ˜ j (θ lj ) T˜ [M bb Ns j
j=1 T ˆ1 ˆ¯ Ns δ ]T˜ + ¯ bb δ10 , . . . , K K I (θ l ) = T˜ [K bb Ns 0
nθ
(3.2) ˆ¯ 1 δ , . . . , K ˆ¯ Ns δ ]Th ˜ T [K ˜ j (θ lj ) T bb 1 j bb Ns j
j
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