Krylov subspaces recycling based model order reduction for acoustic BEM systems and an error estimator

Abstract

Boundary Element Method frequency sweep analyses in acoustics are usually accompanied by a vast numerical cost of assembling and solving numerous linear systems. In that context, this work proposes a model order reduction technique to mitigate the resulting computational cost of such analyses. First, a series expansion of the Green's function BEM kernel is leveraged to construct a series of frequency independent matrices. Next, in a model order reduction way, the arising matrices are projected on a reduced basis utilizing a Galerkin projection.By this off-line matrix projection, both the assembly and the solution of the BEM full-size linear systems degenerate into assembling and solving a reduced system for all frequencies. Significant speed-up factors can, thus, be achieved for both operations. The projection basis employed in this model reduction scheme is developed through an Arnoldi algorithm for the BEM systems on a grid of master frequencies. The method is based on Krylov subspaces recycling, as the subspaces produced at master frequencies are recycled to approximate the surface distribution of the acoustic variables on the whole frequency range of interest. Utilizing Krylov subspaces facilitates as well the definition of a robust error estimator that indicates the quality of the reduced system. The performance of the proposed method is assessed for both an exterior and an interior problem for a simple and more complicated geometry respectively.

A subspace iteration eigensolver based on Cauchy integrals for vibroacoustic problems in unbounded domains

“…Once the residual stagnates, the algorithm is terminated. If the accuracy is then deemed to be insufficient, the whole procedure can be repeated with the computed eigenvectors as new initial guess and by incorporating a more accurate solution strategy for the projected EVP in Equation (12). The latter is discussed in Section 3.3, where we will see that the accuracy can be increased while retaining information from the previous run.…”

“…El-Guide et al 32 address this issue by subspace iteration in which the columns of the projection matrix are individually computed by inverse power iteration. In our case, the inflation of the EVP associated with the linearization is mitigated by the projection (12), where the projection matrix (16) and the rational approximation of the EVP (26) can be based on the same contour points. The computational effort for solving the linearized EVP is reduced to an order of  ( (mN p ) 3 ) .…”

“…The BE matrices associated with T ( z j ) are stored in memory in order to avoid re‐assembly in each iteration. In this respect, the use of data‐sparse formats 42 or model order reduction of BE matrices 12 can alleviate the memory requirements. The initial guess of eigenvectors can either be random, or in the case of moderately strong structural acoustic interaction, X (0) can be chosen as the in vacuo modes of the structure.…”

“…Early works on this topic focused on interpolating either the coefficient matrix, 8 the underlying kernel function, 9 or even the resulting response itself. 10 More recent work is concerned with efficient strategies for solving the sequence of linear systems by means of model order reduction [11][12][13] and greedy sampling. 14 Along with frequency sweeps, modal analysis accounts for another fundamental tool to study the dynamical behavior and provides system-inherent properties such as eigenfrequencies and modal damping values.…”

“…Early works on this topic focused on interpolating either the coefficient matrix, 8 the underlying kernel function, 9 or even the resulting response itself 10 . More recent work is concerned with efficient strategies for solving the sequence of linear systems by means of model order reduction 11‐13 and greedy sampling 14 …”