An A Priori Bound for Automated Multilevel Substructuring

Abstract

Abstract. The Automated Multi-Level Substructuring (AMLS) method has been developed to reduce the computational demands of frequency response analysis and has recently been proposed as an alternative to iterative projection methods like Lanczos or Jacobi-Davidson for computing a large number of eigenvalues for matrices of very large dimension. Based on Schur complements and modal approximations of submatrices on several levels AMLS constructs a projected eigenproblem which yields good approximations of eigenvalues at the lower end of the spectrum. Rewriting the original problem as a rational eigenproblem of the same dimension as the projected problem, and taking advantage of a minmax characterization for the rational eigenproblem we derive an a priori bound for the AMLS approximation of eigenvalues. Here the large finite element model is recursively divided into very many substructures on several levels based on the sparsity structure of the system matrices. Assuming that the interior degrees of freedom of substructures depend quasistatically on the interface degrees of freedom, and modeling the deviation from quasistatic dependence in terms of a small number of selected substructure eigenmodes the size of the finite element model is reduced substantially yet yielding satisfactory accuracy over a wide frequency range of interest. Recent studies ([16], [14], e.g.) in vibroacoustic analysis of passenger car bodies where very large FE models with more than one million degrees of freedom appear and several hundreds of eigenfrequencies and eigenmodes are needed have shown that AMLS is considerably faster than Lanczos type approaches.We stress the fact that substructuring does not mean that it is obtained by a domain decomposition of a real structure, but it is understood in a purely algebraic sense, i.e. the dissection of the matrices can be derived by applying a graph partitioner like CHACO [11] or METIS [15] to the matrix under consideration. However, because of its pictographic nomenclature we will use terms like substructure or eigenmode from frequency response problems when introducing the AMLS method.From a mathematical point of view AMLS is a projection method where the ansatz space is constructed exploiting Schur complements of submatrices and truncation of spectral representations of subproblems. In this paper we will take advantage of the facts that the original eigenproblem is equivalent to a rational eigenvalue problem of the same dimension as the projected problem in AMLS, which can be interpreted as exact condensation of the original eigenproblem with respect to an appropriate basis. Its eigenvalues at the lower end of the spectrum can be characterized as minmax values of a Rayleigh functional of this rational eigenproblem. Hence, comparing the Rayleigh quotient of the projected problem and the Rayleigh functional of the rational

On a Non-Symmetric Eigenvalue Problem Governing Interior Structural–Acoustic Vibrations

“…The upper bound in Equation (32) to the relative error has the same structure as the error bound given in [52] for CMS applied to a definite eigenvalue problem Kx = λMx. In the definite case, the lower bound is zero due to the fact that CMS is a projection method, and the eigenvalues under consideration are at the lower end of the spectrum.…”

A Mode Selection Algorithm for the Flexibility-Based Component Mode Synthesis

“…Also, Kim et al [17] have developed another mode selection method based on the eigenvector relation between substructural and global modes, and it shows better performance than previous ones but also is applicable to various CMS methods such as Craig-Bampton (CB) and Automated multi-level substructuring (AMLS) methods. In addition, Kim and Lee recently developed new error estimation methods to directly estimate the relative eigenvalue error [18][19][20] unlike the previous methods [9,21,22]. In this paper, we propose an iterative mode selection algorithm that combines the mode selection and error estimation methods developed by Kim et al [17,20], and its feasibility is studied using the F-CMS method.…”

Solving an elliptic PDE eigenvalue problem via automated multi-level substructuring and hierarchical matrices

“…For this reason, the AMLS method is often analysed in literature (e.g., in [28,30,67]) only in a pure algebraic setting. …”

“…In the literature [28,67] several heuristic approaches have been derived on how to select eigenpairs. These heuristics are based purely on the analysis of the algebraic eigenvalue problem ( K, M ) without using any geometry information of the underlying partial differential equation (4.1).…”