Generalized finite element methods for quadratic eigenvalue problems

TL;DR

This paper introduces a localized orthogonal decomposition-based generalized finite element method to efficiently solve large-scale quadratic eigenvalue problems arising in structural mechanics, especially with rapidly varying material data.

Contribution

It develops a novel GFE space construction that reduces computational cost while maintaining accuracy for large-scale QEPs with complex material variations.

Findings

Proves convergence rates for the proposed method.

Numerical experiments confirm reduced computational cost.

Maintains accuracy comparable to fine-scale models.

Abstract

We consider a large-scale quadratic eigenvalue problem (QEP), formulated using P1 finite elements on a fine scale reference mesh. This model describes damped vibrations in a structural mechanical system. In particular we focus on problems with rapid material data variation, e.g., composite materials. We construct a low dimensional generalized finite element (GFE) space based on the localized orthogonal decomposition (LOD) technique. The construction involves the (parallel) solution of independent localized linear Poisson-type problems. The GFE space is then used to compress the large-scale algebraic QEP to a much smaller one with a similar modeling accuracy. The small scale QEP can then be solved by standard techniques at a significantly reduced computational cost. We prove convergence with rate for the proposed method and numerical experiments confirm our theoretical findings.