Nonlinear Reduction using the Extended Group Finite Element Method

TL;DR

This paper introduces a nonlinear reduction framework based on the extended group finite element method, combining model order reduction and complexity reduction techniques to enable efficient real-time solutions for nonlinear finite element problems.

Contribution

It extends the group finite element method to quasilinear problems and integrates proper orthogonal decomposition and empirical interpolation for improved efficiency.

Findings

Reduced computational overhead in nonlinear finite element problems

Effective model order reduction for real-time applications

Superior performance demonstrated in three benchmark problems

Abstract

In this paper, we develop a nonlinear reduction framework based on our recently introduced extended group finite element method. By interpolating nonlinearities onto approximation spaces defined with the help of finite elements, the extended group finite element formulation achieves a noticeable reduction in the computational overhead associated with nonlinear finite element problems. However, the problem's size still leads to long solution times in most applications. Aiming to make real-time and/or many-query applications viable, we apply model order reduction and complexity reduction techniques in order to reduce the problem size and efficiently handle the reduced nonlinear terms, respectively. For this work, we focus on the proper orthogonal decomposition and discrete empirical interpolation methods. While similar approaches based on the group finite element method only focus on semilinear problems, our proposed framework is also compatible with quasilinear problems. Compared to existing methods, our reduced models prove to be superior in many different aspects as demonstrated in three numerical benchmark problems.