A localized reduced basis approach for unfitted domain methods on parameterized geometries

TL;DR

This paper presents a novel reduced basis framework for efficiently solving parameterized PDEs on unfitted geometries, overcoming challenges posed by geometrical parameters and demonstrating significant computational savings.

Contribution

It introduces a localization-based reduced basis method with snapshot extension for unfitted domain discretizations, applicable to 2D and 3D problems.

Findings

Significant reduction in online computational cost.

Method is accurate and discretization-agnostic.

Effective for 2D spline-based and 3D elastic problems.

Abstract

This work introduces a reduced order modeling (ROM) framework for the solution of parameterized second-order linear elliptic partial differential equations formulated on unfitted geometries. The goal is to construct efficient projection-based ROMs, which rely on techniques such as the reduced basis method and discrete empirical interpolation. The presence of geometrical parameters in unfitted domain discretizations entails challenges for the application of standard ROMs. Therefore, in this work we propose a methodology based on i) extension of snapshots on the background mesh and ii) localization strategies to decrease the number of reduced basis functions. The method we obtain is computationally efficient and accurate, while it is agnostic with respect to the underlying discretization choice. We test the applicability of the proposed framework with numerical experiments on two model problems, namely the Poisson and linear elasticity problems. In particular, we study several benchmarks formulated on two-dimensional, trimmed domains discretized with splines and we observe a significant reduction of the online computational cost compared to standard ROMs for the same level of accuracy. Moreover, we show the applicability of our methodology to a three-dimensional geometry of a linear elastic problem.