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

TL;DR

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 that is computationally efficient and accurate, while it is agnostic with respect to the underlying discretization choice.

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.

Figures (19)

• Figure 1: Geometrical setting: the geometrically parameterized domain $\Omega(\bm{\mu})$, built through subtraction operations, is embedded in the background domain $\Omega_0$.
• Figure 2: Univariate B-spline basis for different parameters $\mu_1,\mu_2,\mu_3$ defining the trimming location. The functions depicted in blue are fully active, in dotted blue are trimmed active and in grey are inactive.
• Figure 3: Example 6.1.1: Exemplary solution snapshots for $\mu=[0.5,0.9,1.5]$.
• Figure 4: Example 6.1.1: Error decay of global DEIM approximations in $L^{\infty}$-error norm for right-hand side vector (a) and matrix (b) based on POD tolerance of $\epsilon_{POD}^d= 10^{-7}$.
• Figure 5: Example 6.1.1: Comparison of singular values decay between global and local DEIM approximations for right-hand side vector (a) and matrix (b) using different number of clusters.