Consistent Parametric Model Order Reduction by Matrix Interpolation for Varying Underlying Meshes

TL;DR

This work tackles the challenge of parametric MOR for geometric parameters where FE meshes vary with the parameter, by introducing a framework that represents reduced bases as displacement fields on a reference mesh. Mesh morphing (spring-analogy with elastic hardening and radial basis function morphing) and basis interpolation enable consistent representation of sampled bases on a common discretization, permitting meaningful interpolation of reduced operators even for non-matching meshes. The approach detects subspace inconsistencies via subspace angles and uses adaptive sampling to cluster parameter regions, delivering high accuracy in both 1D and 2D geometric parameter studies and outperforming existing methods that transform reduced operators or pad bases. Offline morphing and interpolation are computationally manageable, with RBF morphing offering particularly small offline costs, enabling scalable application to more complex, multi-parameter, and 3D problems.

Abstract

Parametric model order reduction (pMOR) is a powerful tool for accelerating finite element (FE) simulations while maintaining parametric dependencies. For geometric parameters, pMOR by matrix interpolation is a well-suited approach because it does not require an affine representation of the parametric dependency, which is often not available for geometric parameters. However, the method requires that the underlying FE mesh has the same number of degrees of freedom and the same topology for all parameter configurations. This requirement can be difficult or even impossible to achieve for large parameter ranges or when automatic meshing is used. In this work, we propose a novel framework for pMOR by matrix interpolation for varying underlying meshes. The key idea is to understand the sampled reduced bases as continuous displacement fields that can be represented in different discretizations. By using mesh morphing and basis interpolation, the sampled reduced bases described in varying meshes can all be represented in terms of one reference mesh. This not only allows for performing pMOR by matrix interpolation, but also enables comparing the subspaces that the reduced bases span, which is important to detect strong changes that could lead to inconsistencies in the reduced operators. For mesh morphing, two strategies, namely morphing by spring analogy with elastic hardening and radial basis function morphing, were implemented and tested. Numerical experiments on a beam-shaped plate and a plate with a hole for one- and two-dimensional parameter spaces show that the proposed framework achieves high accuracy for both morphing methods and performs significantly better than two existing approaches for pMOR by matrix interpolation for varying underlying meshes.

Figures (12)

• Figure 1: Visualization of the characteristic features for a plate with a circular hole.
• Figure 2: Visualization of mesh morphing.
• Figure 3: Visualization of representing a basis vector described in one of the sampled meshes in terms of the reference mesh.
• Figure 4: Triangular element with lineal and torsional springs.
• Figure 5: Beam-shaped plate.