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

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.