Multi-mesh reduced-order basis method for finite element analysis

Abstract

Reduced order modeling of differential equations parametrized over a parameter space can be used to accelerate optimization and parameter estimation problems. The method of snapshots or reduced order basis is well established among researchers as a tool to build reduced order models of ordinary differential equations. The reduced order basis method has been utilized for numerical solution of parametric PDE problems by researchers in recent years and has many advantages over response surface methods. The application of ROB to finite element analysis has been restricted to using a fixed mesh for snapshots. In this work, a new method is developed for construction of ROB from a set of snapshots defined over various meshes. Consistent inner product is defined for the finite dimensional functional spaces and a new general purpose geometric intersection algorithm is developed to enable the inner product computations for all dimensions. Compatible inner products are used to construct the multi-mesh proper orthogonal decomposition method. The newly developed multi-mesh POD method removes the restriction of fixed mesh from ROB method and it can also be applied outside the context of finite element analysis.