A Reduced Order Approach for the Embedded Shifted Boundary FEM and a Heat Exchange System on Parametrized Geometries

TL;DR

This paper introduces a reduced order modeling approach combined with the Shifted Boundary Method for efficient simulation of parametrized heat transfer problems, avoiding complex mesh updates and cut cell treatments.

Contribution

It presents a novel combination of model order reduction with the Shifted Boundary Method, enabling efficient and accurate simulations on parametrized geometries without mesh updates.

Findings

Effective reduction in computational cost for parametrized heat transfer problems

No need for mesh updates when geometries change

Smooth transition of reduced modes across boundaries

Abstract

A model order reduction technique is combined with an embedded boundary finite element method with a POD-Galerkin strategy. The proposed methodology is applied to parametrized heat transfer problems and we rely on a sufficiently refined shape-regular background mesh to account for parametrized geometries. In particular, the employed embedded boundary element method is the Shifted Boundary Method (SBM) recently proposed. This approach is based on the idea of shifting the location of true boundary conditions to a surrogate boundary, with the goal of avoiding cut cells near the boundary of the computational domain. This combination of methodologies has multiple advantages. In the first place, since the Shifted Boundary Method always relies on the same background mesh, there is no need to update the discretized parametric domain. Secondly, we avoid the treatment of cut cell elements, which usually need particular attention. Thirdly, since the whole background mesh is considered in the reduced basis construction, the SBM allows for a smooth transition of the reduced modes across the immersed domain boundary. The performances of the method are verified in two dimensional heat transfer numerical examples.