A Reduced Basis approach for PDEs on parametrized geometries based on the Shifted Boundary Finite Element Method and application to a Stokes Flow

TL;DR

This paper introduces a combined reduced basis and Shifted Boundary Method approach for efficiently solving parametrized PDEs on complex geometries, demonstrated on 2D Stokes flow problems.

Contribution

It presents a novel integration of SBM with POD-Galerkin for efficient PDE solutions on complex, parametrized geometries without remeshing or morphing.

Findings

Efficient solution of complex parametrized geometries using SBM.

Enhanced stability and performance of ROM with SBM.

Successful application to 2D Stokes flow problems.

Abstract

We propose a model order reduction technique integrating the Shifted Boundary Method (SBM) with a POD-Galerkin strategy. This approach allows to treat more complex parametrized domains in an efficient and straightforward way. The impact of the proposed approach is threefold. First, problems involving parametrizations of complex geometrical shapes and/or large domain deformations can be efficiently solved at full-order by means of the SBM, an unfitted boundary method that avoids remeshing and the tedious handling of cut cells by introducing an approximate surrogate boundary. Second, the computational effort is further reduced by the development of a reduced order model (ROM) technique based on a POD-Galerkin approach. Third, the SBM provides a smooth mapping from the true to the surrogate domain, and for this reason, the stability and performance of the reduced order basis are enhanced. This feature is the net result of the combination of the proposed ROM approach and the SBM. Similarly, the combination of the SBM with a projection-based ROM gives the great advantage of an easy and fast to implement algorithm considering geometrical parametrization with large deformations. The transformation of each geometry to a reference geometry (morphing) is in fact not required. These combined advantages will allow the solution of PDE problems more efficiently. We illustrate the performance of this approach on a number of two-dimensional Stokes flow problems.