Reduced Basis, Embedded Methods and Parametrized Levelset Geometry

TL;DR

This paper explores reduced order modeling techniques for parametrized geometrical heat exchange and fluid flow problems using embedded finite element methods like SBM and CutFEM, emphasizing stability and efficiency improvements.

Contribution

It introduces a parameterized Nitsche shifted boundary formulation combined with POD-Galerkin reduced order models for geometrically parametrized problems, highlighting advantages over classical approaches.

Findings

Reduced order models significantly decrease computation time.

Stability issues can be mitigated with supremizer enrichment.

Numerical experiments confirm efficiency across multiple dimensions.

Abstract

In this chapter we examine reduced order techniques for geometrical parametrized heat exchange systems, Poisson, and flows based on Stokes, steady and unsteady incompressible Navier-Stokes and Cahn-Hilliard problems. The full order finite element methods, employed in an embedded and/or immersed geometry framework, are the Shifted Boundary (SBM) and the Cut elements (CutFEM) methodologies, with applications mainly focused in fluids. We start by introducing the Nitsche's method, for both SBM/CutFEM and parametrized physical problems as well as the high fidelity approximation. We continue with the full order parameterized Nitsche shifted boundary variational weak formulation, and the reduced order modeling ideas based on a Proper Orthogonal Decomposition Galerkin method and geometrical parametrization, quoting the main differences and advantages with respect to a reference domain approach used for classical finite element methods, while stability issues may overcome employing supremizer enrichment methodologies. Numerical experiments verify the efficiency of the introduced ``hello world'' problems considering reduced order results in several cases for one, two, three and four dimensional geometrical kind of parametrization. We investigate execution times, and we illustrate transport methods and improvements. A list of important references related to unfitted methods and reduced order modeling are [11, 8, 9, 10, 7, 6, 12].