Simple, accurate, and efficient embedded finite element methods for fluid–solid interaction

Abstract

This work presents a new approach to implementing a recently proposed optimal order Cut Finite Element Method (CutFEM) for problems with moving embedded solid structures in viscous incompressible flows. This new approach uses the notion of equivalent polynomials, introduced previously in the context of the eXtended Finite Element Methods (XFEM), to implement exact integration for terms involving products of polynomials with Heaviside and Dirac distributions. Combining CutFEM and equivalent polynomials results in a method for fluid–structure interaction that (1) has the same number of degrees of freedom as the underlying conforming Galerkin method on the fixed background mesh, which is independent of the configuration of non-conforming interfaces, (2) has the same element assembly structure as classical FEM on the background mesh—with standard quadrature rules, and (3) retains the convergence properties, indeed the precise theoretical structure, of the original CutFEM method. The result is a method that is robust, accurate, and efficient for interacting particles, and which provides a convenient approach for upgrading legacy finite element codes to include embedded boundaries with CutFEM or similar formulations that can retain the same asymptotic accuracy as the underlying boundary conforming finite element method.