On the efficiency of explicit and semi-explicit immersed boundary finite element methods for wave propagation problems

TL;DR

This paper evaluates the efficiency of explicit and semi-explicit immersed boundary finite element methods for wave propagation, focusing on their computational performance and stability in complex geometries with badly cut elements.

Contribution

It compares implicit-explicit schemes and stabilization methods in terms of computational time and stability for wave problems with complex geometries.

Findings

Implicit-explicit schemes improve critical time step sizes.

Stabilization methods enhance numerical stability.

Efficiency depends on discretization and matrix sparsity.

Abstract

Immersed boundary methods have attracted substantial interest in the last decades due to their potential for computations involving complex geometries. Often these cannot be efficiently discretized using boundary-fitted finite elements. Immersed boundary methods provide a simple and fully automatic discretization based on Cartesian grids and tailored quadrature schemes that account for the geometric model. It can thus be described independently of the grid, e.g., by image data obtained from computed tomography scans. The drawback of such a discretization lies in the potentially small overlap between certain elements in the grid and the geometry. These badly cut elements with small physical support pose a particular challenge for nonlinear and/or dynamic simulations. In this work, we focus on problems in structural dynamics and acoustics and concentrate on solving them with explicit time-marching schemes. In this context, badly cut elements can lead to unfeasibly small critical time step sizes. We investigate the performance of implicit-explicit time marching schemes and two stabilization methods developed in previous works as potential remedies. While these have been studied before with regard to their effectiveness in increasing the critical time step size, their numerical efficiency has only been considered in terms of accuracy per degree of freedom. In this paper, we evaluate the computation time required for a given accuracy, which depends not only on the number of degrees of freedom but also on the selected spatial discretization, the sparsity patterns of the system matrices, and the employed time-marching scheme.