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.

翻译：浸入边界法因其在处理复杂几何结构计算中的潜力，在过去数十年间引起了广泛关注。对于这类几何结构，采用边界拟合有限元往往难以实现高效离散。浸入边界法基于笛卡尔网格和针对几何模型定制的数值积分方案，提供了一种简单且全自动的离散方法。因此，几何模型可以独立于网格进行描述，例如通过计算机断层扫描获得的图像数据。这种离散方法的缺点在于网格中某些单元与几何模型之间可能存在极小的重叠区域。这些物理支撑域狭小的"不良切割单元"对非线性和/或动态仿真提出了特殊挑战。本研究聚焦于结构动力学和声学领域的问题，重点探讨采用显式时间推进格式求解此类问题的方法。在此背景下，不良切割单元可能导致临界时间步长过小而无法实际应用。我们研究了隐式-显式时间推进格式以及先前工作中开发的两种稳定化方法的性能，将其作为潜在的解决方案。虽然已有研究从提高临界时间步长的角度评估过这些方法的有效性，但对其数值效率的考量仅局限于每个自由度的精度。本文通过评估达到给定精度所需的计算时间来全面衡量计算效率，该指标不仅取决于自由度数量，还与所选空间离散格式、系统矩阵的稀疏模式以及采用的时间推进格式密切相关。