Wells David, Vadala-Roth Ben, Lee Jae H, Griffith Boyce E
Department of Mathematics, University of North Carolina, Chapel Hill, NC, USA.
U.S. Army Corps of Engineers, Engineer Research and Development Center, Coastal, and Hydraulic Laboratory, Vicksburg, MS, USA.
J Comput Phys. 2023 Mar 15;477. doi: 10.1016/j.jcp.2022.111890. Epub 2023 Jan 13.
The immersed finite element-finite difference (IFED) method is a computational approach to modeling interactions between a fluid and an immersed structure. The IFED method uses a finite element (FE) method to approximate the stresses, forces, and structural deformations on a and a finite difference (FD) method to approximate the momentum and enforce incompressibility of the entire fluid-structure system on a The fundamental approach used by this method follows the immersed boundary framework for modeling fluid-structure interaction (FSI), in which a force spreading operator prolongs structural forces to a Cartesian grid, and a velocity interpolation operator restricts a velocity field defined on that grid back onto the structural mesh. With an FE structural mechanics framework, force spreading first requires that the force itself be projected onto the finite element space. Similarly, velocity interpolation requires projecting velocity data onto the FE basis functions. Consequently, evaluating either coupling operator requires solving a matrix equation at every time step. Mass lumping, in which the projection matrices are replaced by diagonal approximations, has the potential to accelerate this method considerably. This paper provides both numerical and computational analyses of the effects of this replacement for evaluating the force projection and for the IFED coupling operators. Constructing the coupling operators also requires determining the locations on the structure mesh where the forces and velocities are sampled. Here we show that sampling the forces and velocities at the nodes of the structural mesh is equivalent to using lumped mass matrices in the IFED coupling operators. A key theoretical result of our analysis is that if both of these approaches are used together, the IFED method permits the use of lumped mass matrices derived from nodal quadrature rules for any standard interpolatory element. This is different from standard FE methods, which require specialized treatments to accommodate mass lumping with higher-order shape functions. Our theoretical results are confirmed by numerical benchmarks, including standard solid mechanics tests and examination of a dynamic model of a bioprosthetic heart valve.
浸入式有限元-有限差分（IFED）方法是一种用于模拟流体与浸入式结构之间相互作用的计算方法。IFED方法使用有限元（FE）方法来近似结构上的应力、力和结构变形，使用有限差分（FD）方法来近似动量并确保整个流固系统的不可压缩性。该方法所采用的基本方法遵循用于模拟流固相互作用（FSI）的浸入边界框架，其中力扩散算子将结构力扩展到笛卡尔网格上，速度插值算子将定义在该网格上的速度场限制回结构网格上。在有限元结构力学框架下，力扩散首先要求将力本身投影到有限元空间。同样，速度插值要求将速度数据投影到有限元基函数上。因此，在每个时间步评估任何一个耦合算子都需要求解一个矩阵方程。质量集中法，即将投影矩阵替换为对角近似，有可能显著加速该方法。本文提供了关于这种替换对评估力投影和IFED耦合算子的影响的数值和计算分析。构建耦合算子还需要确定在结构网格上采样力和速度的位置。在这里我们表明，在结构网格的节点处采样力和速度等同于在IFED耦合算子中使用集中质量矩阵。我们分析的一个关键理论结果是，如果同时使用这两种方法，IFED方法允许对任何标准插值单元使用从节点求积规则导出的集中质量矩阵。这与标准有限元方法不同，标准有限元方法需要专门处理以适应高阶形状函数的质量集中。我们的理论结果通过数值基准得到了证实，包括标准固体力学测试和对生物人工心脏瓣膜动态模型的研究。
