The immersed finite element-finite difference (IFED) method is a computational approach to modeling interactions between a fluid and an immersed structure. This method uses a finite element (FE) method to approximate the stresses and forces on a structural mesh and a finite difference (FD) method to approximate the momentum of the entire fluid-structure system on a Cartesian grid. 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. Force spreading and velocity interpolation both require projecting data onto the finite element space. 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. 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 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.

翻译：浸入有限元-有限差分（IFED）方法是一种计算流体与浸入结构相互作用的数值方法。该方法采用有限元（FE）方法近似结构网格上的应力与力，并采用有限差分（FD）方法在笛卡尔网格上近似整个流固耦合系统的动量。该方法的基本框架遵循浸入边界法对流体-结构相互作用（FSI）建模的思路，其中力延拓算子将结构力传递至笛卡尔网格，速度插值算子将定义在该网格上的速度场约束回结构网格。力延拓与速度插值均需将数据投影至有限元空间。因此，每步时间推进中求解耦合算子均需解算矩阵方程。质量集中法——即用对角近似替代投影矩阵——有望显著提升该方法效率。构建耦合算子还需确定结构网格上力和速度的采样位置。本文证明，在结构网格节点处采样力与速度等价于在耦合算子中采用集中质量矩阵。分析的一个核心理论结果是：若同时使用这两种方法，IFED方法允许对任意标准插值单元采用由节点求积法则导出的集中质量矩阵。这与标准有限元方法不同——后者需要特殊处理才能兼容高阶形函数的质量集中。数值基准测试验证了理论结果，涵盖标准固体力学测试及生物瓣膜动力学模型检验。