用于隐式浸入边界方程多重网格预处理的耦合感知万卡光滑化方法
Coupling-Aware Vanka Smoothing for Multigrid Preconditioning of the Implicit Immersed Boundary Equations
AI总结：
本文提出耦合感知万卡光滑化策略，实现隐式浸入边界方程的有效多重网格预处理，经测试其收敛效率高且网格适应性好，是首个适用于时变隐式IB公式的鲁棒多重网格策略。
AI中文摘要：
浸入边界（IB）方法分别采用结构力学和流体动力学的自然拉格朗日及欧拉公式来建模流固相互作用，但IB力的显式时间离散会对时间步长施加与刚度相关的上限。将这些力隐式处理可消除该限制，但所得耦合线性系统会随结构刚度增大而愈发难以求解。通过代数消去拉格朗日自由度可得到简化的欧拉速度-压力IB系统。本文引入一种耦合感知万卡（CAV）光滑化策略，以实现该系统的有效多重网格预处理。CAV块由标准压力中心万卡块的并集构建，欧拉弹性矩阵的图决定哪些块需合并。在网格细化时，CAV块大小保持有界，因此每个多重网格循环的计算成本随欧拉自由度数量线性增长。采用目标点、膜和梁力定律的测试表明，经CAV预处理的FGMRES可在9至15次迭代内将相对残差降低10个数量级，且在网格细化时几乎无增长。在模拟柔性纤维绕流的非线性基准测试中，当欧拉网格从32×32单元细化至256×256单元时，每次牛顿求解的平均FGMRES迭代次数仅从8.6增至9.5。据所知，CAV是首个用于时变隐式IB公式的鲁棒多重网格策略。
英文摘要：
The immersed boundary (IB) method models fluid--structure interaction using the natural Lagrangian and Eulerian formulations of structural mechanics and fluid dynamics, respectively, but explicit time discretization of the IB force imposes a stiffness-dependent upper bound on the time-step size. Treating these forces implicitly removes this restriction, but the resulting coupled linear systems become increasingly difficult to solve as the structural stiffness increases. Algebraically eliminating the Lagrangian degrees of freedom yields a reduced Eulerian velocity--pressure IB system. Here we introduce a coupling-aware Vanka (CAV) smoothing strategy to enable effective multigrid preconditioning of this system. CAV patches are built as unions of standard pressure-centered Vanka patches, with the graph of the Eulerian elasticity matrix determining which patches are combined. Under grid refinement, CAV patch sizes remain bounded, and the computational cost of each multigrid cycle thereby grows linearly with the number of Eulerian degrees of freedom. Tests using target-point, membrane, and beam force laws show that CAV-preconditioned FGMRES reduces the relative residual by ten orders of magnitude in $9--15$ iterations, with little growth under grid refinement. In a nonlinear benchmark modeling flow past a flexible fiber, the average number of FGMRES iterations per Newton solve increases only from $8.6$ to $9.5$ as the Eulerian grid is refined from $32\times32$ to $256\times256$ cells. To our knowledge, CAV provides the first robust multigrid strategy for time-dependent implicit IB formulations.
