The volume penalization (VP) or the Brinkman penalization (BP) method is a diffuse interface method for simulating multiphase fluid-structure interaction (FSI) problems in ocean engineering and/or phase change problems in thermal sciences. The method relies on a penalty factor (which is inversely related to body's permeability $\kappa$) that must be large to enforce rigid body velocity in the solid domain. When the penalty factor is large, the discrete system of equations becomes stiff and difficult to solve numerically. In this paper, we propose a projection method-based preconditioning strategy for solving volume penalized (VP) incompressible and low-Mach Navier-Stokes equations. The projection preconditioner enables the monolithic solution of the coupled velocity-pressure system in both single phase and multiphase flow settings. In this approach, the penalty force is treated implicitly, which is allowed to take arbitrary large values without affecting the solver's convergence rate or causing numerical stiffness/instability. It is made possible by including the penalty term in the pressure Poisson equation. Solver scalability under grid refinement is demonstrated. A manufactured solution in a single phase setting is used to determine the spatial accuracy of the penalized solution. Second-order pointwise accuracy is achieved for both velocity and pressure solutions. Two multiphase fluid-structure interaction (FSI) problems from the ocean engineering literature are also simulated to evaluate the solver's robustness and performance. The proposed solver allows us to investigate the effect of $\kappa$ on the motion of the contact line over the surface of the immersed body. It also allows us to investigate the dynamics of the free surface of a solidifying metal

翻译：体积惩罚（VP）或布林克曼惩罚（BP）方法是一种模拟海洋工程中多相流固耦合（FSI）问题和/或热科学中相变问题的扩散界面方法。该方法依赖于惩罚因子（与物体渗透率$\kappa$成反比），该因子必须足够大以在固体域中强制执行刚体速度。当惩罚因子较大时，离散方程组变得刚性且难以数值求解。本文提出了一种基于投影法的预处理策略，用于求解体积惩罚（VP）不可压缩和低马赫纳维-斯托克斯方程。该投影预处理器能够在单相流和多相流设置中实现耦合速度-压力系统的整体求解。在该方法中，惩罚力被隐式处理，允许其取任意大值而不会影响求解器的收敛速率或导致数值刚度/不稳定。这通过将惩罚项纳入压力泊松方程得以实现。展示了求解器在网格细化下的可扩展性。利用单相设置中的制造解确定了惩罚解的空间精度，速度和压力解均达到了二阶逐点精度。同时模拟了海洋工程文献中的两个多相流固耦合（FSI）问题，以评估求解器的鲁棒性和性能。所提出的求解器使我们能够研究$\kappa$对浸没物体表面接触线运动的影响，并能够探究凝固金属自由表面的动力学行为。