Partitioned solutions to fluid-structure interaction problems often employ a Dirichlet-Neumann decomposition, where the fluid equations are solved subject to Dirichlet boundary conditions on velocity from the structure, and the structure equations are solved subject to forces from the fluid. In some scenarios, such as an elastic balloon filling with air, an incompressible fluid domain may have pure Dirichlet boundary conditions, leading to two related issues which have been described as the incompressibility dilemma. First, the Dirichlet boundary conditions must satisfy the incompressibility constraint for a solution to exist. However, the structure solver is unaware of this constraint and may supply the fluid solver with incompatible velocities. Second, the constant fluid pressure mode lies in the null space of the fluid pressure solver, but must be determined to apply to the structure. Previously proposed solutions to the incompressibility dilemma have included modifying the fluid solver, the structure solver, or the Dirichlet-Neumann coupling interface between them. In this paper, we present a boundary pressure projection method which alleviates the incompatibility while maintaining the Dirichlet-Neumann structure of the decomposition and without modification of the fluid or solid solvers. Our method takes incompatible velocities from the structure solver and projects them to be compatible while in the process computing the constant pressure modes for the Dirichlet regions. The compatible velocities are then used as Dirichlet boundary conditions for the fluid while the constant pressure modes are added to the fluid-solver-computed pressures to be applied to the structure. The intermediate computation performed in the boundary pressure projection method is small, with the number of unknowns equal to the number of Dirichlet regions. We show that the boundary pressure projection method can be used to solve a variety of scenarios including inflation of an elastic balloon and the action of a hydraulic press. We also demonstrate the method on multiple coupled Dirichlet regions. The method offers a simple approach to overcome the incompressibility dilemma using a small intermediate computation that requires no additional knowledge of the black box fluid and solid solvers.

流体-结构相互作用问题的分区解决方案通常采用Dirichlet-Neumann分解，其中流体方程受Dirichlet边界条件的约束，其速度取决于结构的速度，结构方程则受流体力的影响。在某些情况下，例如充满空气的弹性气球，不可压缩的流体域可能具有纯Dirichlet边界条件，从而导致两个相关的问题，这些问题被描述为不可压缩的困境。首先，狄利克雷边界条件必须满足不可压缩性约束才能存在解。但是，结构求解器没有意识到这一限制，可能会为流体求解器提供不兼容的速度。其次，恒定流体压力模式位于流体压力求解器的零空间内，但必须确定要适用于该结构。先前针对不可压缩性难题提出的解决方案包括修改流体求解器，结构求解器或它们之间的Dirichlet-Neumann耦合接口。在本文中，我们提出了一种边界压力投影方法，该方法可减轻不相容性，同时保持分解的Dirichlet-Neumann结构且无需修改流体或固体求解器。我们的方法从结构求解器获取不兼容的速度，并将它们投影为兼容的，同时在计算Dirichlet区域的恒定压力模式的过程中。然后将兼容速度用作流体的Dirichlet边界条件，同时将恒压模式添加到要应用到结构的流体溶解器计算压力中。在边界压力投影方法中执行的中间计算量很小，未知数等于Dirichlet区域数。我们表明，边界压力投影方法可用于解决各种情况，包括弹性气球的充气和液压机的作用。我们还演示了在多个耦合Dirichlet区域上的方法。该方法提供了一种简单的方法，可使用小的中间计算来克服不可压缩性难题，而无需额外了解黑匣子流体和固体求解器。我们表明，边界压力投影方法可用于解决各种情况，包括弹性气球的充气和液压机的作用。我们还演示了在多个耦合Dirichlet区域上的方法。该方法提供了一种简单的方法，可使用小的中间计算来克服不可压缩性难题，而无需额外了解黑匣子流体和固体求解器。我们表明，边界压力投影方法可用于解决各种情况，包括弹性气球的充气和液压机的作用。我们还演示了在多个耦合Dirichlet区域上的方法。该方法提供了一种简单的方法，可使用小的中间计算来克服不可压缩性难题，而无需额外了解黑匣子流体和固体求解器。