In this paper, we develop a geometrically flexible technique for computational fluid-structure interaction (FSI). The motivating application is the simulation of tri-leaflet bioprosthetic heart valve function over the complete cardiac cycle. Due to the complex motion of the heart valve leaflets, the fluid domain undergoes large deformations, including changes of topology. The proposed method directly analyzes a spline-based surface representation of the structure by immersing it into a non-boundary-fitted discretization of the surrounding fluid domain. This places our method within an emerging class of computational techniques that aim to capture geometry on non-boundary-fitted analysis meshes. We introduce the term "immersogeometric analysis" to identify this paradigm. The framework starts with an augmented Lagrangian formulation for FSI that enforces kinematic constraints with a combination of Lagrange multipliers and penalty forces. For immersed volumetric objects, we formally eliminate the multiplier field by substituting a fluid-structure interface traction, arriving at Nitsche's method for enforcing Dirichlet boundary conditions on object surfaces. For immersed thin shell structures modeled geometrically as surfaces, the tractions from opposite sides cancel due to the continuity of the background fluid solution space, leaving a penalty method. Application to a bioprosthetic heart valve, where there is a large pressure jump across the leaflets, reveals shortcomings of the penalty approach. To counteract steep pressure gradients through the structure without the conditioning problems that accompany strong penalty forces, we resurrect the Lagrange multiplier field. Further, since the fluid discretization is not tailored to the structure geometry, there is a significant error in the approximation of pressure discontinuities across the shell. This error becomes especially troublesome in residual-based stabilized methods for incompressible flow, leading to problematic compressibility at practical levels of refinement. We modify existing stabilized methods to improve performance. To evaluate the accuracy of the proposed methods, we test them on benchmark problems and compare the results with those of established boundary-fitted techniques. Finally, we simulate the coupling of the bioprosthetic heart valve and the surrounding blood flow under physiological conditions, demonstrating the effectiveness of the proposed techniques in practical computations.

中文翻译：

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流体-结构相互作用的浸入式几何变分框架：在生物人工心脏瓣膜中的应用

在本文中，我们开发了一种用于计算流固耦合 (FSI) 的几何灵活技术。激励应用是在整个心动周期中模拟三叶生物假体心脏瓣膜功能。由于心脏瓣膜瓣叶的复杂运动，流体域会发生大变形，包括拓扑结构的变化。所提出的方法通过将其浸入周围流体域的非边界拟合离散化中来直接分析结构的基于样条的表面表示。这将我们的方法置于一类新兴的计算技术中，这些技术旨在捕获非边界拟合分析网格上的几何形状。我们引入术语“浸入式几何分析”来识别这种范式。该框架从 FSI 的增强拉格朗日公式开始，该公式通过拉格朗日乘数和惩罚力的组合来强制执行运动学约束。对于浸入的体积物体，我们通过替代流固界面牵引力来正式消除乘数场，从而得出 Nitsche 的方法，用于在物体表面上强制执行狄利克雷边界条件。对于在几何上建模为曲面的浸入式薄壳结构，由于背景流体解空间的连续性，来自相对侧的牵引力抵消，留下惩罚方法。应用于生物假体心脏瓣膜，其中在传单上有很大的压力跳跃，揭示了惩罚方法的缺点。为了抵消通过结构的陡峭压力梯度，而不会出现伴随强惩罚力的条件问题，我们重新启动了拉格朗日乘子场。此外，由于流体离散化不是根据结构几何形状定制的，因此壳上压力不连续性的近似值存在重大误差。在用于不可压缩流动的基于残差的稳定方法中，这个错误变得特别麻烦，导致在实际细化水平上存在问题的可压缩性。我们修改现有的稳定方法以提高性能。为了评估所提出方法的准确性，我们在基准问题上对其进行测试，并将结果与​​已建立的边界拟合技术的结果进行比较。最后，