We study, both theoretically and numerically, the equilibrium of a hinged rigid leaflet with an attached rotational spring, immersed in a stationary incompressible fluid within a rigid channel. Through a careful investigation of the properties of the domain functional describing the angular momentum exerted by the fluid on the leaflet (which depends on both the leaflet angular position and its thickness), we identify sufficient conditions on the spring stiffness function for the existence (and uniqueness) of equilibrium positions. This study resorts to techniques from shape differential calculus. We propose a numerical technique that exploits the mesh flexibility of the Virtual Element Method (VEM). A (polygonal) computational mesh is generated by cutting a fixed background grid with the leaflet geometry, and the problem is then solved with stable VEM Stokes elements of degrees 1 and 2 combined with a bisection algorithm. We prove quasi-optimal error estimates and present a large array of numerical experiments to document the accuracy and robustness with respect to degenerate geometry of the proposed methodology.

中文翻译：

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虚拟单元法对浸入式刚性瓣叶的平衡分析

我们从理论上和数值上研究了铰接刚性瓣叶与连接的旋转弹簧的平衡，该弹簧浸入刚性通道内的静止不可压缩流体中。通过仔细研究描述流体在小叶上施加的角动量的域泛函的性质（这取决于小叶角位置及其厚度），我们确定了弹簧刚度函数存在的充分条件（和唯一性）的平衡位置。本研究采用形状微积分技术。我们提出了一种利用虚拟元素方法 (VEM) 的网格灵活性的数值技术。（多边形）计算网格是通过使用传单几何切割固定的背景网格来生成的，1和2结合二分算法。我们证明了准最优误差估计，并提出了大量的数值实验，以记录所提出方法的退化几何的准确性和鲁棒性。