一种非拟合径向基函数生成的有限差分法在膈肌模拟中的应用
An unfitted radial basis function generated finite difference method applied to thoracic diaphragm simulations
摘要：
胸隔膜是驱动人类呼吸循环的肌肉。使用一个偏微分方程组（PDE）模型的线性弹性，我们计算的位移和应力在二维截面的隔膜在其收缩状态。边界数据由位移和牵引条件的混合组成。如果这些是强加的，并且条件不兼容，这会导致解决方案的平滑性降低。因此，边界数据首先使用最小二乘径向基函数生成的有限差分（RBF-FD）框架进行平滑。然后将边界条件转化为具有光滑系数的Robin边界条件。相同的框架也用于基于从计算机断层摄影（CT）扫描的切片获得的数据来近似隔膜截面的边界曲线。为了解决偏微分方程，我们采用非拟合最小二乘径向基函数-FD方法。这使得更容易处理薄且非凸的隔膜的几何形状。我们数值计算表明，我们的解决方案收敛与高阶对一个有限元解决方案上的细网格评估。通过这个简化的数值模型，我们还可以在接近更复杂的三维模型之前深入了解与隔膜几何形状和边界条件相关的挑战。
英文摘要：
The thoracic diaphragm is the muscle that drives the respiratory cycle of a human being. Using a system of partial differential equations (PDEs) that models linear elasticity we compute displacements and stresses in a two-dimensional cross section of the diaphragm in its contracted state. The boundary data consists of a mix of displacement and traction conditions. If these are imposed as they are, and the conditions are not compatible, this leads to reduced smoothness of the solution. Therefore, the boundary data is first smoothed using the least-squares radial basis function generated finite difference (RBF-FD) framework. Then the boundary conditions are reformulated as a Robin boundary condition with smooth coefficients. The same framework is also used to approximate the boundary curve of the diaphragm cross section based on data obtained from a slice of a computed tomography (CT) scan. To solve the PDE we employ the unfitted least-squares RBF-FD method. This makes it easier to handle the geometry of the diaphragm, which is thin and non-convex. We show numerically that our solution converges with high-order towards a finite element solution evaluated on a fine grid. Through this simplified numerical model we also gain an insight into the challenges associated with the diaphragm geometry and the boundary conditions before approaching a more complex three-dimensional model.
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