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.

胸膈是驱动人类呼吸循环的肌肉。使用模拟线性弹性的偏微分方程 (PDE) 系统，我们计算收缩状态下隔膜二维横截面的位移和应力。边界数据由位移和牵引条件的混合组成。如果这些都是按原样强加的，并且条件不兼容，则会导致解决方案的平滑度降低。因此，首先使用最小二乘径向基函数生成的有限差分 (RBF-FD) 框架对边界数据进行平滑处理。然后将边界条件重新表述为具有平滑系数的 Robin 边界条件。基于从计算机断层扫描 (CT) 扫描切片获得的数据，相同的框架也用于近似隔膜横截面的边界曲线。为了解决偏微分方程，我们采用了非拟合最小二乘 RBF-FD 方法。这使得处理薄且非凸的隔膜的几何形状变得更容易。我们用数值表明，我们的解决方案以高阶收敛于在精细网格上评估的有限元解决方案。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。这是薄且非凸的。我们用数值表明，我们的解决方案以高阶收敛于在精细网格上评估的有限元解决方案。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。这是薄且非凸的。我们用数值表明，我们的解决方案以高阶收敛于在精细网格上评估的有限元解决方案。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。