A higher-order fictitious domain method (FDM) for Reissner-Mindlin shells is proposed which uses a three-dimensional background mesh for the discretization. The midsurface of the shell is immersed into the higher-order background mesh and the geometry is implied by level-set functions. The mechanical model is based on the Tangential Differential Calculus (TDC) which extends the classical models based on curvilinear coordinates to implicit geometries. The shell model is described by PDEs on manifolds and the resulting FDM may typically be called Trace FEM. The three standard key aspects of FDMs have to be addressed in the Trace FEM as well to allow for a higher-order accurate method: (i) numerical integration in the cut background elements, (ii) stabilization of awkward cut situations and elimination of linear dependencies, and (iii) enforcement of boundary conditions using Nitsche's method. The numerical results confirm that higher-order accurate results are enabled by the proposed method provided that the solutions are sufficiently smooth.

中文翻译：

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壳的高阶痕量有限元方法

提出了一种用于 Reissner-Mindlin 壳的高阶虚拟域方法 (FDM)，该方法使用三维背景网格进行离散化。壳的中面浸入高阶背景网格中，几何由水平集函数隐含。机械模型基于切线微分 (TDC)，它将基于曲线坐标的经典模型扩展到隐式几何。壳模型由流形上的偏微分方程描述，生成的 FDM 通常称为 Trace FEM。FDM 的三个标准关键方面也必须在 Trace FEM 中解决，以允许使用更高阶的精确方法：(i) 切割背景元素中的数值积分，(ii) 尴尬的切割情况的稳定性和线性的消除依赖，(iii) 使用 Nitsche 方法强制执行边界条件。数值结果证实，只要解足够平滑，所提出的方法就可以实现更高阶的准确结果。