Surface Mesh Generation for Generalized FEM-Techniques

已建立的(曲面)区域离散化方法通常旨在生成由三角形或四边形(或两者的混合)组成的下层曲面的镶嵌，因为这些方法与标准的有限元基函数兼容。然而，将在本研究组中研究的广义离散化技术允许相对于单个单元的(局部)形状以及整体网格的(全局)结构具有更大的灵活性。例如，由于SBFEM和VEM方法可以处理任意的凸形或星形多边形单元，因此它们在单元的调整、它们与相关结构的对准以及它们的(各向异性)大小方面提供了更大的灵活性。在这个子项目中，我们将系统地探索自由曲面网格生成中的这些额外自由度，并开发新的技术来高效、灵活地生成具有可控属性的多边形单元分解。我们的目标是开发高效和灵活的网格生成技术，考虑到特定的限制，并利用广义离散化方法增加的自由度。特别是，我们将系统地探索诸如(1)不同大小、各向异性和形状的单元的特性，(2)通过将现有节点移动到需要更精细分辨率的区域来实现无需局部细化的网格自适应性，以及(3)强制壳的无条件(“内置”)单元平面性。通过开发能够有效更新网格结构或顶点分布的灵活网格方案，计算分析过程的模拟和网格划分阶段可以交错和集成以获得最佳性能。

Established approaches for (surface) domain discretization usually aim to generate tessellations of the underlying surface consisting of triangles or quadrilaterals (or mixtures of both) since these are compatible with standard FEM basis functions. The generalized discretization techniques that will be investigated in this research group, however, allow for more flexibility with respect to the (local) shape of the individual cells as well as the (global) structure of the overall mesh. Since, e.g., SBFEM and VEM methods can deal with arbitrary convex or star-shaped polygonal cells, they provide more flexibility concerning the adjustment of the cells, their alignment to relevant structures, as well as their (anisotropic) sizing. In this sub-project we will systematically explore these additional degrees of freedom in mesh generation for freeform surfaces and develop novel techniques for the efficient and flexible generation of polygonal cell decompositions with controllable properties. We aim at developing efficient and flexible mesh generation techniques that take into account the specific restrictions and exploit the added freedom of generalized discretization approaches. In particular, we will systematically explore properties such as (1) varying size, anisotropy, and shape of cells, (2) mesh adaptivity without local refinement by shifting existing nodes into regions that require finer resolution and (3) enforcing unconditional ("built-in") cell planarity for shells. By developing flexible meshing schemes that enable the efficient update of the mesh structure or vertex distribution, the simulation and meshing stages of a computational analysis process can be interleaved and integrated for optimal performance.