Finite Elements in Analysis and Design ( IF 3.1 ) Pub Date : 2022-01-25 , DOI: 10.1016/j.finel.2021.103700 Nils Zander 1 , Hadrien Bériot 2 , Claus Hoff 1 , Petr Kodl 1 , Leszek Demkowicz 3
This paper extends the multi-level hp-approach—previously introduced for the isotropic refinement of quadrilateral and hexahedral elements—to the anisotropic refinement of quadrilateral and triangular meshes. To this end, this work first introduces anisotropic split operators for the hierarchical refine-by-superposition concept. For quadrilateral cells, the well-known anisotropic -split is used. For the anisotropic refinement of triangular cells, a novel split along the barycentric mid-lines is introduced, which minimizes the number of new cells without distorting their shape. Second, it is shown that the isotropic multi-level hp-approach of handling hanging nodes is not directly applicable in the case of directional refinement. For this reason, this work analyses the requirements of a valid fe-basis on anisotropically refined, irregular multi-level meshes and by this generalizes the isotropic multi-level hp-concept. On this basis, an algorithmic realization is derived demonstrating that the applied refine-by-superposition idea carries the implementational simplicity and the native support of arbitrary irregular meshes to anisotropic refinements. A systematic study of benchmarks featuring vertex-, edge-, and edge-vertex singularities as well as boundary layers shows that these advantages come neither at the expense of the approximation quality, nor the sparsity or conditioning of the final equation system. Further, an ad hoc edge-based indication scheme is introduced that reliably guides the place and the direction of the anisotropic refinement process even on unstructured meshes. The presented results qualify the proposed approach as an automatic, anisotropic multi-level hp-refinement method for quadrilateral and triangular meshes, coming at a marginal implementational complexity and without any restrictions due to arbitrary hanging nodes and the associated dead-lock problems.
中文翻译:
四边形和三角形网格的各向异性多级 hp 细化
本文将多级hp方法(之前介绍用于四边形和六面体单元的各向同性细化)扩展到四边形和三角形网格的各向异性细化。为此,这项工作首先为分层提炼叠加概念引入了各向异性拆分算子。对于四边形单元,众所周知的各向异性-split 被使用。对于三角形单元的各向异性细化,引入了一种沿重心中线的新分裂,它最大限度地减少了新单元的数量而不扭曲它们的形状。其次,表明各向同性多级hp处理悬挂节点的方法在定向细化的情况下并不直接适用。出于这个原因,这项工作分析了有效的 fe 基础对各向异性细化、不规则多级网格的要求,并由此概括了各向同性多级hp-概念。在此基础上,推导出算法实现,证明所应用的叠加细化思想具有实现的简单性和对任意不规则网格对各向异性细化的原生支持。对具有顶点、边和边顶点奇点以及边界层的基准的系统研究表明,这些优势既不以牺牲近似质量为代价,也不以最终方程系统的稀疏性或条件为代价。此外,引入了一种特殊的基于边缘的指示方案,即使在非结构化网格上也能可靠地引导各向异性细化过程的位置和方向。所呈现的结果使所提出的方法成为一种自动的、各向异性的多级hp细化方法对于四边形和三角形网格,由于任意悬挂节点和相关的死锁问题,实现的复杂性很小,并且没有任何限制。