Anisotropic multi-level hp-refinement for quadrilateral and triangular meshes

Abstract

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 h 2 -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. • Generalization of multi-level hp-method to anisotropic refinement of triangles and quadrilaterals. • Dead-lock free directional refinement without limitations due to hanging nodes. • Formulation of an anisotropic split of triangles that is free of element distortions. • Automatically driven anisotropic refinement of unstructured, mixed meshes.