Hexagon-based all-quadrilateral mesh generation with guaranteed angle bounds

• Published in: Computer methods in applied mechanics and engineering Vol. 200; no. 23; pp. 2005 - 2020
• Main Authors: Liang, Xinghua, Zhang, Yongjie
• Format: Journal Article
• Language: English
• Published: Kidlington Elsevier B.V 01.06.2011; Elsevier
• Subjects: All-quadrilateral mesh; Angles (geometry); Applied sciences; Artificial intelligence; Boundaries; Computer science; control theory; systems; Construction; Curvature; Exact sciences and technology; Finite element method; Guaranteed angle bound; Hexagon-tree; Mathematical analysis; Mathematics; Mesh generation; Methods of scientific computing (including symbolic computation, algebraic computation); Narrow region; Numerical analysis. Scientific computation; Pattern recognition. Digital image processing. Computational geometry; Sciences and techniques of general use; Sharp feature; Stiffness matrix; Sharp feature; Guaranteed angle bound; Hexagon-tree; All-quadrilateral mesh; Narrow region; Stiffness matrix; Tree(graph); Quadrilateral finite element; Hexagonal shape; Preservation; Buffer system; Jacobi matrix; Automatic mesh generation; Modeling; Finite element method; Lakes; Mesh method; Polygon; Curvature; Mesh generation
• ISSN: 0045-7825; 1879-2138
• DOI: 10.1016/j.cma.2011.03.002

In this paper, we present a novel hexagon-based mesh generation method which creates all-quadrilateral (all-quad) meshes with guaranteed angle bounds and feature preservation for arbitrary planar domains. Given any planar curves, an adaptive hexagon-tree structure is constructed by using the curvature of the boundaries and narrow regions. Then a buffer zone and a hexagonal core mesh are created by removing elements outside or around the boundary. To guarantee the mesh quality, boundary edges of the core mesh are adjusted to improve their formed angles facing the boundary, and two layers of quad elements are inserted in the buffer zone. For any curve with sharp features, a corresponding smooth curve is firstly constructed and meshed, and then another layer of elements is inserted to match the smooth curve with the original one. It is proved that for any planar smooth curve all the element angles are within [60° − ε, 120° + ε] ( ε ⩽ 5°). We also prove that the scaled Jacobians defined by two edge vectors are in the range of [ sin (60° − ε), sin 90°], or [0.82, 1.0]. The same angle range can be guaranteed for curves with sharp features, with the exception of small angles in the input curve. Furthermore, an approach is introduced to match the generated interior and exterior meshes with a relaxed angle range, [30°, 150°]. We have applied our algorithm to a set of complicated geometries, including the China map, the Lake Superior map, and a three-component air foil with sharp features. In addition, all the elements in the final mesh are grouped into five types, and most elements only need a few flops to construct the stiffness matrix for finite element analysis. This will significantly reduce the computational time and the required memory during the stiffness matrix construction.