Automated mesh generation on parametric surface

Abstract Curved shapes designed by CAD system are usually defined by parametric surfaces, for example, NURBS. This page proposes a mesh generation method for parametric surfaces. This method tightly packs elliptic bubbles in parameter space and generates triangular mesh in parametric space by using anisotropic Delaunay triangulation. This method can generate a mesh satisfying the requirements listed below. meshing for trimmed surface meshing for complex surface shape well-graded meshing well-shaped meshing Background Related works are listed below. Quad-tree method This method subdivides parameter space into four sub-domains recursively. But if given surface is trimmed surface, distorted elements may be generated on the trimmed boundary. The distorted elements cause the error of finite element analysis. Delaunay triangulation in parameter space Delaunay triangulation in parameter space does not generate a well-shaped mesh on surface. If parameter distribution of given surface is extremely non-uniform, dostorted elements are generated on surface. Delaunay-like triangulation on surface Delaunay triangulation selects one of two triangle pairs composed of four points on a plane, which pair satisfies the confition that circumcircle of three points does not include the other point inside. This Delaunay-like method substitutes circumsphere of three points on surface for circumcircle. But if a shape where two points parametrically located further are located close in real space is given, this method fails triangulation. Automated mesh generation on parametric surface The overview of the entire algorithm is same as original bubble mesh method . But three terms listed below are different from the original one. 1) Caluculate elliptic bubbles in parameter space from circular bubbles on surface This method generates elliptic bubbles in parameter space from circular bubbles given on surface. Suppose two coordinate systems A and B on a tangent plane of surface, where the coordinate system A is defined by orthogonal unit vectors and the coordinate system B is defined by tangent vectos on surface. See figure. When a circle on surface is defined by coordinate system A, a ellipse in parameter space is calculated by using tensor transformation from coordinate system A to B. 2) Packing elliptic bubbles in parameter space This method tightly packs elliptic bubbles in parameter space instead of packing circle on surface. Same algorithm as anisotropic mesh generation is applied here. 3) Triangulation in parameter space This method generates triangular mesh in parameter space insted of generating mesh on surface. Same algorithm as anisotropic mesh generation is also applied here. See the animation (128KB) showing bubble conversing process in parametric space and in real space. Figures on Left side show a mesh in real space and figures on right side show a mesh in parameter space. Results of mesh generation on surface Packed bubbles on surface Triangular mesh on surface Packed bubbles in parametric space Triangular mesh in parametric space To the top page of meshing project

Last modified 30 June 1998