An Adaptive Parametric Surface Mesh Generation Method Guided by Curvatures

This work presents an adaptive mesh generation strategy for parametric surfaces. The proposed strategy is controlled by curvatures and the error measured between the analytical and discrete curvatures guides the adaptive process. The analytical curvature is a mathematical representation that models the domain, whereas the discrete curvature is an approximation of that curvature and depends directly on the used mesh. The proposed strategy presents the following aspects: it is able to refine and coarsen regions of the mesh; it considers the local error measures to ensure good global quality; it ensures good transition of the mesh and it deals with any type of parametric surfaces since it works in the parametric space.

[1]  P. George,et al.  Parametric surface meshing using a combined advancing-front generalized Delaunay approach , 2000 .

[2]  Guoliang Xu,et al.  Convergence analysis of a discretization scheme for Gaussian curvature over triangular surfaces , 2006, Comput. Aided Geom. Des..

[3]  Ehud Rivlin,et al.  A comparison of Gaussian and mean curvature estimation methods on triangular meshes of range image data , 2007, Comput. Vis. Image Underst..

[4]  Dereck S. Meek,et al.  On surface normal and Gaussian curvature approximations given data sampled from a smooth surface , 2000, Comput. Aided Geom. Des..

[5]  Xiangmin Jiao,et al.  An analysis and comparison of parameterization-based computation of differential quantities for discrete surfaces , 2009, Comput. Aided Geom. Des..

[6]  S. H. Lo,et al.  Mesh generation over curved surfaces with explicit control on discretization error , 1998 .

[7]  Vincent Borrelli,et al.  Error term in pointwise approximation of the curvature of a curve , 2010, Comput. Aided Geom. Des..

[8]  S. H. Lo,et al.  Finite element mesh generation over analytical curved surfaces , 1996 .

[9]  Steve Oudot,et al.  Provably good sampling and meshing of surfaces , 2005, Graph. Model..

[10]  W. Jeong,et al.  LOD Generation with Discrete Curvature Error Metric , 2001 .

[11]  D. Levin,et al.  Optimizing 3D triangulations using discrete curvature analysis , 2001 .

[12]  Mark Meyer,et al.  Discrete Differential-Geometry Operators for Triangulated 2-Manifolds , 2002, VisMath.

[13]  Tamal K. Dey,et al.  Sampling and Meshing a Surface with Guaranteed Topology and Geometry , 2007, SIAM J. Comput..

[14]  David F. Rogers,et al.  Mathematical elements for computer graphics , 1976 .

[15]  Steven J. Owen,et al.  Advancing Front Surface Mesh Generation in Parametric Space Using a Riemannian Surface Definition , 1998, IMR.

[16]  K. R. Grice,et al.  Robust, geometrically based, automatic two‐dimensional mesh generation , 1987 .

[17]  Luiz Fernando Martha,et al.  Surface mesh regeneration considering curvatures , 2008, Engineering with Computers.

[18]  Geoff Wyvill,et al.  Towards an understanding of surfaces through polygonization , 1998, Proceedings. Computer Graphics International (Cat. No.98EX149).

[19]  Luiz Fernando Martha,et al.  Mesh Generation on High-Curvature Surfaces Based on a Background Quadtree Structure , 2002, IMR.