A unified method for hybrid subdivision surface design using geometric partial differential equations

Surface subdivision gains popularity in surface design owing to its flexible applications for geometry with complicated topology and simple computational scheme, and the geometric partial differential equation (GPDE) method is an advanced technology for constructing high-quality smooth surfaces. In this paper, we composite these two ingredients to form a unified method for freeform surface design. We choose the mean curvature flow and Willmore flow as our driven GPDEs, and the finite element method coupled with a hybrid Loop and Catmull-Clark subdivision algorithm as the numerical simulation method. This research presents a novel technique to evaluate the finite element basis functions and the first attempt for constructing the GPDE subdivision surface with hybrid control meshes consisting of triangles and quadrilaterals. Numerical experiments show that the construction method is efficient and robust, yielding high-quality hybrid subdivision surfaces.

[1]  M. Sabin,et al.  Behaviour of recursive division surfaces near extraordinary points , 1978 .

[2]  Guillermo Sapiro,et al.  Area and Length Preserving Geometric Invariant Scale-Spaces , 1995, IEEE Trans. Pattern Anal. Mach. Intell..

[3]  M. Ortiz,et al.  Subdivision surfaces: a new paradigm for thin‐shell finite‐element analysis , 2000 .

[4]  Guoliang Xu,et al.  Construction of quadrilateral subdivision surfaces with prescribed G1 boundary conditions , 2011, Comput. Aided Des..

[5]  Charles T. Loop,et al.  Quad/Triangle Subdivision , 2003, Comput. Graph. Forum.

[6]  R. Bryant.  A duality theorem for Willmore surfaces , 1984 .

[7]  Yousef Saad,et al.  Iterative methods for sparse linear systems , 2003 .

[8]  Guoliang Xu,et al.  G2 surface modeling using minimal mean-curvature-variation flow , 2007, Comput. Aided Des..

[9]  George Merrill Chaikin,et al.  An algorithm for high-speed curve generation , 1974, Comput. Graph. Image Process..

[10]  Jos Stam,et al.  Exact evaluation of Catmull-Clark subdivision surfaces at arbitrary parameter values , 1998, SIGGRAPH.

[11]  Bobby Bodenheimer,et al.  Synthesis and evaluation of linear motion transitions , 2008, TOGS.

[12]  Joe Warren,et al.  Subdivision Methods for Geometric Design: A Constructive Approach , 2001 .

[13]  W. Mullins.  Two‐Dimensional Motion of Idealized Grain Boundaries , 1956 .

[14]  Olga Sorkine-Hornung,et al.  Mixed Finite Elements for Variational Surface Modeling , 2010, Comput. Graph. Forum.

[15]  J. Escher,et al.  The volume preserving mean curvature flow near spheres , 1998 .

[16]  Guoliang Xu,et al.  Construction of minimal subdivision surface with a given boundary , 2011, Comput. Aided Des..

[17]  Wang Yuguo,et al.  Piecewise Smooth Surfaces Reconstruction for Finite Element Mixed Meshes , 2010, 2010 International Conference on Digital Manufacturing & Automation.

[18]  J. Clark,et al.  Recursively generated B-spline surfaces on arbitrary topological meshes , 1978 .

[19]  Ahmad H. Nasri,et al.  Designing Catmull-Clark subdivision surfaces with curve interpolation constraints , 2002, Comput. Graph..

[20]  Henning Biermann,et al.  Piecewise smooth subdivision surfaces with normal control , 2000, SIGGRAPH.

[21]  E. Catmull,et al.  Recursively generated B-spline surfaces on arbitrary topological meshes , 1978 .

[22]  Leif Kobbelt,et al.  A variational approach to subdivision , 1996, Comput. Aided Geom. Des..

[23]  Tony DeRose,et al.  Piecewise smooth surface reconstruction , 1994, SIGGRAPH.

[24]  Michael M. Kazhdan,et al.  Can Mean‐Curvature Flow be Modified to be Non‐singular? , 2012, Comput. Graph. Forum.

[25]  N. Dyn,et al.  A butterfly subdivision scheme for surface interpolation with tension control , 1990, TOGS.

[26]  Leif Kobbelt,et al.  Geometric fairing of irregular meshes for free-form surface design , 2001, Comput. Aided Geom. Des..

[27]  Charles T. Loop,et al.  Smooth Subdivision Surfaces Based on Triangles , 1987 .

[28]  Joe D. Warren,et al.  Discrete bi-Laplacians and biharmonic b-splines , 2012, ACM Trans. Graph..

[29]  J. Warren,et al.  Subdivision methods for geometric design , 1995 .

[30]  Hong Qin,et al.  A novel FEM-based dynamic framework for subdivision surfaces , 2000, Comput. Aided Des..

[31]  Malcolm A. Sabin,et al.  Behaviour of recursive division surfaces near extraordinary points , 1998 .

[32]  Jörg Peters,et al.  Combining 4- and 3-direction subdivision , 2004, ACM Trans. Graph..

[33]  Ahmad H. Nasri,et al.  Surface interpolation on irregular networks with normal conditions , 1991, Comput. Aided Geom. Des..

[34]  L. Simon.  Existence of surfaces minimizing the Willmore functional , 1993 .

[35]  Malcolm I. G. Bloor,et al.  Representing PDE surfaces in terms of B-splines , 1990, Comput. Aided Des..

[36]  Charles T. Loop,et al.  Approximating Catmull-Clark subdivision surfaces with bicubic patches , 2008, TOGS.

[37]  William J. Firey,et al.  Shapes of worn stones , 1974 .