An Art Gallery Approach to Submap Meshing

Abstract The Art Gallery Theorem and its lemmas have been used since the 1990s in the area of computer graphics to decompose orthogonal polygons into convex sub-regions. This paper extends and improvises Art Gallery concepts to non-orthogonal, generic polygons to perform a multiblock decomposition of faces into a set of maximal, single-loop, convex sub-faces. Concepts such as staircase, dent and notch are used to categorize face concavities and virtual vertices are inserted on smoother concave boundary bends. Multiblocking is performed without the need for Delaunay triangulation with the aid of a notch diagram. A light-weight, mesher-native topology builder that uses virtual topological elements is proposed to construct a virtual topology network from a notch diagram. Subsequently, virtual faces are processed for transfinite meshing. Results indicate the advantage of the proposed technique over existing procedures in terms of orthogonality and anisotropy.

[1]  Nilanjan Mukherjee.  A Combined Subdivision and Advancing Loop-Front Surface Mesher (Triangular) for Automotive Structures , 2010 .

[2]  Marilyn Breen.  Staircase kernels in orthogonal polygons , 1992 .

[3]  A. Aggarwal.  The art gallery theorem: its variations, applications and algorithmic aspects , 1984 .

[4]  T. Vassilev,et al.  Visibilty: Finding the Staircase Kernel in Orthogonal Polygons , 2012 .

[5]  Steve Fisk,et al.  A short proof of Chvátal's Watchman Theorem , 1978, J. Comb. Theory, Ser. B.

[6]  Nilanjan Mukherjee,et al.  High Quality Bi-Linear Transfinite Meshing with Interior Point Constraints , 2006, IMR.

[7]  Nilanjan Mukherjee,et al.  A Transfinite Meshing Approach for Body-In-White Analyses , 2010, IMR.

[8]  J. O'Rourke.  Art gallery theorems and algorithms , 1987 .

[9]  Nilanjan Mukherjee,et al.  Flattening 3D Triangulations for Quality Surface Mesh Generation , 2008, IMR.

[10]  Marilyn Breen.  Staircasek-kernels for orthogonal polygons , 1992 .

[11]  T. Tam,et al.  2D finite element mesh generation by medial axis subdivision , 1991 .

[12]  V. Chvátal.  A combinatorial theorem in plane geometry , 1975 .

[13]  Frank Hoffmann,et al.  The art gallery theorem for polygons with holes , 1991, [1991] Proceedings 32nd Annual Symposium of Foundations of Computer Science.

[14]  Joseph C. Culberson,et al.  Covering a simple orthogonal polygon with a minimum number of orthogonally convex polygons , 1987, SCG '87.

[15]  J. Kahn,et al.  Traditional Galleries Require Fewer Watchmen , 1983 .

[16]  M. Whiteley,et al.  Two and three-quarter dimensional meshing facilitators , 2005, Engineering with Computers.