Summary
A new indirect way of producing all-quad meshes is presented. The method takes advantage of a well known algorithm of the graph theory, namely the Blossom algorithm that computes the minimum cost perfect matching in a graph in polynomial time. Then, the triangulation itself is taylored with the aim of producing right triangles in the domain. This is done using the infinity norm to compute distances in the meshing process. The alignement of the triangles is controlled by a cross field that is defined on the domain. Meshes constructed this way have their points aligned with the cross field direction and their triangles are almost right everywhere. Then, recombination with our Blossom-based approach yields quadrilateral meshes of excellent quality.
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References
Blacker, T.D., Stephenson, M.B.: Paving: A new approach to automated quadrilateral mesh generation. International Journal for Numerical Methods in Engineering 32, 811–847 (1991)
Frey, P.J., Marechal, L.: Fast adaptive quadtree mesh generation. In: Proceedings of the Seventh International Meshing Roundtable, Citeseer (1998)
Lee, C.K., Lo, S.H.: A new scheme for the generation of a graded quadrilateral mesh. Computers and Structures 52, 847–857 (1994)
Borouchaki, H., Frey, P.J.: Adaptive triangular–quadrilateral mesh generation. International Journal for Numerical Methods in Engineering 45(5), 915–934 (1998)
Owen, S.J., Staten, M.L., Canann, S.A., Saigal, S.: Q-morph: An indirect approach to advancing front quad meshing. International Journal for Numerical Methods in Engineering 9, 1317–1340 (1999)
Edmonds, J.: Maximum matching and a polyhedron with vertices. J. of Research at the National Bureau of Standards 69B, 125–130 (1965)
Edmonds, J., Johnson, E.L., Lockhart, S.C.: Blossom I: A computer code for the matching problem. In: Watson, T.J., Edmonds, J., Johnson, E.L., Lockhart, S.C. (eds.) IBM T. J. Watson Research Center, Yorktown Heights (1969)
Lévy, B., Liu, Y.: Lp centroidal voronoi tesselation and its applications. In: ACM Transactions on Graphics (SIGGRAPH Conference Proceedings) (2010)
Rebay, S.: Efficient unstructured mesh generation by means of delaunay triangulation and bowyer-watson algorithm. Journal of Computational Physics 106(1), 125–138 (1993)
Remacle, J.-F., Lambrechts, J., Seny, B., Marchandise, E., Johnen, A., Geuzaine, C.: Blossom-quad: a non-uniform quadrilateral mesh generator using a minimum cost perfect matching algorithm. International Journal for Numerical Methods in Engineering (submitted 2011)
Edmonds, J.: Paths, trees, and flowers. Canad. J. Math. 17, 449–467 (1965)
Lawler, E.L.: Combinatorial Optimization: Networks and Matroids. Holt, Rinehart, and Winston, New York, NY (1976)
Gabow, H.: Implementation of Algorithms for Maximum Matching on Nonbipartite Graphs. PhD thesis, Stanford University (1973)
Gabow, H., Galil, Z., Micali, S.: An o(ev log v) algorithm for finding a maximal weighted matching in general graphs. SIAM J. Computing 15, 120–130 (1986)
Gabow, H.N.: Data structures for weighted matching and nearest common ancestors with linking. In: Proceedings of the 1st Annual ACM-SIAM Symposium on Discrete Algorithms, pp. 434–443 (1990)
Cook, W., Rohe, A.: Computing minimum-weight perfect matchings. INFORMS Journal on Computing 11(2), 138–148 (1999)
Pébay, P.P.: Planar quadrangle quality measures. Engineering with Computers 20(2), 157–173 (2004)
Bommes, D., Zimmer, H., Kobbelt, L.: Mixed-integer quadrangulation. In: SIGGRAPH 2009: ACM SIGGRAPH 2009 Papers, pp. 1–10. ACM Press, New York (2009)
Lévy, B., Petitjean, S., Ray, N., Maillot, J.: Least squares conformal maps for automatic texture atlas generation. ACM Transactions on Graphics 21(3), 362–371 (2002)
Watson, D.F.: Computing the n-dimensional delaunay tessellation with application to voronoi polytopes. The Computer Journal 24(2), 167–172 (1981)
Marchandise, E., de Wiart, C.C., Vos, W.G., Geuzaine, C., Remacle, J.F.: High-quality surface remeshing using harmonic maps–part ii: Surfaces with high genus and of large aspect ratio. International Journal for Numerical Methods in Engineering 86, 1303–1321 (2011)
Frey, P.J., George, P.-L.: Mesh Generation - Application To Finite Elements. Wiley (2008)
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Remacle, J.F. et al. (2011). A Frontal Delaunay Quad Mesh Generator Using the L ∞ Norm. In: Quadros, W.R. (eds) Proceedings of the 20th International Meshing Roundtable. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-24734-7_25
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DOI: https://doi.org/10.1007/978-3-642-24734-7_25
Publisher Name: Springer, Berlin, Heidelberg
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