# zbMATH — the first resource for mathematics

Meshfree, probabilistic determination of point sets and support regions for meshless computing. (English) Zbl 0993.65009
Summary: New algorithms are presented for the determination of point sets and associated support regions that can then be used in meshless computing methods. The algorithms are probabilistic in nature so that they are totally meshfree, i.e., they do not require, at any stage, the use of any coarse or fine boundary conforming or superimposed meshes. Computational examples are provided that show, for both uniform and non-uniform point distributions, that the algorithms result in high-quality point sets and high-quality support regions. Furthermore, the algorithms lend themselves well to parallelization.

##### MSC:
 65C05 Monte Carlo methods 65Y05 Parallel numerical computation 65D18 Numerical aspects of computer graphics, image analysis, and computational geometry
Full Text:
##### References:
 [1] Arya, S.; Mount, D.; Netanyahu, N.; Silverman, R.; Wu, A., An optimal algorithm for approximate nearest neighbor searching, J. ACM, 45, 891-923, (1998) · Zbl 1065.68650 [2] Babuska, I.; Melenk, J., The partition of unity finite element method: basic theory and applications, Comput. methods appl. mech. engrg., 139, 289-314, (1996) · Zbl 0881.65099 [3] Bentley, J., Multidimensional binary search trees used for associative searching, Commun. ACM, 18, 509-517, (1975) · Zbl 0306.68061 [4] Belytschko, T.; Krongauz, Y.; Organ, D.; Fleming, M.; Krysl, P., Meshless methods: an overview and recent developments, Comput. methods appl. mech. engrg., 139, 3-47, (1996) · Zbl 0891.73075 [5] Belytschko, T.; Lu, Y.; Gu, L., Element-free Galerkin methods, Int. J. numer. meth. engrg., 37, 229-256, (1994) · Zbl 0796.73077 [6] Choi, Y.; Kim, S., Node generation scheme for meshfree method by Voronoi diagram and weighted bubble packing, () [7] Du, Q.; Faber, V.; Gunzburger, M., Centroidal Voronoi tessellations: applications and algorithms, SIAM rev., 41, 637-676, (1999) · Zbl 0983.65021 [8] Q. Du, M. Gunzburger, Grid generation and optimization based on centroidal Voronoi tessellations, Appl. Math. Comput., to appear · Zbl 1024.65118 [9] Q. Du, M. Gunzburger, L.-L. Ju, Probablistic methods for centroidal Voronoi tessellations and their parallel implementations, Parallel Comput., submitted · Zbl 1014.68202 [10] Duarte, C.; Oden, J.T., Hp clouds – a meshless method to solve boundary value problems, Numer. meth. PDE, 12, 673-705, (1996) · Zbl 0869.65069 [11] M. Griebel, M. Schweitzer, A particle-partition of unity method for the solution of elliptic, parabolic and hyperbolic PDEs, SIAM J. Sci. Comput. 22 (2001) 853-890 · Zbl 0974.65090 [12] Liu, W.K.; Jun, S.; Zhang, Y.F., Reproducing kernel particle methods, Int. J. numer. meth. fluids, 20, 1081-1106, (1995) · Zbl 0881.76072 [13] X.-Y. Li, S.-H. Teng, A. Ungor, Biting: advancing front meets sphere packing, Int. J. Numer. Meth. Engrg. 49 (2000) 61-91 · Zbl 0966.65096 [14] Lloyd, S., Least square quantization in PCM, IEEE trans. inform. theory, 28, 129-137, (1982) · Zbl 0504.94015 [15] MacQueen, J., Some methods for classification and analysis of multivariate observations, (), 281-297 · Zbl 0214.46201 [16] Okabe, A.; Boots, B.; Sugihara, K., Spatial tessellations: concepts and applications of Voronoi diagrams, (1992), Wiley Chichester · Zbl 0877.52010 [17] Samet, H., The design and analysis of spatial data structures, (1990), Addison-Wesley Reading, MA [18] Spanier, J.; Gelbard, E., Monte Carlo principles and neutron transport problems, (1969), Addison-Wesley New York · Zbl 0244.62004 [19] J. Swegle, S. Attaway, F. Mello, D. Hicks, An analysis of the smoothed particle hydrodynamics, Tech. Rep. SAND93-2513, Sandia National Laboratories, Albuquerque, 1994 · Zbl 0818.76071
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.