An Introduction to Moving Least Squares Meshfree Methods
Keywords:
Meshfree Methods, Meshless Methods, Moving Least Squares, Diffuse ElementsAbstract
We deal here with some fundamental aspects of a category of meshfree methods based on Moving Least Squares (MLS) approximation and interpolation. These include EFG, RKPM and Diffuse Elements. In this introductory text, we discuss different formulations of the MLS from the point of view of numerical precision and stability. We talk about the issues of both “diffuse” and “full” derivation and we give proof of convergence of both approaches. We propose different algorithms for the computation of MLS based shape functions and we give their explicit forms in 1D, 2D and 3D. The topics of weight functions, the interpolation property with or without singular weights, the domain decomposition and the numerical integration are also discussed. We formulate the integration constraint, necessary for a method to satisfy the linear patch test. Finally, we develop a custom integration scheme, which satisfies this integration constraint.
Downloads
References
Babuska I., Banerjee U., Osborn J.E., Meshless and generalized finite element methods: a
survey of some major results, TICAM Report 02-03, University of Texas at Austin,
January 2002.
Babuska I., Melenk J. M., “The Partition of Unity Method”, International Journal for
Numerical Methods in Engineering, 40, 727-758, 1997.
Barnhill, R.E., Representation and approximation of surfaces in Mathematical Software, ed.
J.R. Rice, Academic Press, New York, 69-120, 1977.
Beissel S., Belytschko T., “Nodal Integration of the Element-Free Galerkin Method”,
Computer Methods in Applied Mechanics and Engineering, 139, 49-74, 1996.
Belytschko T., Krongauz Y., Organ D., Fleming M., and Krysl P., “Meshless Methods: An
Overview and Recent Developments”, Computer Methods in Applied Mechanics and
Engineering, 139, 3-47, 1996.
Belytschko T., Lu Y.Y., Gu L., “Element-free Galerkin Methods”, International Journal for
Numerical Methods in Engineering, 37, 229-256, 1994.
Belytschko T., Gu L., Lu. Y. Y., “Fracture and crack growth by element-free Galerkin
methods. Modelling and Simulation”, Material Science and Engineering, 2, 519-534,
(a).
Bonet J., Lok T.-S. L., “Variational and momentum preservation aspects of smooth particle
hydrodynamics formulation”, Computer Methods in Applied Mechanics and Engineering,
, 97-115, 1999.
Breitkopf P., Rassineux A., Touzot G., Villon P., “Explicit form and efficient computation of
MLS shape functions and their derivatives”, International Journal for Numerical Methods
in Engineering, 48, 451-456, 2000.
Breitkopf P., Rassineux A., Villon P., Saannouni K., Cherouat H., “Meshfree operators for
consistent field transfer in large deformation plasticity”, ECCOMAS-ECCM-2001,
Cracow, Poland, 26-29 June 2001.
Breitkopf P., Touzot G., Villon P., “Double Grid Diffuse Collocation Method”,
Computational Mechanics, 25, No 2/3, 199-206, 2000.
Chen J-S, Han W., You Y., Meng X., “A Reproducing Kernel Method with Nodal
Interpolation Property”, International Journal for Numerical Methods in Engineering, in
press, 2002.
Chen J. S., Wu C. T., Yoon S., You Y., “Nonlinear Version of Stabilized Conforming Nodal
Integration for Galerkin Meshfree Methods”, International Journal for Numerical
Methods in Engineering, 53, 2587-2615, 2002.
Cleveland, W.S., “Robust Locally Weighted Regression and Smoothing Scatterplots”,
Journal of the American Statistical Association, December, Vol, 74, No. 368, 829-836,
De S., Bathe K.J., “The Method of Finite Spheres”, Computational Mechanics, 25, 329-345,
Dolbow J., Belytschko T., “Numerical integration of the Galerkin weak form in meshfree
methods”, Computational Mechanics, 23(3), 219-230, 1999.
Gordon William J., Wixom James A., “Shepard's method of metric interpolation to bivariate
and multivariate interpolation”, Math. Comp. 32(141), 253-264, 1978.
Huerta A., Vidal Y., Villon P., “Locking in the Incompressible Limit: Pseudo-Divergence-
Free Element Free Galerkin”, Proceedings of the Fifth World Congress on Computational
Mechanics (WCCM V), July 7-12, 2002, Vienna, Austria.
Krige, D.G. Two-dimensional weighted moving average trend surfaces for ore evaluation.
Journal of the South African Institute of Mining & Metallurgy, 67, 13-79, 1966.
Lancaster P., Salkauskas K., Curve and Surface Fitting: an Introduction, Academic Press,
London, Orlando, 1986.
Lancaster P., Salkauskas K., “Surfaces generated by moving least squares methods”, Math.
Comp. 37, 141-158, 1981.
Lin H., Atluri S.N., “Meshless Local Petrov-Galerkin (MLPG) Method for Convection -
Diffusion Problems”, Computer Modeling in Engineerng & Sciences, 1(2), 45-60, 2000.
Liszka T., Orkisz J., “The Finite Difference Method at Arbitrary Irregular Grids and its
Application in Applied Mechanics”, Computers and Structures, 11, 83-95, 1980.
Liu W. K., Chen Y., Jun S., Chen J. S., Belytschko T., Pan C., Uras R. A. Chang C. T.,
“Overview and Applications of the Reproducing Kernel Particle Methods”, Archives of
Computational Methods in Engineering: State of the art reviews, 3, 3-80, 1996.
Lu Y.Y., Belytschko T., Gu L., “A New Implementation of the Element Free Galerkin
Method”, Computer Methods in Applied Mechanics and Engineering, 113, 397-414,
Lucy L. B., “A numerical approach to the testing of the fission hypothesis”, Astronomical
Journal 82, 1013-1024, 1977.
Mac Lain DH, “Drawing Contours with Arbitrary Data Points”, The Computer Journal, 17(4),
-324, 1974.
Matheron G., “Principles of Geostatistics”, Economic Geology, 58, 1246- 1266, 1963.
Mukherjee Y.X., Mukherjee S., “On boundary conditions in Element-Free Galerkin Method”,
Computational Mechanics, 11, 1997, 264-270.
Nayroles B., Touzot G., Villon P., “Generalizing the Finite Element Method: Diffuse
Approximation and Diffuse Elements”, Computational Mechanics, 10, 307-318, 1992.
Oñate E., Idelsohn S.R., “A mesh-free finite point method for advective-diffusive transport
and fluid flow problems”, Computational Mechanics, 21, 283-292, 1998.
Savignat J-M, Approximation diffuse Hermite et ses applications, Thèse de Doctorat, Ecole
des Mines de Paris, October 2000.
Shepard D., “A two-dimensional interpolation function for irregularly spaced data”, Proc.
rd National Conference ACM, 517-524, 1968.
Sukumar N., Moës N., Moran B. Belytschko T., “Extended Finite Element Method for Three-
Dimensional Crack Modeling”, International Journal for Numerical Methods in
Engineering, 48(11), 1549-1570, 2000.
Sulsky D., Schreyer HL, The Particle-In-Cell Method as a Natural Impact Algorithm, Sandia
National Laboratories, Contract No. AC-1801, 1993.
Syczewski M., Tribillo R., “Singularities of Sets Used in the Mesh Method”, Computers and
Structures, 14(5-6), 509-511, 1981.
Villon P, Contribution à l’Optimisation, Thèse de Docteur d’Etat, Université de Technologie
de Compiègne, France, 1991.
Wyatt M.J., Davies G., Snell C., “A New Difference Based Finite Element Method”, Instn.
Engineers, 59(2), 395-409, 1975.
Zhang X., Liu X-H., Song K-Z., Lu M-W., Least-squares collocation meshless method,
International Journal for Numerical Methods in Engineering, 51, 1089-1100, 2001.
