Meshfree Methods, Encyclopedia of Computational Mechanics, Vol
Huerta, A., Belytschko, T. Fernández-Méndez, S. and Rabczuk, T., Meshfree Methods, Encyclopedia of Computational Mechanics, Vol. 1, Chapter 10, pp. 279-309, 2004 Meshfree Methods Antonio Huerta2, Ted Belytschko1, Sonia Fern¶andez-M¶endez2, Timon Rabczuk1 1 Department of Mechanical Engineering, Northwestern University, 2145 Sheridan Road, Evanston, IL 60208, USA. 2 Laboratori de C`alculNum`eric,Universitat Polit`ecnica de Catalunya, Jordi Girona 1, E-08034 Barcelona, Spain. ABSTRACT The aim of this chapter is to provide an in-depth presentation and survey of meshfree particle methods. Several particle approximations are reviewed; the SPH method, corrected gradient methods and the moving least squares (MLS) approximation. The discrete equations are derived from a collocation scheme or a Galerkin method. Special attention is paid to the treatment of essential boundary conditions. A brief review of radial basis functions is given because they play a signi¯cant role in meshfree methods. Finally, di®erent approaches for modelling discontinuities in meshfree methods are described. key words: Meshfree, meshless, moving least squares, smooth particle hydrodynamics, element-free Galerkin, reproducing kernel particle methods, partitions of unity, radial basis functions, discontinuous functions, coupling with ¯nite elements, incompressibility. 1. INTRODUCTION As the range of phenomena that need to be simulated in engineering practice broadens, the limitations of conventional computational methods, such as ¯nite elements, ¯nite volumes or ¯nite di®erence methods, have become apparent. There are many problems of industrial and academic interest which cannot be easily treated with these classical mesh-based methods: for example, the simulation of manufacturing processes such as extrusion and molding, where it is necessary to deal with extremely large deformations of the mesh, or simulations of failure, where the simulation of the propagation of cracks with arbitrary and complex paths is needed.
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