Recent developments of meshfree and particle methods and their applications in applied mechanics are surveyed. Three major methodologies have been reviewed. First, smoothed particle hydrodynamics ~SPH! is discussed as a representative of a non-local kernel, strong form collocation approach. Second, mesh-free Galerkin methods, which have been an active research area in recent years, are reviewed. Third, some applications of molecular dynamics ~MD! in applied mechanics are discussed. The emphases of this survey are placed on simulations of finite deformations, fracture, strain localization of solids; incompressible as well as compressible flows; and applications of multiscale methods and nano-scale mechanics. This review article includes 397 references. @DOI: 10.1115/1.1431547#

Meshfree and particle methods and their applications

SUMMARY In this paper, conforming finite elements on polygon meshes are developed. Polygonal finite elements provide greater flexibility in mesh generation and are better-suited for applications in solid mechanics which involve a significant change in the topology of the material domain. In this study, recent advances in meshfree approximations, computational geometry, and computer graphics are used to construct different trial and test approximations on polygonal elements. A particular and notable contribution is the use of meshfree (natural-neighbour, nn) basis functions on a canonical element combined with an affine map to construct conforming approximations on convex polygons. This numerical formulation enables the construction of conforming approximation on n-gons (n 3), and hence extends the potential applications of finite elements to convex polygons of arbitrary order. Numerical experiments on second-order elliptic boundary-value problems are presented to demonstrate the accuracy and convergence of the proposed method. Copyright 2004 John Wiley & Sons, Ltd.

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Conforming polygonal finite elements

This paper gives an overview over Meshfree Methods Starting point is an extended and modified classification of Meshfree Methods due to three aspects: The construction of a partition of unity, the choice of an approximation either with or without using an extrinsic basis and the choice of test functions, resulting into a collocation, Bubnov-Galerkin or Petrov-Galerkin Meshfree Method Most of the relevant Meshfree Methods are described taking into account their different origins and viewpoints as well as their advantages and disadvantages Typical problems arising in meshfree methods like integration, treatment of essential boundary conditions, coupling with mesh-based methods etc~are discussed Some valuing comments about the most important aspects can be found at the end of each section This text was revised in 2004

Classification and Overview of Meshfree Methods

This paper is an overview of recent developments in the construction of finite element interpolants, which areC
 0-conforming on polygonal domains. In 1975, Wachspress proposed a general method for constructing finite element shape functions on convex polygons. Only recently has renewed interest in such interpolants surfaced in various disciplines including: geometric modeling, computer graphics, and finite element computations. This survey focuses specifically on polygonal shape functions that satisfy the properties of barycentric coordinates: (a) form a partition of unity, and are non-negative; (b) interpolate nodal data (Kronecker-delta property), (c) are linearly complete or satisfy linear precision, and (d) are smooth within the domain. We compare and contrast the construction and properties of various polygonal interpolants—Wachspress basis functions, mean value coordinates, metric coordinate method, natural neighbor-based coordinates, and maximum entropy shape functions. Numerical integration of the Galerkin weak form on polygonal domains is discussed, and the performance of these polygonal interpolants on the patch test is studied.

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Recent advances in the construction of polygonal finite element interpolants

Fil: Idelsohn, Sergio Rodolfo. Consejo Nacional de Investigaciones Cientificas y Tecnicas. Centro Cientifico Tecnologico Conicet - Santa Fe. Instituto de Desarrollo Tecnologico para la Industria Quimica. Universidad Nacional del Litoral. Instituto de Desarrollo Tecnologico para la Industria Quimica; Argentina

The meshless finite element method

SUMMARY Natural neighbour co-ordinates (Sibson co-ordinates) is a well-known interpolation scheme for multivariate data tting and smoothing. The numerical implementation of natural neighbour co-ordinates in a Galerkin method is known as the natural element method (NEM). In the natural element method, natural neighbour co-ordinates are used to construct the trial and test functions. Recent studies on NEM have shown that natural neighbour co-ordinates, which are based on the Voronoi tessellation of a set of nodes, are an appealing choice to construct meshless interpolants for the solution of partial dierential equations. In Belikov et al. (Computational Mathematics and Mathematical Physics 1997; 37(1):9{15), a new interpolation scheme (non-Sibsonian interpolation) based on natural neighbours was proposed. In the present paper, the nonSibsonian interpolation scheme is reviewed and its performance in a Galerkin method for the solution of elliptic partial dierential equations that arise in linear elasticity is studied. A methodology to couple nite elements to NEM is also described. Two signicant advantages of the non-Sibson interpolant over the Sibson interpolant are revealed and numerically veried: the computational eciency of the non-Sibson algorithm in 2-dimensions, which is expected to carry over to 3-dimensions, and the ability to exactly impose essential boundary conditions on the boundaries of convex and non-convex domains. Copyright ? 2001 John Wiley & Sons, Ltd.

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A. Yu Semenov

Papers

Natural neighbour Galerkin methods