Meshless Methods: an Overview and Recent Developments

Meshless approximations based on moving least squares kernels and partitions of unity are examined It is shown that the three methods are in most cases iden tical except for the important fact that partitions of unity enable p adaptivity to be achieved Methods for constructing discontinuous approximations and approx imations with discontinuous derivatives are also described Next several issues in implementation are reviewed discretization collocation and Galerkin quadrature in Galerkin and fast ways of constructing consistent moving least square approxima tions The paper concludes with some sample calculations Introduction The problems of computational mechanics grow ever more challenging For example in the simulation of manufacturing processes such as extrusion and molding it is necessary to deal with extremely large deformations of the mesh while in computations of castings the propagation of interfaces between solids and liquids is crucial In simulations of failure processes we need to model the propagation of cracks with arbitrary and complex paths In the development of advanced materials methods which can track the growth of phase boundaries and extensive microcracking are required These problems are not well suited to conventional computational methods such as nite element nite volume or nite di erence methods The underlying structure of these methods which originates from their reliance on a mesh is not well suited to the treatment of discontinuities which do not coincide with the original mesh lines Thus the most viable strategy for dealing with moving discontinuities in methods based on meshes is to remesh in each step of the evolution so that mesh lines remain coincident with the discontinuities throughout the evolution of the problem This can of course introduce numerous di culties such as the need to project between meshes in successive stages of the problem which lead to degradation of accuracy and complexity in the computer program not to mention the burden associated with a large number of remeshings The objective of meshless methods is to eliminate at least part of this structure by constructing the approximation entirely in terms of nodes Although in many meshless methods recourse must be taken to meshes in at least parts of the method moving discontinuities can usually be treated without remeshing with minor costs in accuracy degradation Thus it becomes possible to solve large classes of problems which are very awkward with mesh based methods Although meshless methods originated about twenty years ago the research e ort devoted to them until recently was miniscule The starting point which seems to have the longest continuous history is the smooth particle hydrodynamics SPH method Lucy who used it for modeling astrophysical phenomena without boundaries such as exploding stars and dust clouds Compared to other methods in these times of proli c academicians the rate of publications was very modest for many years and is mainly re ected in the papers of Monaghan and coworkers Monaghan Monaghan In these papers the method was explained as a kernel estimate to provide a more rational basis However except for some estimates on the accuracy of the kernel estimate which is not directly relevant to the accuracy of the method in the solution of partial di erential equations there was little in the way of estimation of the accuracy of the method Recently there has been substantial improvement in these methods Swegle Hicks and Attaway have shown the origin of the so called tensile instabil ity through a dispersion analysis of the linearized equations and proposed a viscosity