Journal ArticleDOI
Element‐free Galerkin methods
Ted Belytschko,Y. Y. Lu,L. Gu +2 more
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In this article, an element-free Galerkin method which is applicable to arbitrary shapes but requires only nodal data is applied to elasticity and heat conduction problems, where moving least-squares interpolants are used to construct the trial and test functions for the variational principle.Abstract:
An element-free Galerkin method which is applicable to arbitrary shapes but requires only nodal data is applied to elasticity and heat conduction problems. In this method, moving least-squares interpolants are used to construct the trial and test functions for the variational principle (weak form); the dependent variable and its gradient are continuous in the entire domain. In contrast to an earlier formulation by Nayroles and coworkers, certain key differences are introduced in the implementation to increase its accuracy. The numerical examples in this paper show that with these modifications, the method does not exhibit any volumetric locking, the rate of convergence can exceed that of finite elements significantly and a high resolution of localized steep gradients can be achieved. The moving least-squares interpolants and the choices of the weight function are also discussed in this paper.read more
Citations
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Derivative recovery and a posteriori error estimate for extended finite elements
Stéphane Bordas,Marc Duflot +1 more
TL;DR: In this paper, a simple and effective local a posteriori error estimate for partition of unity enriched finite element methods such as the extended finite element method (XFEM) is proposed, in which near-tip asymptotic functions are added to the MLS basis.
Journal ArticleDOI
Overview and recent advances in natural neighbour galerkin methods
TL;DR: A survey of the most relevant advances in natural neighbour Galerkin methods is presented in this article, where the Sibson and the Laplace (non-Sibsonian) interpolation schemes are used as trial and test functions.
Journal ArticleDOI
Element-Free Galerkin solutions for Helmholtz problems: fomulation and numerical assessment of the pollution effect
Ph. Bouillard,S. Suleaub +1 more
TL;DR: In this article, the Element-Free Galerkin Method (EFGM) is examined in its application to acoustic wave propagation addressed by the Helmholtz equation, and numerical tests on two-dimensional problems focus on the parameters governing the EFGM.
Journal ArticleDOI
Trefftz Methods for Time Dependent Partial Differential Equations
TL;DR: In this paper, a mesh-free approach to numerically solving a class of second order time dependent partial differential equations which in-clude equations of parabolic, hyperbolic and parabolic-hyperbolic types is presented.
Journal ArticleDOI
The arbitrary local mesh replacement method: An alternative to remeshing for crack propagation analysis
TL;DR: In this paper, a finite element-based method is proposed to accommodate the arbitrary motion of a crack in the context of general two-dimensional configurations of loading and body geometry, and a weak statement of compatibility is enforced on the interface between the two meshes, thereby introducing a small set of additional unknowns.
References
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Journal ArticleDOI
Smoothed particle hydrodynamics: Theory and application to non-spherical stars
R. A. Gingold,Joseph J Monaghan +1 more
Journal ArticleDOI
Theory of Elasticity (3rd ed.)
Journal ArticleDOI
Surfaces generated by moving least squares methods
Peter Lancaster,K. Salkauskas +1 more
TL;DR: In this article, an analysis of moving least squares (m.l.s.) methods for smoothing and interpolating scattered data is presented, in particular theorems concerning the smoothness of interpolants and the description of m. l.s. processes as projection methods.
Journal ArticleDOI
Generalizing the finite element method: Diffuse approximation and diffuse elements
TL;DR: The diffuse element method (DEM) as discussed by the authors is a generalization of the finite element approximation (FEM) method, which is used for generating smooth approximations of functions known at given sets of points and for accurately estimating their derivatives.
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