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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Error analysis of collocation method based on reproducing kernel approximation
TL;DR: This work discusses a reproducingkernel collocation method, where the reproducing kernel (RK) shape functions with compact support are used as approximation functions and shows that using RK shape function for collocation of strong form, the degree of polynomial basis functions has to be larger than one for convergence, which is different from the condition for weak formulation.
Journal ArticleDOI
Treatment of material discontinuity in two meshless local Petrov-Galerkin (MLPG) formulations of axisymmetric transient heat conduction
TL;DR: In this article, the convergence of the H 0 and H 1 error norms for the four numerical solutions with an increase in the number of equally spaced nodes and in the quadrature points is scrutinized.
Journal ArticleDOI
Numerical method for shape optimization using meshfree method
TL;DR: In this article, a numerical method for continuum-based shape design sensitivity analysis and optimization using the mesh-free method is proposed, where the reproducing kernel particle method is used for domain discretization in conjunction with the Gauss integration method.
Journal ArticleDOI
Development of a novel meshless Local Kriging (LoKriging) method for structural dynamic analysis
TL;DR: In this article, a meshless approach called Local Kriging (LoKriging) method is presented to study the dynamic responses of structures, where the Krigging interpolation is employed to construct the field approximation function and meshless shape functions, and the local weak form of partial differential governing equations is derived by the weighted residual technique.
Journal ArticleDOI
Dispersion error reduction for acoustic problems using the edge‐based smoothed finite element method (ES‐FEM)
TL;DR: In this article, a detailed analysis on the numerical dispersion error in solving 2D acoustic problems governed by the Helmholtz equation using the edge-based smoothed finite element method (ES-FEM), in comparison with the standard FEM, is presented.
References
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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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