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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A point interpolation meshless method based on radial basis functions
Jianguo Wang,Gui-Rong Liu +1 more
TL;DR: In this article, a point interpolation meshless method is proposed based on combining radial and polynomial basis functions, which makes the implementation of essential boundary conditions much easier than the meshless methods based on the moving least-squares approximation.
Journal ArticleDOI
The design and analysis of the Generalized Finite Element Method
TL;DR: The GFEM is introduced as a combination of the classical Finite Element Method (FEM) and the Partition of Unity Method (PUM) to solve problems in domains with complex geometry with less error and less computer resources.
Journal ArticleDOI
Meshfree and particle methods and their applications
Shaofan Li,Wing Kam Liu +1 more
TL;DR: A survey of mesh-free and particle methods and their applications in applied mechanics can be found in this article, where the emphasis is 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.
Journal ArticleDOI
A finite point method in computational mechanics. applications to convective transport and fluid flow
TL;DR: In this article, the finite point method (FPM) is proposed for solving partial differential equations, which is based on a weighted least square interpolation of point data and point collocation for evaluating the approximation integrals.
Journal ArticleDOI
Reproducing kernel particle methods for structural dynamics
TL;DR: Numerical and theoretical results show the proposed reproducing kernel interpolation functions satisfy the consistency conditions and the critical time step prediction; furthermore, the RKPM provides better stability than Smooth Particle Hydrodynamics (SPH) methods.
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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