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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Upper bound solution to elasticity problems: A unique property of the linearly conforming point interpolation method (LC-PIM)
TL;DR: In this paper, the linearly conforming point interpolation method (LC-PIM) is used to obtain an upper bound solution in energy norm for elasticity problems, which is a very important and unique property of the PIM.
Journal ArticleDOI
A finite point method for elasticity problems
TL;DR: In this paper, the basis of the finite point method (FPM) for the fully meshless solution of elasticity problems in structural mechanics is described and a stabilization technique based on a finite calculus procedure is used to improve the quality of the numerical solution.
Journal ArticleDOI
The material point method in large strain engineering problems
TL;DR: The material point method as discussed by the authors is a variant of the finite element method formulated in an arbitrary Lagrangian-Eulerian description of motion, where the motion of material points, representing subregions of the analysed continuum, is traced against a background of the computational element mesh.
Journal ArticleDOI
The intrinsic XFEM: a method for arbitrary discontinuities without additional unknowns
TL;DR: In this article, a new method for treating arbitrary discontinuities in a finite element (FE) context is presented, which constructs an approximation space consisting of mesh-based, enriched moving least-squares (MLS) functions near the point of interest and standard FE shape functions elsewhere.
Journal ArticleDOI
Three-dimensional fracture propagation with numerical manifold method
TL;DR: The NMM is developed to analyze three dimensional (3D) fracture propagation and the maximum tensile stress criterion is implemented to determine whether the fracture will propagate and the direction of fracture propagation.
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
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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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