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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Dynamic analysis of sandwich beams with functionally graded core using a truly meshfree radial point interpolation method
TL;DR: In this paper, the displacement field is approximated by the radial point interpolation method (RPIM) regardless of predefined mesh, and the domain integrals are evaluated by the so-called Cartesian transformation method (CTM) to obviate the need for a background cell.
Journal ArticleDOI
Smoothing, enrichment and contact in the element-free Galerkin method
Ted Belytschko,Mark Fleming +1 more
TL;DR: In this paper, three methods for smoothing meshless approximations near nonconvex boundaries such as cracks are reviewed and compared: diffraction method, which wraps the nodal domain of influence a short distance around a point of discontinuity, such as a crack tip; transparency method, gradually severs the domains of influence near crack tips; and see-through method, or continuous line criterion.
Journal ArticleDOI
Multiquadric and its shape parameter—A numerical investigation of error estimate, condition number, and round-off error by arbitrary precision computation
TL;DR: In this paper, the authors used arbitrary precision arithmetic in the computation of radial basis functions to test conditional positive definiteness, error estimate, optimal shape parameter, traditional and effective condition numbers, round-off error, derivatives of interpolator, and the edge effect of interpolation.
Journal ArticleDOI
Treatment of material discontinuity in the Element-Free Galerkin method
L.W. Cordes,Brian Moran +1 more
TL;DR: In this article, a method for application of the Element-Free Galerkin method (EFG) to solid mechanics problems containing material discontinuities is presented, where the trial and test functions for the weak form are constructed with moving least square interpolants in each material domain.
Journal ArticleDOI
Harmonic reproducing kernel particle method for free vibration analysis of rotating cylindrical shells
TL;DR: In this article, a mesh-free approach is proposed for the free vibration analysis of rotating cylindrical shells, where the reproducing kernel particle estimation is employed in hybridized form with harmonic functions, to approximate the two-dimensional displacement field.
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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