Umbrella spherical integration: a stable meshless method for non‐linear solids

Kucherov, Leonid; Tadmor, Ellad B.; Miller, Ronald Mellado

doi:10.1002/nme.1871

Cited by 6 publications

(6 citation statements)

References 39 publications

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“…those which use integration on sub-domains [3,7,8,11,[13][14][15][16][17] and those which do not [4,9,10,12,[18][19][20][21][22][23][24]. The methods in the latter category are cheaper and faster for implementation when compared with those in the former category.…”

Section: Introductionmentioning

confidence: 91%

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The generalized finite point method

2009

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In this paper we propose a new mesh-less method based on a sub-domain collocation approach. By reducing the size of the sub-domains the method becomes similar to the well-known finite point method (FPM) and thus it can be regarded as the generalized form of finite point method (GFPM). However, unlike the FPM, the equilibrium equations are weakly satisfied on the sub-domains. It is shown that the accuracy of the results is dependent on the sizes of the sub-domains. To find an optimal size for a sub-domain we propose a patch test procedure in which a set of polynomials of higher order than those chosen for the approximations/interpolations are used as the exact solution and a suitable error norm is minimized through a size tuning procedure. In this paper we have employed the GFPM in elasto-static problems. We give the results of the size optimization in a series of tables for further use. Also the results of the integrations on a generic sub-domain are given as a series of library functions for those who want to use GFPM as a cheap and fast integral-based mesh-less method. The performance of GFPM has been demonstrated by solving several sample problems.

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Section: Introductionmentioning

confidence: 91%

“…The explicit form ofḠ c is given in the "Appendix". Another point to note is that the order of differentiation of shape functions needed in GFPM is less than that of FPM [compare (17) with (22)]. Moreover, in the case of presence of point loads Eq.…”

Section: Discretization and Formulation In Gfpmmentioning

confidence: 96%

The generalized finite point method

2009

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“…Unlike in [31] where the adapted nodal influence domains are obtained by interpolating the distances to the natural neighbors depending on direction, in the present work the compact support formed by the convex hull of the first and second ring of natural neighbors has cubic or exponential weight function defined over a mapped domain which is a unit disc. The use of a mapping procedure to get umbrella type shape functions [25] and for extended parametric meshless Galerkin method [5] has also been recently used as step to achieve truly mesh free methods. There has also been use of such conformal mapping methods together with Voronoi cell finite element model for analyzing irregular inhomogeneities and inclusions [47].…”

Section: Construction Of Polygonal Compact Supports Based Onmentioning

confidence: 99%

Mesh free Galerkin method based on natural neighbors and conformal mapping

2008

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In this work, a robust mesh free method has been presented for the analysis of two dimensional problems. An efficient natural neighbor algorithm for construction of polygonal support domains has been used in this method that takes into account the nonuniform nodal discretization in the element free Galerkin formulation. The use of natural neighbors for determining the compact support is shown to overcome some of the shortcomings of the conventional distance metric based methods. For nonuniform nodal discretization there is a need for evaluating weights that have anisotropic compact supports. The smoothness and conformance of the weight function to the support domain obtained from natural neighbor algorithm is achieved through an efficient conformal mapping procedure such as Schwarz-Christoffel mapping. Numerical examples demonstrate that the proposed mesh free method gives good estimates of the stress/strain fields.

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“…They showed mathematically that the zero row sum condition is necessary for convergence; this condition is related to the divergence-free integral condition. Kucherov et al [24] introduced what they call an 'umbrella' integration, which is truly mesh free and substantially faster than the previous truly mesh-free methods.…”

Section: Introductionmentioning

confidence: 98%

A new support integration scheme for the weakform in mesh‐free methods

Liu

Belytschko

2009

Numerical Meth Engineering

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SUMMARYA new support integration technique is proposed, which is similar to those used in truly mesh-free methods. The contribution of this paper is to exploit the divergence-free condition for the support integrals to construct quadrature formulas that only require three integration points per particle in two dimensions. Numerical examples show that the proposed integration method can achieve results that agree with manufactured closed-form solutions.

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Umbrella spherical integration: a stable meshless method for non‐linear solids

Cited by 6 publications

References 39 publications

The generalized finite point method

The generalized finite point method

Mesh free Galerkin method based on natural neighbors and conformal mapping

A new support integration scheme for the weakform in mesh‐free methods

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