Convergent meshfree approximation schemes of arbitrary order and smoothness

Bompadre, Agustín; Perotti, Luigi E.; Cyron, Christian J.; Ortiz, Michael

doi:10.1016/j.cma.2012.01.020

Cited by 23 publications

(30 citation statements)

References 21 publications

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“…We have observed the same behavior in the numerical approximation of a fourth-order phase-field model for biomembranes [74]. This suggests further mathematical analysis of the method [75]. The proposed method can be easily enhanced to account for internal connections or non-manifold shells [72,76].…”

Section: Discussionsupporting

confidence: 63%

Nonlinear manifold learning for meshfree finite deformation thin‐shell analysis

Millán

Rosolen

Arroyo

2012

Numerical Meth Engineering

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SUMMARYCalculations on general point-set surfaces are attractive because of their flexibility and simplicity in the preprocessing but present important challenges. The absence of a mesh makes it nontrivial to decide if two neighboring points in the three-dimensional embedding are nearby or rather far apart on the manifold. Furthermore, the topology of surfaces is generally not that of an open two-dimensional set, ruling out global parametrizations. We propose a general and simple numerical method analogous to the mathematical theory of manifolds, in which the point-set surface is described by a set of overlapping charts forming a complete atlas. We proceed in four steps: (1) partitioning of the node set into subregions of trivial topology; (2) automatic detection of the geometric structure of the surface patches by nonlinear dimensionality reduction methods; (3) parametrization of the surface using smooth meshfree (here maximum-entropy) approximants; and (4) gluing together the patch representations by means of a partition of unity. Each patch may be viewed as a meshfree macro-element. We exemplify the generality, flexibility, and accuracy of the proposed approach by numerically approximating the geometrically nonlinear Kirchhoff-Love theory of thin-shells. We analyze standard benchmark tests as well as point-set surfaces of complex geometry and topology.

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Section: Discussionsupporting

confidence: 63%

Nonlinear manifold learning for meshfree finite deformation thin‐shell analysis

Millán

Rosolen

Arroyo

2012

Numerical Meth Engineering

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“…In [58] this formulation is extended to Stokes flow in two dimensions and three-dimensional incompressible elasticity and its stability is demonstrated through inf-sup numerical tests. High order max-ent schemes [59][60][61][62] would be another option in order to employ richer approximants for the velocities. However, such approximation schemes are not guaranteed to be LBBcompliant if coupled with constant or linear approximants for the pressure.…”

Section: Introductionmentioning

confidence: 99%

A stabilized formulation with maximum entropy meshfree approximants for viscoplastic flow simulation in metal forming

et al. 2014

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The finite element method is the reference technique in the simulation of metal forming and provides excellent results with both Eulerian and Lagrangian implementations. The latter approach is more natural and direct but the large deformations involved in such processes require remeshing-rezoning algorithms that increase the computational times and reduce the quality of the results. Meshfree methods can better handle large deformations and have shown encouraging results. However, viscoplastic flows are nearly incompressible, which poses a challenge to meshfree methods. In this paper we propose a simple model of viscoplasticity, where both the pressure and velocity fields are discretized with maximum entropy approximants. The inf-sup condition is circumvented with a numerically consistent stabilized formulation that involves the gradient of the pressure. The performance of the method is studied in some benchmark problems including metal forming and orthogonal cutting.

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“…As an additional bonus, signed shape functions enable the consideration of general-not necessarily convex-domains. These extensions are pursued in a follow-up publication [4], where LME-type approximation schemes of arbitrary order and smoothness are derived and their convergence properties are analyzed using the general analysis framework developed in this paper.…”

Section: Discussionmentioning

confidence: 99%

Convergence Analysis of Meshfree Approximation Schemes

Bompadre¹,

Schmidt²,

Ortiz³

2012

SIAM J. Numer. Anal.

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This work is concerned with the formulation of a general framework for the analysis of meshfree approximation schemes and with the convergence analysis of the Local MaximumEntropy (LME) scheme as a particular example. We provide conditions for the convergence in Sobolev spaces of schemes that are n-consistent, in the sense of exactly reproducing polynomials of degree less or equal to n ≥ 1, and whose basis functions are of rapid decay. The convergence of the LME in W 1,p loc (Ω) follows as a direct application of the general theory. The analysis shows that the convergence order is linear in h, a measure of the density of the point set. The analysis also shows how to parameterize the LME scheme for optimal convergence. Because of the convex approximation property of LME, its behavior near the boundary is singular and requires additional analysis. For the particular case of polyhedral domains we show that, away from a small singular part of the boundary, any Sobolev function can be approximated by means of the LME scheme. With the aid of a capacity argument, we further obtain approximation results with truncated LME basis functions in H 1 (Ω) and for spatial dimension d > 2.

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Convergent meshfree approximation schemes of arbitrary order and smoothness

Cited by 23 publications

References 21 publications

Nonlinear manifold learning for meshfree finite deformation thin‐shell analysis

Nonlinear manifold learning for meshfree finite deformation thin‐shell analysis

A stabilized formulation with maximum entropy meshfree approximants for viscoplastic flow simulation in metal forming

Convergence Analysis of Meshfree Approximation Schemes

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