Smooth, second order, non‐negative meshfree approximants selected by maximum entropy

Cyron, Christian J.; Arroyo, Marino; Ortiz, Michael

doi:10.1002/nme.2597

Cited by 63 publications

(81 citation statements)

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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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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 above program is convex, smooth and feasible for any spa tial dimension d (as long as x 2 convX), and produces C 1 meshfree non negative functions p a ðxÞ [1]. Moreover, the constraints (con sistency conditions) guarantee solutions that reproduce exactly af fine functions (see [6,7,28,29] for higher order approaches). Duality methods provide an efficient route to solving the optimiza tion problem and computing almost explicitly p a ðxÞ at each evalu ation point x.…”

Section: Maximum Entropy Approximation Schemesmentioning

confidence: 99%

“…The non negativity and first order reproducing conditions endow these approximants with the structure of convex geometry [1], like linear finite ele ment, natural neighbor method [3], subdivision approximants [4], or B Spline and Non Uniform Rational B Splines (NURBS) basis functions [5]. Max ent approximants have been extended to second order [6,7], and to arbitrary order by dropping non negativity [8].…”

Section: Introductionmentioning

confidence: 99%

“…The resulting approximation scheme automatically retains the non negativity and smoothness of the B Spline and LME parents. Although max ent approximants can be extended to higher order consistency, at the expense of a more involved formulation [6,7], numerical experiments show that first order consistent approximants perform very well, even in high order partial differential equations. In [7], we showed that first order LME approximants attain the same accuracy as 5th or der B Splines for structural vibrations, and are comparable to sec ond order max ent approximation schemes in a fourth order phase field model [23], or in thin shell problems [24,25], where they also compete with subdivision finite elements.…”

Section: Introductionmentioning

confidence: 99%

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Blending isogeometric analysis and local maximum entropy meshfree approximants

Rosolen

Arroyo

2013

Computer Methods in Applied Mechanics and Engineering

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a b s t r a c tWe present a method to blend local maximum entropy (LME) meshfree approximants and isogeometric analysis. The coupling strategy exploits the optimization program behind LME approximation, treats iso geometric and LME basis functions on an equal footing in the reproducibility constraints, but views the former as data in the constrained minimization. The resulting scheme exploits the best features and over comes the main drawbacks of each of these approximants. Indeed, it preserves the high fidelity boundary representation (exact CAD geometry) of isogeometric analysis, out of reach for bare meshfree methods, and easily handles volume discretization and unstructured grids with possibly local refinement, while maintaining the smoothness and non negativity of the basis functions. We implement the method with B Splines in two dimensions, but the procedure carries over to higher spatial dimensions or to other non negative approximants such as NURBS or subdivision schemes. The performance of the method is illus trated with the heat equation, and linear and nonlinear elasticity. The ability of the proposed method to impose directly essential boundary conditions in non convex domains, and to deal with unstructured grids and local refinement in domains of complex geometry and topology is highlighted by the numerical examples.

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“…Compact support shape functions were then derived using Gaussian weight functions (or priors) in [35], work which was extended in [36] to any weight function (or generalized prior). Firstorder consistent max-ent shape functions [36] were then extended to second order in [37] and higher order in [38] and max-ent was used in [39] for the automatic calculation of the nodal domain of influence within a meshless method.…”

Section: Maximum Entropy Shape Functionsmentioning

confidence: 99%

An adaptive finite element/meshless coupled method based on local maximum entropy shape functions for linear and nonlinear problems

Ullah

Coombs

Augarde

2013

Computer Methods in Applied Mechanics and Engineering

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'An adaptive nite element/meshless coupled method based on local maximum entropy shape functions for linear and nonlinear problems.', Computer methods in applied mechanics and engineering., 267 . pp. 111-132. Further information on publisher's website:http://dx.doi.org/10.1016/j.cma.2013.07.018Publisher's copyright statement: NOTICE: this is the author's version of a work that was accepted for publication in Computer Methods in Applied Mechanics and Engineering. Changes resulting from the publishing process, such as peer review, editing, corrections, structural formatting, and other quality control mechanisms may not be re ected in this document. Changes may have been made to this work since it was submitted for publication. A de nitive version was subsequently published in Computer Methods in Applied Mechanics and Engineering, 267, 2013, 10.1016/j.cma.2013.07.018. Additional information:Use policyThe full-text may be used and/or reproduced, and given to third parties in any format or medium, without prior permission or charge, for personal research or study, educational, or not-for-pro t purposes provided that:• a full bibliographic reference is made to the original source • a link is made to the metadata record in DRO • the full-text is not changed in any way The full-text must not be sold in any format or medium without the formal permission of the copyright holders.Please consult the full DRO policy for further details. AbstractIn this paper, an automatic adaptive coupling procedure is proposed for the finite element method (FEM) and the element-free Galerkin method (EFGM) for linear elasticity and for problems with both material and geometrical nonlinearities. In this new procedure, initially the whole of the problem domain is modelled using the FEM. During an analysis, those finite elements which violate a predefined error measure are automatically converted to an EFG zone. This EFG zone can be further refined by adding nodes, thus avoiding computationally expensive FE remeshing. Local maximum entropy shape functions are used in the EFG zone of the problem domain for two reasons: their weak Kronecker delta property at the boundaries allows straightforward imposition of essential boundary conditions and also provides a natural way to couple the EFG and FE regions compared to the use of moving least squares basis functions. The Zienkiewicz & Zhu error estimation procedure with the superconvergent patch method for strains and stresses recovery is used in the FE region of the problem domain, while the Chung & Belytschko error estimation procedure is used in the EFG region.

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Smooth, second order, non‐negative meshfree approximants selected by maximum entropy

Cited by 63 publications

References 20 publications

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

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

Blending isogeometric analysis and local maximum entropy meshfree approximants

An adaptive finite element/meshless coupled method based on local maximum entropy shape functions for linear and nonlinear problems

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