On the optimum support size in meshfree methods: A variational adaptivity approach with maximum‐entropy approximants

Rosolen, A.; Millán, Daniel; Arroyo, Marino

doi:10.1002/nme.2793

Cited by 58 publications

(67 citation statements)

References 26 publications

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“…where the set of non-negative nodal parameters {β a = γ a /h 2 a } a=1,...,N defines the locality of the approximants [25,26]. The dimensionless parameter γ a characterizes the degree of locality of the basis function associated to the node x a , while h a represents the nodal spacing.…”

Section: The Local Maximum Entropy Approximantsmentioning

confidence: 99%

“…Thanks to this property essential boundary conditions can be easily imposed on the boundary of convex domains. In addition the support of the basis functions can be flexibly controlled [25,26] and their evaluation is fast and robust using duality methods [25]. In a recent work [27] it was shown that max-ent approximants can be blended with other convex approximants such as B-splines or NURBS basis functions in the vicinity of the boundary of the domain.…”

Section: Introductionmentioning

confidence: 99%

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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: The Local Maximum Entropy Approximantsmentioning

confidence: 99%

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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“…We consider here LME basis functions. See [39,40,38] for the LME formulation, properties, and the evaluation of the basis functions and their derivatives. Then, let p a (ξ) denote the LME approximants associated to the point-set Ξ κ on a domain A κ ⊂ R 2 , a subset of the convex hull of the reduced node set conv Ξ κ .…”

Section: Numerical Representation Of the Surfacesmentioning

confidence: 99%

“…Recently, nonlinear manifold learning techniques have been exploited to parametrize 2D sub-domains of a point-set surface, which are then used as parametric patches and glued together with a partition of unity [38,21]. Here, we combine this methodology with local maximum-entropy (LME) meshfree approximants [39,40,5] because of their smoothness, robustness, and relative ease of quadrature compared with other meshfree approximants.…”

Section: Introductionmentioning

confidence: 99%

Phase-field modeling of fracture in linear thin shells

Amiri

Millán

Shen

et al. 2014

Theoretical and Applied Fracture Mechanics

285

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We present a phase-field model for fracture in Kirchoff-Love thin shells using the local maximumentropy (LME) meshfree method. Since the crack is a natural outcome of the analysis it does not require an explicit representation and tracking, which is advantage over techniques as the extended finite element method that requires tracking of the crack paths. The geometric description of the shell is based on statistical learning techniques that allow dealing with general point set surfaces avoiding a global parametrization, which can be applied to tackle surfaces of complex geometry and topology. We show the flexibility and robustness of the present methodology for two examples: plate in tension and a set of open connected pipes.

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“…For the quartic polynomial, q = x a − x /ρ a and ρ a = γh a is the radius of the basis function support at node a. In a recent study on max-ent meshfree methods [41], it has been shown that substantial improvements in accuracy are realized by letting the supportwidth parameters as unknowns and solving for them through the variational structure (minimizing principle) of the continuum problem.…”

Section: Maximum-entropy Basis Functionsmentioning

confidence: 99%

Maximum-entropy meshfree method for incompressible media problems

Ortiz

Puso

Sukumar

2011

Finite Elements in Analysis and Design

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A novel maximum-entropy meshfree method that we recently introduced in Ref.[1] is extended to Stokes flow in two dimensions and to three-dimensional incompressible linear elasticity. The numerical procedure is aimed to remedy two outstanding issues in meshfree methods: the development of an optimal and stable formulation for incompressible media, and an accurate cell-based numerical integration scheme to compute the weak form integrals.On using the incompressibility constraint of the standard u-p formulation, a u-based formulation is devised by nodally averaging the hydrostatic pressure around the nodes. A modified Gauss quadrature scheme is employed, which results in a correction to the stiffness matrix that alleviates integration errors in meshfree methods, and satisfies the patch test to machine accuracy.The robustness and versatility of the maximum-entropy meshfree method is demonstrated in three-dimensional computations using tetrahedral background meshes for integration. The meshfree formulation delivers optimal * Corresponding author Email address: alortiz@ucdavis.edu (A. Ortiz) December 25, 2010 rates of convergence in the energy and L 2 -norms. Inf-sup tests are presented to demonstrate the stability of the maximum-entropy meshfree formulation for incompressible media problems. Preprint submitted to Elsevier

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On the optimum support size in meshfree methods: A variational adaptivity approach with maximum‐entropy approximants

Abstract: Rosolen, A,; Millán, D. and Arroyo, M., On the optimum support size in meshfree methods: a variational adaptivity approach with maximum entropy approximants,

Cited by 58 publications

References 26 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

Phase-field modeling of fracture in linear thin shells

Maximum-entropy meshfree method for incompressible media problems

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