Maximum-entropy meshfree method for compressible and near-incompressible elasticity

Ortiz, A.; Puso, Michael A.; Sukumar, N.

doi:10.1016/j.cma.2010.02.013

Cited by 66 publications

(63 citation statements)

References 67 publications

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“…In the max-ent framework a stable formulation where max-ent basis functions are used to approximate the displacements and piece-wise linear FE shape functions are used for the pressure was introduced in [56] for compressible and near-incompressible elasticity. In order to ensure stability the displacement approximation is enhanced with an extra node in the interior of each integration triangle, resembling the MINI finite elements [57].…”

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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“…In Ref. [1], we have shown that the use of (15) whose support is the intersection of the support of φ a and φ b and as such can differ appreciably from the union of the cells used in the numerical integration. As a consequence, significant numerical errors can be expected from the numerical integration using (15) or (16).…”

Section: Discrete Weak Formmentioning

confidence: 99%

“…and performing row-sum in the pressure term leads to Now, solving for p a in (13), the following volume-averaged nodal pressure is obtained [1]:…”

Section: Discrete Weak Formmentioning

confidence: 99%

“…[32] a recipe to correct the computed stiffness matrix is proposed to eliminate the inaccuracy in the numerical integration. In Ortiz et al [1], we introduced a modified integration scheme which reduced integration errors in general and suppressed them entirely for the constant strain patch test.…”

Section: Introductionmentioning

confidence: 99%

“…[1] to two-dimensional Stokes flow and to three-dimensional analysis of incompressible elastic solids. In a nutshell, the method consists of developing a displacement or velocity-based formulation by nodally averaging the hydrostatic pressure around the nodes in the divergence-free constraint.…”

Section: Introductionmentioning

confidence: 99%

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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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B-bar based algorithm applied to meshfree numerical schemes to solve unconfined seepage problems through porous media

Navas

López‐Querol

2015

Int. J. Numer. Anal. Meth. Geomech.

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SUMMARYThe object of this work is to establish a meshfree framework for solving coupled, steady and transient problems for unconfined seepage through porous media. The Biot's equations are formulated in displacements (or u w) assuming an elastic solid skeleton. The free surface location and its evolution in time are obtained by interpolation of pore water pressures throughout the domain. Shape functions based on the principle of local maximum entropy are chosen for the meshfree approximation schemes. In order to avoid the locking involved in the fluid phase of the porous media, a B-bar based algorithm is devised to compute the average volumetric strain in a patch composed of various integration points. The efficiency of such an implementation for one phase problems is shown through the Benchmark problem, Cook's membrane loaded by a distributive shear load. The proposed methodology is firstly applied to various classical examples in unconfined steady seepage problems through earth dams, then to the dynamic consolidation of a soil column. The results obtained for both problems are quite satisfactory and demonstrate the feasibility of the proposed method in solving coupled problems in porous media.

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Maximum-entropy meshfree method for compressible and near-incompressible elasticity

Cited by 66 publications

References 67 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

Maximum-entropy meshfree method for incompressible media problems

B-bar based algorithm applied to meshfree numerical schemes to solve unconfined seepage problems through porous media

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