Meshfree co‐rotational formulation for two‐dimensional continua

Computer Methods in Applied Mechanics and Engineering

Arroyo

2015

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In this paper, we devise cell-based maximum-entropy (max-ent) basis functions that are used in a Galerkin method for the solution of partial differential equations. The motivation behind this work is the construction of smooth approximants with controllable support on unstructured meshes. In the variational scheme to obtain max-ent basis functions, the nodal prior weight function is constructed from an approximate distance function to a polygonal curve in R 2 . More precisely, we take powers of the composition of R-functions via Boolean operations. The basis functions so constructed are nonnegative, smooth, linearly complete, and compactly-supported in a neighborring of segments that enclose each node. The smoothness is controlled by two positive integer parameters: the normalization order of the approximation of the distance function and the power to which it is raised. The properties and mathematical foundations of the new compactly-supported approximants are described, and its use to solve two-dimensional elliptic boundary-value problems (Poisson equation and linear elasticity) is demonstrated. The sound accuracy and the optimal rates of convergence of the method in Sobolev norms are established.

Section: Minimum Relative Entropy Approximantsmentioning

confidence: 99%

Cell-based maximum-entropy approximants

Millán

Computer Methods in Applied Mechanics and Engineering

Arroyo

2015

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“…Once the converged λ * is found, the basis functions are computed from (3) and the gradient of the basis functions is [40] …”

Section: Maximum-entropy Basis Functionsmentioning

confidence: 99%

Maximum-entropy meshfree method for incompressible media problems

Ortiz

Puso

Finite Elements in Analysis and Design

2011

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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

“…Recently, new applications of max-ent meshfree basis functions have emerged: co-rotational formulation is presented in Ref. [58] and second-order max-ent approximants are proposed in Ref. [59].…”

Section: Maximum-entropy Basis Functionsmentioning

confidence: 99%

“…Examples of prior weight functions include Gaussian radial basis functions [13] and quartic polynomials [58]:…”

Section: Maximum-entropy Basis Functionsmentioning

confidence: 99%

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

Ortiz

Puso

Computer Methods in Applied Mechanics and Engineering

2010

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Numerical integration errors and volumetric locking in the near-incompressible limit are two outstanding issues in Galerkin-based meshfree computations. In this paper, we present a modified Gaussian integration scheme on background cells for meshfree methods that alleviates errors in numerical integration and ensures patch test satisfaction to machine precision. Secondly, a lockingfree small-strain elasticity formulation for meshfree methods is proposed, which draws on developments in assumed strain methods and nodal integration techniques. In this study, maximum-entropy basis functions are used; however, the generality of our approach permits the use of any meshfree approximation. Various benchmark problems in two-dimensional compressible and near-incompressible small strain elasticity are presented to demonstrate the accuracy and optimal convergence in the energy norm of the maximumentropy meshfree formulation.