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

By A. Rosolen, Daniel Millán, Marino Arroyo. 2010

70 citations recorded by the source corpus.

We present a method for the automatic adaption of the support size of meshfree basis functions in the context of the numerical approximation of boundary value problems stemming from a minimum principle. The method is based on a variational approach, and the central idea is that the variational principle selects both the discretized physical fields and the discretization parameters, here those defining the support size of each basis function. We consider local maximum‐entropy approximation schemes, which exhibit smooth basis functions with respect to both space and the discretization parameters (the node location and the locality parameters). We illustrate by the Poisson, linear and non‐linear elasticity problems the effectivity of the method, which produces very accurate solutions with very coarse discretizations and finds unexpected patterns of the support size of the shape functions. Copyright © 2009 John Wiley & Sons, Ltd.

Subfields: Computational Mechanics, Mechanics of Materials, Statistical and Nonlinear Physics

View Paper: https://www.semanticscholar.org/paper/c7aeaf56ef4de209c3e6e3996b3d41823ea34958

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In Computational Mechanics

In Mechanics of Materials

In Statistical and Nonlinear Physics

Citations in the corpus

Exact imposition of boundary conditions with distance functions in physics-informed deep neural networks (2021, background): “[55] proposed a variational adaptivity formulation to find optimal values of {βi}i=1 that minimize the potential energy functional for Poisson equation and nonlinear elasticity.”
Implementation of regularized isogeometric boundary element methods for gradient‐based shape optimization in two‐dimensional linear elasticity (2016, background): “For recent progress in this direction, the interested readers can refer to the work in [14], where a variational adaptivity approach was proposed to optimize the support domain size of meshfree shape functions.”
Phase‐field modeling and simulation of fracture in brittle materials with strongly anisotropic surface energy (2015, unknown): “0 in all examples [49].”
Phase-field modeling of fracture in linear thin shells (2014, unknown): “See [39, 40, 38] for the LME formulation, properties, and the evaluation of the basis functions and their derivatives.”