An Element Free Galerkin Method Based on the Modified Moving Least Squares Approximation

Habibullah Amin Chowdhury, Adam Wittek, Karol Miller, Grand Roman Joldes

    Research output: Contribution to journalArticle

    6 Citations (Scopus)
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    Abstract

    This paper demonstrates that the recently developed modified moving least squares (MMLS) approximation possess the necessary properties which allow its use as an element free Galerkin (EFG) approximation method. Specifically, the consistency and invariance properties for the MMLS are proven. We demonstrate that MMLS shape functions form a partition of unity and the MMLS approximation satisfies the patch test. The invariance properties are important for the accurate computation of the shape functions by using translation and scaling to a canonical domain. We compare the performance of the EFG method based on MMLS, which uses quadratic base functions, to the performance of the EFG method which uses classical MLS with linear base functions, using both 2D and 3D examples. In 2D we solve an elasticity problem which has an analytical solution (bending of a Timoshenko beam) while in 3D we solve an elasticity problem which has an exact finite element solution (unconstrained compression of a cube). We also solve a complex problem involving complicated geometry, non-linear material, large deformations and contacts. The simulation results demonstrate the superior performance of the MMLS over classical MLS in terms of solution accuracy, while shape functions can be computed using the same nodal distribution and support domain size for both methods.

    Original languageEnglish
    Pages (from-to)1197-1211
    Number of pages15
    JournalJournal of Scientific Computing
    VolumeDec 2016
    DOIs
    Publication statusPublished - 28 Dec 2016

    Fingerprint

    Least squares approximations
    Moving Least-squares Approximation
    Element-free Galerkin Method
    Moving Least Squares
    Galerkin methods
    Shape Function
    Elasticity Problem
    Basis Functions
    Invariance
    Element-free Galerkin
    Demonstrate
    Elasticity
    Patch Test
    Partition of Unity
    Square Functions
    Timoshenko Beam
    Galerkin Approximation
    Finite Element Solution
    Large Deformation
    Quadratic Function

    Cite this

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    abstract = "This paper demonstrates that the recently developed modified moving least squares (MMLS) approximation possess the necessary properties which allow its use as an element free Galerkin (EFG) approximation method. Specifically, the consistency and invariance properties for the MMLS are proven. We demonstrate that MMLS shape functions form a partition of unity and the MMLS approximation satisfies the patch test. The invariance properties are important for the accurate computation of the shape functions by using translation and scaling to a canonical domain. We compare the performance of the EFG method based on MMLS, which uses quadratic base functions, to the performance of the EFG method which uses classical MLS with linear base functions, using both 2D and 3D examples. In 2D we solve an elasticity problem which has an analytical solution (bending of a Timoshenko beam) while in 3D we solve an elasticity problem which has an exact finite element solution (unconstrained compression of a cube). We also solve a complex problem involving complicated geometry, non-linear material, large deformations and contacts. The simulation results demonstrate the superior performance of the MMLS over classical MLS in terms of solution accuracy, while shape functions can be computed using the same nodal distribution and support domain size for both methods.",
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    An Element Free Galerkin Method Based on the Modified Moving Least Squares Approximation. / Chowdhury, Habibullah Amin; Wittek, Adam; Miller, Karol; Joldes, Grand Roman.

    In: Journal of Scientific Computing, Vol. Dec 2016, 28.12.2016, p. 1197-1211.

    Research output: Contribution to journalArticle

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