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

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) meshless 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 that 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). 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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“…The right-hand-side edge is rigidly constrained. Detailed description of this example is provided in Chowdhury et al (2017). The model was implemented using Meshless Total Lagrangian Explicit Dynamics (MTLED) method developed by Horton et al (2010), Chowdhury et al (2015), Joldes et al (2015a, c) with the Modified Moving Least Square (MMLS) shape functions.…”

“…This problem is not limited Fig. 15 Computation of deformations of cantilever beam subjected to bending using Galerkin-type Meshless Total Lagrangian Explicit Dynamics (MTLED) method developed by Horton et al (2010), Chowdhury et al (2015), Joldes et al (2015a, c), Chowdhury et al (2017) with the Modified Moving Least Squarse (MMLS) shape functions by Chowdhury et al (2017). The figure shows the differences (in mm) between the analytical solution from Timoshenko and Goodier (1970) and the deformations computed using MTLED.…”

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