### 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 language | English |
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Pages (from-to) | 1197-1211 |

Number of pages | 15 |

Journal | Journal of Scientific Computing |

Volume | Dec 2016 |

DOIs | |

Publication status | Published - 28 Dec 2016 |

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*Journal of Scientific Computing*, vol. Dec 2016, pp. 1197-1211. https://doi.org/10.1007/s10915-016-0337-z

**An Element Free Galerkin Method Based on the Modified Moving Least Squares Approximation.** / Chowdhury, Habibullah Amin; Wittek, Adam; Miller, Karol; Joldes, Grand Roman.

Research output: Contribution to journal › Article

TY - JOUR

T1 - An Element Free Galerkin Method Based on the Modified Moving Least Squares Approximation

AU - Chowdhury, Habibullah Amin

AU - Wittek, Adam

AU - Miller, Karol

AU - Joldes, Grand Roman

PY - 2016/12/28

Y1 - 2016/12/28

N2 - 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.

AB - 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.

KW - Element free Galerkin

KW - Meshless method

KW - Modified moving least squares

KW - Numerical methods

UR - http://www.scopus.com/inward/record.url?scp=85007415857&partnerID=8YFLogxK

U2 - 10.1007/s10915-016-0337-z

DO - 10.1007/s10915-016-0337-z

M3 - Article

VL - Dec 2016

SP - 1197

EP - 1211

JO - Journal of Scientific Computing

JF - Journal of Scientific Computing

SN - 0885-7474

ER -