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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, nonlinear 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 

Pages (fromto)  11971211 
Number of pages  15 
Journal  Journal of Scientific Computing 
Volume  71 
Issue number  3 
DOIs  
Publication status  Published  1 Jun 2017 
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 2 Finished

Biomechanics Meets Robotics: Methods for Accurate and Fast Needle Targeting
Wittek, A., Singh, S., Miller, K., Hannaford, B. & Fichtinger, G.
ARC Australian Research Council
1/01/16 → 31/03/22
Project: Research

Neuroimage as Biomechanical Model  New Real Time Computational Biomechanics of the Brain
Miller, K., Wittek, A., Carey, G. & Kikinis, R.
ARC Australian Research Council
1/01/12 → 31/12/14
Project: Research