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# A Fast Element-Free Galerkin Method for 3D Elasticity Problems

Zhijuan Meng1, Yanan Fang1, Yumin Cheng2,*
1 School of Applied Science, Taiyuan University of Science and Technology, Taiyuan, 030024, China
2 Shanghai Key Laboratory of Mechanics in Energy Engineering, Shanghai Institute of Applied Mathematics and Mechanics, School of Mechanics and Engineering Science, Shanghai University, Shanghai, 200072, China
* Corresponding Author: Yumin Cheng. Email:
(This article belongs to this Special Issue: Numerical Methods in Engineering Analysis, Data Analysis and Artificial Intelligence)

Computer Modeling in Engineering & Sciences 2022, 132(1), 55-79. https://doi.org/10.32604/cmes.2022.019828

Received 18 October 2021; Accepted 15 December 2021; Issue published 02 June 2022

### Abstract

In this paper, a fast element-free Galerkin (FEFG) method for three-dimensional (3D) elasticity problems is established. The FEFG method is a combination of the improved element-free Galerkin (IEFG) method and the dimension splitting method (DSM). By using the DSM, a 3D problem is converted to a series of 2D ones, and the IEFG method with a weighted orthogonal function as the basis function and the cubic spline function as the weight function is applied to simulate these 2D problems. The essential boundary conditions are treated by the penalty method. The splitting direction uses the finite difference method (FDM), which can combine these 2D problems into a discrete system. Finally, the system equation of the 3D elasticity problem is obtained. Some specific numerical problems are provided to illustrate the effectiveness and advantages of the FEFG method for 3D elasticity by comparing the results of the FEFG method with those of the IEFG method. The convergence and relative error norm of the FEFG method for elasticity are also studied.

### Keywords

Improved element-free Galerkin method; dimension splitting method; finite difference method; fast element-free Galerkin method; elasticity

Meng, Z., Fang, Y., Cheng, Y. (2022). A Fast Element-Free Galerkin Method for 3D Elasticity Problems. CMES-Computer Modeling in Engineering & Sciences, 132(1), 55–79.

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