Acta Mechanica Sinica

, Volume 34, Issue 3, pp 462–474

# The dimension split element-free Galerkin method for three-dimensional potential problems

• Z. J. Meng
• H. Cheng
• L. D. Ma
• Y. M. Cheng
Research Paper

## Abstract

This paper presents the dimension split element-free Galerkin (DSEFG) method for three-dimensional potential problems, and the corresponding formulae are obtained. The main idea of the DSEFG method is that a three-dimensional potential problem can be transformed into a series of two-dimensional problems. For these two-dimensional problems, the improved moving least-squares (IMLS) approximation is applied to construct the shape function, which uses an orthogonal function system with a weight function as the basis functions. The Galerkin weak form is applied to obtain a discretized system equation, and the penalty method is employed to impose the essential boundary condition. The finite difference method is selected in the splitting direction. For the purposes of demonstration, some selected numerical examples are solved using the DSEFG method. The convergence study and error analysis of the DSEFG method are presented. The numerical examples show that the DSEFG method has greater computational precision and computational efficiency than the IEFG method.

## Keywords

Dimension split method Improved moving least-squares (IMLS) approximation Improved element-free Galerkin (IEFG) method Finite difference method (FDM) Dimension split element-free Galerkin (DSEFG) method Potential problem

## Notes

### Acknowledgements

This work was supported by the National Natural Science Foundation of China (Grants 11571223, 51404160) and Shanxi Province Science Foundation for Youths (Grant 2014021025-1).

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© The Chinese Society of Theoretical and Applied Mechanics; Institute of Mechanics, Chinese Academy of Sciences and Springer-Verlag GmbH Germany, part of Springer Nature 2018

## Authors and Affiliations

• Z. J. Meng
• 1
• 2
• H. Cheng
• 3
• L. D. Ma
• 4
• Y. M. Cheng
• 1
1. 1.Shanghai Institute of Applied Mathematics and Mechanics, Shanghai Key Laboratory of Mechanics in Energy EngineeringShanghai UniversityShanghaiChina
2. 2.School of Applied ScienceTaiyuan University of Science and TechnologyTaiyuanChina
3. 3.Department of Civil EngineeringShanghai UniversityShanghaiChina
4. 4.School of Materials Science and EngineeringTaiyuan University of Science and TechnologyTaiyuanChina