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REVIEW

Advances in the Improved Element-Free Galerkin Methods: A Comprehensive Review

Heng Cheng¹, Yichen Yang¹, Yumin Cheng^2,*

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:

Computer Modeling in Engineering & Sciences 2025, 145(3), 2853-2894. https://doi.org/10.32604/cmes.2025.073178

Received 12 September 2025; Accepted 20 November 2025; Issue published 23 December 2025

Abstract

The element-free Galerkin (EFG) method, which constructs shape functions via moving least squares (MLS) approximation, represents a fundamental and widely studied meshless method in numerical computation. Although it achieves high computational accuracy, the shape functions are more complex than those in the conventional finite element method (FEM), resulting in great computational requirements. Therefore, improving the computational efficiency of the EFG method represents an important research direction. This paper systematically reviews significant contributions from domestic and international scholars in advancing the EFG method. Including the improved element-free Galerkin (IEFG) method, various interpolating EFG methods, four distinct complex variable EFG methods, and a series of dimension splitting meshless methods. In the numerical examples, the effectiveness and efficiency of the three methods are validated by analyzing the solutions of the IEFG method for 3D steady-state anisotropic heat conduction, 3D elastoplasticity, and large deformation problems, as well as the performance of two-dimensional splitting meshless methods in solving the 3D Helmholtz equation.

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