The Complex Variable Element-Free Galerkin Methods: A Comprehensive Review and Recent Advances

As a significant improvement and supplement to the element-free Galerkin (EFG) method, the complex variable element-free Galerkin (CVEFG) method and its improved versions introduce complex variable basis functions when constructing shape functions. These methods are used not only to approximate vector functions but also to approximate scalar functions by using only the real part of the shape functions. They demonstrate higher accuracy than the EFG method in solving linear and nonlinear mechanics problems, positioning them as an important direction for improving the EFG method. This paper reviews the advances in the complex variable moving least-squares (CVMLS) approximation and its further improvements, as well as those of the CVEFG methods based on these CVMLS approximations. The formulas for the CVMLS approximation and its improvements are provided, along with those for the CVEFG and improved methods. Several numerical examples are presented to demonstrate the effectiveness and advantages of the improved CVEFG methods. Finally, relevant conclusions and future prospects are discussed.