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Based on element-free Galerkin (EFG) method and the complex variable moving least-squares (CVMLS) approximation, the complex variable element-free Galerkin (CVEFG) method for two-dimensional elasticity problems is presented in this paper. With the CVMLS approximation, the trial function of a two-dimensional problem is formed with a one-dimensional basis function. The number of unknown coefficients in the trial function of the CVMLS approximation is less than in the trial function of moving least-squares (MLS) approximation, and we can thus select fewer nodes in the meshless method that is formed from the CVMLS approximation than are required in the meshless method of the MLS approximation with no loss of precision. The formulae of the CVEFG method for two-dimensional elasticity problems is obtained. Compared with the conventional meshless method, the CVEFG method has a greater precision and computational efficiency. For the purposes of demonstration, some selected numerical examples are solved using the CVEFG method.

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Based on the complex variable moving least-square (CVMLS) approximation, the complex variable element-free Galerkin (CVEFG) method for two-dimensional viscoelasticity problems under the creep condition is presented in this paper. The Galerkin weak form is employed to obtain the equation system, and the penalty method is used to apply the essential boundary conditions, then the corresponding formulae of the CVEFG method for two-dimensional viscoelasticity problems under the creep condition are obtained. Compared with the element-free Galerkin (EFG) method, with the same node distribution, the CVEFG method has higher precision, and to obtain the similar precision, the CVEFG method has greater computational efficiency. Some numerical examples are given to demonstrate the validity and the efficiency of the method.

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In this paper, an interpolating complex variable moving least-squares (ICVMLS) method is presented. In the ICVMLS method, the trial function of a two-dimensional problem is formed with a one-dimensional basis function, and the shape function of the ICVMLS method satisfies the property of Kronecker δ function. The ICVMLS method has greater computational efficiency than the moving least-squares (MLS) approximation. Then combining the ICVMLS method with the Galerkin weak form of temperature field problems, an interpolating complex variable element-free Galerkin (ICVEFG) method is proposed. In the ICVEFG method, we can obtain the equation system by applying the essential boundary conditions directly. Compared with the element-free Galerkin (EFG) method and the complex variable element-free Galerkin (CVEFG) method, the ICVEFG method in this paper has higher accuracy and efficiency.