The Interpolating Complex Variable Element-Free Galerkin Method for Temperature Field Problems

Abstract

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.

Mixed discrete least squares meshless method for solving the linear and non-linear propagation problems

“…(15) leads to the following system of equations to be solved for C n+1 and q n+1 : C n+1 + t(V n :q n+1 r:q n+1 ) = g n ; rC n+1 q n+1 = 0: (17) The set of Eq. (17) can be written in a compact form as:…”

“…The Galerkin procedure was utilized to discretize the governing PDEs, leading to symmetric coe cient matrices. This method was successfully used to investigate various engineering problems such as static and dynamic analyses of shell structures [16], temperature eld problems [17], 2D fracture problems [18], and unsteady non-linear heat transfer [19]. The rate of convergence of the EFG method was shown to be higher than that of FEM [19].…”

Steady heat conduction analyses using an interpolating element-free Galerkin scaled boundary method

“…Based on different approximation functions, many kinds of meshless methods have been proposed. Some representative examples are the element-free Galerkin (EFG) method [15,16], the meshless local Petrov-Galerkin method (MLPG) [17,18], the reproducing kernel particle method (RKPM) [19] and the point interpolation method (PIM) [20].…”

“…To use the MLS approximation, only an array of nodes is required in the domain under consideration. The EFG method exhibits some advantages such as increasing accuracy and rate of convergence, requiring no post-processing to obtain a smooth gradient field [15,16]. In order to unite advantages of the scaled boundary method and the EFG method, a novel element-free Galerkin scaled boundary method (EFG-SBM) [21][22][23] is proposed through the incorporation of the EFG approach into the scaled boundary method.…”

The complex variable meshless local Petrov–Galerkin method for elasticity problems of functionally graded materials

“…Consequently, the efficiency is improved. The CVEFG method in the published literature has been demonstrated to be quite successful in solving science and engineering problems [21][22][23][24][25][26][27][28][29][30][31][32][33]. Introducing the complex variable theory into the reproducing kernel particle method, Chen et al [34] proposed the complex variable reproducing kernel particle method (CVRKPM) and take advantage of the CVRKPM in the analysis of transient heat conduction problems [35], potential problems [36], elastoplasticity problems [37].…”