Complex Variable Moving Least-squares Research Articles

A COMPLEX VARIABLE MESHLESS MANIFOLD METHOD FOR FRACTURE PROBLEMS

Based on the complex variable moving least-squares (CVMLS) approximation and the finite cover theory, the complex variable meshless manifold method (CVMMM) for fracture problems is presented in this paper. The CVMMM employs two cover systems which are the mathematical cover system and the physical cover system. The shape function in the CVMMM is derived with the CVMLS approximation and the finite cover theory. The finite cover theory is used to model cracks which lead to interior discontinuous displacements. At the tip of a crack of a problem, we use the analytical solution near the tip of a crack to extend the trial function of the CVMMM, then the corresponding approximation function is obtained. From the minimum potential energy principle, the corresponding formulae of the CVMMM for fracture problems are presented. Some numerical examples are presented to demonstrate the method in this paper.

Complex variable boundary element-free method for two-dimensional elastodynamic problems

We proposed a new direct meshless boundary integral equation technique – the complex variable boundary element-free method (CVBEFM) based on the complex variable moving least-squares (CVMLS) approximation and the boundary element-free method (BEFM), to study the two-dimensional elastodynamic problems. 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 that in the trial function of the moving least-squares (MLS) approximation. Therefore it requires fewer nodes in the meshless method which formed from the CVMLS approximation than that formed from the MLS approximation with no lose of precision. The Laplace transform is used to formulate the boundary integral equations of the two-dimensional elastodynamics and then the formulae of the CVBEFM for two-dimensional elastodynamic problems are derived. The CVBEFM is a direct numerical method in which the basic unknown quantities are the real solutions of the nodal variables. Moreover in the CVBEFM, the boundary conditions can be applied directly and easily that leads to a greater computational precision. In this paper, we selected a few numerical examples to illustrate the applicability of the CVBEFM.

THE COMPLEX VARIABLE ELEMENT-FREE GALERKIN (CVEFG) METHOD FOR TWO-DIMENSIONAL ELASTICITY PROBLEMS

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.

Complex variable moving least‐squares method: a meshless approximation technique

AbstractBased on the moving least‐squares (MLS) approximation, we propose a new approximation method—the complex variable moving least‐squares (CVMLS) approximation. 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 the 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 meshless method that is derived from the CVMLS approximation also has a greater computational efficiency. From the CVMLS approximation, we propose a new meshless method for two‐dimensional elasticity problems—the complex variable meshless method (CVMM)—and the formulae of the CVMM for two‐dimensional elasticity problems are obtained. Compared with the conventional meshless method, the CVMM has a greater precision and computational efficiency. For the purposes of demonstration, some selected numerical examples are solved using the CVMM. Copyright © 2006 John Wiley & Sons, Ltd.