Combining the local boundary integral equation with the improved moving least-squares (IMLS) approximation, an improved local boundary integral equation (ILBIE) method for two-dimensional potential problems is presented in this paper. In the IMLS approximation, the weighted orthogonal functions are used as basis functions. The IMLS approximation has greater computational efficiency and precision than the existing moving least-squares (MLS) approximation and does not lead to an ill-conditioned equations system. The corresponding formulae of the ILBIE method are obtained. Comparing with the conventional local boundary integral equation (LBIE) method, the ILBIE method is a direct meshless boundary integral equation method in which the basic unknown quantity is the real solution of the nodal variables, and the boundary conditions can be implemented directly and easily as in the finite element method. The ILBIE method has greater computational efficiency and precision. Some numerical examples to demonstrate the efficiency of the method are presented in this paper.
Based on the complex variable moving least-square (CVMLS) approximation and a local symmetric weak form, the complex variable meshless local Petrov—Galerkin (CVMLPG) method of solving two-dimensional potential problems is presented in this paper. In the present formulation, 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-square (MLS) approximation. The essential boundary conditions are imposed by the penalty method. The main advantage of this approach over the conventional meshless local Petrov—Galerkin (MLPG) method is its computational efficiency. Several numerical examples are presented to illustrate the implementation and performance of the present CVMLPG method.
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