Direct Meshless Method Research Articles

A Direct Meshless Method for Solving Two-Dimensional Second-Order Hyperbolic Telegraph Equations

In this paper, a direct meshless method (DMM), which is based on the radial basis function, is developed to the numerical solution of the two-dimensional second-order hyperbolic telegraph equations. Since these hyperbolic telegraph equations are time dependent, we present two schemes for the basis functions from radial and nonradial aspects. The first scheme is fulfilled by considering time variable as normal space variable to construct an “isotropic” space-time radial basis function. The other scheme considered a realistic relationship between space variable and time variable which is not radial. The time-dependent variable is treated regularly during the whole solution process and the hyperbolic telegraph equations can be solved in a direct way. Numerical experiments performed with the proposed numerical scheme for several two-dimensional second-order hyperbolic telegraph equations are presented with some discussions, which show that the DMM solutions are converging very fast in comparison with the various existing numerical methods.

A Burton-Miller boundary element-free method for Helmholtz problems

A Burton-Miller boundary element-free method is developed by using the Burton-Miller formulation for meshless and boundary-only analysis of Helmholtz problems. The method can produce a unique solution at all wavenumbers and is valid for Dirichlet, Neumann and mixed problems simultaneously. An efficient numerical integration procedure is presented to handle both strongly singular and hypersingular boundary integrals directly and uniformly. Numerical results reveal that this direct meshless method only involves boundary nodes and can deal with Helmholtz problems at extremely large wavenumbers.

An interpolating boundary element-free method for three-dimensional potential problems

This paper begins by discussing an improved interpolating moving least-square (IIMLS) method and the properties of its shape function. In the IIMLS method, the shape function is of delta function property and the weight function is nonsingular, so it overcomes the drawbacks in both the moving least-square approximation and the interpolating moving least-square method. Then combining boundary integral equations with the IIMLS method, an interpolating boundary element-free method (IBEFM) is developed for three-dimensional potential problems. In the IBEFM, only a nodal data structure on the boundary face of a domain is required. Unlike the boundary node method (BNM), the IBEFM is a direct meshless method in which the primary unknown quantities are real solutions of nodal variables, and boundary conditions can be imposed directly and easily, which leads to a greater computational precision. Besides, the number of both unknowns and system equations in the IBEFM is only half of that in the BNM, and thus the computing speed and efficiency are increased. Numerical examples on curve/surface fittings and potential problems indicate that the efficiency and convergence rate of the present methods is high.