Abstract
By introducing the dimension splitting method (DSM) into the improved interpolating moving least-squares (IMLS) method with nonsingular weight function, a dimension splitting–interpolating moving least squares (DS-IMLS) method is first proposed. Since the DSM can decompose the problem into a series of lower-dimensional problems, the DS-IMLS method can reduce the matrix dimension in calculating the shape function and reduce the computational complexity of the derivatives of the approximation function. The approximation function of the DS-IMLS method and its derivatives have high approximation accuracy. Then an improved interpolating element-free Galerkin (IEFG) method for the two-dimensional potential problems is established based on the DS-IMLS method. In the improved IEFG method, the DS-IMLS method and Galerkin weak form are used to obtain the discrete equations of the problem. Numerical examples show that the DS-IMLS and the improved IEFG methods have high accuracy.
Highlights
The construction of approximation functions in the meshless method is only related to nodes and independent of the mesh, so the meshless method has the advantages of no grid reconstruction and high computational accuracy [1,2,3,4]
This paper aims to propose a new hybrid method to obtain the shape function of the meshless method by incorporating the dimension splitting method (DSM) into the improved interpolating moving least-squares (IMLS) method
(1) By coupling the DSM and improved IMLS methods, this paper aims to propose a new hybrid method to obtain the shape function of the meshless method, which is called the “dimension splitting
Summary
The construction of approximation functions in the meshless method is only related to nodes and independent of the mesh, so the meshless method has the advantages of no grid reconstruction and high computational accuracy [1,2,3,4]. The meshless method has become an essential numerical method in scientific and engineering calculation problems [5,6,7,8]. Many meshless methods have been proposed based on different construction methods of the shape function or discretization approach of the problem to be solved. The smoothed particle hydrodynamics (SPH) method [9], moving least squares (MLS) approximation [10], point interpolation method (PIM) [11], and radial basis function [12,13,14,15] are the widely used method to construct the meshless approximation. The major shortcoming of the MLS is a lack of Kronecker delta function property. The approximation of the PIM satisfies the properties of delta function. When the node arrangement is not very suitable, the matrix singularity is likely to occur in the calculation of the shape function for PIM
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