Abstract
The interpolating moving least-squares (IMLS) method is discussed in detail, and a simpler formula of the shape function of the IMLS method is obtained. Then, based on the IMLS method and the Galerkin weak form, an interpolating element-free Galerkin (IEFG) method for two-point boundary value problems is presented. The IEFG method has high computing speed and precision. Then error analysis of the IEFG method for two-point boundary value problems is presented. The convergence rates of the numerical solution and its derivatives of the IEFG method are presented. The theories show that, if the original solution is sufficiently smooth and the order of the basis functions is big enough, the solution of the IEFG method and its derivatives are convergent to the exact solutions in terms of the maximum radius of the domains of influence of nodes. For the purpose of demonstration, two selected numerical examples are given to confirm the theories.
Highlights
Conventional computational methods, such as the finite element method (FEM) and the boundary element method (BEM), cannot be applied well to some engineering problems
The interpolating moving least-squares (IMLS) method is discussed in detail
Since the shape function of the IMLS method satisfies the property of Kronecker δ function, the interpolating element-free Galerkin (IEFG) method can apply the essential boundary condition directly
Summary
Conventional computational methods, such as the finite element method (FEM) and the boundary element method (BEM), cannot be applied well to some engineering problems. Most and Bucher enhanced the regularized weighting function to obtain a true interpolation MLS approximation [59] Another possible solution for this problem is the interpolating moving leastsquares (IMLS) method presented by Lancaster and Salkauskas [60]. In the IEFG method, the essential boundary conditions are applied directly and and the number of unknown coefficients in the trial function of the IMLS method is less than that in the trial function of the MLS approximation. Based on the IMLS method of this paper and the Galerkin weak form, an IEFG method for two-point boundary value problems is presented. Since the shape function of the IMLS method satisfies the property of Kronecker δ function, the IEFG method can apply the essential boundary condition directly. For the purpose of demonstration, some selected numerical examples are given to confirm the theory
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