Abstract
Radial basis function (RBF) has been widely used in many scientific computing and engineering applications, for instance, multidimensional scattered data interpolation and solving partial differential equations. However, the accuracy and stability of the RBF methods often strongly depend on the shape parameter. A coupled RBF (CRBF) method was proposed recently and successfully applied to solve the Poisson equation and the heat transfer equation (Appl. Math. Lett., 2019, 97: 93–98). Numerical results show that the CRBF method completely overcomes the troublesome issue of the optimal shape parameter that is a formidable obstacle to global schemes. In this paper, we further extend the CRBF method to solve the elastostatic problems. Discretization schemes are present in detail. With two elastostatic numerical examples, it is found that both numerical solutions of the CRBF method and the condition numbers of the discretized matrices are almost independent of the shape parameter. In addition, even if the traditional RBF methods take the optimal shape parameter, the CRBF method achieves better accuracy.
Highlights
A radial function φ(rj) is a function of the Euclidean norm rj ‖x − xj‖2, where x ∈ Rn is the center point and xj ∈ Rn is a point in the influence domain of x
In the past few decades, the radial function has been used as a special basis function for solving interpolation problems and discretizing partial differential equations (PDEs) by means of collocation techniques [1,2,3], which is referred to as the radial basis function (RBF) method in this paper. e interest in the RBF method has three principal reasons [4]: (i) the approximate value by using RBF can be estimated without using meshes; (ii) RBF gives very accurate results both for interpolation problems and for solving partial differential equations; (iii) there is enough flexibility in the choice of basis functions
Our findings show that the coupled RBF (CRBF) method has better accuracy and convergence rate than the traditional RBF method even if the latter takes the optimal shape parameter
Summary
A radial function φ(rj) is a function of the Euclidean norm rj ‖x − xj‖2, where x ∈ Rn is the center point and xj ∈ Rn is a point in the influence domain of x. For the former group, the infinitely smooth RBFs (such as Multiquadric (MQ), Inverse Multiquadric (IMQ), and Gaussian (GA)) lead to spectral convergence, which is a great advantage in actual applications These types of RBFs contain a user-defined positive shape parameter, called c, which controls the stability and accuracy of the RBFs approximation. Many researchers tried to present some empirical formulas based on the number and distribution of data points to select a good value for the shape parameter; see [12,13,14] for Mathematical Problems in Engineering. Our findings show that the CRBF method has better accuracy and convergence rate than the traditional RBF method even if the latter takes the optimal shape parameter.
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