Bending analysis of quasicrystal plates using adaptive radial basis function method

A novel mesh-free kernel method is presented for solving function interpolation problems and partial differential equations (PDEs). Despite the simplicity and accuracy of the radial basis function (RBF) method, two main difficulties have been identified: the selection of the scale or shape parameter and the ill-conditioning issue. To address these challenges, an effective condition number (ECN) strategy is proposed for the scale parameter selection, utilizing the ill-conditioning issue to minimize errors. The study demonstrates that this method performs exceptionally well with severely ill-conditioned matrices, making it suitable challenging engineering problems. However, when the problem is severely ill-conditioned, an accurate and efficient method for determining the minimal singular values used in the ECN formulation is required. To tackle this, we utilize an efficient algorithm called minsv, which specifically computes the minimal singular value. The significance of accurately computing the smallest singular value is demonstrated through two interpolation examples, involving non-smooth and smooth test functions with various RBFs and data points. Furthermore, the RBF method is applied to solve a quasicrystal problem, which presents a challenging engineering problem with four unknown parameters, and excellent results are obtained. Finally, we illustrate that using MATLAB's svd function can potentially produce completely incorrect minimal singular values. The codes implemented are available at https://doi.org/10.24433/CO.8716692.v1.