Mathematical Problems in Engineering
Volume 2012 (2012), Article ID 632072, 20 pages
Research Article

An Improved Collocation Meshless Method Based on the Variable Shaped Radial Basis Function for the Solution of the Interior Acoustic Problems

1State Key Laboratory of Digital Manufacturing Equipment and Technology, School of Mechanical Science and Engineering, Huazhong University of Science and Technology, Wuhan 430074, China
2Reactor Engineering Experiment Technology Center, China Nuclear Power Technology Research Institute, Shenzhen, Guangzhou 518026, China

Received 3 August 2012; Accepted 13 September 2012

Academic Editor: Jyh Horng Chou

Copyright © 2012 Shuang Wang et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.


As an efficient tool, radial basis function (RBF) has been widely used for the multivariate approximation, interpolating continuous, and the solution of the particle differential equations. However, ill-conditioned interpolation matrix may be encountered when the interpolation points are very dense or irregularly arranged. To avert this problem, RBFs with variable shape parameters are introduced, and several new variation strategies are proposed. Comparison with the RBF with constant shape parameters are made, and the results show that the condition number of the interpolation matrix grows much slower with our strategies. As an application, an improved collocation meshless method is formulated by employing the new RBF. In addition, the Hermite-type interpolation is implemented to handle the Neumann boundary conditions and an additional sine/cosine basis is introduced for the Helmlholtz equation. Then, two interior acoustic problems are solved with the presented method; the results demonstrate the robustness and effectiveness of the method.