Application of MQ-RBF method for solving seepage problems with a new algorithm for optimization of the shape parameter

Document Type : Research Article

Authors

University of Qom

Abstract

The accuracy of the meshless method, Multiquadric, depends completely on the choice of its optimal shape parameter. The purpose of this research is proposing a new algorithm for determining the optimal shape parameter. It resolves some of the previous difficulties, such as depending on the number of computational nodes or an exact solution of the problem, high cost and low accuracy of calculations, being experimental, convergence of classical optimization methods to local optimal points and so on. For this purpose, in addition to introducing a new objective function, Genetic Algorithm(GA) was used and for speeding up the process of its solution, lower bound and upper bound of the shape parameter are suggested as minimum (when the coefficient matrix is not singular) and maximum radius of computational nodes, respectively. The algorithm consists of four steps: 1) producing initial shape parameters by GA in the proposed range, 2) introducing the MQ function with a few numbers of computational points, 3) introducing the MQ function with a large number of computational points, and 4) minimizing the difference between solutions of two functions obtained from the two preceding steps. In the meta-heuristic algorithm, uniform and non-uniform regular distributions of computational nodes have been successfully applied and it was shown that with this approach, an optimal constant shape parameter independent of the number of computational points could be obtained for arbitrary geometries. For verification, examples of homogeneous, inhomogeneous and anisotropic types of the seepage phenomena were solved so that domain decomposition technique was used for inhomogeneous problems and complex geometries. A comparison of results with other exact and numerical solutions showed the high ability and accuracy of the proposed algorithm.

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