Application of RBF Multiquadric method for solving seepage problems using a new algorithm for Optimization of the shape parameter

Accuracy of the Multiquadric method depends strongly on the choice of the shape parameter. This research proposes a new algorithm for determining the optimal shape parameter. It resolves some of the difficulties such as dependence on the number of computational nodes, existence of an analytical solution of the problem, high cost and low accuracy of calculations, convergence of classical optimization methods to local optimal points and so on. In this regard, the Genetic Algorithm (GA) has been used and, lower and upper bounds of the shape parameter are suggested as minimum (when the coefficient matrix is not singular) and maximum of Euclidean radius, for speeding up the solution process.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 of solutions of two functions obtained from the two preceding steps. In the meta-heuristic algorithm, distribution of points is uniform such that each three points are the vertices of equilateral triangles in the problem domain.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. The study shows the high capability and accuracy of the proposed algorithm. In this approach, the optimal shape parameter can be achieved independent of the number of computational points for arbitrary geometries.