Metodologias baseadas no mínimo residuo para escolha do parâmetro de forma utilizado em RBF'S aplicadas no método assimétrico livre de malhas (MESHLESS)

Abstract

The main objective of this research is to propose a new methodology to determine the shape parameter c in the Meshless numerical method using examples in one and two dimensional. For each dimension, it was developed two computers codes using Scilab 5.3.3. It was used a general format that represents various kinds of partial differential equations. The Meshless method choosen was the Asymmetric Method using Multiquadric RBF’s and Inverse Multiquadric RBF’s. It was proposed to minimize the residues of the domain and the boundary in order to find the better c parameter. It was used two methods: Method 1 uses both domain and contour residues while Method 2 uses both residues too, but contour residues is calculated like it was domain residue. For 1-dimension, three equations with yours exacts solutions were used. For 2-dimensions, the Laplace equation was used to compare the numerical results from model in three cases: For Cases 1 and 2 (hypothetic cases), the exact solution was known, therefore it was possible to compare the analytical solution with the solution achieved by using the residue methodology proposed. Case 3 is a real scenario for which the methodology was applied to simulate the hydraulic head situation in a small region in the Araripe Sedimentary Basin, situated in the state of Ceará-Brazil. For 1-dimension cases, method 2 had great numerical results when they were compared with the analytic solution. Although, in 2-dimension problems, method 1 had good numerical results, it was the best one, most cases the obtained shape parameter was the same as the best one obtained with the exact solution. In the real case application, the results indicate good potential for methodology. The percentage of errors was about 5%. The quantity of nodes and contour condition influence the numerical results.