Meshfree finite element techniques using novel families of compactly supported radial basis functions.

Fernando César Meira Menandro¹; Wesley Ferrari¹; Nikolas Garcia Repsold¹

¹ Universidade Federal do Espírito Santo

doi:10.20906/CPS/CILAMCE2015-0434

Resumo

Abstract The use of compactly supported radial basis functions (CSRBF) has spread into the scientific approximation community for their localization properties and their mathematical simplicity, since they just depend on a single variable (radius) and, unless the radial basis is not compactly supported, affect the results only on a neighborhood (support) of the given center [1,2]. However, the mathematical derivation of continuous functions for different continuity requirements can be quite cumbersome [3]. In this paper two new families of CSRBFs are presented for which the continuity requirements are greatly simplified. The first function family proposed consists of a combination of different inverse functions, similar to inverse quadric functions, and is thus called inverse family. The second family is called the rational family, in which the functions are obtained as a ratio of two polinomial functions. The proposed functions are used on a meshfree finite element software which takes advantage of their simplicity. The results seem to demonstrate the accuracy and convergence of the proposed functions when compared to some CSRBFs presented on the subject literature[4]. References: [1] Iske, A., Multiresolution Methds in Scattered Data Modeling, Lecture Notes in Computational Science and Engineering, Vol. 37, Springer Verlag, Berlin, Germany, 2004. [2] Fasshauer, G. E., Meshfree Approximation Methods with MATLAB, Interdisciplinary Mathematical Sciences, Vol. 6, World Scentific Publishing, London, UK, 2007. [3] Wendland, H., Scattered Data Approximation, Cambridge Monographs on Applied and Computational Mathematics, Vol. 17, Cambridge University Press, Cambridge, UK, 2005. [4] Menandro, F.C.M., Loeffler Neto, C.F., and Bulcao, A. Meshfree finite element solution of differential equations by radial basis functions interpolation. In Proceedings of the XXVI CILAMCE, 2011.

Palavras-chave: Meshfree finite element methods; Compactly supported radial basis functions