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Fast calculation of electro thermo static and elasticity fields in 3D-medium with isolated inclusions using application of Gaussian approximating functions

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The problem of calculation of electro and thermo static fields in an infinite homogeneous medium with a heterogeneous isolated inclusion (Kanaun et al) has shown to be reduced to the solution of integral equations for the fields inside the inclusion using Gaussian functions (V. Mazya) for the approximation of the unknown fields. Using this approach coefficients of the matrix of the discretized system will be obtained in closed analytical forms. Only information necessary to carry out the method is the coordinates of the centers of the Gaussian functions (nodes) in the region occupied by the inclusion ie. a mesh-free method. Using a regular grid of nodes the matrix of the discretized problem will have a Teoplitzs structure. Hence Fast Fourier Transform (FFT) technique can be used for the calculation of the matrix-vector products within an iterative solution of the system of linear algebraic equations of the discretized problem. The proposed algorithm is simple fast and does not require much computer memory. In practice this has led to over ten folds reduction in the required computational time and the allocated memory space and enabling consideration of very fine grids not possible with other tried solution methods. Comparisons of the numerical and exact solutions for electrostatic fields inside spherical inclusions with step changing properties are presented here. Second boundary value problem of elasticity for 3D-bodies with cracks is another problem where this approach has been applied successfully. References:S. Kanaun and S. Babaii A numerical method for the solution of thermo and electro static problems for a medium with isolated inclusions Journal of Computational Physics 192 471-493 (2003).V. Mazya Approximate approximation in The Mathematics of Finite Elements and Applications. highlights 1993 edited by J.R. Whitman 77 Wiley Chichester (1994).S. Kanaun A. Markov and S. Babaii An efficient numerical method for the solution of the second boundary value problem of elasticity for 3D- bodies with cracks Int J Fract DOI 10.1007/s10704-013-9885-5