5 - Numerical solution of volume integral equations for static fields in heterogeneous media

Abstract

This chapter is devoted to an efficient numerical method for solution of volume integral equations for static fields in heterogeneous media. Approximation of the fields by linear combinations of Gaussian radial functions reduces the problems to systems of linear algebraic equations for the coefficients of the approximation (the discretized problems). For the Gaussian functions, the elements of the matrices of the discretized problems are calculated in explicit analytical forms. Because of large dimensions of the discretized problems, only iterative methods can be used for the solution. The algorithms of the appropriate iterative methods are presented, and FFT algorithms for fast calculation of the matrix-vector products in the process of iterations are adopted. Examples of numerical solutions of the volume integral equations for electrostatic, elastic, thermo-elastic, and elasto-plastic fields in heterogeneous media are considered. Accuracy of the method is demonstrated by comparison with other numerical methods and exact solutions in particular cases.