Keywords: perturbation, inclusion, composite medium, integral equation, Toeplitz, boundary point method, fast Fourier transform, meshless.

Currently a number of computational methods to estimate mechanical fields inside the composite materials are available. Usually, the main role is played by the finite element method (FEM) based on differential equations supplied by the assumption of periodic fields. On the other hand, there are a number of applications in engineering practice where a strongly non-periodic character of the composite medium can also be encountered (e.g. biological materials, foams, concrete, etc.). In such a case the problem formulation via integral equations is more convenient. This article briefly presents the boundary point method (BPM) discretization of certain integral operators used to solve a simple two-dimensional problem of infinite medium with separated inclusions. The method is meshless. So, it can be used in the framework of strongly perturbed materials much more easily than the conventional FEM.

Similarly to the previously published works by Maz'ya [1] or Kanaun [2] the theory of integral equations and discretization of their solution using Gauss approximating functions is presented. This numerical algorithm is based on the Toeplitz form of the resulting matrix and the fast Fourier transform (FFT) technique used to solve some steps of employed iterative solvers.

This contribution briefly presents the basic theoretical and practical aspects of the BPM. Particularly, it starts from the formulation of basic equilibrium equation of an isotropic elastic deformable body. Consequently, the governing integral equations of infinite composite medium in terms of displacements and strains, respectively, are introduced and followed by the boundary point discretization of the strain field. This step is followed by the main topic of this contribution, which is the domain decomposition and application of the balance iterative procedure for the solution of a problem with multiple inclusions. Calculation of particular values of the discretized operators, whose densities denote the basis functions, is carried out using a minimal residual (MRM) iterative solver. Clearly, the final linearized system can be established in many different ways but in the view of the BPM the regular discretization grid is convenient to use because of the dependence of that system only on an equidistant distance h. Then, the FFT technique can be employed with a great benefit to solve the resulting system of algebraic equations. The accuracy of such an approach is checked by comparing the calculated local strains with the FEM predictions for a two-dimensional three-inclusion problem.

1

V. Maz'ya, "A new approximation method and its application to the calculation of volume potentials", Boundry point method, 3. DFG-koloqium des DFG-Forschungs schwerpunktes "Randelelementmethoden", 1991.

2

S.K. Kanaun, S.B. Kochekseraii, "A Numerical Method for the Solution of 3D-integral Equations of Electro-static Theory Based on Gaussian Approximating Functions", Applied Mathematics and Computation, Elsevier, 184, 754-768, 2007. doi:10.1016/j.amc.2006.05.175

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