Computational & Technology Resources
an online resource for computational,
engineering & technology publications 

CivilComp Proceedings
ISSN 17593433 CCP: 89
PROCEEDINGS OF THE SIXTH INTERNATIONAL CONFERENCE ON ENGINEERING COMPUTATIONAL TECHNOLOGY Edited by: M. Papadrakakis and B.H.V. Topping
Paper 127
Calculation of Elastic Stresses and Strains inside a Medium with Multiple Isolated Inclusions J. Novák
Department of Mechanics, Faculty of Civil Engineering, Czech Technical University in Prague, Czech Republic , "Calculation of Elastic Stresses and Strains inside a Medium with Multiple Isolated Inclusions", in M. Papadrakakis, B.H.V. Topping, (Editors), "Proceedings of the Sixth International Conference on Engineering Computational Technology", CivilComp Press, Stirlingshire, UK, Paper 127, 2008. doi:10.4203/ccp.89.127
Keywords: perturbation, inclusion, composite medium, integral equation, Toeplitz, boundary point method, fast Fourier transform, meshless.
Summary
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 nonperiodic 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 twodimensional 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 twodimensional threeinclusion problem. References
purchase the fulltext of this paper (price £20)
go to the previous paper 
