This paper deals with homogenization of non linear fibre-reinforced composites in the coupled thermomechanical field. For this kind of structures, i.e. inclusions randomly dispersed in a matrix, the self consistent methods are particularly suitable to describe the problem. Usually, in the framework of the self consistent scheme the homogenized material behaviour is obtained with a symbolic approach. For the non linear case, that method may become tedious. This paper presents a different, fully numerical procedure. The effective properties are determined by minimizing a functional expressing the difference (in some chosen norm) between the solution of the heterogeneous problem and the equivalent homogenous one. The heterogeneous problem is solved with the Finite Element method, while the second one has its analytical solution. The two solutions are written as a function of the (unknown) effective parameters, so that the final global solution is found by iterating between the two single solutions. Further, it is shown that the considered homogenization scheme can be seen as an inverse problem and Artificial Neural Networks are used to solve it.

*Keywords:* Generalized Self-Consistent-Like method, non-linear homogenization, Artificial Neural Networks, inverse problems, thermo-mechanical analysis, multiscale modelling, unsmearing.

E. Szymczyk, A. Derewońko, G. Sławiński, J. Jachimowicz. Numerical estimation of internal stress relieving in destructive test. (Received in the final form January 18, 2010) CAMES 2009 (16)

The paper deals with the analysis of residual stress fields in the riveted joint and the estimation of the internal stress magnitude releasing by partial and complete removing of the rivet material. Stress relieving causes deformations around the rivet hole, which can be measured and compared to the deformation state before removal. Numerical FE simulations of the upsetting process are carried out to determine the residual stress and strain fields. The contact with friction is defined between the mating parts of the joint. Non-destructive testing methods are used in combination with numerical calculations.

*Keywords:* riveted joint, FEM local model, destructive and non-destructive methods.

A. Maciąg. Solving thermoelasticity problems by means of Trefftz functions. (Received in the final form January 15, 2010) CAMES 2009 (16)

The paper presents a new method of approximate solving of the two- and three-dimensional thermoelasticity problems in a finite body. The method presented here can be used for solving direct and inverse problems as well. System of thermoelasticity equations is reduced to the system of wave equations where the temperature occurs as inhomogeneity in one of them. The thermal field is approximated by linear combination of heat polynomials (Trefftz functions for heat conduction equation). The system of wave equations is solved by means of wave polynomials (Trefftz functions for wave equation). Convergence of the T-functions method is proved. The procedure of solving direct and inverse thermoelasticity problems by means of Trefftz functions is tested on an example. Sensitiveness of the method according to data disturbance was checked.

M. Sokała. Solution of 2D non-homogenous wave equation by using polywave functions. (Received in the final form February 9, 2010) CAMES 2009 (16)

The paper presents a specific technique of solving the non-homogenous wave equation with the use of Trefftz functions for the wave equation. The solution was presented as a sum of a general integral and a particular integral. The general integral was expressed in the form of a linear combination of Trefftz functions for the wave equation. In order to obtain the particular integral polywave functions were used. They were generated by using the inverse operator L^{-1} of the equation taking into consideration the Trefftz functions.

*Keywords:* polywave functions, Trefftz functions, wave polynomials, wave equation.

J. Ptaszny and P. Fedeliński. Stress analysis of linear elastic structures by the fast multipole boundary element method. (Received in the final form January 15, 2010) CAMES 2009 (16)

In this paper a fast multipole boundary element method (FMBEM) analysis of internal stress in twodimensional linear elastic structures is presented. The expansions of the potentials occurring in the stress integral equation are obtained by the differentiation of local series built for the displacement equation potentials, and application of the strain-displacement and stress-strain relations. Results of the analysis are presented. To illustrate the accuracy of the method a stress concentration problems are considered, which are a square plate with a circular hole under tension, and a gear. The application of the FMBEM can reduce the analysis time in relation to the conventional BEM case, providing similar accuracy. Presented method can be applied in the BEM analysis of non-linear structures, which requires the evaluation of internal strains or stresses.

*Keywords:* linear elasticity, stress analysis, boundary element method, fast multipole method.

T. Krykowski. Modelling of concrete cover degradation caused by rebar?s corrosion- multicomponent media theory approach. (Received in the final form January 23, 2010) CAMES 2009 (16)

This paper presents the model based on the theory of multicomponent media that allows modelling of rebar corrosion processes. The presented model extends and consolidates the dissertations that have been described in the papers [8-10]. The aim of the present work is a creation of the model consistent with the thermodynamics of multicomponent media with internal parameters, allowing description and numerical modelling of reinforced concrete structures degradation as the result of corrosion by using FEM.

*Keywords:* theory of plasticity, corrosion of reinforced concrete, mass transport, concrete cover splitting.

K. Grysa, A. Maciąg, B. Maciejewska. Trefftz function for solving a quasi-static inverse problem of thermal stresses. (Received in the final form February 11, 2010) CAMES 2009 (16)

The problem of thermal stresses in a hollow cylinder is considered. The problem is two-dimensional and the cross-section of the hollow cylinder is approximated as a long and thin rectangle as the ratio of the inner and outer radiuses is close to one. On the outer boundary of the hollow cylinder the heat source moves with a constant velocity. In the case of the rectangle the heat source moves on the upper side and the conditions of equality of temperatures and heat fluxes are assumed on the left and right boundaries. The stresses are to be found basing on the temperature measured inside the considered region, which means that an inverse problem is considered. Both for the temperature field and the displacements and stresses the finite element method is used. Thermal displacement potentials are introduced to find displacements and stresses. In order to construct the base functions in each element the Trefftz functions are used. For the temperature field the time-space finite elements are used and for the thermal displacement potentials the spatial elements are applied. Thanks to the use of the Trefftz functions a low-order approximation has given a solution very close to the exact one.

*Keywords:* Trefftz function, finite element method, thermal stresses, inverse problem, heat polynomials.

M. Jasiński. Sensitivity analysis of transient bioheat transfer with perfusion rate dependent on tissue injury. (Received in the final form January 18, 2010) CAMES 2009 (16)

The sensitivity analysis of transient temperature field in the tissue domain with respect to its thermophysical parameters is discussed. In particular, the influence of tissue specific heat, thermal conductivity, perfusion rate and metabolic heat source on the temperature distribution is considered. In order to determine the influence of variations of these parameters on temperature distribution the direct approach of sensitivity analysis is applied. Perfusion rate is treated as dependent on tissue injury which is estimated on the basis of Arrhenius integral. On the stage of numerical realization the boundary element method is used. In the final part of paper the results obtained are shown.

*Keywords:* bioheat transfer, Arrhenius scheme, tissue necrosis, boundary element method.

M. M. Kamiński, J. Szafran Eigenvalue analysis for high telecommunication towers with lognormal stiffness by the response function method and SFEM. (Received in the final form February 28, 2010) CAMES 2009 (16)

The main aim of this paper is to demonstrate the application of the generalized stochastic perturbation technique to model the lognormal random variables in structural mechanics. This is done to study probabilistic characteristics of the eigenvibrations for the high telecommunication towers with random stiffness, which are modeled as the linear elastic 3D trusses. The generalized perturbation technique based on the Taylor expansion is implemented using the Stochastic Finite Element Method in its Response Function version. The main difficulty here, in a comparison to this technique previous applications, is a necessity of both odd and even order terms inclusion in all the Taylor expansions. The hybrid numerical approach combines the traditional FEM advantages with the symbolic computing and its visualization power and it enables for a verification of probabilistic convergence of the entire computational procedure.

*Keywords:* stochastic dynamics, Stochastic Finite Element Method, response function method, stochastic perturbation technique.

S. Czarnecki. Application of the Strongin-Sergeyev global optimization method in the compliance minimization of latticed shells. (Received in the final form January 18, 2010) CAMES 2009 (16)

The aim of the present paper is to revisit some known truss optimization problems by applying the genuine Strongin and Sergeyev's algorithm of the global search [8]. By employing the space-filling Hilbert-Peanotype curves, the wide class of non-convex and multidimensional constrained global optimization problems is reduced to one-dimensional ones. Then, the global minimum of the objective function in one-dimensional problem can be effectively found by means of Multivariate Index Method (MIM) that can be treated as a special version of one-dimensional Global Search Algorithm (GSA) over the set of open intervals adopted to constrained problems.

E. Błazik-Borowa. The quality of numerical modelling of the flow around two buildings with different heights. (Received in the final form February 9, 2010) CAMES 2009 (16)

The paper is devoted to the problems with quality of numerical modeling for two-dimensional incompressible flow around two models of buildings with different heights. The calculations have been made with use of the turbulence model k-ε in the standard version and with the Finite Volume Method. The quality evaluation for the calculation is based on the comparison of the results with measurements in a wind tunnel. Hence, in this paper there have been presented the graphs of averaged velocities which are results of author's own measurements, as well as the graphs presenting the error in the calculated flow velocities. The main conclusion drawn from the research is that the flow around two models is more complicated than the flow around the single one and therefore the calculation results for the set of models are less accurate in comparison with the ones obtained for the single model.