We investigate the evolution of Tollmien-Schlichting waves in boundary layers in the presence of moderate buoyancy arising from the heating or cooling of a compliant wall. We exploit the multi-deck structure of the flow in the limit of large Reynolds numbers to make an asymptotic analysis of the pertubed flow, along the upper-branch of the neutral stability curve, to derive linear neutral results. These results are discussed and are compared to rigid wall results. Also, a brief parametric study, based on the linear neutral results, is presented and the results are discussed.
*Keywords:* linear stability, heating and cooling, compliant surface.

S. Kaliszky and J. Lógó. Layout optimization of disks by the use of rigid-plastic element model. CAMES 2002 (9)

In the paper, the layout optimization of rigid-plastic disks is presented. The method is based on a model where a disk is subdivided into rectangular elements interconnected by normal and shear forces along their edges. Using this model statically admissible stress fields are constructed and the static theorem of limit analysis is applied. Following the concept of porous materials the design variables are the unknown densities of the elements with variable yield stress expressed in terms of the densities. Two complementary optimum design problems are presented. The load intensity is maximized at given intensity of the load and the total amount of material is minimized at prescribed amount of material, respectively. Both problems are expressed in the forms of nonlinear mathematical programming. The application is illustrated by two examples.

T. Toyoda and E. Kita. Structural optimization method based on cellular automata simulation. CAMES 2002 (9)

This paper describes the topology and the shape optimization scheme of the continuum structures using the cellular automata simulation. The design domain is divided into small square cells. By considering the cells as the elements, the stress analysis of the structure is carried out by finite element method. Then, the design variables are updated according to the local rule and the stress distribution. The rule is defined as the simple relationship between a cell whose design variable is updated and its neighborhood cells. In this paper, we will discuss the formulation to analytically derive the rules from the optimization problems. The special constraint condition named as ``CA-constraint condition'' is introduced first and then, the global optimization problem for the whole structure is divided into the local problem for some neighboring cells. The derived rules are applied to the same numerical example in order to discuss the theoretical validity of the formulation and the feature of the rules.
*Keywords:* topology and shape optimization, cellular automata (CA), local rule, 2D elastic problem, finite element method (FEM).

Z. Nowak and A. Stachurski. Global optimization in material functions identification for voided media plastic flow. CAMES 2002 (9)

The aim of this paper is to present an application of the global optimization method of Boender et al. to a material function identification in a mechanical problem. These material functions are found in the evolution equation for a volume void fraction parameter describing nucleation and growth of microvoids in the flow of porous ductile solids and they play an important role in proper constitutive modelling of postcritical behaviour and fracture. In the evolution equation a plastic strain controlled nucleation process is simulated and uniaxial tension deformation history is considered. In nonlinear regression the minimization of the mean squares functional is assumed. The problem is treated directly as a global optimization one. The necessity of the use of a global optimization approach follows from the hypothesis that there can exist many local minima in the considered problem. The possibility of the existence of many local minima is not usually taken into account. The global optimization method of Boender et al. was applied to minimize the least squares functional. We determine the material functions parameters on the basis of the given Fischer's experimental data set. This data set has been obtained for axisymmetric tension of steel specimens. The results of numerical calculations presented in the paper proved the validity of the hypothesis about the existence of many local minima.
*Keywords:* plastic flow of voided media, material functions identification, global optimization, nonlinear regression, nonlinear programming.

M. Stanuszek. Computer modelling of cable reinforced membranes. CAMES 2002 (9)

This paper is intended to summarise the actual work in the area of large deformations of tension systems. The previously conducted research adds significant contributions to the understanding of the response characteristics of pneumatics and suspended membranes with wrinkling allowed. Here the attention will be focused on the applications of the one-dimensional tensioned cable systems. Two- and three-dimensional tensile structures, will be forced to work with compressed members. Since cables can not transmit any compressive forces a verified numerical algorithm, based on the monitoring of the load displacement path is proposed. The special computer code NAFDEM (Nonlinear Analysis by Finite Difference and Element Methods) was adapted to solve undertaken highly nonlinear problems. Calculated results were verified numerically and compared with the solutions obtained by the numerical integration technique.

T. Skiepko. Heat transmission across fouled tubes analysed by Trefftz approach. CAMES 2002 (9)

The paper presents application of the Trefftz method for analysis a case of heat conduction problem across two coupled regions - fouling layer of complex form deposited onto the tube wall at its outer periphery. Taken into considerations modes of heat transfer from a hot gas to the outer surface of the fouling layer are: either by pure convection or radiation, and by both the modes combined. A fluid flowing inside the tube exchanges heat by only convection at constant transfer coeffcient. Based on the variational principle and Trefftz method the boundary weighted residual approach has been developed providing in turn an equation system for analysis of the problem under study. Then, results of series of systematic numerical experiments illustrating convergence and accuracy of the approach when applied to the case in point have been shown for a specific input data set assumed. To emphasise practical significance of the method, a calculated temperature distribution 3D chart and thermal resistance of 2D fouling deposits conclude the paper.

R. Trebinski. A monotone predictor-corrector scheme for advection. CAMES 2002 (9)

A monotone predictor-corrector finite difference scheme solving the advection equation has been proposed. A geometrical interpretation of the Burstein scheme forms a basis for construction of the new scheme. The main idea consists in defining a proper limitation algorithm in the predictor step preventing formation of new extremes of the solution profile. Various variants of the scheme have been tested for the linear advection equation and an optimum version has been chosen for further developments. Extensions to the nonlinear case and inhomogenous, solution independent velocity field have been made. Application of the time splitting procedure enables the scheme to be applied for multidimensional advection problems. For chosen test problems the scheme behaves better than schemes proposed in the literature.
*Keywords:* finite difference schemes, monotone schemes, advection equation.

I. Kozák and T. Szabó. Stability of a circular ring in postcritical equilibrium states with two deformation-dependent loads and geometricla imperfections. CAMES 2002 (9)

The circular ring is linearly elastic and its cross-section is rectangular. Two deformation dependent distributed loads, that is follower loads, are applied simultaneously on the outer surface of the ring. The first load is a uniform pressure on the whole outer surface. The second load is uniform normal traction exerted on two surface parts situated in axially symmetric positions. Both loads are selfequilibrated independently from each other. A nonlinear FE program with 3D elements is used for the numerical analysis of a geometrically perfect and two imperfect rings. Displacement control is used in the equilibrium iterations. Equilibrium surfaces are determined in the space of three parameters such as one characteristic displacement coordinate, and two load factors. The stability analysis is performed in the knowledge of the equilibrium surfaces.
*Keywords:* deformation dependent loads, two-parameter loads, geometric imperfection, displacement control, equilibrium surface, limit point, bifurcation point, unstable region.