In this work, the application of an indirect Trefftz collocation method to the analysis of bending of thin plates (Kirchhoff's theory) is described. The deflection field approximation is obtained with the use of a set of functions satisfying a *a priori* the homogeneous part of the differential equation of the problem. Each of the approximating functions is derived from a known thick plate solution. The boundary conditions are imposed by means of continuous (integral) and discrete (collocation) least squares methods. Numerical examples are presented and the accuracy of the proposed technique is assessed.
*Keywords:* Trefftz method, thin plates, point collocation, complex trial functions, least squares.

J. Bielski and M. Radwanska. Computational problems of FE-analysis of elastic-plastic surface structures. CAMES 2001 (8)

The paper contains a review of problems connected with numerical analysis of elastic-plastic surface structures. Given is detailed information about finite elements as well as about the algorithm of physically non-linear analysis using the incremental-iterative Newton-Raphson method with the consistent modular matrix. The main goal of the paper is to compare numerical results obtained with elements based on either the volume or area approach to the formulation of physical relations. The presented examples are obtained with the use of computer code *MANKA*. They illustrate some numerical problems induced by elastic-plastic deformation of chosen types of plates.
*Keywords:* material non-linearity, FEM, volume/area approach

M. Younes and A. Potiron. Shape optimization of mechanical structure by an adjoint variables method and genetic algorithm. CAMES 2001 (8)

The shape optimization of machine elements or structures consists in searching the optimal form satisfying the imposed mechanical, technological and geometrical criteria. In this paper two methods, developed for shape optimization of uni and bidimensional mechanical structures are offered. The first one, known as the adjoint variables method, is based upon the evaluation of the sensitivity or the derivatives of the functional with respect to the evolution of the structure shape. It requires the use of a mathematical optimization code in order to converge towards the solution. The second method deals with Genetic Algorithms whose principle rises from the evolution of individuals living in nature. Within the framework of structures optimization, a new Genetic Algorithm has been developed. The analysis is carried out by the finite element method. The first part of this article is devoted to optimal shape research of unidimensional structures such as beams while the second treats the shape optimization of bidimensional parts. To show the effectiveness of each of the two methods, examples are presented, and the numerical results obtained show that a good convergence was obtained in each case.

N. Sczygiol. Object-oriented analysis of the numerical modelling of castings solidification. CAMES 2001 (8)

The paper concerns a method of implementation for the numerical modelling of the solidification process in which the finite element method was used. Modern techniques of software engineering were applied to reach the aim. The decomposition of the problem domain, for the needs of object-oriented analysis, was carried out. The relationships between parts of the analysed problem were discussed. At first, the object-oriented analysis was investigated in general for the wide range of problems solved by the finite element method, e.g. thermomechanic problems of castings, and then it was investigated in detail for the solidification process. The most important specialisation of classes for object implementation of the solidification model were also discussed. The enthalpy solidification formulations were used in the numerical modelling. The three models of solid phase growth, used for solidification modelling of two-component alloys, were described. The method for determining the dependence of enthalpy in relation to temperature and the formula for calculating the solid fraction were shown for each of the three models.

N. Sczygiol and G. Szwarc. Application of enthalpy formulations for numerical simulation of castings solidification. CAMES 2001 (8)

The paper deals with a numerical modelling of solidification in which enthalpy formulations were used. The finite element method (FEM) was applied for computer simulation of solidification. This is the most common numerical method used in the simulation of physical processes. The enthalpy formulations are more convenient to use than temperature formulations in the multidimensional problems in which FEM is applied. The paper concentrated on two enthalpy formulations: the apparent heat capacity formulation and the basic enthalpy formulation. The time integration schemes and the numerical realisation of boundary conditions were discussed. The models of solid phase growth and the implementation details used in this paper were shown in the first author's paper right above. The presented results of computer simulations contain: temperature fields, solidification kinetics, cooling velocities and calculated distributions of equiaxed grain size.

M. Klisinski. Inconsistency errors of constant velocity multi-time step integration algorithms. CAMES 2001 (8)

Previously known multi-time step integration methods for finite element computations in structural dynamics have been shown to be unstable due to interpolation error propagation. New algorithms of multi-time step integration based on constant velocity during subcycling are investigated. The assumption of constant velocity gives linear variation of displacements so the errors connected to interpolation at the interface between different time step partitions are eliminated. As a consequence, the new constant velocity algorithms give bounded solutions and have been shown to be conditionally stable by their authors. However, numerical investigation demonstrates that if time steps close to the stability limit are used, the errors for higher natural modes are so huge that the obtained solutions can only be considered as incorrect. The main reason for this behaviour is that the constant velocity time integration algorithms are inconsistent. Displacements can be calculated either by direct integration or from the equation of motion leading to different solutions. Based on the numerical results it is concluded that use of time steps below stability limit is insufficient to assure proper solutions. Therefore, significant time step reductions are often required to assure acceptable error levels. As a consequence, the new subcycling algorithms can be more expensive than ordinary time integration. Because they also lead to larger errors the constant velocity subcycling algorithms are useless from practical point of view. Since subcycling is available as an option in LS-DYNA a serious warning is issued to potential users.

M. Berkovic, D. Mijuca and I. Grozdanovic. Some continuous stress mixed formulations and inf-sup test. CAMES 2001 (8)

The most frequent motivation for the use of mixed methods is their robustness in the presence of certain limiting and extreme situations. At variance, the main goal of the present paper is to reconsider the use of mixed formulation as a tool for wider application, i.e., to study the stability of the proposed procedure treating problems in elasticity otherwise well suited for the solution by the usual displacement method. Computational procedure for the inf-sup test is outlined, and the results are given.

W. Szemplinska-Stupnicka, E. Tyrkiel and A. Zubrzycki On the stability "in the large'' and unsafe disturbances in a nonlinear oscillator. CAMES 2001 (8)

The problem of the stability "in the large'' and the unsafe disturbances of the equilibrium position is studied for the structures whose dynamics is governed by the equation of motion of the pendulum with parametric excitation. The system displays a variety of nonlinear and chaotic phenomena, so that the study requires the use of theoretical concepts of the mathematics of chaos. Detailed explorations are performed by the aid of the nonlinear software package *Dynamics*.

Z. Iwanow, J. Bauer and M. Jelinska-Kazimierczuk. Splines in physicochemical studies of liquid binary mixtures. CAMES 2001 (8)

Physicochemical studies of the organic-water mixtures show that their properties are not linear functions of their concentration but depend on the mixture composition in various ways. The evaluation of the measurement results requires an interpolation of the experimental data and the derivatives of a mixture property with respect to concentration should be known as well. The results of measurements are disturbed by experimental error which causes the scatter of the approximated function values and oscillations of the approximating function derivatives. In the paper an application of the 3rd degree splines for the calculation of derivative and the interpolation for a non-uniform mesh are considered. Smoothing methods of an approximating function by means of splines are proposed. Some numerical examples illustrating the efficiency of the smoothing method and its applications are presented.
*Keywords:* splines, approximation, smoothing.

A. Stankiewicz and J. Pamin. Simulation of instabilities in non-softening Drucker-Prager plasticity. CAMES 2001 (8)

The problem of instability and strain localization in a hardening non-associative Drucker-Prager plasticity theory is analyzed. The classical and gradient-enhanced versions of the theory are reviewed and instability indicators are summarized. The regularizing properties of the gradient-enhancement are shown. The classical plane strain biaxial compression test is analyzed in terms of the analytical prediction of ellipticity loss and numerical simulation of the process of shear band formation and evolution. The influence of material model parameters, especially of the degree of non-associativity and the gradient influence, on the instability properties is demonstrated.