The paper presents a general numerical model for the analysis of prestressed concrete with the application to beam, thin shell and volume type of prestressed structures. Discrete, embedded approach is used to model curved, bonded or unbonded tendons. Also a partial bond may be introduced by the friction between the tendon and the surrounding body. In the finite element model, two types of elements are obtained. One is a typical finite element for the kind of structure modeled, and the other is an embedded, three noded, subparametric tendon element. Equations of the finite element method have been obtained from the incremental form of the principle of virtual work providing geometrical linearity and possibility of nonlinear physical relations. Numerical examples illustrate application to modeling of beam, thin shell and volume type of prestressed structures as well as the impact of the friction on the axial force distribution in prestressing tendon.

H.J. Antunez. Sensitivity analysis of transient metal forming with incompressible linear elements. CAMES 2000 (7)

On the basis of a recently developed method which allows the use of linear elements for metal forming simulation within the flow approach, sensitivity analysis is carried out. Aiming at large, industrial problems, attention is focused on the explicit version, which is considered more effective for such problems, although implicit time integration is possible as well. By time step splitting a stabilization sub-matrix is obtained, which allows the use of equal interpolation for velocity and pressure. Specifically, linear triangles and tetrahedra have been used, which are easily generated by automatic meshers. Sensitivity analysis is carried out by the Direct Differentiation Method, with which similar analyses have been performed by the author for the flow approach within a direct solution scheme.
*Keywords:* sensitivity analysis, metal forming, explicit time integration, split algorithm, flow approach.

J. Bauer and J. Latalski. Manufacturing tolerances of truss members' lengths in minimum weight design. CAMES 2000 (7)

In most cases a safety of optimal construction may be limited by the violation of stress, buckling or displacement constraints. An unexpected exceed of these constraints may be caused by manufacturing tolerances of structural elements (differences between assumed and obtained dimensions). This requires an incorporation of tolerance problem in optimum design. One may deal with two different tolerances - the first case is when it's related to the members' cross-section variations, whereas the second notion represents the variation of elements' lengths. Considering operation conditions and manufacturing techniques the second case of tolerance seems to be more important. This approach states the problem of minimum weight design of a structure with initial distortions. A standard solution algorithm with the Kuhn-Tucker theorem was used with the adjoint variable method. Necessary optimality conditions have the form of equations and inequalities. The equality constraints were put forward for the average values of design variables *l*, while tolerances *t_j* were introduced into inequality equations i.e. the limit values of stresses and displacements were diminished by the positive products of appropriate sensitivities and tolerances. The method was next illustrated by an example of a ten bar bench-mark problem - a typical one for testing algorithms in structural optimization. The idea presented in this paper may be used not only for truss structures but it can be easily extended to other kinds of structures like frames, composites etc.
*Keywords:* structural optimization, manufacturing tolerances, nonlinear optimization.

G. Bognar. Finite element method for a nonlinear problem. CAMES 2000 (7)

We consider the nonlinear eigenvalue problem of a nonlinear partial differential equation under Dirichlet boundary condition in a two-dimensional space. The classical solutions are given for rectangular domains. We give numerical solutions obtained by finite element method for the first eigenvalue and eigenfunctions and we analyze the error in the approximate finite element solutions.

W. Cecot and W. Rachowicz. Adaptive solution of problems modeled by unified state variable constitutive equations. CAMES 2000 (7)

The objective of the work was an efficient, numerical implementation of one of the unified, internal-state-variable constitutive models. Such models are general and convenient in numerical applications since they describe elastic, plastic, viscous, damage phenomena and they do not require neither yielding condition nor loading/unloading criterion. However, they result in so called stiff initial-boundary value problems. Therefore, an efficient numerical implementation demand adaptive techniques, both in space and in time. The paper presents application of such an adaptation approach. It uses an improved version of the semi-implicit Euler method with automatic time step control and the *h* refinement of the FEM meshes based on the interpolation error estimate and on the reliable, selfequilibrated, implicit, *a posteriori*i estimate. Selected problems were solved and both the efficiency and reliability of the unified model were confirmed.

D. Dent, M. Paprzycki and A. Kucaba-Pietal. Recent advances in solvers for nonlinear alegebraic equations. CAMES 2000 (7)

In this paper the performance of four solvers for systems of nonlinear algebraic equations applied to a number of test problems with up to 250 equations is discussed. These problems have been collected from research papers and from the Internet and are often recognized as "standard'' tests. Solver quality is assessed by studying their convergence and sensitivity to simple starting vectors. Experimental data is also used to categorize the test problems themselves. Future research directions are summarized.

Z. Dziopa. Rocket missile lift-off and flight simulator program. CAMES 2000 (7)

The subject of this paper is a movable antiaircraft short-range rocket set. The control algorithm for a target homing short-range rocket missile by means of the proportional approach method is performed by an executive system comprising the co-ordinator and automatic pilot devices. The co-ordinator is equipped with a gyro-device and a system determining working out of the ideal, desirable signal. The automatic pilot device has a pair of double-position external vanes and internal gas control engine. A missile control, in accordance with the implemented algorithm, is achieved by using a rocket rotating motion around its longitudinal axis.

L. Fedorowicz and J. Fedorowicz. Structure-subsoil contact task - an iterative engineering realisation. CAMES 2000 (7)

The analytical formulation of an iterative procedure applied for structure-subsoil systems is presented in the paper. A physical and engineering interpretation has been given for the presented algorithm.

S. Fialko. High-performance aggregation element-by-element Ritz-gradient method for structure dynamic response analysis. CAMES 2000 (7)

The article presents the high-performance Ritz-gradient method for the finite element (FE) dynamic response analysis. It is based on the generation of the orthogonal system of the basis vectors. The gradient approach with two-level aggregation preconditioning on the base of element-by-element technique is applied to minimize the Rayleigh quotient for the preparation of each basis vector. It ensures the evolution of the regular basis vector toward the lowest eigenmode without aggregating and decomposing the large-scale stiffness matrix. Such method often happens to be more effective for dynamic response analysis, when compared to the classical modal superposition method, especially for seismic response analysis of the large-scale sparse eigenproblems. The proposed method allows one to apply arbitrary types of finite elements due to aggregation approach, and ensures fast problem solution without considerable exigencies concerning the disk storage space required, which is due to the use of EBE technique. This solver is implemented in commercial programs RobotV6 and Robot97 (software firm RoboBAT) for the seismic analysis of large-scale sparse problems and it is particularly effective when the consistent mass matrix is used.

M. Jasinski and A. Pownuk. Modelling of heat transfer in biological tissue by interval FEM. CAMES 2000 (7)

In this paper, an algorithm of calculation of extreme values of temperature based on interval arithmetic is presented. Many mechanical systems with uncertain parameters *lambda in Lambda* can be described by a parameter dependent system of linear equations *K(lambda)T=B(lambda)*. Using natural interval extension of a real function, one can transform the system of linear equations into the system of linear interval equations **K(lambda)T=B(lambda)**. Solution of the system of linear interval equations always contains the exact solution of the parameter dependent system of equations. A new method of computation of extreme values of mechanical quantities based on the monotonicity test is introduced. This method can give exact solution of a parameter dependent system of equations.

A. Karafiat. A-priori estimates of the hp-adaptive BEM in elastic scattering of acoustic waves. CAMES 2000 (7)

In the paper some a-priori *hp-*-adaptive error estimates, applied to the problem of acoustic wave scattering on an elastic body in the 2D space, solved by the Boundary Element Method, are presented. The estimate includes both the function- and boundary approximation errors.
*Keywords:* Boundary Element Method, acoustic scattering, hp- adaptive method, a-priori error estimate.

A. Kawalec. Numerical modelling and robust parametric estimation of surfaces useful for representation of tooth flanks. CAMES 2000 (7)

This paper concerns modelling of surfaces resulting from measurements or from digital simulation of surfaces e.g., tooth flanks, based on the theory of gearing, with intentionally introduced modification usually defined in discrete form. Computational methods of modelling of curves and surfaces are briefly reviewed. Problems of stability of geometric modelling and related problems of parametric estimation of mathematical models, representing curves and surfaces are discussed. An analysis of multicollinearity of the measurement matrix is performed. A method of regularization of matrix, containing coordinates of nodes from considered surface, is proposed. This method allows to improve the robustness of parametric estimation and is specially helpful for on-line parametric estimation of surface models utilized during measurements.

A. Kierklo, R. Tribillo and A. Walendziuk. Computer simulation of the effort states of the teeth's hard tissues. CAMES 2000 (7)

The study presents a class of effort of brittle media problems. The model of human tooth with non-carious cervical lesion was analysed. Regions of the most disadvantageous loads was determined.

Z. Koruba. Optimisation of construction parameters of gyroscope system on elastic suspension. CAMES 2000 (7)

In the paper are derived the equations of motion of a gyroscope on elastic suspension, mounted on a movable base. An algorithm for the selection of the optimum construction parameters and the matrices of amplifications of the gyroscope regulator is presented. The latter is aimed at the quickest transitory process damping and also at the minimisation of errors resulting from the friction in frame bearings, base angular motion and non-linearity of the impact.

J. Kucwaj. Generation of hybrid grids over plane domains. CAMES 2000 (7)

The paper presents an algorithm of coverage 2-D multiconnected domain by trianlges and quadrilaterals. The zone covered with quadrilaterals is structured, thus the zone covered with triangles is unstructured. The density of the structured grid is controlled only on one of the surrounding loop, on which the points are generated with mesh density function. Unstructured zone is triangulated by using Delaunay advancing front technique triangulation of points previously generated.

J. Kucwaj. The algorithm of adaptation by using graded meshes generator. CAMES 2000 (7)

An algorithm of remeshing based on graded meshes generator is presented. The algorithm starts with an initial grid, which is iteratively improved taking into account error estimate. Mesh density functions are used to generate grid over domain on which boundary value problem is solved. It is observed, that succussive meshes are convergent and especially they become denser near singularities. For unstructured grid generation the advancing front technique combined with Delaunay triangulation is used. The boundary of 2-D domain may be represented by B-spline curves. It may be multiconnected.

Z. Kulpa and K. Roslaniec. Solution sets for systems of linear interval equations. CAMES 2000 (7)

The paper discusses various classes of solution sets for *linear interval systems of equations*, and their properties. Interval methods constitute an important mathematical and computational tool for modelling real-world systems (especially mechanical) with (bounded) uncertainties of parameters, and for controlling rounding errors in computations. They are in principle much simpler than general probabilistic or fuzzy set formulation, while in the same time they conform very well with many practical situations. Linear interval systems constitute an important subclass of such interval models, still in the process of continuous development. Two important problems in this area are discussed in more detail - the classification of so-called *united solution sets*, and the problem of overestimation of interval enclosures (in the context of linear systems of equations called also a *matrix coefficient dependence problem*).

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R. Lackner and H.A. Mang. Adaptive ultimate load analysis of RC shells. CAMES 2000 (7)

The aim of the present paper is a synthesis of both realistic modelling of the structural behavior of reinforced concrete (RC) shells and an adaptive finite element (FE) calculation tool suitable for the solution of nonlinear problems involving strain-hardening and softening plasticity. In the context of incremental-iterative analysis, an incremental error estimator is introduced. It is based on the rate of work. The reference solution required for error estimation is obtained by means of a recovery scheme applied to stress resultants. If the estimated error exceeds a prespecified threshold value, a new mesh is designed. Mesh generation is performed in the 2D parametric space of the shell. After mesh refinement, the state variables are transferred from the old to the new mesh and the calculation is restarted at the load level which was attained by the old mesh. The usefulness of the developed adaptive analysis scheme is demonstrated by a numerical analysis of an RC cooling tower.

E. Majchrzak and M. Dziewonski. Numerical simulation of freezing process using the BEM. CAMES 2000 (7)

The boundary element method is applied for numerical simulation of the freezing process proceeding in biological tissue under the influence of cylindrical cryoprobe. From the mathematical point of view the problem discussed belongs to the group of moving boundaries ones for which the mushy zone sub-region (intermediate phase) is considered. In this paper the mathematical model of the process is formulated using the fixed domain approach and a parameter called the substitute thermal capacity determines the evolution of latent heat. On a stage of numerical computations the generalized variant of the alternating phase truncation method (APTM) is applied and the basic mathematical model is rebuilt by the introduction of the enthalpy function. The boundary element method together with APTM leads to the simple and effective numerical algorithm because the difficulties connected with the non-linear problem modelling can be omitted. In the final part of the paper the results of computations are shown.

D. Mazur. Analysis of damages on induction motors by the finite element method. CAMES 2000 (7)

The results of computations of forces and moments in induction motor made with OPERA 2D program and possibilities of application of FEA methods to consider the failure states during the calculations of induction machine dynamics are discussed in this paper. A mathematics model of an induction motor makes it possible to perform the calculations and analysis of stator currents and electromagnetic torque when the rotor rotates eccentrically, as well as when a bar or rotor ring is broken. A tool has been created which enables to predict preliminarily the results of the defects, which might be introduced at the motor manufacturing stage. To calculate 3D free vibration frequencies of the rotor system of the motor under consideration the PATRAN package has been used. It allowed us to examine the resonance frequencies of the rotor. For the diagnostic measurements the MotorMonitor(TM) program has been applied. It was possible to attain high computational effectiveness due to parallel programming on CONVEX computers of ACC ``CYFRONET''.

A. Pownuk. Optimization of mechanical structures using interval analysis. CAMES 2000 (7)

The problem of optimal design consists in finding the optimum parameters according to a specified optimality criterion. Existing optimization methods [1,2] usually are not reliable or cannot use the nondifferentiable, not continuous objective functions or constraints. An interval global optimization method is very stable and robust, universally applicable and fully reliable. The interval algorithm guarantees that all stationary global solutions have been found. In this paper the algorithm is applied to optimization of mechanical systems, calculation of extreme values of mechanical quantities and to optimization of structures with uncertain parameters. [1] http://solon.cma.univie.ac.at/~neum/glopt.html [2] http://plato.la.asu.edu/guide.html

J. Rakowski and P. Litewka. An efficient 3D curved beam finite element. CAMES 2000 (7)

The purpose of this paper is to create an efficient finite element for the static analysis of 3D arch structures. It is the circural in-plane element with possibility of out-of-plane action. In the element the influences of shear and axial forces on arch displacements are taken into account. The base is the set of exact shape functions, which fulfil the differential equilibrium equation of 3D arch. These shape functions allow us to obtain the exact element stiffness matrix. The element was tested in several numerical examples, results were compared with available analytical solutions and other numerical results. A very good agreement of the results was obtained.

N. Sczygiol. Approaches to enthalpy approximation in numerical simulation of two-component alloy solidification. CAMES 2000 (7)

The paper deals with the numerical modelling of solidification of two-component metal alloys. The numerical model was worked out using the enthalpy formulation of solidification and the finite element method. The work concentrated on one enthalpy formulation, namely the basic enthalpy formulation [1].The models of solid phase growth as well as implementation details are shown in the paper [2].A comparison of the results of the numerical simulation of solidification was made for three approaches to enthalpy approximation as a function of temperature. The three approaches were called: complete, incomplete and linear. The results of simulation, for incomplete enthalpy approximation were almost identical to the results of complete approximation. The computing time for incomplete approximation was substantially lower than the computing time for complete approximation, and comparable to the computing time for linear approximation.
[1] Ph. Thevoz, J.L. Desbiolles, M. Rappaz. Modeling of equiaxed microstructure formation in casting.Metall. Trans. A, 20A: 311-322, 1989.
[2] N. Sczygiol. Object-oriented analysis of the numerical modelling of castings solidification. **8**: 79-98, 2001.

G. Thierauf and J. Cai. Parallel evolutionary optimization of structures. CAMES 2000 (7)

By introducing a variable coding technique, a parallel optimization method based on a combination of GAs and ESs is presented. The advantages of both GAs and ESs, like coding of genetic information and adaptation of optimization parameters, are enhanced by this new method.

E. Tyrkiel, W. Szemplinska-Stupnicka and A. Zubrzycki On the boundary crises of chaotic attractors in nonlinear oscillators. CAMES 2000 (7)

In nonlinear dissipative mechanical systems, bifurcations of chaotic attractors called *boundary crises* appear to be the cause of most sudden changes in chaotic dynamics. They result in a sudden loss of stability of chaotic attractor, together with destruction of its basin of attraction and its disappearance from the phase portrait. Chaotic attractor is destroyed in the collision with an unstable orbit (*destroyer
saddle*) sitting on its basin boundary, and the structure of the saddle defines the type of the crisis - *regular* or *chaotic* one. In the paper we exemplify both types of the boundary crisis by using a mathematical model of the symmetric twin-well Duffing oscillator; we consider the regular boundary crisis of the cross-well chaotic attractor, and the chaotic boundary crisis of the single-well chaotic attractor. Our numerical analysis makes use of the underlying topological structure of the phase space, namely the geometry of relevant invariant manifolds, as well as the structure of basins of attraction of the coexisting attractors. The study allows us to establish some relevant relations between the properties of the regular and chaotic boundary crisis, and to outline the differences that result mainly in the post-crisis system behavior.

Z. Waszczyszyn and E. Pabisek. Application of a Hopfield type neural network to the analysis of elastic problems with unilateral constraints. CAMES 2000 (7)

On the base of Hopfield-Tank neural network the Panagiotopoulos approach is briefly discussed. The approach is associated with the analysis of quadratic programming problem with unilateral constraints. Then modifications of this approach are proposed. The original Panagiotopoulos approach is illustrated by the analysis of crack detachment in an elastic body [1]. Efficiency of the proposed modifications is shown on a numerical example of an angular plate. Finally some special conclusions are expressed. [1] P.S. Theocaris, P.D. Panagiotopoulos. Neural networks for computing in fracture mechanics - methods and prospects of applications. Comp. Meth. Appl. Mech. Eng., 106: 213-228, 1993.

G.N. Wells and L.J. Sluys. Embedded discontinuities for 3D mode-I and mode-II failure modelling. CAMES 2000 (7)

The incorporation of displacement discontinuities in finite elements is examined. The incorporation of displacement discontinuities allows the use of discrete constitutive models in a continuum framework in order to avoid the mesh sensitivity of classical continuum models when strain softening is introduced. The procedure for building discontinuities into finite elements is examined, as well as two classes of constitutive models for mode-I and mode-II failure analysis. The performance of the model is illustrated with three-dimensional examples.

L. Ziemianski and B. Miller. Dynamic model updating using neural networks. CAMES 2000 (7)

The paper presents an application of Artificial Neural Networks for updating a mathematical model of the structure based on dynamic parameters. Neural networks which predict the value of selected stiffness or concentrated masses on the basis of Frequency Response Function (FRF) have been built. Two types of neural networks have been used for this task: multi-layer feed-forward (MLFF) networks with different learning algorithms and networks with radial basis function (RBF). Preceding the update, the FRF is compressed in order to reduce the number of input values necessary for updating the model.