Paper summarizes the first results of two-dimensional (2D) numerically modelled expansion and flow of compressible and non-viscous gas in typical parts of air jet weaving system; namely in main nozzle designed as an ejector with various shapes of the mixing zone, in relay (auxiliary) nozzle with substantial flow separation in the rash flow bend directly before the nozzle outlet, and the influence of the reed dent edges shape on the free stream reflection and penetration through reed gaps along a real ``porous'' wall. The used Euler's equations are solved by a Finite Volumes Method (FVM) with automatic mesh generation and optimization of unstructured triangle mesh. Graphical results show 2D isolines of all gas state values, further Mach number, entropy and velocity vectors. 1D profiles of all quantities along chosen cross-sections or surfaces can be obtained, too. They give to the designer a large and quick review about the problem. The coincidence with experiment, measuring and real weaving tests is very good. The advantage of numerical modelling consists in the very quick, simple and user-friendly operation.

L. Baranyi and M. Shirakashi. Numerical solution for laminar unsteady flow about fixed and oscillating cylinders. CAMES 1999 (6)

This paper presents a finite-difference solution of the two-dimensional, time dependent incompressible Navier-Stokes equations for laminar flow about fixed and oscillating cylinders placed in an otherwise uniform flow. Using boundary fitted coordinates, the equations are transformed to a non-inertial reference frame fixed to the cylinder. The primitive variable formulation is used for the solution of the problem. A special transformation provides a fine grid scale near the cylinder walls and a coarse grid in the far field. Forward difference is used in time, fourth order central difference in space except for convective terms for which a modified third-order upwind scheme is used. Velocity values are obtained explicitly, and the successive over-relaxation (SOR) method yields the pressure distribution. Computed drag coefficients and dimensionless vortex shedding values were compared with experimental results for rigid cylinders and a very good agreement has been obtained. Amplitude bounds of locked-in vortex shedding due to forced crossflow oscillation of a circular cylinder are also determined for Re=180.
*Keywords:* incompressible flow, Navier-Stokes equations, unsteady flow, laminar flow, bluff body, lock-in.

F. Biondini, F. Bontempi and P.G. Malerba. Optimal strut-and-tie models in reinforced concrete structures. CAMES 1999 (6)

A general solution of the diffusion problems which concern the R.C. structures, may be deduced by the limit analysis, by means of truss schemes suitable to model the load transfer mechanism. In particular such schemes allow us to share the carrying functions between concrete and steel reinforcement. Latest developments call this kind a solution Strut-and-Tie (S&T) modellization. In this paper a procedure for the automatic search for optimal S&T models in R.C. elements is proposed. A highly indeterminate pin-jointed framework of a given layout is generated within the assigned geometry of the concrete element and an optimum truss is found by the minimization of a suitable objective function. Such a function allows us to search for the optimum truss according to a reference behaviour (the principal stress field) deduced through a F.E.A. and assumed as representative of the given continuum. After having explained the theoretical principles and the mathematical formulation, some examples show the pratical application of the procedure and its capability in handling complex stress paths, through schemes which result rational and suitable for a consistent design.

N.K. Birnbaum, N.J. Francis and B.I. Gerber. Coupled techniques for the simulation of fluid-structure and impact problems. CAMES 1999 (6)

The ability to model the different regimes of a physical event using different numerical techniques has a number of advantages. Instead of applying the same general solver to all domains of a problem, a solver optimized for a particular regime of material behavior may be used. Thus, in a single analysis, one type of solver may be used for fluid behavior while another type is used for solid/structural response. The various domains in the problem are then coupled together in space and time to provide an efficient and accurate solution. Examples in the use of Eulerian, Lagrangian, Arbitrary Lagrange Eulerian (ALE), Structural, and Smooth Particle Hydrodynamic (SPH) techniques, in various combinations, will be applied to general problems in fluid-structure interaction and impact problems. The relative advantages and limitations of such coupled approaches will be discussed. Examples include the following: fluid dynamics, blast, impact/penetration (Euler); hypervelocity impact onto spacecraft shields (Lagrange-Lagrange); buckling of a thin walled structure (Structural); impact and penetration of a projectile onto concrete (SPH-Lagrange, Lagrange-Lagrange, Euler-Lagrange). Numerical results are presented with comparison against experiment where available.

A. Csébfalvi. Discrete optimal weight design of geometrically nonlinear truss-structures. CAMES 1999 (6)

In this paper, we introduce a new branch-and-bound type method, for discrete minimal weight design of geometrically nonlinear truss structure subject to constraints on member stresses and nodal displacements when the member cross-sectional areas are available from a discrete set of a given catalogue. The discrete optimization problem can be formulated as a tree search procedure. The initial - unfeasible - node of the searching tree is obtained by decreasing the relaxed cross-sectional areas to the closest discrete catalogue value. Each node of the branch-and-bound searching tree is characterized by the weight of the structure, the actual value of the infeasibility penalty function, and the minimal relaxed additional weight, that is needed to obtain a feasible structure from the given state. The proposed method involves an exterior point method to determine the relaxed solution of the minimum weight design problem. The algorithm seems computationally attractive and has been tested on a large number of examples. Numerical results are presented for a well-known test problem.

T. Czibere and L. Kalmár. Determination of the field of constrain forces representing the effect of the impeller's blading. CAMES 1999 (6)

The aim of this paper is to present a numerical method to determine a field of constrain forces which hydrodynamically represents the effect of the blading of the impeller of a radial-flow pump. The field of constrain forces are perpendicular to the stream surfaces of the relative velocity field and congruent to the blade surface of the impeller. The calculation of the constrain force field is based on the solution of the inverse problem of the hydrodynamic cascade theory. In the determination of the constrain force it is supposed that the frictionless and incompressible fluid flow is completely attached to the blade surfaces. The constrain force field can be calculated by the change of the moment of momentum in the absolute inviscid fluid flow which depends on the state of the pump. Knowing the constrain force field it is possible to calculate the distributions of the relative velocity, pressure and energy loss on the mean stream surface (F) of the impeller by solving the governing equations of the viscous relative flow. By calculating the energy loss belonging to different volume rates an approximate real head-discharge characteristic of the impeller also can be computed.

Z. Gáspár, N. Radics and A. Recski. Rigidity of square grids with holes. CAMES 1999 (6)

Bolker and Crapo gave a graph theoretical model of square grid frameworks with diagonal rods of certain squares. Using this model there are very fast methods for connected planar square grid frameworks to determine their (infinitesimal) rigidity when we can use diagonal rods, diagonal cables or struts, long rods, long cables or struts. But what about square grids containing some kind of holes? We will show that the model can be extended to the problem of holes too.
*Keywords:* grids, rigidity, frameworks, graphs.

G. Haase and E.H. Lindner. Advanced solving techniques in optimization of machine components. CAMES 1999 (6)

We consider the optimal design of a machine frame under several stress constraints. The included shape optimization is based on a Quasi-Newton Method and requires the solving of the plain stress state equations in a complex domain for each evaluation of the objective therein. The complexity and robustness of the optimization depends strongly on the solver for the pde. Therefore, solving the direct problem requires an iterative and adaptive multilevel solver which detects automatically the regions of interest in the changed geometry. Although we started with a perfected type frame we achieved another 10% reduction in mass.

P. Helnwein, H.A. Mang and B. Pichler. Ab initio estimates of stability limits on nonlinear load-displacement paths: potential and limitations CAMES 1999 (6)

In order to avoid a fully nonlinear prebuckling analysis by the finite element method for the mere purpose of obtaining the stability limit in the form of a bifurcation or a snap-through point, this limit may be estimated by means of the solution of a suitable linear eigenvalue problem. What seems to be most suitable in this context, is a consistent linearization of the mathematical formulation of the static stability condition. It can be interpreted as the stability criterion for the tangent to the load-displacement diagram at a known equilibrium state in the prebuckling domain. Based on this linearization, higher-order estimates of the stability limit can be obtained from scalar postcalculations. Unfortunately, the order of such an estimate is only defined in an asymptotic sense. Nevertheless, for many engineering structures the geometric nonlinearity in the prebuckling domain is moderate. In this case, the general information from asymptotic analysis is frequently relevant for the entire prebuckling domain. This allows good ab initio estimates of stability limits based on nonlinear load-displacement paths. The nucleus of this article is the discussion of the potential and the limitations of determination of stability limits based on ab initio estimates of nonlinear load-displacement paths. The theoretical findings are corroborated by the results from a comprehensive numerical study.

H. Köppe, U. Gabbert, H. Köppe, F. Laugwitz and Ch.-T. Weber. Comparison of numerical and experimental results of structronic plate and shell structures. CAMES 1999 (6)

In the paper a general purpose finite element software for the simulation of piezoelectric materials and structronic (structure and electronic) systems is presented. The equations of coupled electromechanical problems are given in a weak form, which are the basis of the development of 1D, 2D, 3D as well as multilayered composite shell elements. The smart structures finite element code includes static and dynamic analysis, where also controlled problems can be simulated. Two test examples are presented to compare the numerical results with measurements.

I. Kozák, F. Nándori and T. Szabó. FE analysis of geometrically nonlinear static problems with follower loads. CAMES 1999 (6)

We have considered a linearly elastic body loaded by tractions inward normal to the instantaneous surface. Due to the increment of the surface element vector there is a contribution to the tangent stiffness matrix referred to as load correction stiffness matrix. The goal of the numerical experiments is to determine the bifurcation point on the fundamental equilibrium path. Linear eigenvalue problems with follower loads are also analysed.
*Keywords:* follower loads, finite element method, limit of elastic stability, eigenvalue problem.

F. Kun, G.A. D'Addetta, H.J. Herrmann and E. Ramm. Two-dimensional dynamic simulation of fracture and fragmentation of solids. CAMES 1999 (6)

We present a two-dimensional discrete model of solids that allows us to follow the behavior of the solid body and of the fragments well beyond the formation of simple cracks. The model, consisting of polygonal cells connected via beams, is an extension of discrete models used to study granular flows. This modeling is particularly suited for the simulation of fracture and fragmentation processes. After calculating the macroscopic elastic moduli from the cell and beam parameters, we present a detailed study of an uniaxial compression test of a rectangular block, and of the dynamic fragmentation processes of solids in various experimental situations. The model proved to be successful in reproducing the experimentally observed subtleties of fragmenting solids.

I.-W. Lee, H.-J. Jung and M.-Ch. Kim. Efficient free vibration analysis of large structures with close or multiple natural frequencies. Part I: Undamped structures. CAMES 1999 (6)

An efficient numerical method which can calculate the eigenproblem for the large structural system with multiple or close natural frequencies is presented. The method is formulated by the accelerated Newton-Raphson method to the transformed problem. The method can calculate the natural frequencies and mode shapes without any numerical instability which may be encountered in the well-known methods such as the subspace iteration method or the determinant search method which has been widely used for solving eigenvalue problem. The efficiency of the method is verified by comparing convergence and solution time for numerical examples with those of the subspace iteration method and the determinant search method.
*Keywords:* free vibration analysis, accelerated Newton-Raphson method, multiple or close natural frequencies.

I.-W. Lee, H.-J. Jung and M.-Ch. Kim. Efficient free vibration analysis of large structures with close or multiple natural frequencies. Part II: damped structures. CAMES 1999 (6)

An efficient solution method is presented to solve the eigenvalue problem arising in the dynamic analysis of nonproportionally damped structural systems with close or multiple eigenvalues. The proposed method is obtained by applying the modified Newton-Raphson technique and the orthonormal condition of the eigenvectors to the linear eigenproblem format through matrix augmentation of the quadratic eigenvalue problem. In the iteration methods such as the inverse iteration method and the subspace iteration method, singularity may be occurred during the factorizing process when the shift value is close to an eigenvalue of the system. However, even if the shift value is an eigenvalue of the system, the proposed method guarantees nonsingularity, which is analytically proved. The initial values of the proposed method can be taken as the intermediate results of iteration methods or results of approximate methods. Two numerical examples are also presented to demonstrate the effectiveness of the proposed method and the results are compared with those of the well-known subspace iteration method and the Lanczos method.
*Keywords:* quadratic eigenproblem, eigenvalue, non-proportional damped system.

T. Liszkai and J. Farkas. Minimum cost design of ring and stringer stiffened cylindrical shells. CAMES 1999 (6)

In this paper the optimum design of bent cylindrical shells with welded ring or stringer stiffeners are treated. The objective function is the cost of the structure and the constraints are related to overall and local stability. The problem is solved by MathCad 7.plus software and presented also graphically for ring stiffened cylinders and by CFSQP optimization software for stringer stiffened cylinders.

E. Majchrzak and B. Mochnacki. Numerical model of heat transfer between blood vessel and biological tissue. CAMES 1999 (6)

The thermal processes proceeding within a perfused tissue in the presence of a vessel are considered. The Pennes bio-heat transfer equation determines the steady state temperature field in tissue sub-domain, while the ordinary differential equation resulting from the energy balance describes the change of blood temperature along the vessel. The coupling of above equations results from the boundary condition given on the blood vessel wall. The problem is solved using the combined numerical algorithm, in particular the boundary element method (for the tissue sub-domain) and the finite differences method (for blood vessel sub-domain).

B. Miller, G. Piatkowski and L. Ziemianski. Beam yielding load identification by neural networks. CAMES 1999 (6)

The paper presents the application of Artificial Neural Networks for the identification of the load causing a partial yielding in the cross-section of a simple supported beam. The identification of the load was based on a change of the dynamic parameters (eigenfrequencies) of the partially yielding structure. On this basis and using neural networks a tool for the location and evaluation of the load causing the deformation was built. The optimum network architecture, learning algorithm, number of epochs, and the minimum number of eigenfrequencies have been found. In order to come to the final conclusions, a wide variety of network architectures (from simple networks with four neurons in one hidden layer to complex networks consisting of two or three simple networks), learning algorithms and different numbers of learning epochs have been tested.

D.T. Nguyen, W.H. Tang, Y.K. Tung and H.B. Runesha Nonlinear constrained optimizer and parallel processing for golden block line search. CAMES 1999 (6)

Generalized exponential penalty functions are constructed for the multiplier methods in solving nonlinear programming problems. The non-smooth extreme constraint Gext is replaced by a single smooth constraint Gs by using the generalized exponential function (base a>1). The well-known K.S. function is found to be a special case of our proposed formulation. Parallel processing for Golden block line search algorithm is then summarized, which can also be integrated into our formulation. Both small and large-scale nonlinear programming problems (up to 2000 variables and 2000 nonlinear constraints) have been solved to validate the proposed algorithms.

B. Paláncz and G. Popper. Mixed algorithm for solving boundary value problem. CAMES 1999 (6)

Symbolic computation has been applied to Runge-Kutta technique in order to solve two-point boundary value problem. The unknown initial values are considered as symbolic variables, therefore they will appear in a system of algebraic equations, after the integration of the ordinary differential equations. Then this algebraic equation system can be solved for the unknown initial values and substituted into the solution. Consequently, only one integration pass is enough to solve the problem instead of using iteration technique like shooting-method. This procedure is illustrated by solving the boundary value problem of the mechanical analysis of a liquid storage tank. Computation was carried out by MAPLE V. Power Edition package.

Z. Szabó. Parametric excitation of pipes through fluid flow. CAMES 1999 (6)

In this paper the dynamic behaviour of a continuum inextensible pipe containing fluid flow is investigated. The fluid is considered to be incompressible, frictionless and its velocity relative to the pipe has the same but time-periodic magnitude along the pipe at a certain time instant.
The equations of motion are derived via Lagrangian equations and Hamilton's principle. The system is non-conservative, and the amount of energy carried in and out by the flow appears in the model. It is well-known, that intricate stability problems arise when the flow pulsates and the corresponding mathematical model, a system of ordinary or partial differential equations, becomes time-periodic.
The method which constructs the state transition matrix used in Floquet theory in terms of Chebyshev polynomials is especially effective for stability analysis of systems with multi-degree-of-freedom. The stability charts are created w.r.t. the forcing frequency omega, the perturbation amplitude nu and the average flow velocity U.
*Keywords:* pulsatile flow, Floquet theory, Chebyshev polynomials.

G. Szász and G.T. Flowers. Stability considerations in controller designs developed using the LF transformation. CAMES 1999 (6)

Controlling time periodic systems is a significant engineering challenge. One innovative approach that seems to be especially promising involves application of the Lyapunov Floquet (LF) transformation to eliminate time periodic terms from the system state matrices. Traditional control design techniques are then applied and the resulting gains transformed back to the original domain. Typically, the controller design process involves the use of an auxiliary control matrix and (assuming that the actual control matrix is time-varying and non-invertable) a pseudo inverse (which introduces approximations into the procedure). The degree to which the desired control results are achieved depends very strongly upon the impact of these approximations on the actual system dynamics. The research effort described below is concerned with investigating the performance of this LF control strategy and the existence of situations in which application of the procedure may produce undesirable behaviors.