The variational theory is the theoretical basis of the finite element method, meshfree particle methods and other modern numerical techniques. The present paper establishes a family of variational principles for nonlinear piezoelectricity. A new constitutive relation is suggested, which is deduced as a stationary condition of a generalized variational principle.
*Keywords:* variational theory, piezoelectricity, constitutive equations.

T. Stręk. The Lyapunov exponents for the partitioned-pipe mixer. CAMES 2003 (10)

This paper presents a mechanical model of the partitioned-pipe mixer (PPM) in case where pipe of the static mixer rotates with angular periodic velocity. Mixing becomes more efficient if the forcing of fluid mixing process is time periodic. Chaos in duct flows can be achieved by time modulation or by spatial changes along the duct axis. The values of Lyapunov exponents for flow in PPM are calculated.
*Keywords:* partitioned-pipe mixer, chaos, Lyapunov exponents.

I. Skalna. Methods for solving systems of linear equations of structure mechanics with interval parameters. CAMES 2003 (10)

Interval analysis permits to calculate guaranteed a posteriori bounds for the solutions of problems with uncertain (interval) input data. Most of the methods of interval analysis assume that all input data vary independently within the given lower and upper bounds. In many practical applications it need not be a case, and the assumption of independence may lead to large overestimation of the set of solutions.
The subject of this work is the problem of solving systems of linear interval equations with coefficients linearly dependent on a set of interval parameters called *coefficient dependence problem*. The purpose of this work is to present methods producing sharp bounds for the set of solutions of systems with dependent input data.
The paper starts with an introduction to systems of linear interval equations and the problem of data dependencies in such systems. A parametric formulation of the coefficient dependence problem follows next.
Finally, three algorithms to calculate tighter bounds for problems with linearly dependent coefficients, namely the Rump's method, its improved version developed by the author, and the IPM method based on the results from Neumaier [8] are presented and discussed. The algorithms are evaluated and compared using some examples of truss structure analysis.

M. Paszyński. Object-oriented software system that performs FPM simulation in the area with moving boundary, and its application to the blood flow problem. CAMES 2003 (10)

The paper presents the project of object-oriented software system for modeling flows of fluids inside the area with moving boundary. The flow of a fluid is modeled by using Fluid Particle Model. Finite Elements mesh is generated on the boundary of the area, to allow calculations of stresses on the boundary, arisen from interaction of the fluid with the boundary. Here was presented the application of the system, that simulates the phenomenon of energy and mass transport in large arteries in man. The validity of the computational method was established by comparing the numerical results to medical measurements data. The architecture and results of Fluid Particle Model were presented to compare with the architecture and results of Finite Element models for blood flow in arteries, described in other publications.

M. Klisiński, Ch. Luo and E. Postek. Discussion on application of piece-wise linear weight functions in 2D contact problems. CAMES 2003 (10)

Standard higher order finite elements often perform unsatisfactory in contact problems. The major difficulties are caused by uneven distribution of nodal forces resulting in oscillating contact pressures. The paper presents a new approach that eliminates this drawback. The weight functions are chosen in such a way that even distributions of nodal forces are obtained. It is achieved by applying piece-wise linear functions. Two new 2D isoparametric quadratic elements are derived: 6-node triangle and 8-node quadrilateral, and tested in many examples. The new elements have unsymmetric stiffness matrices, but the provided examples show their good performance in contact problems.

E. Kita, Y. Ikeda and N. Kamiya. Application of Trefftz method to steady-state heat conduction problem in functionally gradient materials. CAMES 2003 (10)

This paper describes the application of Trefftz method to the steady-state heat conduction problem on the functionally gradient materials. Since the governing equation is expressed as the non-linear Poisson equation, it is difficult to apply the ordinary Trefftz method to this problem. For overcoming this difficulty, we will present the combination scheme of the Trefftz method with the computing point analysis method. The inhomogeneous term of the Poisson equation is approximated by the polynomial of the Cartesian coordinates to determine the particular solution related to the inhomogeneous term. The solution of the problem is approximated with the linear combination of the particular solution and the *T*-complete functions of the Laplace equation. The unknown parameters are determined so that the approximate solution will satisfy the boundary conditions by means of the collocation method. Finally, the scheme is applied to some numerical examples.
*Keywords:* Trefftz method, computing point analysis method, steady-state heat conduction, functionally gradient materials.

R. Będziński, K. Ścigała. FEM analysis of strain distribution in tibia bone and relationship between strains and adaptation of bone tissue. CAMES 2003 (10)

The purpose of the research is the estimation of strain distribution in tibia bone. Resultant strain distribution constitutes necessary data for the calculations made in the process of simulation of bone tissue adaptation. Estimation of strain distribution in proximal part of tibia bone is made for different load conditions (including the one following total knee arthroplasty and a surgical correction of lower limb with the application of high tibial osteotomy). The model of tibia bone and soft tissues, prepared for finite element analysis, was made with the use of Ansys 5.6. The geometry of bone was estimated by 3-D digitalisation of a physical model of bone. Displacements distribution obtained from the simulation was compared with the measurements of the physical model of a knee joint. In the research the holographic interferometry method was applied. The results of this calculation are helpful in the estimation of boundary conditions for a simulation of bone tissue functional adaptation in the region of a knee joint. It has been found out that there are differences in strain distribution in different load conditions. However, the perfect agreement of experimental and numerical results for a simple static load indicates that the numerical model is valid for this simulation in a certain range of the applied load.
*Keywords:* tibia bone, bone tissue adaptation, high tibial osteotomy, knee joint arthroplasty.