The paper deals with the application of the differential quadrature method based on a piecewise polynomial to the nonlinear vibration analysis of beams. The initial-boundary-value problem is solved to study the computational stability of the method. The results are compared with those, obtained by the conventional differential quadrature. The effects of the spline degree, the number of nodes and the distribution of sampling points on the convergence and stability is also presented. The nonlinear free vibration analysis is carried out to verify the accuracy of the method.

S.P. Anjali Devi, B. Ganga. MHD nonlinear flow and heat transfer over a stretching porous surface of constant heat flux. CAMES 2008 (15)

MHD nonlinear steady flow and heat transfer over a porous surface stretching with a power-law velocity and of constant heat flux is investigated. The governing nonlinear partial differential equations are reduced to nonlinear ordinary differential equations by using similarity transformation. As the presented solution method requires the magnetic field to vary in space in a specific manner, a special form for the variable magnetic field is chosen. Resulting equations are numerically solved by Runge-Kutta shooting method. Values of skin-friction and rate of heat transfer are obtained. The effect of magnetic field, stretching parameter, magnetic interaction parameter, suction parameter and Prandtl number over a flow field and other physical quantities have been discussed in detail.

K.A. Oladejo, O.A. Koya, L.O. Adekoya. A computational model for static and dynamic balancing of masses on rotating shafts. CAMES 2008 (15)

Balancing is the process of improving the mass distribution of a body so that it rotates in its bearings without unbalanced centrifugal forces. It is thus critical to the performance of any high speed equipment. The problem is mathematically modeled and a genetic algorithm is presented for obtaining optimal solutions for balancing problems on rotating shafts. This is eventually converted into computer package titled BALANCER, developed using the VisualBASIC platform. Examples are presented to illustrate implementation of the methodology. The model was tested by using typical problems, correctly solved in the literature using conventional methods. The results of the three examples gave same match with those obtained from analytical approach. The accuracy of analysis using the model and the students' feedback suggest that integration of the software tool will be beneficial for improving students' performance in any dynamics course.
*Keywords:* modeling, balancing, rotating shaft, off-line, imbalance.

P. Dłużewski, M. Maździarz, P. Traczykowski, G. Jurczak, K. Niihara, R. Nowak, K. Kurzydłowski. A hybrid atomistic-continuum finite element modelling of nanoindentation and experimental verification for copper crystal. CAMES 2008 (15)

Problem of locally disordered atomic structure is solved by using a hybrid formulation in which nonlinear elastic finite elements are linked with discrete atomic interaction elements. The continuum approach uses nonlinear hyperelasticity based upon the generalized strain while the atomistic approach employs the Tight-Binding Second-Moment Approximation potential to create new type of elements. The molecular interactions yielding from constitutive models of TB-SMA were turned into interactions between nodes to solve a boundary value problem by means of finite element solver. In this paper we present a novel way of modelling materials behaviour where both discrete (molecular dynamics) and continuum (nonlinear finite element) methods are used. As an example, the nanoindentation of a copper sample is modelled numerically by applying a hybrid formulation. Here, the central area of the sample subject to nanoindentation process is discretised by an atomic net where the remaining area of the sample far from indenters tip is discretised by the use of a nonlinear finite element mesh.

E. Kita, Y. Hirayama. Application of Trefftz method for temperature rise analysis on human skin exposed to radiation. CAMES 2008 (15)

This paper describes the application of the Trefftz method to the temperature rise in human skin exposed to radiation from a cellular phone. A governing equation is given as the Poisson equation. An inhomogeneous term of the equation is approximated with a polynomial function in Cartesian coordinates. The use of the approximated term transforms the original boundary-value problem to that governed with a homogeneous differential equation. The transformed problem can be solved by the traditional Trefftz formulation. Firstly, the present method is applied to a simple numerical example in order to confirm the formulation. The temperature rise in a skin exposed to radiation is considered as a second example.
*Keywords:* Trefftz method, Poisson equation, polynomial function.

T. Winkler, J. Tokarczyk, S. Bojara. Use of reverse engineering method in verification of virtual prototypes. CAMES 2008 (15)

The prototypes of *Falling-Object Protective Structures* (FOPS), which are used in self-propelled mining machines, undergo obligatory laboratory strength tests. The stand tests should be preceded with virtual prototyping of computer models to reduce the number of prototypes to an indispensable minimum. The Finite Element Method was used for the virtual prototyping. At the same time, after tests, the real protective structure was transformed by the Reverse Engineering method into a computer model. The deformations were compared on both models, at the same cross sections. The anthropometric model of operator was placed inside the reconstructed model.
*Keywords:* falling-object protective structures (FOPS), virtual prototyping, reverse engineering, human body modelling, finite element method.

E. Pabisek. Self-learning FEM/NMM approach to identification of equivalent material models for plane stress problem. CAMES 2008 (15)

The autoprogressive and cumulative algorithms, basing on `on line' formulation of patterns and the training of NMM (Neural Material Model), are evaluated to be comparable in case of uniaxial stress state problems. It is shown in the paper that for the plane stress boundary value problems the autoprogressive algorithm, in which NMM is trained for each load increment, is superior to the cumulative algorithm. In order to formulate a small NMM and accelerate the convergence of the iteration of computed equilibrium paths to the monitored paths, a smaller number of inputs NMM is discussed and a modified selection of the training patterns is applied. A new approach is proposed with respect to the designing of NMMs, combining the 'on line' and 'off line' training of neural networks. The discussed problems are illustrated with two study cases. They are related to the formulation of NMMs for the identification of equivalent materials in plane trusses made of the Ramberg-Osgood material and for elasto-plastic plane stress boundary value problems.
*Keywords:* neural networks; material model; constitutive modelling.