A new approach called the "Variational Theory of Complex Rays'' has been developed in order to calculate the vibrations of slightly damped elastic plates in the medium-frequency range. The solution of a small system of equations, which does not result from a fine spatial discretization of the structure, leads to the evaluation of effective quantities (deformation energy, vibration amplitude, ...). Here we extend this approach, which was already validated for assemblies of homogeneous substructures, to the case of heterogeneous substructures.
*Keywords:* vibrations, medium-frequency range, complex rays, heterogeneous structures.

I. A. Brigadnov. Duality method in limit analysis problem of non-linear elasticity. CAMES 2003 (10)

The limit analysis problem (LAP) for estimation of mechanical durability for non-linear elastic solids is examined. The appropriate dual problem is formulated. After the standard piecewise linear continuous finite-element approximation, the dual LAP is transformed into the problem of mathematical programming with linear limitations as equalities. This finite dimensional problem is solved by the standard method of gradient projection.
*Keywords:* non-linear elastic solid, limit analysis problem, duality method.

S. Abo Diab. Quadrilateral folded plate structure elements of reduced Trefftz type. CAMES 2003 (10)

General strategy for developing finite elements of general geometric shape explained on quadrilateral folded plate structure element ensuring invariance properties is presented in this paper. The basic idea of this strategy consists in using the natural coordinate system only for defining the element geometry and performing the element integration in a mapped biunit square. For defining the approximation functions a suitable local Cartesian coordinate system defined from the directions of the covariant base vectors and the perpendicular contravariant base vectors is used. The origin of the local coordinate system is located at the element centroid (centre of gravity). Hybrid and boundary finite elements of reduced Trefftz type for analysing the folded plate structures are also presented. The folded plate structure element is a combination of a plate bending element and a plane stress element.

N. A. Dumont. Variationally-based hybrid boundary element methods. CAMES 2003 (10)

The hybrid stress boundary element method (HSBEM) was introduced in 1987 on the basis of the Hellinger-Reissner potential, as a generalization of Pian's hybrid finite element method. This new two-field formulation makes use of fundamental solutions to interpolate the stress field in the domain of an elastic body, which ends up discretized as a superelement with arbitrary shape and arbitrary number of degrees of freedom located along the boundary. More recently, a variational counterpart - the hybrid displacement boundary element method (HDBEM) - was proposed, on the basis of three field functions, with equivalent advantages. The present paper discusses these methods as well as the traditional, collocation boundary element method (CBEM). The mechanical properties of the resulting matrix equations are investigated and a series of concepts in both HDBEM and CBEM that have not been properly considered by previous authors, particularly in which concerns body forces, are redefined. This is not a review paper, but rather a theoretical, comparative analysis of three methods, with many physical considerations, some innovations and a few academic illustrations.
*Keywords:* boundary element methods, generalized inverse matrices, variational methods.

N. A. Dumont, R. A. P. Chaves. General time-dependent analysis with the frequency-domain hybrid boundary element method. CAMES 2003 (10)

The paper presents an attempt to consolidate a formulation for the general analysis of the dynamic response of elastic systems. Based on the mode-superposition method, a set of coupled, higher-order differential equations of motion is transformed into a set of uncoupled second order differential equations, which may be integrated by means of standard procedures. The first motivation for these theoretical developments is the hybrid boundary element method, a generalization of T. H. H. Pian's previous achievements for finite elements which, requiring only boundary integrals, yields a stiffness matrix for arbitrary domain shapes and any number of degrees of freedom. The method is also an extension of a formulation introduced by J. S. Przemieniecki, for the free vibration analysis of bar and beam elements based on a power series of frequencies, that handles constrained and unconstrained structures, non-homogeneous initial conditions given as nodal values as well as prescribed domain fields (including rigid body movement), forced time-dependent displacements, and general domain forces (other than inertial forces).

J. A. T. de Freitas. Time integration and the Trefftz Method Part I - First-order and parabolic problems. CAMES 2003 (10)

The finite element method is applied in the time domain to establish formulations for the integration of first-order and parabolic (transient) problems. The modal decomposition concept is applied using two distinct approaches. The first is based on modal decomposition in the space domain to recover the well-established method for uncoupling the parabolic system of equations. To overcome the limitations of this approach in the implementation of large-scale, non-linear problems, the second approach that is reported consists in inducing uncoupling through modal decomposition in the time domain without using the periodic approximation that characterise analyses in the frequency domain. The methods of modal decomposition are related with the implementation of the Trefftz concept in both time and space.
*Keywords:* time integration, first-order problems, parabolic problems, Trefftz method.

J. A. T. de Freitas. Time integration and the Trefftz Method Part II - Second-order and hyperbolic problems. CAMES 2003 (10)

The finite element method is applied in the time domain to establish formulations for the integration of second-order and hyperbolic (dynamic) problems. Modal decomposition in the space domain is used to recover the well-established method for uncoupling the equations of motion, which is extended to include general time approximation bases. The limitations of this approach in the implementation of large-scale, non-linear problems while preserving the uncoupling of the equations of motion are overcome by using the alternative concept of modal decomposition in the time domain. Both single- and double-field formulations are presented and the associated Trefftz formulations are established.
*Keywords:* Time integration, second-order problems, hyperbolic problems, Trefftz method

B. van Hal, W. Desmet, D. Vandepitte, P. Sas. Hybrid Finite Element - Wave Based Method for acoustic problems. CAMES 2003 (10)

The finite element method (FEM) is widely accepted for the steady-state dynamic response analysis of acoustic systems. It exhibits almost no restrictions with respect to the geometrical features of these systems. However, its application is practically limited to the low-frequency range. An alternative method is the wave based method, which is an indirect Trefftz method. It exhibits better convergence properties than the FEM and therefore allows accurate predictions at higher frequencies. However, the applicability is limited to systems of moderate geometrical complexity. The coupling between both methods is proposed. Only the parts of the problem domain with a complex geometry are modelled using the FEM, while the remaining parts are described with a wave based model. The proposed hybrid method has the potential to cover the mid-frequency range, where it is still difficult for currently existing (deterministic) techniques to provide satisfactory prediction results within a reasonable computational time.

I. Herrera. Unified theory of Trefftz methods and numerical implications. CAMES 2003 (10)

In the 1^{st} International Workshop, devoted to Trefftz Method, the author presented an indirect approach to Trefftz Method (Trefftz-Herrera Method), while in the Second one some of the basic ideas of how to integrate different approaches to Trefftz method were introduced. The present Plenary Lecture, corresponding to the 3^{rd} International Workshop of this series, is devoted to show that Trefftz Method, when formulated in an suitable framework, is a very broad concept capable of incorporating and unifying many numerical methods for partial differential equations. In this manner, the unified theory of Trefftz Method that was announced in the second publication of this series, has been developed. It includes Direct Trefftz Methods (Trefftz-Jirousek) and Indirect Trefftz Methods (Trefftz-Herrera). At present, the unified theory is fully developed and an overview is given here, as well as a brief description of its numerical implications.

L. Jakubovicová, M. Vasko, V. Kompis. Trefftz functions using the fundamental solution with the singularity outside the domain. CAMES 2003 (10)

Two types of Trefftz (T-) functions are often used - fundamental solutions with their singularities outside the given region and general solutions of homogenous differential equations. For elasticity problems the general solution of the homogeneous differential equation (equilibrium equation in displacements known as Lame-Navier equations) can be found in the polynomial form.
In this paper we present the first type of T-functions. The paper deals with the investigation of accuracy and stability of the resulting system of discretized equations in relation to the position of the source (singularity) point. In this way non-singular reciprocity based boundary integral equations relate the boundary tractions and the boundary displacements of the searched solution to corresponding quantities of the known solutions.
It was found that there exist an optimal relation of the distance of the singularity to the distance of the collocation points where both the integration accuracy and numerical stability are good.
*Keywords:* point and line Hertzian contact, infinitesimal displacements, large element/sub-domain concept, FEM/BEM technique.

A. A. Javadi, T. P. Tan, M. Zhang. Neural network for constitutive modelling in finite element analysis. CAMES 2003 (10)

Finite element method has, in recent years, been widely used as a powerful tool in analysis of engineering problems. In this numerical analysis, the behavior of the actual material is approximated with that of an idealized material that deforms in accordance with some constitutive relationships. Therefore, the choice of an appropriate constitutive model, which adequately describes the behavior of the material, plays a significant role in the accuracy and reliability of the numerical predictions. Several constitutive models have been developed for various materials. Most of these models involve determination of material parameters, many of which have no physical meaning [1, 2].
In this paper a neural network-based finite element analysis will be presented for modeling engineering problems. The methodology involves incorporation of neural network in a finite element program as a substitute to conventional constitutive material model. Capabilities of the presented methodology will be illustrated by application to practical engineering problems. The results of the analyses will be compared to those obtained from conventional constitutive models.
*Keywords:* finite element, neural network, constitutive modelling, soil.

E. A. W. Maunder. Hybrid equilibrium plate elements of high degree. CAMES 2003 (10)

This paper concerns the modelling of plate bending problems governed by Reissner-Mindlin theory when hybrid equilibrium elements of high polynomial degree are used. The fields of statically admissible stress-resultants are categorised into three types according to the nature of their incompatibilities, i.e. pure Trefftz or strongly compatible, weakly compatible, and hyperstatic or strongly incompatible. The effects of this categorisation are reflected in the element formulation. Incompatibilities are quantified in terms of local discontinuities which also account for transverse twist terms. The construction of bases for the three corresponding subspaces of stress-resultants by numerical and/or algebraic means is reviewed. The potential use of a reformulated element is considered in the context of glass plate structures where residual or hyperstatic stresses play an important role.

E. A. W. Maunder. Trefftz in translation. CAMES 2003 (10)

This paper reviews the important concepts presented by Trefftz in 1926 regarding bounds to solutions, error estimation, and hybrid fields for use with domain decomposition. Observations are offered from the perspective of today's relatively mature state of the art in finite element methods. The numerical examples presented by Trefftz are also reviewed with the benefit of `exact' solutions made available from current commercial finite element methods. The accuracies of the solutions given by Trefftz are quantified and compared, and the effectivity indices of Trefftz's proposed error estimates are also quantified. An English translation of the original German version of Trefftz's paper is included for reference in an Appendix.

O. J. B. A. Pereira, J. P. M. de Almeida. Adaptive refinement for a local error bound based on duality. CAMES 2003 (10)

This paper presents the basis of an adaptive mesh refinement technique aimed at reducing a local error, i.e. the error in a local quantity, which is defined as the integral of a stress or a displacement in a given subregion. Two pairs of dual solutions, one corresponding to the applied load and the other to the virtual action, dual of the local quantity, are used to bound the local error and to provide the element error indicators for the adaptive process. A test case is used to exemplify the behaviour of the technique.

J. Petrolito. Analytical formulation of hybrid-Trefftz thick plate elements. CAMES 2003 (10)

This paper discusses a domain-based formulation for hybrid-Trefftz\ thick plate elements. The formulation can be applied to triangular and quadrilateral elements, and enables an analytical formulation of these elements to be used. Techniques for improving the computational efficiency are explored, and a simple and accurate triangular element is formulated in this manner.

R. Piltner. Solution representations for Trefftz-type finite elements. CAMES 2003 (10)

Solution representations are available for several differential equations. For elasticity problems some of the solution representations are considered in this paper. The solution representations can be used for a systematic construction of Trefftz functions for the derivation of Trefftz-type finite elements. For the example of a thick plate a set of Trefftz functions is presented.
*Keywords:* Trefftz functions, Trefftz-type finite elements.

R. Piltner. The derivation of special purpose element functions using complex solution representations. CAMES 2003 (10)

For several elasticity problems, solution representations for the displacements and stresses are available. The solution representations are given in terms of ``arbitrary'' complex valued functions. For any choice of the complex functions, the governing differential equations are automatically satisfied. Complex solution representations are therefore useful for applications of the Trefftz method. For the analysis of local stress concentrations, due to the local geometry of the boundary curve, such solution representations can be very helpful in the construction of appropriate series of Trefftz functions. In this paper, a few examples are given to demonstrate how to construct Trefftz functions for special purpose finite elements, which include the local solution behavior around a stress concentration or stress singularity.
*Keywords:* elasticity, complex solution representations, stress singularities, Trefftz functions, Trefftz-type finite elements.

S. Y. Reutskiy. Trefftz type method for 2D problems of electromagnetic scattering from inhomogeneous bodies. CAMES 2003 (10)

A new numerical method for scattering from inhomogeneous bodies is presented. The cases of E and H-polarizated incident wave scattered by an infinite 2D cylinder are considered. The scattered field is looked for in two different domains. The first one is a bounded region inside the scattering body with an inhomogeneous permittivity ε(x,y). The second one is an unbounded homogeneous region outside the scatterer. An approximate solution for the scattered field inside the scatterer is looked for by applying the QTSM technique. The method of discrete sources is used to approximate the scattered field in the unbounded region outside the scattering body. A comparison of the numerical and analytic solutions is performed.

K.Y. Sze, W.G. Jin, N. Sheng, J. Li. Trefftz methods for plane piezoelectricity. CAMES 2003 (10)

Starting from the governing equations, the general solution and the complete solution set for plane piezoelectricity are derived in this paper. Subsequently, the Trefftz collocation method (TCM) is formulated. TCM falls into the category of Trefftz indirect methods which adopt the truncated complete solution set as the trial functions. Similar to the boundary element method, the solution procedure of TCM requires only boundary discretization. Numerical examples are presented to illustrate the efficacy of the formulation.
*Keywords:* Trefftz, piezoelectricity, boundary element, collocation.

B. Szybiński, A.P. Zieliński, M. Karaś. Folded-plate structures with openings - analysis and optimization. CAMES 2003 (10)

This paper is a continuation and development of the dissertation[1]. Complex folded-plate structures with holes are analyzed using the Trefftz-type finite elements, which appears very effective. The shape functions of these elements (Trefftz functions) fulfill respective differential equations. Then, a certain optimization algorithm is proposed, in which an optimized structure can have a large number of parameters used as optimization variables. Therefore, in particular stages of the proposed procedure, less important variables can be eliminated. The choice of the active variable set is based on investigation of sensitivity of the objective function and constraints on small changes of these variables.
*Keywords:* Trefftz-type finite elements, hybrid elements, folded-plate structures, optimization of structures.

C.M. Tiago, L.M.S.S. Castro, V.M.A. Leităo. Trefftz and RBF-based formulations for concrete beams analysis using damage models. CAMES 2003 (10)

In this work a hybrid-Trefftz formulation and a *meshless* approach based on the use of radial basis functions (RBF) are applied to the analysis of reinforced concrete beams. Resorting to the Mazars model, the concrete is represented by an elastic medium with progressive damage. In the hybrid-Trefftz formulation a stress field that satisfies *a priori* the equilibrium equations on the domain is used. The displacements on the static boundary are independently approximated, resulting in a governing system where the operators have to be integrated over the domain of the problem. In what concerns the *meshless* approach, radial basis functions are used to approximate the displacement fields but, as a collocation procedure is used, no integrations are required. A numerical example illustrates the results obtained with both techniques.

A. Uściłowska-Gajda, J.A. Kołodziej, M. Ciałkowski, A. Frąckowiak. Comparison of two types of Trefftz method for the solution of inhomogeneous elliptic problems. CAMES 2003 (10)

The solution of inhomogeneous elliptic problems by the Trefftz method has become increasingly more popular during the last decade [2, 3, 4]. One method of solution uses the fundamental solutions as trial functions and the inhomogeneous part is expressed by radial basis functions (RBFs). The purpose of this paper is to solve several boundary value problems that have exact solutions. Two error criteria are used for comparison of the exact solutions and the approximated solutions. The first is the mean least square global error. The second has a local character, as it measures the absolute maximal error.