Variational formulations that can be employed in the approximation of boundary value problems involving essential and natural boundary conditions are presented in this paper. They are based on trial functions so chosen as to satisfy a priori the governing differential equations of the problem. The essential boundary conditions are used to construct the displacement approximation basis at finite element level. The natural boundary conditions are enforced on average and their integral forms constitute the variational expression of the finite element approach. The shape functions contain both homogeneous and particular terms, which are related through the interpolation technique used. The application in the framework of the finite element method of the approach proposed here is not trouble free, particularly in what concerns the inter-element continuity condition. The Gauss divergence theorem is used to enforce the essential boundary conditions and the continuity conditions at the element boundary. An alternative but equivalent boundary technique developed for the same purpose is presented also. It is shown that the variational statement of the Trefftz approach is recovered when the Trefftz trial functions are so chosen as to satisfy the essential boundary conditions of the problem.

I.A. Brigadnov. Numerical analysis of dielectrics in powerful electrical fields. CAMES 2001 (8)

The limiting analysis problem for dielectrics in nonhomogeneous powerful electrical fields is considered. In the framework of this problem the external charges for which the appropriate electrostatical variational problem has no solution are calculated, that solution is treated as a beginning of the electrical puncture of dielectric. From the mathematical point of view the limiting analysis problem is non-correct and needs a relaxation. This is achieved using a partial relaxation based on a special discontinuous finite-element approximation.

F.L.S. Bussamra, P.M. Pimenta and J.A.T. Freitas. Hybrid-Trefftz stress elements for three-dimensional elastoplasticity. CAMES 2001 (8)

The stress model of the hybrid-Trefftz finite element formulation is applied to the elastoplastic analysis of solids. The stresses and the plastic multipliers in the domain of the element and the displacements on its boundary are approximated. Harmonic and orthogonal hierarchical polynomials are used to approximate the stresses, constrained to solve locally the Beltrami governing differential equation. They are derived from the associated Papkovitch-Neuber elastic displacement solution. The plastic multipliers are approximated by Dirac functions defined at Gauss points. The finite element equations are derived directly from the structural conditions of equilibrium, compatibility and elastoplasticity. The non-linear governing system is solved by the Newton method. The resulting Hessian matrices are symmetric and highly sparse. All the intervening arrays are defined by boundary integral expressions or by direct collocation. Numerical applications are presented to illustrate the performance of the model.

M.J. Cialkowski. Trefftz functions as basic functions of FEM in application to solution of inverse heat conduction problem. CAMES 2001 (8)

The work presents the application of heat polynomials for solving an inverse problem. The heat polynomials form the Trefftz Method for non-stationary heat conduction problem. They have been used as base functions in Finite Element Method. Application of heat polynomials permits to reduce the order of numerical integration as compared to the classical Finite Element Method with formulation of the matrix of system of equations.

J.F. Debongnie and P. Beckers. On a general decomposition of the error of an approximate stress field in elasticity. CAMES 2001 (8)

The errors of finite element approximations are analysed in a general frame, which is completely independent from the way through which the approximate solution was obtained. It is found that the error always admits decomposition in two terms, namely the equilibrium error and the compatibility error, which are orthogonal. Each of these admits upper and lower bounds that can be computed in a post-processing scheme.

W. Desmet, P. Sas and D. Vandepitte. An indirect Trefftz method for the steady-state dynamic analysis of coupled vibro-acoustic systems. CAMES 2001 (8)

A new prediction technique, based on the indirect Trefftz method, has been developed for the steady-state dynamic analysis of coupled vibro-acoustic systems. In contrast with the finite element method, in which the dynamic field variables within each element are expanded in terms of local, non-exact shape functions, the dynamic field variables are expressed as global wave function expansions, which exactly satisfy the governing dynamic equations. The contributions of the wave functions to the coupled vibro-acoustic response result from a weighted residual formulation of the boundary conditions. This paper discusses the basic principles and convergence properties of the new prediction technique and illustrates its performance for some two-dimensional validation examples. A comparison with the finite element method indicates that the new prediction method has a substantially higher convergence rate. This makes the method suitable for accurate coupled vibro-acoustic predictions up to much higher frequencies than the finite element method.

J.A.T. Freitas and C. Cismasiu. Developments with hybrid-Trefftz stress and displacement elements. CAMES 2001 (8)

The paper reports on the work on hybrid-Trefftz finite elements developed by the Structural Analysis Research Group, ICIST, Technical University of Lisbon. A dynamic elastoplastic problem is used to describe the technique used to establish the alternative stress and displacement models of the hybrid-Trefftz finite element formulations. They are derived using independent time, space and finite element bases, so that the resulting solving systems are symmetric, sparse, naturally *p*-adaptive and particularly well suited to parallel processing. The performance of the hybrid-Trefftz stress and displacement models is illustrated with a number of representative static and dynamic applications of elastic and elastoplastic structural problems.
*Keywords:* Hybrid-Trefftz elements, elasticity, elastoplasticity, dynamics.

J.C. García Orden and J.M. Goicolea. An energy-momentum algorithm for flexible multibody systems with finite element techniques. CAMES 2001 (8)

A unified approach for the treatment of the non-linear dynamics of multibody systems (MBS) composed of both rigid and elastic bodies is proposed. Large displacements and rotations, large strains and non-linear elastic material response are considered for the elastic bodies. The proposed formulation exploits three key ingredients: the use of a dependent set of inertial coordinates of selected points of the system; the use of a basic constraint library enforced through the penalty method; the use of the energy-momentum method to integrate the equations. The proposed algorithm is set in the framework of a non-conventional finite element formulation, which combine naturally the displacement-based discretisation of the deformable bodies with rigid body mechanics. Two key performance features are achieved. The exact conservation of total momentum and total energy in conservative systems is ensured. The major drawback of the penalty method, namely numerical ill-conditioning that leads to stiff equation systems, is overcome.

I. Herrera. On Jirousek method and its generalizations. CAMES 2001 (8)

Professor Jirousek has been a very important driving force in the modern development of Trefftz method, contributing to its application in many different fields such as elasticity, shells and plates theory, Poisson equation and transient heat analysis. This article is dedicated to him. The focus of the paper is to incorporate Jirousek method into a very general framework of Trefftz method which has been introduced by Herrera. Usually finite element methods are developed using splines, but a more general point of view is obtained when they are formulated in spaces of fully discontinuous functions - i.e., spaces in which the functions together with their derivatives may have jump discontinuities - and in the general context of boundary value problems with prescribed jumps. Two broad classes of Trefftz methods are obtained: direct (Trefftz-Jirousek) and indirect (Trefftz-Herrera) methods. In turn, each one of them can be divided into overlapping and non-overlapping.

J. Jirousek, A.P. Zielinski and A. Wróblewski. T-element analysis of plates on unilateral elastic Winkler-type foundation. CAMES 2001 (8)

This paper presents a hybrid-Trefftz finite element algorithm designated as *fictitious load approach*. Its originality resides in the formulation and practical application of concepts which make it possible to account for the unilateral contact conditions of a plate without modification of the finite element mesh. To reach this aim, the approach allows the movable interface between the contact and non-contact parts of the plate to travers any finite element subdomain. The adjustments are confined to fictitious load dependent terms, while the element stiffness matrices remain unchanged during the whole iterative process. Several numerical examples are analysed to assess the effectivity of the T-element algorithm and to compare it with some of the existing solutions of the same problem.
*Keywords:* finite elements, Trefftz method, contact problem.

J.A. Kolodziej and A.C. Mendes. T-functions for 2-D creeping flow on domains with circular cylinders, corners, and possessing symmetry CAMES 2001 (8)

The purpose of the paper is to propose of a way of constructing trial functions for the indirect Trefftz method as applied to 2-D creeping (Stokes) flow problems. The considered cases refer to the problems of flow around fixed and rotating circular cylinders, in corners with two walls fixed, or one wall moving, and flow possessing particular symmetry. The trial functions, proposed and systematically constructed fulfil exactly not only the governing equation, like T-complete Herrera functions, but also certain given boundary conditions and conditions resulting from assumed symmetry. A list of such trial functions, unavailable elsewhere, is presented. The derived functions can be treated as a subset of T-complete Herrera functions, which can be used for solving typical boundary-value problems.

V. Kompis, F. Konkol' and M. Vasko. Trefftz-polynomial reciprocity based FE formulations. CAMES 2001 (8)

The paper contains a general procedure for obtaining of Trefftz polynomials of arbitrary order for 2D or 3D problems by numerical or analytical way. Using Trefftz polynomials for displacement and tractions the unknown displacements and tractions are related by non-singular boundary integral equations. For a multi-domain (element) formulation we suppose the displacements to be continuous between the sub-domains and the tractions are connected in a weak (integral) sense by a variational formulation of inter-element equilibrium. The stiffness matrix defined in this way is nonsymmetric and positive semi-definite. The finite elements can be combined with other well known elements. The form of the elements can be, however, more general (the multiply connected form of the element is possible, transition elements which can be connected to more elements along one side are available). It is also very easy and simply possible to assess the local errors of the solution from the traction incompatibilities (the inter-element equilibrium, which is satisfied in a weak sense only, is the only incompatibility in the solution of the linear problem). The stress smoothing is a very useful tool in the post-processing stage. It can improve the accuracy of the stress field by even one order or more comparing to the simple averaging, if the stress gradients in the element are large. Also the convergence of the so obtained stress field increases. The examples with high order gradient field and crack modelling document the efficiency of this FEM formulation. The extension to the solution of other field problems is very simple.

V.M.A. Leitao. Comparison of Galerkin and collocation Trefftz formulations for plane elasticity. CAMES 2001 (8)

The purpose of this work is to compare and assess, more in terms of computational efficiency than in terms of accuracy, three alternative implementations of a boundary formulation based on the Trefftz method for linear elastostatics, namely a collocation-based and two Galerkin-based approaches. A finite element approach is used in the derivation of the formulation for the Galerkin-based alternative implementations. The coefficients of the structural matrices and vectors are defined either by regular boundary integral expressions or determined by direct collocation of the trial functions. Numerical tests are performed to assess the relative performance of the different alternative implementations.

E.A.W. Maunder. A Trefftz patch recovery method for smooth stress resultants and applications to Reissner--Mindlin equilibrium plate models. CAMES 2001 (8)

A method is proposed for smoothing approximate fields of stress-resultants in patches of finite elements. The method is based on combining Trefftz fields of stress-resultants in a p-version so as to obtain a closest fit using the strain energy norm as a measure. The local systems of equations are formulated from boundary integrals. The method is applied to a problem of a square plate modelled by hybrid equilibrium plate elements using Reissner--Mindlin theory. Results for the problem indicate that the smooth solution for stresses can be in close agreement with the analytic solution in the interior of a patch. Proposals are also included to aid the visualization of tensor and vector continuous fields as stress trajectories.

S. Ohnimus, M. Rüautter and E. Stein. General aspects of Trefftz method and relations to error estimation of finite element approximations. CAMES 2001 (8)

In this paper a guaranteed upper bound of the global discretization error in linear elastic finite element approximations is presented, based on a generalized Trefftz functional. Therefore, the general concept of complementary energy functionals and the corresponding approximation methods of Ritz, Trefftz, the method of orthogonal projection and the hypercircle method are briefly outlined. Furthermore, it is shown how to use a generalized Trefftz functional to solve a Neumann problem in linear elasticity. Based on an implicit a posteriori error estimator within the finite element method, using equilibrated local Neumann problems, the generalized Trefftz functional yields a computable guaranteed upper bound of the discretization error without multiplicative constants.

O.J.B.A. Pereira and J.P.M. Almeida. A posteriori error estimation for equilibrium finite elements in elastostatic problems. CAMES 2001 (8)

Equilibrated solutions, locally satisfying all the equilibrium conditions, may be obtained by using a special case of the hybrid finite element formulation. Equilibrium finite element solutions will normally present compatibility defaults, which may be directly used to estimate the error of the solution, *a posteriori*. Another approach is to construct a compatible solution using the stresses and displacements available from the hybrid solution. From this dual solution, an upper bound for the global error is obtained. In this paper, the hybrid equilibrium element formulation, the occurrence of spurious kinematic modes and the use of super-elements, in *2D* and *3D*, are briefly reviewed. Compatibility defaults for *2D* and *3D* are presented, together with an expression for an element error indicator explicitly based on such defaults. A local procedure for recovering conforming displacements from the equilibrium finite element solution is also presented. The *h*-refinement procedure is adapted to prevent irregular refinement patterns.

S. Suleau and P. Bouillard. Accurate acoustic computations using a meshless method. CAMES 2001 (8)

It is well known today that the standard finite element method (FEM) is unreliable to compute approximate solutions of the Helmholtz equation for high wavenumbers due to the pollution effect, consisting mainly of the dispersion, i.e. the numerical wavelength is longer than the exact one. Unless highly refined meshes are used, FEM solutions lead to unacceptable solutions in terms of precision. The paper presents an application of the Element-Free Galerkin Method (EFGM) leading to extremely accurate results in comparison with the FEM. Moreover, the present meshless formulation is not restricted to regular distribution of nodes as some stabilisation methods and a simple but real-life problem is investigated in order to show the improvement in the accuracy of the numerical results, as compared with FEM results.

B. Szybinski, A. Wróblewski and A.P. Zielinski. General strategy of h-p adaptive solutions in structural Trefftz-type element analysis. CAMES 2001 (8)

The paper deals with a strategy of reliable application of the Trefftz elements in the linear analysis of complex engineering structures with stress concentrators. The standard *p*-adaptivity is suggested in the low gradient areas. For selected large-gradient local zones certain specific T-element substructures are proposed. The *h-p* adaptive procedures for optimization of parameters of the substructures are numerically investigated.

A.P. Zielinski, H. Sanecki and M. Karas. Effectiveness of the Trefftz method in different engineering optimization procedures. CAMES 2001 (8)

In a large class of linear, mathematically modelled engineering problems the Trefftz algorithms give accurate solutions in a relatively short computational time. Moreover, the Trefftz functions, fulfilling the governing differential equations, can be used as shape functions of finite elements (T-elements), also with openings and notches. This suggested the authors to investigate the advantages and limitations of the method in optimization of structures with the stress concentrators, e.g. perforated plates. Certain auxiliary object functions, which included simultaneously the objective of the optimization and the constraints, were introduced and investigated. Different optimization strategies were also taken into consideration. To improve the optimization task in case of a large number of variables the authors suggested an algorithm, which used the engineering sensitivity analysis to eliminate less important variables in the particular stages of the procedure.