In the paper the stationary 2D inverse heat conduction problems are considered. To obtain an approximate solution of the problems three variants of the FEM with harmonic polynomials (Trefftz functions for Laplace equation) as base functions were used: the continuous FEMT, the non-continuous FEMT and the nodeless FEMT. In order to ensure physical sense of the approximate solution, one of the aforementioned physical aspects is taken into account as a penalty term in the functional, which is to be minimized in order to solve the problem. Three kinds of physical aspects that can smooth the solution were used in the work. The first is the minimization of heat flux jump between the elements, the second is the minimization of the defect of energy dissipation and third is the minimization of the intensity of numerical entropy production. The quality of the approximate solutions was verified on two test examples. The method was applied to solve inverse problem of stationary heat transfer in a rib.

H.-T. Huang, Z.-C. Li, A.H.-D. Cheng. Coupling techniques of Trefftz methods. CAMES 2008 (15)

The Trefftz method pioneered by Trefftz in 1926 is described as follows: The particular solutions or the fundamental solutions are chosen, a linear combination of those functions is regarded as an approximate solution of partial differential equations (PDEs), and their expansion coefficients are sought by satisfying the interior and exterior boundary conditions. When the solution domain is not rectangular or sectors, the piecewise particular solutions may be chosen in different subdomains, and some coupling techniques must be employed along their interior boundary conditions. In Li *et al.* [1], the collocation method is used for the Trefftz method, to lead to the collocation Trefftz method (i.e., the indirect Trefftz method). In this paper, we will also discuss other four coupling techniques: (1) the simplified hybrid techniques, (2) the hybrid plus penalty techniques, (3) the Lagrange multiplier techniques for the direct Trefftz method, and (4) the hybrid Trefftz method of Jirousek [2] and Qin [3]. Error bounds are derived in detail for these four couplings, to achieve exponential convergence rates. Numerical experiments are carried out, and comparisons are also made.
[1] Z.C. Li, T.T. Lu, H.Y. Hu, A.H.-D. Cheng, Trefftz and Collocation Methods, WITpress 2008.
[2] J. Jirousek, Comput. Meth. Appl. Mech. Engrg., 14: 65-92, 1978.
[3] Q.H. Qin, The Trefftz Finite and Boundary Element Methods, WITpress 2000.

H. Jopek, J.A. Kołodziej. Application of genetic algorithms for optimal positions of source points in the method of fundamental solutions. CAMES 2008 (15)

This paper describes the application of the method of fundamental solutions for 2-D harmonic and biharmonic problems. Also, genetic algorithm is presented as a numerical procedure used for the determination of source points positions. Choosing good locations of source points is crucial in the MFS as it has a great impact on the quality of the solution. Genetic algorithm is applied in order to find such an arrangement of source points, which provides the solution of sufficient accuracy.
*Keywords:* method of fundamental solutions, genetic algorithm, multicriteria optimization, Motz problem, biharmonic problem.

J.A. Kołodziej, M. Mierzwiczak, H. Tylicki. Effective flexural rigidity of perforated plates by means of Trefftz method with using special purpose Trefftz functions. CAMES 2008 (15)

The main purpose of this paper is the investigation of the boundary effect in bending problem of perforated plates and its influence on the effective flexural rigidity. The considered strip plate is loaded by constant uniformly distributed load and has square penetration pattern. The boundary value problem for determination of deflection repeated element of structure is solved by means of boundary collocation method with a use of the special purpose Trefftz functions. These functions fulfil exactly not only governing equation but also boundary conditions on holes and some symmetry conditions. The number of perforations is discussed on effective rigidity.
*Keywords:* Trefftz method, special purpose Trefftz functions, perforated plate, effective flexural rigidity.

V. Kompią, M. ©tiavnický, M. ®mindák, Z. Murčinková. Trefftz radial basis functions (TRBF). CAMES 2008 (15)

The TRBF's are radial functions satisfying governing equation in the domain. They can be used as interpolation functions of the field variables especially in boundary methods. In present paper discrete dipoles are used to simulate composite material reinforced by stiff particles using with boundary point collocation method which does not require any meshing and any integration. The better the interpolation function satisfies also the boundary conditions, the more efficient it is. In examples it is shown that a triple dipole (which is a TRBF) located into the center of the particle can approximate the inter-domain boundary conditions very good, if the particles are not very close to each other and their size is not very different. In general problem the model can be used as very good start point for international improvements in refined model. Composite reinforced by short fibres with very large aspect ratio continuous TRBF were developed. They enable to reduce problem considerably and to simulate complicated interactions for investigation such composites.
*Keywords:* fibre reinforced composites, meshless method, Trefftz Radial Basis Functions, continuous source functions.

C.-Y. Lee, Q.-H. Qin, H. Wang. Trefftz functions and application to 3D elasticity. CAMES 2008 (15)

When solving complex boundary value problems, the primary advantage of the Trefftz method is that Trefftz functions a priori satisfy the governing differential equations. For the treatment of three-dimensional isotropic elasticity problems, it is proposed that the bi-harmonic solutions in Boussinesq's method can be expressed as half-space Fourier series to bypass the difficulties of integration. A total of 29 Trefftz terms for each component of the displacement vectors are derived from the general solutions of the elasticity system. Numerical assessments on the proposed formulations are performed through two examples (a cubic and a cylindrical body). Results are compared with those from the method of fundamental solutions (MFS) and the commercial finite element method (FEM) software STRAND 7, suggesting that Trefftz functions can provide pseudo-stability, faster convergence and reduced error margins.

A. Maci±g, B. Maciejewska, M. Sokała. 2D wave polynomials as base functions in modified FEM. CAMES 2008 (15)

The paper presents solutions of a two-dimensional wave equation by using Trefftz functions. Two ways of obtaining different forms of these functions are shown. The first one is based on a generating function for the wave equation and leads to recurrent formulas for functions and their derivatives. The second one is based on a Taylor series expansion and additionally uses the inverse Laplace operator. Obtained wave functions can be used to solve the wave equation in the whole considered domain or can be used as base functions in FEM. For solving the problem three kinds of modified FEM are used: nodeless, continuous and discontinuous FEM. In order to compare the results obtained with the use of the aforementioned methods, a problem of membrane vibrations has been considered.
*Keywords:* Trefftz functions, wave functions, inverse operations, FEM.

E.A.W. Maunder. On the stability of hybrid equilibrium and Trefftz finite element models for plate bending problems. CAMES 2008 (15)

This paper is concerned with hybrid stress elements in the context of modelling the behaviour of plates subject to out of plane loading and based on Reissner?Mindlin assumptions. These elements are considered as equilibrium elements with statically admissible stress fields of which Trefftz fields form a special case. The existence of spurious kinematic modes in star patches of triangular elements is reviewed when boundary displacement fields are defined independently for each side. It is shown that for elements of moment degree > 1, the spurious modes for stars only exist at specific locations and/or for certain configurations. The kinematic properties of these modes are used to define sufficient conditions for the stability of a complete mesh of triangular elements. A method is proposed to check mesh stability, and introduce local modifications to ensure overall stability.

I.D. Moldovan, J.A. Teixeira de Freitas. Hybrid-Trefftz stress and displacement elements for dynamic analysis of bounded and unbounded saturated porous media. CAMES 2008 (15)

The displacement and stress models of the hybrid-Trefftz finite element formulation are applied to the dynamic analysis of two-dimensional bounded and unbounded saturated porous media problems. The formulation develops from the classical separation of variables in time and space. A finite element approach is used for the discretization in time of the governing differential equations. It leads to a series of uncoupled problems in the space dimension, each of which is subsequently discretized using either the displacement or the stress model of the hybrid-Trefftz finite element formulation. As typical of the Trefftz methods, the domain approximation bases are constrained to satisfy locally all domain equations. An absorbing boundary element is adopted in the extension to the analysis of unbounded media. The paper closes with the illustration of the application of the alternative hybrid-Trefftz stress and displacement elements to the solution of bounded and unbounded consolidation problems.

Z. Murčinková, V. Kompią, M. ©tiavnický. Trefftz functions for 3D stress concentration problems. CAMES 2008 (15)

The paper deals with solution of 3D problems with stress concentration using the Trefftz functions. The modelled stress concentrators are holes and cavities of spherical and ellipsoidal shapes. Moreover, the random spherical cavity microstructure is modelled. The Method of External Finite Element Approximation (MEFEA) is applied to simulate detailed stress state of mentioned stress concentrators. This boundary-type method was developed to build special approximation functions that are associated with surface which causes the stress concentration. The method does not need discretization by classical finite elements, however, instead of elements the domain is divided into Trefftz type subdomains. The displacement and force boundary conditions are met only approximately whereas the governing equations are fulfilled exactly in the volume for linear elasticity, making it possible to assess accuracy in terms of error in boundary conditions.

Q.-H. Qin, K.-Y. Wang. Application of hybrid-Trefftz finite element method to frictional contact problems. CAMES 2008 (15)

A contact algorithm, based on the hybrid-Trefftz (HT) finite element method (FEM), is developed for the solution of contact problems with Coulomb friction. Contact conditions are directly imposed with the aid of a direct constraint approach. On the other hand, static condensation technique is used to reduce the contact system to a smaller one which involves nodes within the potential contact surfaces only so that it may save computing time significantly. The final contact interface equation is constructed by considering contact conditions as additional equations. An incremental-iterative algorithm is introduced to determine proper load increments and find correct contact conditions. The applicability and accuracy of the proposed approach are demonstrated through three numerical problems.

J. Rejlek, B. Pluymers, A. Hepberger, H.-H. Priebsch, W. Desmet. Application of the Wave Based Technique for steady-state semi-infinite sound radiation analysis. CAMES 2008 (15)

This paper reports on the development of a novel wave based prediction technique for the steady-state sound radiation analysis of three-dimensional semi-infinite problems. Instead of simple polynomial shape functions, this method adopts an indirect Trefftz approach, in which it uses the exact solutions of the governing differential equation for the field variables approximation. Since a fine discretization is no longer required, the resulting wave based models are substantially smaller than the element-based counterparts. Application of the proposed approach to various validation examples illustrates an enhanced computational efficiency as compared with element-based methods.

H. Sanecki, A.P. Zieliński. The Trefftz finite elements modelling crack propagation. CAMES 2008 (15)

Investigation of crack propagation can sometimes be a crucial stage of engineering analysis. The T-element method presented in this work is a convenient tool to deal with it. In general, T-elements are the Trefftz-type finite elements, which can model both continuous material and local cracks or inclusions. The authors propose a special T-element in a form of a pentagon with shape functions analytically modelling the vicinity of the crack tip. This relatively large finite element can be surrounded by even larger standard T-elements. This enables easy modification of the rough element grid while investigating the crack propagation. Numerical examples proved that the "moving pentagon" concept enables easy automatic generation of the T-element mesh, which facilitates observation of crack propagation even in very complicated structures with many possible crack initiators occurring for example in material fatigue phenomena.

M. Sukhorolsky, T. Shopa. The vibration of rectangular orthotropic plate with massive inclusions. CAMES 2008 (15)

The problem on proper and forced vibrations of the loosely leant rectangular orthotropic plate with massive circular inclusion is considered in the paper. The flexure of the plate is described by modified equations of Timoshenko's theory of plates. Numerical solution of the problem is found by the indirect method of boundary elements based on the sequential approach to constructing generalized functions and on collocation method. The problem can be generalized on the case of arbitrary located inclusion and the arbitrary number of them. The influence of the mass of the massive circular inclusion on the proper frequencies of the plate is investigated.

A. U¶ciłowska. Application of the Trefftz method to nonlinear potential problems. CAMES 2008 (15)

In this paper some types of nonlinear potential problems are discussed and some of these problems are solved by the Trefftz method. The attention is paid to Fundamental Solutions Method (FSM) supported by Radial Basis Functions (RBF) approximation. Application of FSM to nonlinear boundary problem requires certain modifications and special algorithms. In this paper two methods of treating the nonlinearity are proposed. One on them is Picard iteration. Due to some problems of application of this method the Homotopy Analysis Method (HAM) is implemented for nonlinear boundary-value problems. The results of numerical experiment are presented and discussed. The conclusion is that the method based on FSM for solving nonlinear boundary-value problem gives result with demanded accuracy.