A meshless method for the solution of linear and non-linear Poisson-type problems involving high gradients is presented. The proposed method is based on collocation with 3rd order polynomial radial basis function coupled with the fundamental solution. The linear problem is solved by satisfying the boundary conditions and the governing differential equations over selected points over the boundary and inside the domain, respectively. In the case of the non-linear case, the resulted equations are highly non-linear and therefore, they are solved using an incremental-iterative procedure. The accuracy and efficiency of the method is verified through several numerical examples.
*Keywords:* meshless method, fundamental solution, RBF, high gradients.

T. Czyż and P. Fedeliński. Boundary element formulation for dynamic analysis of inelastic structures. CAMES 2006 (13)

The boundary element formulation for dynamic analysis of inelastic two-dimensional structures subjected to stationary or transient inertial loads is presented. The problem is solved by using simultaneously the displacement and stress integral equations. The numerical solution requires discretization of the boundary displacements and tractions, and stresses in the interior of the body. The boundary is divided into quadratic elements and the domain into constant or quadratic quadrilateral cells. The unknown stresses in the coupled system of equations are computed using an iterative procedure. The mass matrix of the structure is formulated by using the dual reciprocity method. The matrix equation of motion is solved step-by-step by using the Houbolt direct integration method. Several numerical examples show the influence of the discretization on the accuracy and new applications of the method. The solutions are compared to the analytical results or those computed by the finite element method.
*Keywords:* boundary element method (BEM), dual reciprocity method (DRM), inelasticity, elastoplasticity, dynamic analysis.

H. Jopek, J.A. Kołodziej and T. Stręk. Laminar flow in trapezoidal grooves at finite Bond numbers with shear stress at the liquid-vapor interface by the method of fundamental solutions. CAMES 2006 (13)

The problem of determination of Poiseuille number for a steady gravitational flow of liquid in an inclined open trapezoidal groove is addressed. The solution comprises of two parts. First, for a given groove's dimension, liquid-solid contact angle, and the Bond number, the shape of the free surface is determined starting from the Young-Laplace equation. The shooting method is used for solution of a two-point boundary value problem. Then, having determined the shape of the free surface and slope the groove, the fully developed laminar flow is determined. The boundary value problem is solved using the method of fundamental solutions. Given the distribution of liquid velocity, the Poiseuille number, as a function of the other parameters of the model is analysed.
*Keywords:* flow in trapezoidal groove, liquid-vapor interface, method of fundamental solutions, Bond number, Mathematica.

T. Klekiel and J.A. Kołodziej. Trefftz method for large deflection of plates with application of evolutionary algorithms. CAMES 2006 (13)

The large deflection of thin plates by means of Berger equation is considered. An iterative solution of Berger equation by the method of fundamental solutions is proposed. In each iterative step the Berger equation can be considered as an inhomogeneous partial differential equation of the fourth order. The inhomogeneous term is interpolated by radial basis functions using thin plate splines. For the optimal choice of parameters of the fundamental solutions method an evolutionary algorithm is used. Numerical results for square plate with simply supported edges are presented to compare the obtained results with previous solutions.

V. Kompią and M. ©tiavnický. Trefftz functions in FEM, BEM and meshless methods. CAMES 2006 (13)

The paper contains three different multi-domain formulations using Trefftz (T-) displacement approximation/interpolation, namely the hybrid-displacement FEM, reciprocity based FEM (multi-domain BEM) and the Boundary Meshless Method (BMM) for a single and multi-domain (MD) formulation. All three methods can lead to compatible formulation with the isoparametric FEM, when the displacements along the common boundaries are defined by same interpolation function. All three T-formulations enable to define more complicated elements/subdomains (the T-element can be also a multiply connected region) with integration along the element boundaries, only.

B. Pluymers, C. Vanmaele, W. Desmet, D. Vandepitte. Application of a hybrid finite element- Trefftz approach for acoustic analysis. CAMES 2006 (13)

This paper reviews a wave based prediction technique for steady-state acoustic analysis, which is being developed at the K.U. Leuven Noise and Vibration Research group. The method is a deterministic technique based on an indirect Trefftz approach. Due to its enhanced convergence rate and computational efficiency as compared to conventional element based methods, the practical frequency limitation of the technique can be shifted towards the mid-frequency range. For systems of high geometrical complexity, a hybrid coupling between wave based models and conventional finite element (FE) models is proposed in order to combine the computational efficiency of the wave based method with the high flexibility of FE with respect to geometrical complexity of the considered problem domain. The potential to comply with the mid-frequency modelling challenge through the use of the wave based technique or its hybrid variant, is illustrated for some three-dimensional acoustic validation cases.

A. U¶ciłowska and J.A. Kołodziej. Solution of the nonlinear equation for isothermal gas flows in porous medium by Trefftz method. CAMES 2006 (13)

This paper presents numerical solution to a problem of the transient flow of gas within a two-dimensional porous medium. A method of fundamental solution for space variables and finite difference method for time variable are employed to obtain a solution of the non-linear partial differential equation describing the flow of gas. The inhomogeneous term is expressed by radial basis functions at each time steps. Picard iteration is used for treating nonlinearity.
*Keywords:* isothermal gas flow, porous medium, Trefftz method, fundamental solution.

K.Y. Wang, M. Dhanasekar, Q.H. Qin, Y.L. Kang. Contact analysis using Trefftz and interface finite elements. CAMES 2006 (13)

Hybrid-Trefftz (HT) finite element (FE) analysis of two-dimensional elastic contact problems is addressed with the aid of interface elements and an interfacial constitutive relation. This paper presents the formulation of a four-noded HT finite element for discretizing the contacting bodies and a four-noded interface element that could be embedded in the prospective contact zone for simulating the interaction behaviour. Due to the superior performance, the Simpson-type Newton-Cotes integration scheme is utilized to compute interface element formulation numerically. In order to evaluate the applicability of the present approach two benchmark examples are investigated in detail. Comparisons have been made between the results by the present approach and analytical as well as traditional FE solutions using ABAQUS software.

A. Wróblewski and A.P. Zieliński. Structural inverse problems solved by the T-element approach. CAMES 2006 (13)

Any direct boundary-value problem is defined in a certain area $\Omega$ by a system of differential equations and respective set of boundary conditions. In structural inverse problems the above conditions can be partly unknown. Instead, we can measure certain quantities inside the investigated structure and then approximately define the whole boundary-value problem. Usually, the solutions of inverse problems are connected with the minimization of a certain functionals, which results in optimization procedures. The applications of the trial functions identically fulfilling governing partial differential equations of a discussed problem (the Trefftz approach) can considerably improve these procedures. The original idea of Erich Trefftz was based on modelling objects of simple geometry. In the case of more complex structures the division of the whole object into sub-regions (Trefftz elements) is necessary. This kind of formulation is presented in this paper and is illustrated by numerical examples. The properties of the Trefftz finite elements allow the formulation of effective algorithms, which considerably shorten the time of computer calculations in comparison to standard finite element solutions.