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4 Computer Assisted Mechanics and Engineering Sciences

4 2008

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Energetic approach to direct and inverse heat conduction problems with Trefftz functions used in FEM

100%

Grysa K. , Leśniewska R. , Maciąg A.

Computer Assisted Mechanics and Engineering Sciences

2008

tom Vol. 15, No. 3/4

171-182

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.

Trefftz functions and application to 3D elasticity

88%

Lee C. Y. , Qin Q. H. , Wang H.

Computer Assisted Mechanics and Engineering Sciences

2008

tom Vol. 15, No. 3/4

251-263

When solving complex boundary value problems, the primary advantage of the Trefftz method is that Tr-efftz 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 integral ion. A total of 29 Trefftz terms for each component of the displacement vectors are derived from (he general solutions of the elasticity system. Numerical assessments on the proposed formulations arc performed through two examples (a cubic and a cylindrical body). Results arc 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.

Trefftz functions for 3D stress concentration problems

75%

Murčinková Z. , Kompiš V. , Štiavnicky M.

Computer Assisted Mechanics and Engineering Sciences

2008

tom Vol. 15, No. 3/4

305-318

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 fund ions 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.

2D wave polynomials as base functions in modified FEM

75%

Maciąg A. , Maciejewska B. , Sokała M.

Computer Assisted Mechanics and Engineering Sciences

2008

tom Vol. 15, No. 3/4

265-278

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