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1
Content available The usage of the Trefftz method to determine the Biot number
Maciąg A., Walaszczyk M.
Journal of Applied Mathematics and Computational Mechanics
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2017
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Vol. 16, nr 4
47--55
EN
The paper presents a method for determining the Biot number and the heat transfer coefficient based on the Trefftz functions. Firstly, the temperature distribution in the entire domain is calculated and then used for obtaining the heat transfer coefficient. The usefulness of the method is shown in the examples. The data for the examples are calculated by means of a known exact solution or they are given as measurements. The sensitivity of the presented method is checked. Test examples are used to check the method. Next, the heat transfer coefficient is determined for the real data for a rocket engine.
2
Content available The usage of Trefftz functions and Picard's iteration for solving different problems of a two-dimensional wave equation
Krzyszkowska P., Maciąg A., Walaszczyk M.
Journal of Applied Mathematics and Computational Mechanics
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2014
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Vol. 13, nr 3
129--140
EN
The paper presents a method of solving two-dimensional wave equations which describe vibrations of the membrane with variable thickness and with damping. The differential operator is decomposed into two parts. The first one describes vibrations of the membrane with constant thickness without damping. The second contains the rest of the original operator and is treated as inhomogeneity for the first one. Picard’s iterations are used to calculate a successive approximation of the exact solution. Trefftz functions (wave polynomials) are used to solve the problem in each iteration. The presented examples show the usefulness of the method. The approach described in this paper can be used also for solving nonlinear problems for a wave equation.
3
Identyfikacja strumienia ciepła w warstwie płaskiej - rozwiązanie zagadnienia odwrotnego z wykorzystaniem różnych wariantów metody funkcji Trefftza
Maciąg A., Pawińska A.
Zeszyty Naukowe Politechniki Poznańskiej. Budowa Maszyn i Zarządzanie Produkcją
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2011
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Nr 16
119-138
PL
W artykule wykorzystano funkcje Trefftza do rozwiązano jednowymiarowego, prostego i odwrotnego zagadnienia identyfikacji strumienia ciepła. Zaproponowano dobór punktów pomiarowych temperatury względem czasu w węzłach Czebyszewa. Porównano to podejście z wynikami uzyskanymi dla stałego kroku czasowego. W zagadnieniu odwrotnym testowano podejście globalne (cały obszar czasowo - przestrzenny) oraz bezwęzłową metodę elementów skończonych (sub-structuring) z bazowymi funkcjami Trefftza porównując uzyskane wyniki. Również w tych metodach zastosowano dwa sposoby doboru punktów pomiarowych względem czasu. Zbadano wrażliwość metody na losowe zaburzenia wartości pomiarowych.
EN
In this paper a one-dimensional direct and inverse problem was solved. In the direct problem an influence of different temperature measurements locations on the flux estimation error was checked. Two methods of internal responses simulation were presented: for the constant time interval and in Chebyshev nodes. Comparison of substructuring and Trefftz functions for x e (0,1) was done in the inverse problem. In both methods two ways of choosing temperature measurements locations was shown.
4
Content available remote Computation of plane structures eigenfrequencies by hybrid finite element method using Trefftz functions
Bozduganova V., Todorov M.
Zeszyty Naukowe Politechniki Rzeszowskiej. Mechanika
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2010
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z. 80 [274]
25-32
EN
A hybrid stress method for eigenfrequencies analysis is developed using a plane rectangular hybrid element. Complex Trefftz functions which are solutions of elastostatic problem are used. By the complementary energy variational equation a relationship between the stress parameters and the nodal displacements is obtained. The Lagrange's variational equation for the dynamic case gives an expression for computation of eigenfrequencies.
5
Zastosowanie metody współczynników wrażliwości z wykorzystaniem funkcji Trefftza w badaniach wymiany ciepła w minikanale
Piasecka M., Maciejewska B.
Logistyka
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2010
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nr 6
PL
W referacie przedstawiono sposób określenia współczynnika przejmowania ciepła przy wrzeniu w przepływie płynu chłodniczego przez minikanał prostokątny, ogrzewany asymetrycznie. Lokalne wartości współczynnika przejmowania ciepła wyznaczono dzięki rozwiązaniu dwuwymiarowego zagadnienia odwrotnego przewodnictwa ciepła. Zagadnienie to rozwiazano za pomocą metody Beck`a (metody współczynników wrażliwości) w połączeniu z metodą funkcji Trefftza. Dzięki zastosowaniu metody Beck`a zagadnienie odwrotne rozłożono na dwa zagadnienia proste, a następnie rozwiazania tych zagadnień aproksymowano funkcjami Trefftza. Przedstawiono i przeanalizowano otrzymane wyniki przykładowego eksperymentu, w którym do pomiaru temperatury powierzchni grzejnej wykorzystuje się termografię ciekłokrystaliczną.
EN
The paper presents the method for determination of boiling heat transfer coefficient in cooling liquid flow in a rectangular minichannel with asymmetric heating. Local values of the heat transfer coefficient have been determined following the solution of the two-dimensional reverse heat transfer problem. This problem has been solved with Beck`s method (sensitivity coefficients) in combination with Trefftz functions. Thanks to the application of Beck's method, the reverse problem has been decomposed into two simple problems, whose solutions have been approximated with Trefftz functions. The results obtained from a sample experiment have been presented and analysed, the experiment relying on liquid crystal thermography for the measurement of the heating surface temperature.
6
Content available Współczynnik przejmowania ciepła przy wrzeniu w minikanale : badania doświadczalne i metody obliczeniowe z wykorzystaniem funkcji Trefftza
Piasecka M., Hożejowska S., Maciejewska B.
Inżynieria i Aparatura Chemiczna
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2010
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Nr 2
95-96
PL
Przedstawiono metody wyznaczania współczynnika przejmowania ciepła przy wrzeniu w przepływie płynu chłodniczego w minikanale. Zastosowana technika termografli ciekłokrystalicznej pozwala określić wystąpienie inicjacji wrzenia. Zagadnienie rozwiązywane jest za pomocą funkcji Trefftza, którymi aproksymuje się nieznane temperatury. Ponadto funkcje Trefftza służą do tworzenia funkcji bazowych w metodzie elementów skończonych (MEST). Porównano wyniki otrzymane przy wykorzystaniu obu metod. Słowa kluczowe: wymiana ciepła, minikanał, wrzenie w przepływie, termo-grafia ciekłokrystaliczna, funkcje Trefftza, MES
EN
The paper presents methods for the determination of heat transfer coefficient in cooling liquid flow boiling in a minichannel. The applied technique of liquid crystal thermography enables the determination of boiling incipience. To solve this problem, Trefftz functions are used to approximate unknown temperatures. Moreover, Trefftz functions are employed to obtain base functions in the finite element method (FEM). The results obtained using of both methods were compared.
7
Content available remote Trefftz functions for 3D stress concentration problems
Murčinková Z., Kompiš V., Štiavnicky M.
Computer Assisted Mechanics and Engineering Sciences
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2008
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Vol. 15, No. 3/4
305-318
EN
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.
8
Content available remote Trefftz functions and application to 3D elasticity
Lee C. Y., Qin Q. H., Wang H.
Computer Assisted Mechanics and Engineering Sciences
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2008
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Vol. 15, No. 3/4
251-263
EN
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.
9
Content available remote Energetic approach to direct and inverse heat conduction problems with Trefftz functions used in FEM
Grysa K., Leśniewska R., Maciąg A.
Computer Assisted Mechanics and Engineering Sciences
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2008
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Vol. 15, No. 3/4
171-182
EN
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.
10
Content available remote 2D wave polynomials as base functions in modified FEM
Maciąg A., Maciejewska B., Sokała M.
Computer Assisted Mechanics and Engineering Sciences
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2008
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Vol. 15, No. 3/4
265-278
EN
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
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