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1

The application of Beck’s method combined with FEM and Trefftz functions to determine the heat transfer coefficient in a minichannel

Maciejewska B.

Journal of Theoretical and Applied Mechanics

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2017

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Vol. 55 nr 1

103--116

EN

The aim of this study is to determine the heat transfer coefficient between the heated surface and the boiling fluid flowing in a minichannel on the basis of experimental data. The calculation model is based on Beck’s method coupled with the FEM and Trefftz functions. The Trefftz functions used in the Hermite interpolation are employed to construct the shape functions in the FEM. The unknown local values of the heat transfer coefficient at the foil- -fluid contact surface are calculated from Newton’s law. The temperature of the heated foil and the heat flux on the foil surface are determined by solving a two-dimensional inverse heat conduction problem. The study is focused on the identification of the heat transfer coefficients in the subcooled boiling region and the saturated nucleate boiling region. The results are compared with the data obtained through the one-dimensional method. The investigations also reveal how the smoothing of measurement data affects calculation results.

2

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.

3

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.

4

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.

5

Trefftz functions for a plate vibration problem

Maciąg A.

Journal of Theoretical and Applied Mechanics

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2011

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Vol. 49 nr 1

97-116

EN

The paper presents a new method to obtain an approximate solution to plate vibrations problems. The problem is described by a partial differential equation of fourth order. The key idea of the presented approach is to find polynomials (solving functions) that satisfy the considered differential equation identically. In this sense, it is a variant of the Trefftz method. The method is addressed to differential equations in a finite domain. The approach proposed here has some advantages. The first is that the approximate solution (a linear combination of the solving functions) satisfies the equation identically. Secondly, the method is flexible in terms of given boundary and initial conditions (discrete, missing). Thirdly, the solving functions can be used as base functions for several variants of the Finite Element Method. In this case, the approximation is good even for relatively large elements. It means that the approach is suitable for inverse problems. The formulas for solving functions and their derivatives for the plate vibration equation are obtained. The convergence of the method is proved and numerical examples are included.

PL

Artykuł przedstawia nową metodę rozwiązywania problemów drgań płyty, które opisywane są cząstkowym równaniem różniczkowym czwartego rzędu. Idea metody polega na uzyskaniu wielomianów spełniających w sposób ścisły dane równanie różniczkowe (funkcje Trefftza). Za rozwiązanie przyjmuje się kombinację liniową tych funkcji. Współczynniki kombinacji liniowej dobierane są w taki sposób, aby zminimalizować błąd dopasowania rozwiązania do warunków początkowo - brzegowych. Podejście proponowane w pracy ma kilka zalet. Po pierwsze liniowa kombinacja funkcji Trefftza spełnia w sposób ścisły dane równanie różniczkowe. Po drugie metoda jest elastyczna odnośnie warunków początkowo - brzegowych. Mogą one być ciągłe lub dyskretne. Dodatkowo warunki te mogą być niekompletne, co czyni metodę użyteczną w rozwiązywaniu zagadnień odwrotnych. Trzecią zaletą metody jest możliwość zastosowania uzyskanych wielomianów jako funkcji bazowych w różnych wariantach Metody Elementów Skończonych. W pracy uzyskano wzory rekurencyjne na funkcje Trefftza oraz ich pochodne dla równania drgań płyty. Zbadano zbieżność metody oraz zamieszczono przykłady numeryczne.

6

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.

7

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.

8

Wygładzanie danych pomiarowych funkcjami Trefftza przy wyznaczaniu współczynnika przejmowania ciepła

Hożejowska S., Piasecka M.

Logistyka

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2010

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nr 6

PL

Rozpatrywane jest zagadnienie przepływu wrzącej cieczy przez odchylony od pionu minikanał. Zewnętrzną ścianą kanału jest folia stanowiąca źródło ciepła, której temperatura mierzona jest metodą termografii ciekłokrystalicznej. W celu wyselekcjonowania pomiarów o jak najmniejszych błędach pomiarowych wykorzystano rachunek wyrównawczy dla zagadnień zawarunkowanych. Przyjętym warunkiem była zależność między pomiarami temperatury, a kombinacją liniową funkcji Trefftza (T-funkcji). Rachunek wyrównawczy pozwolił na zminimalizowanie błędów pomiarowych oraz zmniejszenie średnich względnych błędów współczynników przejmowania ciepła.

EN

At boiling liquid flow in an inclined minichannel, the temperature on the outer wall of the foil which is the heat source is measured with the liquid crystal thermography method. In order to obtain the most probable foil temperature measurements, data equalisation with condition numbers has been applied. The assumed condition has been the correlation between temperature measurements and the linear combination of Trefftz functions (T-functions). Data equalisation has enabled minimalisation of measurement errors and decrease in mean relative errors of heat transfer coefficients.

9

Trefftz function for solving a quasi-static inverse problem of thermal stresses

Grysa K., Maciąg A., Maciejewska B.

Computer Assisted Mechanics and Engineering Sciences

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2009

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Vol. 16, No. 3/4

251-266

EN

The problem of thermal stresses in a hollow cylinder is considered. The problem is two-dimensional and the cross-section of the hollow cylinder is approximated as a long and thin rectangle as the ratio of the inner and outer radiuses is close to one. On the outer boundary of the hollow cylinder the heat source moves with a constant velocity. In the case of the rectangle the heat source moves on the upper side and the conditions of eąuality of temperatures and heat fluxes are assumed on the left and right boundaries. The stresses are to be found basing on the temperature measured inside the considered region, which means that an inverse problem is considered. Both for the temperature field and the displacements and stresses the finite element method is used. Thermal displacement potentials are introduced to find displacements and stresses. In order to construct the base functions in each element the Trefftz functions are used. For the temperature field the time-space finite elements are used and for the thermal displacement potentials the spatial elements are applied. Thanks to the use of the Trefftz functions a low-order approximation has given a solution very close to the exact one.

10

Solving thermoelasticity problems by means of Trefftz functions

Maciąg A.

Computer Assisted Mechanics and Engineering Sciences

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2009

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Vol. 16, No. 3/4

193-208

EN

The paper presents a new method of approximate solving of the two- and three-dimensional thermoelasticity problems in a finite body. The method presented here can be used for solving direct and inverse problems as well. System of thermoelasticity equations is reduced to the system of wave equations where the temperature occurs as inhomogeneity in one of them. The thermal field is approximated by linear combination of heat polynomials (Trefftz functions for heat conduction equation). The system of wave eąuations is solved by means of wave polynomials (Trefftz functions for wave equation). Convergence of the T-functions method is proved. The procedure of solving direct and inverse thermoelasticity problems by means of Trefftz functions is tested on an example. Sensitiveness of the method according to data disturbance was checked.

11

Solution of 2D non-homogenous wave equation by using polywave functions

Sokała M.

Computer Assisted Mechanics and Engineering Sciences

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2009

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Vol. 16, No. 3/4

209-221

EN

The paper presents a specific technique of solving the non-homogenous wave equation with the use of Trefftz functions for the wave equation. The solution was presented as a sum of a general integral and a particular integral. The general integral was expressed in the form of a linear combination of Trefftz functions for the wave equation. In order to obtain the particular integral polywave functions were used. They were generated by using the inverse operator L -1 of the equation taking into consideration the Trefftz functions.

12

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.

13

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.

14

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.

15

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

16

Solutions of Two-Dimensional Wave Equation using some Form of the Trefftz Functions

Sokała M.

Computational Methods in Science and Technology

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2008

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Vol. 14, nr 2

111-122

EN

The paper presents a specific technique to generate the Trefftz functions for the two-dimensional wave equation. The obtained functions are used to determine approximate solutions of some tested problems. The accuracy of the method is discussed.

17

New Continuum Formulations for Modelling of Fiber Reinforced Composites

Zmindak M., Novák P., Melicher R.

Machine Dynamics Problems

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2006

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Vol. 30, No. 4

190-199

EN

The composites reinforced by rigid particles improve considerably many properties of matrix materials, they poses higher stiffness, strength, better wear resistance, superior electrical properties. The fibers are supposed to be much stiffer than the matrix and their diameter can be much smaller than their length. On the other side the number of the reinforcing particles can be very high and they can be not uniformly distributed over the structure. From material structure point view the particles present inclusions. For of such problems the methods like (finite element methods) FEM and BEM (boundary element methods) are not very efficient. Computational time for FEM is huge for an accurate solution of the mechanical fields. BEM using distributed forces, dislocations and dipoles along the fiber axis (the source points) enable to model these effects much more efficiently. The intensity of the source functions simulates the interaction of the fiber with the other subjects. Because of very close distance of the source points to the fiber boundary and the quasi-singular form of the integrals, the integration is performed analytically in the fiber direction. In this paper we will present two new methods for modelling composites reinforced by fibers: First proposed method is called here the method of continuous source functions (MCSF) which use continuously distributed dipoles as source functions. The second method is Boundary point method (BPM). In this method all internal fields will be approximated by the Trefftz functions (T-functions).