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1
Content available Comparison of heat transfer phenomena for two different cryopreservation methods : slow freezing and vitrification
Skorupa Anna, Piasecka-Belkhayat Alicja
Journal of Applied Mathematics and Computational Mechanics
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2023
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Vol. 22, nr 1
57--69
EN
The purpose of the research is to prepare a mathematical and numerical model for the phenomenon of heat transfer during cryopreservation. In the paper, two popular methods, slow freezing and vitrification, are compared. Furthermore, the basic model of thermal processes is supplemented by the phenomenon of phase transitions. To determine the temperature distribution during cryopreservation processes, one uses the heat transfer equation proposed by Pennes. An integral part of the energy equation is the substitute thermal capacity (STC) performed according to the concept named one domain method (fixed domain method), The numerical model is developed using the finite difference method (FDM) connected with directed interval arithmetic. The final part of the article contains the results of numerical simulations.
2
Content available Granular computational homogenisation of composite structures with imprecise parameters
Beluch W., Hatłas M., Ptaszny J.
Archives of Mechanics
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2023
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Vol. 75, nr 3
271--300
EN
The paper presents the formulation of a granular computational homogenisation problem and the proposition of a method to solve it, which enables multiscale analysis of materials with uncertain microstructure parameters. The material parameters and the geometry, represented by the interval and fuzzy numbers, are assumed to be unprecise. An _-cut representation of fuzzy numbers allows the use of interval arithmetic for epistemic uncertainties. Directed interval arithmetic is used to reduce the effect of interval widening during arithmetic operations. Response surfaces of diverse types, including Artificial Neural Networks, are used as model reduction methods. The finite element method is employed to solve the boundary value problem on a micro scale. Numerical examples are provided to demonstrate the effectiveness of the proposed approach.
3
Content available Modelling of biological tissue damage process with application of interval arithmetic
Korczak Anna, Jasiński Marek
Journal of Theoretical and Applied Mechanics
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2019
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Vol. 57 nr 1
249--261
EN
In the paper, the numerical analysis of thermal processes proceeding in a 2D soft biological tissue subjected to laser irradiation is presented. The transient heat transfer is described by the bioheat transfer equation in Pennes formulation. The internal heat source resulting from the laser-tissue interaction based on the solution of the diffusion equation is taken into account. Thermophysical and optical parameters of the tissue are assumed as directed intervals numbers. At the stage of numerical realization. the interval finite difference method has been applied. In the final part of the paper, the results obtained are shown.
4
Content available Modelling of transient heat transport in two-layered crystalline solid films using the interval lattice Boltzmann method
Belkhayat-Piasecka A., Korczak A.
Journal of Applied Mathematics and Computational Mechanics
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2017
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Vol. 16, nr 4
57--65
EN
In the paper, the numerical modelling of heat transfer in one-dimensional crystalline solid films is considered. A generalized two-layer problem is described by the Boltzmann transport equations transformed in the phonon energy density equations supplemented by the adequate boundary-initial conditions. Such an approach in which the parameters appearing in the problem analysed are treated as the constant values is widely used, but in this paper the interval values of relaxation time and the boundary condition for silicon and diamond are taken into account. The problem formulated has been solved by means of the interval lattice Boltzmann method using the rules of directed interval arithmetic. In the final part of the paper the results of numerical computations are presented.
5
Content available Numerical modelling of the transient heat transport in a thin gold film using the fuzzy lattice Boltzmann method with α-cuts
Belkhayat-Piasecka A., Korczak A.
Journal of Applied Mathematics and Computational Mechanics
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2016
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Vol. 15, nr 1
123--135
EN
In this paper a description of heat transfer in one-dimensional crystalline solids is presented. The fuzzy lattice Boltzmann method based on the Boltzmann transport equation is used to simulate the nanoscale heat transport in thin metal films. The fuzzy coupled lattice Boltzmann equations for electrons and phonons are applied to analyze the heating process of thin metal films via a laser pulse. Such an approach in which the parameters appearing in the problem analyzed are treated as constant values is widely used. Here, the model with fuzzy values of relaxation times and an electron-phonon coupling factor is taken into account. The problem formulated has been solved by means of the fuzzy lattice Boltzmann method using the α-cuts and the rules of directed interval arithmetic. The application of α-cuts allows one to avoid complicated arithmetical perations in the fuzzy numbers set. In the final part of the paper the results of numerical computations are shown.
6
Content available Modelling of transient heat transport in metal films using the interval lattice Boltzmann method
Piasecka-Belkhayat A., Korczak A.
Bulletin of the Polish Academy of Sciences. Technical Sciences
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2016
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Vol. 64, nr 3
599--606
EN
In the paper a description of heat transfer in one-dimensional crystalline solids is presented. The lattice Boltzmann method based on Boltzmann transport equation is used to simulate the nanoscale heat transport in thin metal films. The coupled lattice Boltzmann equations for electrons and phonons are applied to analyze the heating process of thin metal films via laser pulse. Such approach in which the parameters appearing in the problem analyzed are treated as constant values is widely used, but in the paper the interval values of relaxation times and electron-phonon coupling factor are taken into account. The problem formulated has been solved by means of the interval lattice Boltzmann method using the rules of directed interval arithmetic. In the final part of the paper the results of numerical computations are shown.
7
Content available Modelling of transient heat transport in crystalline solids using the interval lattice Bolzmann method (two-dimesnsional model)
Piasecka-Belkhayat A., Korczak A.
Informatyka, Automatyka, Pomiary w Gospodarce i Ochronie Środowiska
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2014
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nr 1
69--71
EN
In the paper the two-dimensional numerical modelling of heat transfer in crystalline solids is considered. In the mathematical description the relaxation time and the boundary conditions are given as interval numbers. The problem formulated has been solved by means of the interval lattice Boltzmann method using the rules of directed interval arithmetic.
PL
W artykule zaprezentowano dwuwymiarowy model numeryczny przepływu ciepła w ciele krystalicznym. W opisie matematycznym czas relaksacji i warunki brzegowe są zdefiniowane jako liczby przedziałowe. Sformułowane zagadnienie rozwiązano za pomocą interwałowej metody siatek Boltzmanna stosując skierowaną arytmetykę interwałową.
8
Content available Application of the interval lattice boltzmann method for a numerical modelling of thin metal films irradiation by ultra short laser pulses
Piasecka-Belkhayat A., Korczak A.
Informatyka, Automatyka, Pomiary w Gospodarce i Ochronie Środowiska
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2014
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nr 4
85--88
EN
In the paper the one-dimensional numerical modelling of heat transfer in thin metal films irradiated by ultra short laser pulses is considered. In the mathematical description the relaxation times and the boundary conditions for phonons and electrons are given as interval numbers. The problem formulated has been solved by means of the interval lattice Boltzmann method using the rules of directed interval arithmetic. The examples of numerical computations are presented in the final part of the paper.
PL
W artykule zaprezentowano jednowymiarowy model numeryczny przepływu ciepła w cienkich warstwach metalowych poddanych. naświetlaniu laserem pulsacyjnym. W opisie matematycznym czasy relaksacji oraz warunki brzegowe dla fononów i elektronów są zdefiniowane jako liczby przedziałowe. Sformułowane zagadnienie rozwiązano za pomocą interwałowej metody siatek Boltzmanna stosując skierowaną arytmetykę interwałową. W końcowej części artykułu przedstawione są przykłady obliczeń numerycznych.
9
Content available remote Interval versions of central-difference method for solving the poisson equation in proper and directed interval arithmetic
Hoffmann T., Marciniak A., Szyszka B.
Foundations of Computing and Decision Sciences
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2013
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Vol. 38, No. 3
193--206
EN
To study the Poisson equation, the central-difference method is often used. This method has the local truncation error of order O(h2 +k2), where h and k are mesh constants. Using this method in conventional floating-point arithmetic, we get solutions including the method, representation and rounding errors. Therefore, we propose interval versions of the central-difference method in proper and directed interval arithmetic. Applying such methods in floating-point interval arithmetic allows one to obtain solutions including all possible numerical errors. We present numerical examples from which it follows that the presented interval method in directed interval arithmetic is a little bit better than the one in proper interval arithmetic, i.e. the intervals of solutions are smaller. It appears that applying both proper and directed interval arithmetic the exact solutions belong to the interval solutions obtained.
10
Content available Interval boundary element method for transient diffusion problem in two-layered domain
Piasecka-Belkhayat A.
Journal of Theoretical and Applied Mechanics
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2011
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Vol. 49 nr 1
265-276
EN
In the paper, the description of an unsteady heat transfer for one- dimensional problem proceeding in a two-layered domain is presented. It is assumed that all thermophysical parameters appearing in the mathematical description of the problem analysed are given as directed interval values. The problem discussed has been solved using the 1st scheme of the interval boundary element method. The interval Gauss elimination method has been applied to solve the obtained interval system of equations. In the final part of the paper, results of numerical computations are shown.
PL
W pracy przedstawiono opis nieustalonego przepływu ciepła dla zadań jednowymiarowych w obszarach dwuwarstwowych. Założono, że wszystkie parametry termofizyczne pojawiające się w opisie matematycznym analizowanego zadania są wartościami przedziałowymi skierowanymi. Omawiane zagadnienie zostało rozwiązane za pomocą pierwszego schematu przedziałowej metody elementów brzegowych. Do rozwiązania otrzymanego interwałowego układu równań zastosowano przedziałową metodę eliminacji Gaussa. W końcowej części pracy pokazano wyniki obliczeń numerycznych.
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