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1
Content available Identification of thermophysical parameters using an artificial immune system
Dziatkiewicz Jolanta, Poteralski Arkadiusz
Computer Assisted Methods in Engineering and Science
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2023
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Vol. 30, no. 4
521--537
EN
In this paper, the identification of thermophysical parameters using the hyperbolic twotemperature model is made. We investigate the influence of ultra-fast laser pulses on the heating of a thin metal film using this model. Two differential equations coupled with the electron-phonon coupling factor G are used. One of these equations concerns electron temperatures and the other addresses lattice temperatures. Appropriate initial and boundary conditions are imposed for this model. The finite difference method with a staggered grid is used to solve this direct problem. Temperatures for even nodes and heat fluxes for odd nodes are calculated. The results of the direct problem and results of the experiment are compared. In the optimization process, an artificial immune system is used.
2
Content available Implicit scheme of the finite difference method for the second-order dual phase lag equation
Majchrzak E., Mochnacki B.
Journal of Theoretical and Applied Mechanics
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2018
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Vol. 56 nr 2
393--402
EN
The second-order dual phase lag equation (DPLE) as a mathematical model of the microscale heat transfer is considered. It is known that the starting point determining the final form of this equation is the generalized Fourier law in which two positive constants (the relaxation and thermalization times) appear. Depending on the order of the generalized Fourier law expansion into the Taylor series, different forms of the DPLE can be obtained. As an example of the problem described by the second-order DPLE equation, thermal processes proceeding in the domain of a thin metal film subjected to a laser pulse are considered. The numerical algorithm is based on an implicit scheme of the finite difference method. At the stage of numerical modeling, the first, second and mixed order of the dual phase lag equation are considered. In the final part of the paper, examples of different solutions are presented and conclusions are formulated.
3
Content available Dual Phase Lag Model of Melting Process in Domain of Metal Film Subjected to an External Heat Flux
Mochnacki B., Ciesielski M.
Archives of Foundry Engineering
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2016
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Vol. 16, iss. 4
85--90
EN
Heating process in the domain of thin metal film subjected to a strong laser pulse are discussed. The mathematical model of the process considered is based on the dual-phase-lag equation (DPLE) which results from the generalized form of the Fourier law. This approach is, first of all, used in the case of micro-scale heat transfer problems (the extremely short duration, extreme temperature gradients and very small geometrical dimensions of the domain considered). The external heating (a laser action) is substituted by the introduction of internal heat source to the DPLE. To model the melting process in domain of pure metal (chromium) the approach basing on the artificial mushy zone introduction is used and the main goal of investigation is the verification of influence of the artificial mushy zone ‘width’ on the results of melting modeling. At the stage of numerical modeling the author’s version of the Control Volume Method is used. In the final part of the paper the examples of computations and conclusions are presented.
4
Content available Analysis of ultrashort laser pulse interactions with metal films using a two-temperature model
Majchrzak E., Dziatkiewicz J.
Journal of Applied Mathematics and Computational Mechanics
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2015
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Vol. 14, nr 2
31--39
EN
In the paper the problem of thin metal film subjected to the action of the high laser fluence and the ultrashort pulse width is considered. The mathematical model consists of the equations describing the electrons and phonons temperatures and the relationships between the heat fluxes and temperature gradients of electrons and phonons. The problem is solved using the explicit scheme of the finite difference method with staggered grid. In the final part the results of computations and conclusions are presented.
5
Content available Estimation of relaxation and thermalization times in microscale heat transfer model
Mochnacki B., Paruch M.
Journal of Theoretical and Applied Mechanics
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2013
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Vol. 51 nr 4
837--845
EN
The energy equation corresponding to the dual phase lag model (DPLM) results from the generalized form of the Fourier law, in which the two ‘delay times’ (relaxation and thermalization time) are introduced. The DPLM should be used in the case of microscale heat transfer analysis, in particular when thermal processes are characterized by extremely short duration (e.g. ultrafast laser pulse), considerable temperature gradients and very small dimensions (e.g. thin metal film). In this paper, the problem of relaxation and thermalization time identification is discussed, at the same time the heat transfer processes proceeding in the domain of a thin metal film subjected to a laser beam are analyzed. The solution presented bases on the application of evolutionary algorithms. The additional information concerning the transient temperature distribution on a metal film surface is assumed to be known. At the stage of numerical realization, the finite difference method (FDM) is used. In the final part of the paper, an example of computations is presented.
6
Content available remote Application of sensitivity analysis in microscale heat transfer
Majchrzak E., Dziatkiewicz J., Kałuża G.
Computer Assisted Methods in Engineering and Science
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2013
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Vol. 20, no. 2
113--121
EN
In the paper, the thin metal film subjected to the ultrashort laser pulse has been analyzed. The heat conduction in the domain considered has been described by two-temperature model consisting of the system of two coupled parabolic equations determining the electron and lattice temperatures. The sensitivity analysis of electron and lattice temperatures with respect to the parameters appearing in mathematical description has been discussed. In particular, the changes of temperatures due to the changes of coupling factor G and the film thickness L have been estimated. At the stage of numerical computations in a case of basic as well as sensitivity problems solutions the explicit scheme of finite difference method has been used. In the final part of the paper the results of computations have been shown.
7
Content available remote Application of the two-temperature model for a numerical study of multiple laser pulses interactions with thin metal films
Majchrzak E., Dziatkiewicz J.
Prace Naukowe Instytutu Matematyki i Informatyki Politechniki Częstochowskiej
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2012
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Vol. 11, nr 2
63-70
EN
A thin metal film irradiated by multiple laser pulses is considered. The microscale heat transfer in the domain considered is described by hyperbolic two-temperature model. This model contains two energy equations determining the heat exchange in the electron gas and the metal lattice. The problem is solved by a explicit scheme of finite difference method. The influence of separation time between two laser pulses on the electrons and lattice temperatures is discussed.
8
Content available remote Identification of electron-phonon coupling factor in a thin metal film subjected to an ultrashort laser pulse
Majchrzak E., Dziatkiewicz J.
Computer Assisted Methods in Engineering and Science
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2012
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Vol. 19, no. 4
383--392
EN
A thin metal film subjected to a laser pulse is considered. The problem is described by the system of energy equations describing the electron gas and lattice temperatures. The thermal interactions between electrons and lattice are determined by the parameter G called the electron-phonon coupling factor. To estimate the unknown parameter G the identification problem is formulated. The additional information necessary to solve an inverse problem is the knowledge of transient measurements of the reflectivity or transmissivity variation which is proportional to the variation of the electron temperature. So, at the stage of inverse problem solution, it is possible to assume the knowledge of electrons temperature on the irradiated surface of the system (x = 0). To solve the identification problem the gradient method basing on the least squares criterion and sensitivity coefficients is used. In the final part of the paper the results of computations are shown
9
Content available remote Two - temperature model of microscale heat transfer
Majchrzak E., Poteralska J.
Computer Methods in Materials Science
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2011
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Vol. 11, No. 2
330-336
EN
Thin metal film subjected to a short-pulse laser heating is considered. The hyperbolic two-temperature model describing the temporal and spatial evolution of the lattice and electrons temperatures is discussed. At a stage of numerical computations the finite difference method is used under the assumption that a part of thermophysical parameters in mathematical model of the process considered is temperature-dependent. In the final part of the paper the examples of computations are shown.
PL
Rozpatrywano cienką warstwę metalową poddaną działaniu lasera. Procesy cieplne w analizowanym obszarze opisano dwu-temperaturowym hiperbolicznym modelem, który uwzględnia zmiany temperatury elektronów i sieci krystalicznej. Zadaniel rozwiązano za pomocą metody różnic skończonych, przy czym uwzględniono zmienne z temperaturą wartości niektórych parametrów termofizycznych. W końcowej części artykułu przedstawiono wyniki obliczeń.
10
Content available remote Modelling of microscale heat transfer in cylindrical domains
Mochnacki B., Majchrzak E.
Computer Methods in Materials Science
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2011
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Vol. 11, No. 2
337-342
EN
Thermal processes in a thin metal film subjected to a short-pulse laser heating are considered (axially-symmetrical 2D problem). Heat transfer through thin films subjected to an ultrafast laser pulse is of vital importance in microtechnology applications and it is a reason that the problems connected with a fast heating of solids have become a very active research area. The heat transfer proceeding in domain analyzed (microscale heat transfer) is described by dual phase lag model (DPLM). According to the newest opinions the DPLM constitutes a very good description of real heat transfer processes proceeding in the micro-scale domains subjected to the strong external heat flux. The base of DPLM formulation is a generalized form of Fourier law (GFL) in which two time parameters tauq, tau appear (the relaxation time and thermalization one, respectively). The acceptation of GFL leads to DPLM equation (Özisik & Tzou, 1994; Smith & Norris, 2003). In the paper the thermal interactions between external heat source qb and cylindrical micro-domain are analyzed. The capacity of external heat source (the Neumann boundary condition) is given by function dependent on spatial coordinates and time. On the boundary beyond the region of laser action, the no-flux condition is assumed. It should be pointed out that the DPL model requires the adequate transformation of boundary conditions which appear in the typical macro heat conduc-tion models. The initial conditions arę also known (initial temperature of domain and initial heating rate). Numerical model of the process discussed bases on a certain variant of FDM presented with full particulars in Chapter 2.
PL
W pracy rozpatruje się proces przepływu ciepła w mikroobszarach poddanych działaniu wiązki laserowej (zadanie osiowo-symetryczne). Analiza problemów tego typu ma istotne znaczenie w szeroko rozumianej mikrotechnologii i stąd wynika duże zainteresowanie badaniami teoretycznymi oraz eksperymentalnymi związanymi z ultraszybkim oddziaływaniem lasera na powierzchnię mikrowarstw wykonanych z różnych materiałów. Przepływ ciepła w mikroskali może być opisany m.in. równaniem różniczkowym energii z dwoma czasami opóźnień (DPL - dual phase lag) i zgodnie z najnowszymi poglądami taki właśnie model stanowi najlepsze przybliżenie rzeczywistych procesów cieplnych zachodzących w tej skali. DPL wynika z uogólnienia znanego prawa Fouriera do którego wprowadza się dwaj dodatkowe parametry tau q tau T (czas relaksacji i czas termalizacji, odpowiednio). Tematem rozważań w prezentowanej pracy jest model przepływu ciepła w jednorodnej warstewce materiału (chromu) poddanej działaniu zewnętrznego źródła ciepła (warunek brzegowy Neumanna) o zadanej zmiennej w czasie i na powierzchni intensywności q b (funkcja typu dzwonowego, co implikuje orientację obiektu we współrzędnych walcowych), na pozostałych powierzchniach umownie ograniczających rozpatrywany obszar przyjęto warunki adiabatyczne. Algorytm modelowania procesów cieplnych bazuje na jawnym schemacie MRS i w końcowej części pracy przedstawiono wyniki symulacji numerycznych.
11
Content available Estimation of dual phase lag model parameters using the evolutionary algorithms
Mochnacki B., Paruch M.
Archives of Foundry Engineering
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2011
|
Vol. 11, iss. 3 spec.
277--281
EN
Generalization of Fourier law, in particular the introduction of two ‘delay times’ (relaxation time τq and thermalization time τT) leads to the new form of energy equation called the dual-phase-lag model (DPLM). This equation should be applied in a case of microscale heat transfer modeling. In particular, DPLM constitutes a good approximation of thermal processes which are characterized by extremely short duration (e.g. ultrafast laser pulse), extreme temperature gradients and geometrical features of domain considered (e.g. thin metal film). The aim of considerations presented in this paper is the identification of two above mentioned positive constants τq, τT. They correspond to the relaxation time, which is the mean time for electrons to change their energy states and the thermalization time, which is the mean time required for electrons and lattice to reach equilibrium. In this paper the DPLM equation is applied for analysis of thermal processes proceeding in a thin metal film subjected to a laser beam. At the stage of computations connected with the identification problem solution the evolutionary algorithms are used. To solve the problem the additional information concerning the transient temperature distribution on a metal film surface is assumed to be known.
12
Content available remote Two-temperature microscale heat transfer model. Part 2, Determination of lattice parameters
Majchrzak E., Poteralska J.
Prace Naukowe Instytutu Matematyki i Informatyki Politechniki Częstochowskiej
|
2010
|
Vol. 9, nr 1
109-119
EN
Two-temperature microscale heat transfer model is presented. This model contains two energy equations determining the heat exchange in the electron gas and the metal lattice. A key issue in the application of this model is the proper description of temperature dependent thermophysical parameters of the material considered and this problem is discussed here. In this part the determination of metal lattice thermophysical parameters is presented.
13
Content available remote Two-temperature microscale heat transfer model. Part 1, Determination of electrons parameters
Majchrzak E., Poteralska J.
Prace Naukowe Instytutu Matematyki i Informatyki Politechniki Częstochowskiej
|
2010
|
Vol. 9, nr 1
99-108
EN
Two-temperature microscale heat transfer model is presented. This model contains two energy equations determining the heat exchange in the electron gas and the metal lattice. A key issue in the application of this model is the proper description of temperature dependent thermophysical parameters of the material considered and this problem is discussed here. In this part the determination of electrons thermophysical parameters is presented.
14
Content available Numerical simulation of thermal processes proceeding in a multi-layered film subjected to ultrafast laser heating
Majchrzak E., Mochnacki B., Suchy J. S.
Journal of Theoretical and Applied Mechanics
|
2009
|
Vol. 47 nr 2
383-396
EN
In the paper, the mathematical model, numerical algorithm and examples of computations concerning thermal processes proceeding in a multi-layered thin film subjected to an ultrafast laser pulse are discussed. The equations describing a course of the analysed process correspond to the dual-phase-lag model and contain both the relaxation time tau q and additionally the thermalization time tau T. At the stage of numerical simulation, the finite difference method has been used. The algorithm is based on an artificial decomposition of the domain considered, while common thermal interactions between successive layers are taken into account using conditions of heat flux and temperature continuity at points corresponding to internal boundaries (1D task has been considered).
PL
W pracy przedstawiono model matematyczny, algorytm numeryczny i przykłady symulacji dotyczących przebiegu procesów cieplnych w wielowarstwowym mikroobszarze nagrzewanym ultraszybkim impulsem generowanym przez laser. Równanie opisujące przebieg procesu odpowiada modelowi z dwoma opóźnieniami wynikającymi z czasu relaksacji i czasu termalizacji. Na etapie obliczeń numerycznych wykorzystano metodę różnic skończonych. Algorytm bazuje na sztucznej dekompozycji obszaru wielowarstwowego, przy czym wzajemne oddziaływania między warstwami uwzględniono poprzez założenie ciągłości strumienia ciepła i pola temperatury na powierzchniach kontaktu. Biorąc pod uwagę geometrię obszaru, rozpatrywano zadanie jednowymiarowe.
15
Content available remote Finite difference model of short-pulse laser interactions with thin metal film
Majchrzak E., Mochnacki B., Suchy J. S.
Computer Methods in Materials Science
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2009
|
Vol. 9, No. 2
316-321
EN
In the paper the mathematical model, numerical algorithm and examples of computations connected with the ultrafast heating of thin films are discussed. The heat transfer proceeding in domain considered (microscale heat transfer) is described by means of the dual-phase-lag-model and the mathematical description of the process bases on the equation in which the relaxation time and thermalization time appear. The pulse laser action is taken into account by an additional term in energy equation corresponding to internal heat source, at the same time along the boundaries limiting the system the no-flux conditions are assumed. On a stage of numerical modelling a three level implicit finite difference scheme has been developed. A geometry of thin film allows to consider a 1D task, and then a solution of only one three diagonal linear system of equations corresponds to transition from time t to the new level of time. In this place the Thomas algorithm has been used. In the final part of the paper the examples of computations are shown.
PL
Rozważano procesy cieplne zachodzące w cienkiej war­stewce metalowej poddanej działaniu lasera o krótkim impulsie. Przepływ ciepła w analizowanym obszarze (w skali mikro) opisuje model podwójnego opóźnienia i opis matematyczny procesu bazuje na równaniu, w którym pojawia się czas relaksacji oraz czas termalizacji. Oddziaływanie lasera uwzględniono w dodatkowym składniku równania energii odpowiadającym wewnętrznej funkcji źródła, równocześnie na powierzchniach ograniczających założono warunki adiabatyczne. Na etapie modelowania numerycznego opracowano trój poziomo wy niejawny schemat metody różnic skończonych. Biorąc pod uwagę geometrię cienkiej warstewki, rozpatrywano zadanie ID (jednowymiarowe) i wówczas dla każdego przejścia od chwili t do chwili t +delta t należy rozwiązać tylko jeden trój diagonalny układ równań. Układ ten rozwiązano wykorzystując algorytm Thomasa. W końcowej części artykułu pokazano przykłady obliczeń.
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