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1

Reconstruction of the heat transfer coefficient in the inverse Stefan problem

Matlak J., Słota D., Zielonka A.

Hutnik, Wiadomości Hutnicze

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2018

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Vol. 85, nr 1

6--9

EN

In the paper we will present the method of finding the heat transfer coefficient in the inverse problem of pure metal solidification. In the considered model the shrinkage of metal and the air-gap between material and mold will be taken into account. The method is based on the algorithm for solution of the direct problem and on the Artificial Bee Colony algorithm. In the algorithm for solving the direct problem we use the finite element method supplemented by the procedure allowing to define the position of the moving interface and the change of material size associated with the shrinkage. To solve the inverse problem, a functional defining the error of approximate solution must be minimized. To minimize this functional we use the Artificial Bee Colony algorithm. Then we present the computational example illustrating precision and stability of the presented method.

PL

W pracy zaprezentowana zostanie metoda wyznaczania współczynnika wnikania ciepła w zagadnieniu odwrotnym krzepnięciem czystego metalu. W rozważanym modelu uwzględniony będzie skurcz metalu oraz szczelina powietrzna pomiędzy odlewem i wlewkiem. Prezentowana metoda wykorzystuje algorytm rozwiązania zagadnienia bezpośredniego oraz algorytm pszczeli. W algorytmie rozwiązania zagadnienia bezpośredniego wykorzystano metodę elementów skończonych uzupełnioną o procedurę pozwalającą określić położenie granicy rozdziału faz oraz zmianę wymiarów wlewka spowodowaną skurczem metalu. W rozwiązaniu zagadnienia odwrotnego należy zminimalizować funkcjonał określający błąd rozwiązania przybliżonego. W tym celu wykorzystano algorytm pszczeli. Przedstawiono także przykład obliczeniowy ilustrujący dokładność i stabilność prezentowanej metody.

2

The Taylor transformation hybrid method applied for solving the Stefan problem

Grzymkowski R., Hetmaniok E., Pleszczyński M.

Silesian Journal of Pure and Applied Mathematics

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2016

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Vol. 6, iss. 1

111--123

EN

The paper presents the analytic-numerical hybrid method using, among others, the Taylor transformation, thanks to which the solution of the Stefan problem is replaced by the solution of a nonlinear system of equations.

3

Sensitivity analysis of phase transformation model based on solution of diffusion equation

Bzowski K., Bachniak D., Pernach M., Pietrzyk M.

Archives of Civil and Mechanical Engineering

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2016

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Vol. 16, no. 2

186--192

EN

Presented work is focused on modelling of the phase transformation during laminar cooling after hot rolling of dual phase steel strips. Conventional FE model describing heat transfer was used in the macroscale. The model based on the solution of the diffusion equation with moving boundary was selected to predict properties of the steel based on phase transformations which occur in microscale. Preliminary observations indicated that results depend on various parameters of the model, such as: diffusion coefficient, boundary velocity factor and cooling rate. Therefore, sensitivity analysis of the model with respect to these parameters was performed in order to enhance the predictive capabilities of the model and to simplify further solution.

4

Numerical modeling of pure metal solidification using the one domain approach

Szopa R.

Journal of Applied Mathematics and Computational Mechanics

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2015

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Vol. 14, nr 3

121--126

EN

Numerical modeling of pure metal solidification on the basis of the well-known Stefan model is rather difficult. The knowledge of temporary solidification front position and the local values of the solidification rate in the normal direction for time t are necessary in order to determine the new position of moving boundary for time t + ∆t. The problem is especially complicated for 2D and 3D tasks. The concept greatly simplifying the modeling of solidification process boils down to the introduction of the artificial region corresponding to the mushy zone sub-domain. For this region the substitute thermal capacity is defined and the mathematical model corresponds to the one domain approach. The artificial mushy zone appears owing to conventional enlargement of solidification point on a certain interval of temperature ∆T. The basic goal of the paper is the numerical analysis of the influence of the interval ∆T on the numerical solution simulating the thermal processes in the domain of the solidifying metal.

5

One-phase Stefan problem with temperature-dependent thermal conductivity and a boundary condition of Robin type

Briozzo A. C., Natale M. F.

Journal of Applied Analysis

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2015

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Vol. 21, nr 2

89--97

EN

We study a one-phase Stefan problem for a semi-infinite material with temperature-dependent thermal conductivity with a boundary condition of Robin type at the fixed face x = 0. We obtain sufficient conditions for data in order to have a parametric representation of the solution of similarity type for t ≥ t0 > 0 with t0 an arbitrary positive time. This explicit solution is obtained through the unique solution of an integral equation with the time as a parameter.

6

Solution of the Pure Metals Solidification Problem by Involving the Material Shrinkage and the Air-Gap Between Material and Mold

Matlak J., Słota D.

Archives of Foundry Engineering

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2015

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Vol. 15, iss. 3 spec.

47--52

EN

In this paper we describe an algorithm for solving the pure metals solidification problem by involving the metal shrinkage and air-gap between material and mold. In this algorithm we use the finite element method supplemented by the procedures allowing to define the position of the moving interface and the change of the material size associated with the shrinkage. We present also an example illustrating the precision of presented method.

7

Numerical scheme for one phase 1D fractional Stefan problem using the similarity variable technique

Błasik M.

Journal of Applied Mathematics and Computational Mechanics

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2014

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Vol. 13, nr 1

13--21

EN

In this paper we present a numerical method to solve a one-dimensional, one-phase extended Stefan problem with fractional time derivative described in the Caputo sense. The proposed method is based on applying a similarity variable for the anomalous-diffusion equation and the finite difference method. In the final part, examples of numerical results are discussed.

8

Effect of Mesh Quality on the Numerical Solution of the Solidification of Pure Metal

Skrzypczak T., Węgrzyn-Skrzypczak E.

Archives of Foundry Engineering

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2013

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Vol. 13, iss. 2 spec.

89--92

EN

The paper presents a method of mathematical and numerical modelling of directional solidification process of pure metal in the two-dimensional region. In this case, the thermal conditions associated with the process favours the occurrence of sharp solidification front. The mathematical description of the process is based on the Stefan formulation with appropriate continuity conditions on the solid-liquid interface. The numerical model is based on the finite element method (FEM). The calculations were made on a fixed mesh with diffused solidification front to avoid the difficulties associated with the discontinuity. Temporary position of the interface was calculated with the use of the level set method (LSM). Effect of the quality of the spatial discretization on the accuracy of numerical solution was investigated. Obtained results of the temporary front position were compared with the analytical solution. The correlation between the quality of the spatial discretization and the accuracy of the results was observed. Methods used in the work had significant impact on the computation time and helped avoid the explicit consideration of discontinuity of heat flux on the front.

9

Numerical scheme for the one-phase 1D Stefan problem using curvilinear coordinates

Błasik M.

Prace Naukowe Instytutu Matematyki i Informatyki Politechniki Częstochowskiej

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2012

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Vol. 11, nr 3

9-14

EN

In this paper we present a new approach to solving a one-dimensional, one-phase Stefan problem. The proposed method is based on choosing (a) suitable curvilinear space coordinate/s for the heat-flow equation and the finite difference method. In the final part of this paper, examples of numerical calculations are shown.

10

Sharp Interface Numerical Modeling of Solidification Process of Pure Metal

Skrzypczak T.

Archives of Metallurgy and Materials

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2012

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Vol. 57, iss. 4

1189-1199

EN

The paper is focused on the study of the solidification process of pure metals, in which the solidification front is smooth. It has the shape of a surface separating liquid from solid in three dimensional space or a curve in 2D. The location and topology of moving interface change over time and its velocity depends on the values of heat fluxes on the solid and liquid side of it. Such a formulation belongs to a group called Stefan problems. A mathematical model of the Stefan problem is based on differential equations of heat conduction and interface motion. This system of equations is supplemented by appropriate initial and boundary conditions as well as the continuity conditions at the solidification interface. The solution involves the determination of temporary temperature field and interface position. Typically, it is impossible to obtain the exact solution of such problem. This paper presents a mathematical model for the two-dimensional problem. The equation of heat conduction is supplemented with Dirichlet and Neumann boundary conditions. Interface motion is described by the level set equation which solution is sought in the form of temporary distribution of the signed distance function. Zero level of the distance field coincides with the position of the front. Values of the signed distance function obtained from the level set equation require systematic reinitialization. Numerical model of the process based on the finite element method (FEM) is also presented. FEM equations are derived and discussed. The explicit time integration scheme is proposed. It helps to avoid solving the system of equations during each time step. The reinitialization procedure of the signed distance function is described in detail. Examples of numerical analysis of the solidification process of pure copper within the complex geometry are presented. Results obtained from the use of constant material properties are compared with those obtained from the use of temperature dependent properties.

PL

W pracy skupiono się na badaniu procesu krzepnięcia czystych metali, podczas którego front krzepnięcia pozostaje płaski. W przypadku trójwymiarowym jest on powierzchnia oddzielająca ciecz od ciała stałego, w przypadku dwuwymiarowym ma postać krzywej. Położenie i topologia frontu krzepnięcia zmienia się w czasie, a prędkość przemieszczania zależy od różnicy wartości strumieni cieplnych po stronie ciała stałego i cieczy. Takie sformułowanie klasyfikuje opisywane zjawisko w grupie tzw. zagadnień Stefana. Model matematyczny tego procesu stanowią równania różniczkowe przewodnictwa ciepła oraz ruchu powierzchni międzyfazowej. Układ ten uzupełniają odpowiednie warunki brzegowe, początkowe oraz warunki ciągłości na froncie. Jego rozwiązanie polega na wyznaczeniu chwilowych pól temperatury oraz położenia frontu. Najczęściej nie da się uzyskać rozwiązania tak sformułowanego problemu w sposób dokładny. W pracy zaprezentowano model matematyczny zagadnienia dla przypadku płaskiego. Równanie różniczkowe przewodnictwa ciepła uzupełniono warunkami brzegowymi Dirichleta oraz Neumanna. Ruch interfejsu międzyfazowego opisano tzw. równaniem poziomic (ang. level set equation), którego rozwiązania poszukiwano w postaci chwilowego rozkładu funkcji dystansu. Izolinia zerowa tego rozkładu pokrywa się z położeniem frontu. Otrzymane wartości funkcji dystansu wymagają systematycznej reinicjalizacji. Przedstawiono również model numeryczny procesu bazujący na metodzie elementów skończonych. Opisano schemat postępowania prowadzący do otrzymania dyskretnych równań MES. Wykorzystano jawny schemat całkowania po czasie, co pozwoliło uniknąć konieczności rozwiazywania układu równań zarówno w przypadku równania przewodnictwa ciepła jak i równania poziomic. Szczegółowo opisano metodę reinicjalizowania funkcji dystansu. Zaprezentowano przykłady analizy numerycznej procesu krzepnięcia czystej miedzi w obszarze o złożonej geometrii. Porównano wyniki otrzymane dla stałych własności materiałowych z wynikami uzyskanymi z wykorzystaniem własności zależnych od temperatury.

11

Identification of the Heat Transfer Coefficient in the Inverse Stefan Problem by Using the ABC Algorithm

Hetmaniok E., Słota D., Zielonka A.

Archives of Foundry Engineering

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2012

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Vol. 12, iss. 2s

27--32

EN

A procedure based on the Artificial Bee Colony algorithm for solving the two-phase axisymmetric one-dimensional inverse Stefan problem with the third kind boundary condition is presented in this paper. Solving of the considered problem consists in reconstruction of the function describing the heat transfer coefficient appearing in boundary condition of the third kind in such a way that the reconstructed values of temperature would be as closed as possible to the measurements of temperature given in selected points of the solid. A crucial part of the solution method consists in minimizing some functional which will be executed with the aid of one of the swarm intelligence algorithms - the ABC algorithm.

12

Stimulating Mathematics-in-Industry

Ockendon J.R.

Matematyka Stosowana : matematyka dla społeczeństwa

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2011

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T. 12/53, nr spe

5-17

PL

Załączony artykuł jest przedrukiem z czasopisma Mathematics Today i jest wykładem, jaki autor, moderator matematyki przemysłowej w Wielkiej Brytanii, wygłosił po ceremonii wręczenia mu w dniu 27 czerwca 2007 r. złotego modelu IMA - Institute of Mathematics and its Applications, Oxford. Wykład poruszał kilka niestandardowych problemów, które wywodziły się z zagadnień stawianych przez szeroko rozumiany przemysł, a następnie były rozwiązywane przez uczestników Study Group. Do nich należały: zagadnienie związane z poprawieniem konstrukcji pantografu, aby zapewnić jego stały kontakt z linią trakcyjną czy problem ze swobodną granicą (problem Stefana) dla metalu występującego w temperaturze przewyższającej temperaturę topnienia superheated. Inne problemy dotyczyły kształtu antenty radaru zapewniającej optymalny odbiór fal rozroszonych czy metody automatycznego pomiaru ilości mleka przepływającego w rurce automatycznej dojarki krów.

13

Solution of the solidification problem by using the variational iteration method

Hetmaniok E., Słota D., Zielonka A.

Archives of Foundry Engineering

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2009

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Vol. 9, iss. 4

63-68

EN

The paper presents the approximated solution of the solidification problem, modelled with the aid of the one-phase Stefan problem with the boundary condition of the second kind, by using the variational iteration method. For solving this problem one needs to determine the distribution of temperature in the given domain and the position of the moving interface. The proposed method of solution consists of describing the considered problem with a system of differential equations in a domain with known boundary, and solving the received system with the aid of VIM method. The accuracy of the obtained approximated solution is verified by comparing it with the analytical solution.

14

Stefan and Kolmogoroff models of solidification. Comparison of numerical solutions

Szopa R., Pawlak E., Siedlecki J.

Prace Naukowe Instytutu Matematyki i Informatyki Politechniki Częstochowskiej

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2008

|

Vol. 7, nr 1

185-192

EN

Problems connected with the mathematical description of pure metals solidification (macro approach) are often called the Stefan ones. The second generation models (micro/ macro approach) discussed in this paper base on a theory presented by Kolmogoroff (Mehl-Johnson-Avrami-Kolmogoroff models). Both macro and micro/macro problems can be analyzed using the numerical methods. The aim of investigations presented here was a comparison of numerical solutions obtained by use of macro and micro/macro approach. On a stage of numerical modelling the finite difference method has been applied.

15

Determination of the continuous casting cross-section with prescribed average temperature

Grzymkowski R.

Archives of Foundry Engineering

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2008

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Vol. 8, iss. 4

51-54

EN

This paper presents the method of determination of the continuous casting cross-section, in which average temperature was equal to a prescribed value. The method proposed here does not require evaluation of temperature distribution. On the basis of input data, a linear or non-linear equation is created (depending on the heat flux form on the region boundaries), which solution enabled determination of the cross-section.

16

Rola założonego kształtu ziaren w symulacji przemiany ferrytycznej

Pernach M., Pietrzyk M.

Rudy i Metale Nieżelazne

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2007

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R. 52, nr 11

866-871

PL

Celem pracy jest określenie wpływu założonego kształtu ziaren ferrytu i austenitu na wyniki symulacji: kinetyki przemiany, ułamka objętości ferrytu, wielkości ziarna ferrytu oraz segregacji węgla przed frontem przemiany. Analizowano model1D (wzrost liniowy), model 2D (koło w kole, sześciokąt foremny w sześciokącie foremnym) oraz 3D (kula w kuli). Modele te oparto na rozwiązaniu II prawa Ficka dla przypadku ruchomej granicy międzyfazowej. Do rozwiązania równania dyfuzji wykorzystano metodę różnic skończonych oraz metodę elementów skończonych. W pracy dokonano porównania wyników symulacji numerycznych z wynikami badań doświadczalnych.

EN

Determination of the influence of ferrite and austenite grain shape on the kinetics of phase transformation, ferrite volume fraction, ferrite grain size and carbon segregation before the front of transformation is the aim of this work. Numerical model in 1D (linear growth), in 2D (the circle in the circle, the regular hexagon in the regular hexagon) and in 3D (the sphere in the sphere) were developed and are presented in the paper. These models are based on the solution of the second Fick law for the case of the moving boundary. The finite difference and finite elements method are used to solve the equation of diffusion. Comparison of the computational results with the experimental data are shown and discussed in this paper.

17

Zagadnienie Stefana dla żebra pracującego przy wrzeniu

Orzechowski T.

Inżynieria i Aparatura Chemiczna

|

2006

|

Nr 6s

179-180

EN

The Stefan problem dealing with the determination of a boundary shape has been presented. A numerical analysis with the use of heat polynomials has been conducted. On the example of a triangular fin a methodology and its effectiveness for boiling heat transfer were shown.

18

Application of generalized finite difference method in numerical modelling of moving boundary problems

Mochnacki B., Lara S.

Prace Naukowe Instytutu Matematyki i Informatyki Politechniki Częstochowskiej

|

2003

|

Vol. 2, nr 1

129--143

EN

The application of generalized finite difference method for numerical modelling of thermal processes proceeding in the solidifying casting domain is presented. The solidification of pure metals and eutectic alloys is considered. In such case the solidification process takes place at the constant temperature (the Stefan problem). From the numerical point of view the solution of this task is very complex; in particular, in the case of 2D or 3D domains; and in literature one can find the procedures enabling to avoid the difficulties with the direct modelling of the problem discussed. The part of them consist in the substitution of the solidification point T* by the certain interval [T* - ΔT, T* + ΔT]. In this way the subdomain of artificial mushy zone is introduced and the fixed domain approach [1] can be used. On the stage of numerical algorithm construction and numerical simulation the generalized finite difference method is used. In the final part of the paper the examples of computations are shown.

19

Stefan problems in non-cylindrical domains arising in Czochralski process of crystal growth

Fukao T., Kenmochi N., Pawłow I.

Control and Cybernetics

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2003

|

Vol. 32, no 2

201-221

EN

In this paper we discuss a two-phase Stefan problem with convection in a non-cylindrical (time-dependent) domain. This work is motivated by phase change phenomenon arising in the Czochralski process of crystal growth. The time-dependence of domain is a mathematical description of the situation in which the material domain changes its shape with time by crystal growth. We consider the so-called enthalpy formulation for it and give its solvability, assuming that the time-dependence of the material domain is prescribed and smooth enough in time, and the convective vector is prescribed, too. Our main idea is to apply the theory of quasi-linear equations of parabolic type.

20

The influence of artificial mushy zone parameters on the numerical solution of the Stefan problem

Mochnacki B., Lara S.

Archiwum Odlewnictwa

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2003

|

R. 3, nr 10

31--36

EN

The solidification of pure metals and eutectic alloys proceed at the constant temperature (the Stefan problem). From the numerical point of view the solution of this task is very complex and in literature one can find the procedures enabling to avoid the difficulties with the direct modelling of the problem discussed. The part of them consist in the substitution of the solidification point T* on the certain interval [T*-ΔT, T*+ΔT]. In this way the subdomain of artificial mushy zone (AMZ) is introduced and the fixed domain approach [1, 2, 3] can be used. In the paper the analysis of AMZ assumed parameters on the results of solidification process simulation is presented. On the stage of numerical computations the generalized FDM has been applied.

PL

Krzepnięcie czystych metali i stopów eutektycznych zachodzi w stałej temperaturze (problem Stefana). Z matematycznego punktu widzenia rozwiązanie takiego zadania jest bardzo skomplikowane i w literaturze można znaleźć procedury umożliwiające ominięcie trudności związanych z bezpośrednim modelowaniem rozważanego problemu. Część z nich bazuje na zastąpieniu temperatury krzepnięcia pewnym przedziałem o szerokości 2ΔT. W ten sposób w modelu pojawia się podobszar nazywany sztuczną strefą dwufazową. W pracy analizowano wpływ parametrów tej strefy na numeryczne rozwiązanie procesu krzepnięcia. Na etapie obliczeń numerycznych wykorzystano uogólnioną metodę różnic skończonych.