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1
Content available Application of the Taylor differential transformation for solving the integro-differential equations
Grzymkowski R., Pleszczyński M.
Silesian Journal of Pure and Applied Mathematics
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2018
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Vol. 8, iss. 1
33--48
EN
A method of solving the integro-differential equations is presented. The discussed equations will be solved by the Taylor differential transformation. By using appropriate properties of this transformation the integro-differential equation will be transformed to a respective recurrence equation. Unfortunately, the high degree of generality and complexity of such defined problem does not allow to obtain the solution in general form. Each equation requires a special method of solution.
2
Content available Application of the Taylor differential transformation in the calculus of variations
Grzymkowski R., Pleszczyński M., Hetmaniok E.
Silesian Journal of Pure and Applied Mathematics
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2017
|
Vol. 7, iss. 1
65--82
EN
The paper presents the method of solving some problems belonging to the area of the calculus of variations, that is the problems of searching for the selected types of functionals which can be transformed to some, nonlinear in general, ordinary differential equations or systems of such equations. The obtained equations are solved on the basis of the Taylor differential transformation.
3
Content available 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.
4
Content available Problem of the moving boundary in continuous casting solved by the analytic-numerical method
Grzymkowski R., Pleszczyński M., Hetmaniok E.
Archives of Foundry Engineering
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2013
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Vol. 13, iss. 1
33--38
EN
Mathematical modeling of thermal processes combined with the reversible phase transitions of type: solid phase – liquid phase leads to formulation of the parabolic or elliptic moving boundary problem. Solution of such defined problem requires, most often, to use some sophisticated numerical techniques and far advanced mathematical tools. The paper presents an analytic-numerical method, especially attractive from the engineer’s point of view, applied for finding the approximate solutions of the selected class of problems which can be reduced to the one-phase solidification problem of a plate with the unknown a priori, varying in time boundary of the region in which the solution is sought. Proposed method is based on the known formalism of initial expansion of a sought function, describing the field of temperature, into the power series, some coefficients of which are determined with the aid of boundary conditions, and on the approximation of a function defining the freezing front location with the broken line, parameters of which are determined numerically. The method represents a combination of the analytical and numerical techniques and seems to be an effective and relatively easy in using tool for solving problems of considered kind.
5
Content available The CC1 project – system for Private Cloud Computing
Chwastowski J., Grzymkowski R., Kruk M., Nabożny M., Natkaniec Z., Olszewski A., Pałka H., Sobocińska Z., Sośnicki T., Szostak M., Syktus P., Wójcik P., Wojtoń T., Zdybał M., Żabiński B., Witek M.
Computer Science
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2012
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Vol. 13 (2)
103-111
EN
The main features of the Cloud Computing system developed at IFJ PAN are described. The project is financed from the structural resources provided by the European Commission and the Polish Ministry of Science and Higher Education (Innovative Economy, National Cohesion Strategy). The system delivers a solution for carrying out computer calculations on a Private Cloud computing infrastructure. It consists of an intuitive Web based user interface, a module for the users and resources administration and the standard EC2 interface implementation. Thanks to the distributed character of the system it allows for the integration of a geographically distant federation of computer clusters within a uniform user environment.
6
Content available remote Reconstruction of the thermal conductivity coefficient by using the harmony search algorithm
Grzymkowski R., Hetmaniok E., Słota D., Zielonka A.
Journal of Achievements in Materials and Manufacturing Engineering
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2011
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Vol. 49, nr 2
299--304
EN
Purpose: of this paper: Aim of this paper is a presentation of the respectively new tool for solving the optimization problems, which is the Harmony Search algorithm in version slightly modified by the authors, used for identifying the thermal conductivity coefficient. Proposed approach is illustrated with an example confirming its usefulness for solving such kinds of problems. Design/methodology/approach: For solving the considered parametric inverse heat conduction problem the approach is applied in which the essential part consists in minimization of the functional representing the differences between the measurement values of temperature and approximate values calculated with the aid of finite difference method. For minimizing the functional the Harmony Search algorithm is used. Findings: The elaboration shows that approaches involving the algorithms of artificial intelligence for solving the inverse heat conduction problems of that kind are efficient and they ensure to receive satisfying results in shorter time in comparison with the classical procedures. Research limitations/implications: Specific properties of the heuristic algorithms require to execute the procedure several times and to average the obtained results because each running of the algorithm can give slightly different results. Each execution of the procedure means the solution of the direct problem associated with the considered inverse problem by using the finite difference method. Practical implications: In spite of the problem described above the approaches involving the heuristic algorithms of artificial intelligence are successful because they are respectively simple and easy to use and they give satisfying results after short time of working. Another advantage of using optimization algorithms of that kind is the fact that they do not need to satisfy any assumptions about the solved problem, oppositely to the classical optimization algorithms. Originality/value: Proposal of the original approach involving the heuristic optimization algorithm for solving the parametric inverse heat conduction problem is discussed in the paper.
7
Content available remote Efficiency of the computer tomography algorithms in examination of the internal structure of materials with non-transparent elements by using the incomplete information
Grzymkowski R., Hetmaniok E., Pleszczyński M.
Journal of Achievements in Materials and Manufacturing Engineering
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2011
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Vol. 49, nr 2
285--298
EN
Purpose: of this paper: The effectiveness of computer tomography algorithms applied for reconstructing the internal structure of objects containing the non-transparent elements is discussed, in conditions of the incomplete information about the examined object. Design/methodology/approach: Problem of the internal structure examination of an object with non-transparent elements, without its destruction, is considered by means of the classical and non-classical algebraic algorithms of computer tomography used in standard approaches and in cases of incomplete projection data. Findings: Computer tomography algorithms, known from literature and designed by the authors, are tested in solving the problems of reconstructing the discrete objects of high contrast with non-transparent elements, with regard to their precision, convergence and utility. Carried out research indicate that the chaotic algorithms are more efficient, for the same values of parameters, in comparison with the corresponding classical algorithms. Practical implications: Problems considered in the paper can arise in some technical issues, for example, in exploring the coal interlayers in looking for the compressed gas reservoirs which can be dangerous for the people’s life and health, in which application of the standard algorithms of computer tomography is impossible for some reasons (like size of the examined object, its localization or its accessibility). Originality/value: In the paper the originally designed by the authors reconstruction algorithms are presented which appear to be more effective than the standard algebraic algorithms adapted for solving problems with the incomplete projection data.
8
Content available Analytic-numerical method of determining the freezing front location
Grzymkowski R., Hetmaniok E., Pleszczyński M.
Archives of Foundry Engineering
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2011
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Vol. 11, iss. 3 spec.
75--80
EN
Mathematical modeling of thermal processes combined with the reversible phase transitions of type: solid phase – liquid phase leads to formulation of the parabolic boundary problems with the moving boundary. Solution of such defined problem requires, most often, to use sophisticated numerical techniques and far advanced mathematical tools. Excellent illustration of the complexity of considered problems, as well as of the variety of approaches used for finding their solutions, gives the papers [1-4]. In the current paper, the authors present the, especially attractive from the engineer point of view, analytic-numerical method for finding the approximate solution of selected class of problems which can be reduced to the one-phase solidification problem of a plate with the unknown a priori, varying in time boundary of the region in which the solution is sought. Proposed method is based on the known formalism of initial expansion of the sought function describing the temperature field into the power series, some coefficients of which are determined with the aid of boundary conditions, and on the approximation of the function defining the location of freezing front with the broken line, parameters of which are numerically determined.
9
Content available remote Application of the homotopy perturbation method for calculation of the temperature distribution in the cast-mould heterogeneous domain
Grzymkowski R., Hetmaniok E., Słota D.
Journal of Achievements in Materials and Manufacturing Engineering
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2010
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Vol. 43, nr 1
299--309
EN
Purpose of this paper: In this paper an application of the new method for solving the heat conduction equation in the heterogeneous cast-mould system, with an assumption of the ideal contact at the cast-mould contact point, is introduced. An example illustrating the discussed approach and confirming its usefulness for solving problems of that kind is also presented in the paper. Design/methodology/approach: For solving the discussed problem the homotopy perturbation method is used, which consists in determining the series convergent to the exact solution or enabling to built the approximate solution of the problem. Findings: The paper shows that the homotopy perturbation method, effective in solving many technical problems, is successful also for examining the considered problem. Research limitations/implications: Solution of the problem is provided with the assumption of an ideal contact between the cast and the mould. In further, research of the discussed method shall be employed to solve problems involving the presence of thermal resistance at the cast-mould contact Practical implications: The method allows to determine the solution in form of the continuous function, which is significant for the analysis of the cast cooling in the mould, in order to avoid the defects formation in the cast. Originality/value: Application of the new method for solving the considered problem.
10
Content available remote Application of the computer tomography in examination of the internal structure of materials by considering the specific conditions of the problem
Grzymkowski R., Hetmaniok E., Pleszczyński M., Zielonka A.
Journal of Achievements in Materials and Manufacturing Engineering
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2010
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Vol. 43, nr 1
288--298
EN
Purpose of this paper: In this paper we present a summary of the results reached in the field of computer tomography applied in some special case – for the problem of incomplete projection data. This particular problem arises in the technical issues in which, for some reasons (like for example size of the examined object, its localization or its accessibility), it is impossible to apply the standard algorithms of computer tomography. Design/methodology/approach: In the paper we discuss the standard algebraic algorithms of computer tomography and, additionally, the new algebraic algorithms (parallel and chaotic), designed by the authors, suitable not only for the case of incomplete projection data but also useful in the standard approach. Findings: The above mentioned algorithms are tested in solving the problems of reconstruction the discrete objects of high-contrast. Moreover, convergence, stability and utility of the algorithms are proved experimentally. Research limitations/implications: Algorithms, created by the authors, are designed for the multiprocessor computers which allow to execute the calculations simultaneously. However, the results compiled in the paper were elaborated by using the one-processor computer. Calculations in which the parallel computing structure will be used are planned for the nearest future.Practical implications: Possibilities of the effective applications of the discussed algorithms in different practical technical problems are showed in the paper. Research, done till now, indicate the chances of applying the proposed algorithms in certain technical problem in which the incomplete projection data appear (like, for example, in searching for the elements in material which cause decreasing of its strength or in looking for the compressed gas reservoirs in the coal bed, which can be dangerous for the people’s life and health). Originality/value: The paper presents the reconstruction algorithms (block and chaotic-bloc), designed by the authors, which appear to be more effective than the standard algebraic algorithms adapted for solving problems with the incomplete projection data.
11
Content available remote Application of the Adomian decomposition method for solving the heat equation in the cast-mould heterogeneous domain
Grzymkowski R., Pleszczyński M., Słota D.
Archives of Foundry Engineering
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2009
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Vol. 9, iss. 4
57-62
EN
The paper is focused on a method for solving the heat equation in a cast-mould heterogeneous domain. The discussed method makes use of the Adomian decomposition method. The derived calculations prove the effectiveness of the method for solving such types of problems.
12
Content available remote 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.
13
Content available remote Numerical method for multi-phase inverse Stefan design problems
Grzymkowski R., Słota D.
Archives of Metallurgy and Materials
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2006
|
Vol. 51, iss. 1
161-172
EN
The inverse problems for differential equations consist of stating the initial conditions, boundary conditions or thermophysical properties of the body. But the insufficiency of input information is compensated by some additional information on the effects of the input conditions. Generally, for the inverse Stefan problem, it is assumed that this additional information is the position of the freezing front, its velocity in normal direction or temperature in selected points of the domain. We may consider the usage of the demanded position of the moving front as the constraint for the cost functional. This kind of problem becomes an inverse design problem. In the paper, the multi-phase inverse Stefan design problems are formulated and described by means of the optimization method. These problems consist of the reconstruction of the function which describes the heat-transfer coefficient, when the positions of the moving interfaces of the phase change are well-known. The method consists of the minimization of a functional, the value of which is the norm of a difference between given position of the moving interface of the phase change and a position reconstructed from the selected function describing the heat-transfer coefficient. In numerical calculations the Nelder-Mead optimization method and the generalized alternating phase truncation method were used.
PL
Modele matematyczne szeregu istotnych zjawisk spotykanych w praktyce prowadzą do różnych typów źle uwarunkowanych zagadnień odwrotnych dla równań fizyki matematycznej, a w szczególności do niepoprawnie postawionych zagadnień dla równania przewodnictwa ciepła. Na ogół zagadnienia te pojawiają się przy próbach odtworzenia przebiegu jakiegoś procesu opisanego zagadnieniem poprawnie postawionym, na podstawie wyników pomiarów, które powinny jednoznacznie określić rozwiązanie, ale nie czynią tego w sposób poprawny. Zagadnienia odwrotne dla równan fizyki matematycznej polegają na określeniu np. warunku początkowego, warunków brzegowych lub parametrów materiału. Brak pewnej części informacji wejściowej, jest uzupełniany dodatkowymi informacjami o konsekwencjach wynikłych z warunków wejściowych. Dla odwrotnego zagadnienia Stefana dodatkowa informacja jest znajomość położenia granicy rozdziału faz, jej predkości w kierunku normalnym lub temperatury w wybranych punktach obszaru. W przypadku, gdy do budowy funkcji celu wykorzystamy zadane położenie granicy rozdziału faz, to tego typu zadanie nosi nazwę odwrotnego zagadnienia projektowego. W pracy będziemy rozważać wielofazowe odwrotne projektowe zagadnienie Stefana, w których dodatkową informacją są położenia granic rozdziału faz. W opisywanej metodzie dobierana będzie wartość (zmiennego w czasie) współczynnika wnikania ciepła, tak aby zminimalizować funkcjonał, którego wartością jest norma różnicy między zadanym położeniem granicy rozdziału faz i położeniem odtworzonym dla wybranego współczynnika wnikania ciepła. W obliczeniach numerycznych wykorzystano metodę optymalizacji Neldera-Meada oraz uogólnioną metodę przemiennej fazy.
14
Content available remote Substitute thermal capacity in two - phase zone for an eutectic Al alloy
Suchy J., Grzymkowski R.
Krzepnięcie Metali i Stopów
|
1998
|
nr 38
67--70
PL
W tym artykule pokazano kolejne rezultaty odtwarzania rozkładu zastępczej pojemności cieplnej w strefie dwufazowej stopów okołoeutektycznych na bazie aluminium. Zastosowano zmienną temperaturę zalewania, kształtującą warunki wymiany ciepła w odlewie. Wyniki tych badań dostarczają informacji użytkowych do symulacji krzepnięcia i krystalizacji stopów odlewniczych, w tym również kompozytów in situ.
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