Method of Minimax Optimization in the Coefficient Inverse Heat-Conduction Problem
Consideration has been given to the inverse problem on identification of a temperature-dependent thermal-conductivity coefficient. The problem was formulated in an extremum statement as a problem of search for a quantity considered as the optimum control of an object with distributed parameters, whi...
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| Veröffentlicht in: | Journal of engineering physics and thermophysics Jg. 89; H. 4; S. 1008 - 1013 |
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| Sprache: | Englisch |
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| Abstract | Consideration has been given to the inverse problem on identification of a temperature-dependent thermal-conductivity coefficient. The problem was formulated in an extremum statement as a problem of search for a quantity considered as the optimum control of an object with distributed parameters, which is described by a nonlinear homogeneous spatially one-dimensional Fourier partial equation with boundary conditions of the second kind. As the optimality criterion, the authors used the error (minimized on the time interval of observation) of uniform approximation of the temperature computed on the object′s model at an assigned point of the segment of variation in the spatial variable to its directly measured value. Pre-parametrization of the sought control action, which a priori records its description accurate to assigning parameters of representation in the class of polynomial temperature functions, ensured the reduction of the problem under study to a problem of parametric optimization. To solve the formulated problem, the authors used an analytical minimax-optimization method taking account of the alternance properties of the sought optimum solutions based on which the algorithm of computation of the optimum values of the sought parameters is reduced to a system (closed for these unknowns) of equations fixing minimax deviations of the calculated values of temperature from those observed on the time interval of identification. The obtained results confirm the efficiency of the proposed method for solution of a certain range of applied problems. The authors have studied the influence of the coordinate of a point of temperature measurement on the exactness of solution of the inverse problem. |
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| AbstractList | Consideration has been given to the inverse problem on identification of a temperature-dependent thermal-conductivity coefficient. The problem was formulated in an extremum statement as a problem of search for a quantity considered as the optimum control of an object with distributed parameters, which is described by a nonlinear homogeneous spatially one-dimensional Fourier partial equation with boundary conditions of the second kind. As the optimality criterion, the authors used the error (minimized on the time interval of observation) of uniform approximation of the temperature computed on the object's model at an assigned point of the segment of variation in the spatial variable to its directly measured value. Pre-parametrization of the sought control action, which a priori records its description accurate to assigning parameters of representation in the class of polynomial temperature functions, ensured the reduction of the problem under study to a problem of parametric optimization. To solve the formulated problem, the authors used an analytical minimax-optimization method taking account of the alternance properties of the sought optimum solutions based on which the algorithm of computation of the optimum values of the sought parameters is reduced to a system (closed for these unknowns) of equations fixing minimax deviations of the calculated values of temperature from those observed on the time interval of identification. The obtained results confirm the efficiency of the proposed method for solution of a certain range of applied problems. The authors have studied the influence of the coordinate of a point of temperature measurement on the exactness of solution of the inverse problem. Consideration has been given to the inverse problem on identification of a temperature-dependent thermal-conductivity coefficient. The problem was formulated in an extremum statement as a problem of search for a quantity considered as the optimum control of an object with distributed parameters, which is described by a nonlinear homogeneous spatially one-dimensional Fourier partial equation with boundary conditions of the second kind. As the optimality criterion, the authors used the error (minimized on the time interval of observation) of uniform approximation of the temperature computed on the object's model at an assigned point of the segment of variation in the spatial variable to its directly measured value. Pre-parametrization of the sought control action, which a priori records its description accurate to assigning parameters of representation in the class of polynomial temperature functions, ensured the reduction of the problem under study to a problem of parametric optimization. To solve the formulated problem, the authors used an analytical minimax-optimization method taking account of the alternance properties of the sought optimum solutions based on which the algorithm of computation of the optimum values of the sought parameters is reduced to a system (closed for these unknowns) of equations fixing minimax deviations of the calculated values of temperature from those observed on the time interval of identification. The obtained results confirm the efficiency of the proposed method for solution of a certain range of applied problems. The authors have studied the influence of the coordinate of a point of temperature measurement on the exactness of solution of the inverse problem. Keywords: coefficient inverse heat-conduction problem, parametric optimization, minimax-optimization method. |
| Audience | Academic |
| Author | Rapoport, É. Ya Diligenskaya, A. N. |
| Author_xml | – sequence: 1 givenname: A. N. surname: Diligenskaya fullname: Diligenskaya, A. N. email: adiligenskaya@mail.ru organization: Samara State Technical University – sequence: 2 givenname: É. Ya surname: Rapoport fullname: Rapoport, É. Ya organization: Samara State Technical University |
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| References | AlifanovOMArtyukhinEARumyantsevSVExtremum Methods of Solution of Ill-Posed Problems and Their Application to Inverse Heat-Transfer Problems [in Russian]1988MoscowNauka0657.35003 StolovichNNMinitskayaNSTemperature Dependences of Thermophysical Properties of Certain Materials [in Russian]1973MinskNauka i Tekhnika TikhonovANArseninVYMethods for Solution of Ill-Posed Problems [in Russian]1979MoscowNauka0499.65030 RapoportEPleshivtsevaYOptimal Control of Induction Heating Processes2007London, New YorkCRC Press, Taylor & Francis Group1308.49022 Yu. Matsevity and S. Lushpenko, An estimation of thermal properties by means of solving internal inverse heat transfer problems, Proc. 2nd Int. Conf. on Inverse Problems in Engineering, June 9–14, 1996, Le Croisic, France; United Engineering Center, New York (1998), pp. 677–685. MorozovVARegular Methods of Solution of Ill-Posed Problems [in Russian]1974MoscowIzd. MGU IvanovVKVasinVVTananaVPThe Theory of Linear Ill-Posed Problems and Its Applications [in Russian]1978MoscowNauka0489.65035 DiligenskayaANRapoportÉYAnalytical methods of parametric optimization in inverse heat-conduction problems with internal heat releaseJ. Eng. Phys. Thermophys.20148751126113410.1007/s10891-014-1114-1 RapoportÉYThe Alternance Method in Applied Optimization Problems [in Russian]2000MoscowNauka1050.90570 AlifanovOMIdentification of the Processes of Heat Transfer of Aircraft and Spacecraft [in Russian]1979MoscowMashinostroenie É. Ya. Rapoport and Ye. É. Pleshivtseva, Special optimization methods in inverse heat-conduction problems, Izv. Ross. Akad. Nauk, Énergetika, No. 5, 144–155 (2002). ButkovskiiAGMalyiSAAndreevYNOptimum Control of Heating of a Metal [in Russian]1972MoscowMetallurgiya ButkovskiiAGMethods for Control of Systems with Distributed Parameters [in Russian]1975MoscowNauka Yu. M. Matsevityi, Inverse Heat-Transfer Problems, in 2 Volumes [in Russian], Naukova Dumka, Kiev (2002). Ye. É. Pleshivtseva and É. Ya. Rapoport, Method of successive parameterization of control actions in boundary-value problems of optimum control of systems with distributed parameters, Izv. Ross. Akad. Nauk, Teor. Sistemy Upravl., No. 3, 22–33 (2009). AlifanovOMInverse Heat-Transfer Problems [in Russian]1988MoscowMashinostroenie0979.80003 AG Butkovskii (1462_CR9) 1972 ÉY Rapoport (1462_CR14) 2000 OM Alifanov (1462_CR1) 1988 1462_CR12 AN Diligenskaya (1462_CR13) 2014; 87 E Rapoport (1462_CR16) 2007 VA Morozov (1462_CR7) 1974 OM Alifanov (1462_CR5) 1979 VK Ivanov (1462_CR8) 1978 1462_CR2 1462_CR3 AN Tikhonov (1462_CR6) 1979 OM Alifanov (1462_CR4) 1988 AG Butkovskii (1462_CR10) 1975 1462_CR11 NN Stolovich (1462_CR15) 1973 |
| References_xml | – reference: MorozovVARegular Methods of Solution of Ill-Posed Problems [in Russian]1974MoscowIzd. MGU – reference: AlifanovOMInverse Heat-Transfer Problems [in Russian]1988MoscowMashinostroenie0979.80003 – reference: Ye. É. Pleshivtseva and É. Ya. Rapoport, Method of successive parameterization of control actions in boundary-value problems of optimum control of systems with distributed parameters, Izv. Ross. Akad. Nauk, Teor. Sistemy Upravl., No. 3, 22–33 (2009). – reference: AlifanovOMArtyukhinEARumyantsevSVExtremum Methods of Solution of Ill-Posed Problems and Their Application to Inverse Heat-Transfer Problems [in Russian]1988MoscowNauka0657.35003 – reference: RapoportÉYThe Alternance Method in Applied Optimization Problems [in Russian]2000MoscowNauka1050.90570 – reference: TikhonovANArseninVYMethods for Solution of Ill-Posed Problems [in Russian]1979MoscowNauka0499.65030 – reference: ButkovskiiAGMalyiSAAndreevYNOptimum Control of Heating of a Metal [in Russian]1972MoscowMetallurgiya – reference: StolovichNNMinitskayaNSTemperature Dependences of Thermophysical Properties of Certain Materials [in Russian]1973MinskNauka i Tekhnika – reference: Yu. Matsevity and S. Lushpenko, An estimation of thermal properties by means of solving internal inverse heat transfer problems, Proc. 2nd Int. Conf. on Inverse Problems in Engineering, June 9–14, 1996, Le Croisic, France; United Engineering Center, New York (1998), pp. 677–685. – reference: ButkovskiiAGMethods for Control of Systems with Distributed Parameters [in Russian]1975MoscowNauka – reference: RapoportEPleshivtsevaYOptimal Control of Induction Heating Processes2007London, New YorkCRC Press, Taylor & Francis Group1308.49022 – reference: Yu. M. Matsevityi, Inverse Heat-Transfer Problems, in 2 Volumes [in Russian], Naukova Dumka, Kiev (2002). – reference: DiligenskayaANRapoportÉYAnalytical methods of parametric optimization in inverse heat-conduction problems with internal heat releaseJ. Eng. Phys. Thermophys.20148751126113410.1007/s10891-014-1114-1 – reference: AlifanovOMIdentification of the Processes of Heat Transfer of Aircraft and Spacecraft [in Russian]1979MoscowMashinostroenie – reference: IvanovVKVasinVVTananaVPThe Theory of Linear Ill-Posed Problems and Its Applications [in Russian]1978MoscowNauka0489.65035 – reference: É. Ya. Rapoport and Ye. É. Pleshivtseva, Special optimization methods in inverse heat-conduction problems, Izv. Ross. Akad. Nauk, Énergetika, No. 5, 144–155 (2002). – volume-title: Temperature Dependences of Thermophysical Properties of Certain Materials [in Russian] year: 1973 ident: 1462_CR15 – volume-title: Identification of the Processes of Heat Transfer of Aircraft and Spacecraft [in Russian] year: 1979 ident: 1462_CR5 – volume-title: Optimal Control of Induction Heating Processes year: 2007 ident: 1462_CR16 – volume-title: Extremum Methods of Solution of Ill-Posed Problems and Their Application to Inverse Heat-Transfer Problems [in Russian] year: 1988 ident: 1462_CR4 – volume-title: Methods for Control of Systems with Distributed Parameters [in Russian] year: 1975 ident: 1462_CR10 – volume-title: The Theory of Linear Ill-Posed Problems and Its Applications [in Russian] year: 1978 ident: 1462_CR8 – ident: 1462_CR3 – volume: 87 start-page: 1126 issue: 5 year: 2014 ident: 1462_CR13 publication-title: J. Eng. Phys. Thermophys. doi: 10.1007/s10891-014-1114-1 – volume-title: The Alternance Method in Applied Optimization Problems [in Russian] year: 2000 ident: 1462_CR14 – ident: 1462_CR2 – volume-title: Optimum Control of Heating of a Metal [in Russian] year: 1972 ident: 1462_CR9 – volume-title: Regular Methods of Solution of Ill-Posed Problems [in Russian] year: 1974 ident: 1462_CR7 – ident: 1462_CR12 – volume-title: Inverse Heat-Transfer Problems [in Russian] year: 1988 ident: 1462_CR1 – volume-title: Methods for Solution of Ill-Posed Problems [in Russian] year: 1979 ident: 1462_CR6 – ident: 1462_CR11 |
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| SubjectTerms | Analysis Classical Mechanics Coefficients Complex Systems Electric properties Engineering Engineering Thermodynamics Heat and Mass Transfer Industrial Chemistry/Chemical Engineering Intervals Inverse problems Mathematical analysis Mathematical models Methods Minimax technique Optimization Parameters Thermal conductivity Thermodynamics |
| Title | Method of Minimax Optimization in the Coefficient Inverse Heat-Conduction Problem |
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