Distributed parameter thermal controllability: a numerical method for solving the inverse heat conduction problem
This paper addresses the inverse heat conduction problem encountered in thermal manufacturing processes. A numerical control algorithm is developed for distributed parameter conduction systems, based on Galerkin optimization of an energy index employing Green's functions. Various temperature pr...
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| Published in | International journal for numerical methods in engineering Vol. 59; no. 7; pp. 945 - 961 |
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| Main Authors | , |
| Format | Journal Article |
| Language | English |
| Published |
Chichester, UK
John Wiley & Sons, Ltd
21.02.2004
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| Subjects | |
| Online Access | Get full text |
| ISSN | 0029-5981 1097-0207 1097-0207 |
| DOI | 10.1002/nme.898 |
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| Abstract | This paper addresses the inverse heat conduction problem encountered in thermal manufacturing processes. A numerical control algorithm is developed for distributed parameter conduction systems, based on Galerkin optimization of an energy index employing Green's functions. Various temperature profiles of variable complexity are studied, using the proposed technique, in order to determine the surface heat input distribution necessary to generate the desired temperature field inside a solid body. Furthermore, the effect of altering the iterative time step and duration of processing time, on the convergence of the solution generated by the aforementioned method is investigated. It is proved that despite the variations in numerical processing, the iterative technique is able to solve the problem of inverse heat conduction in the thermal processing of materials. Copyright © 2004 John Wiley & Sons, Ltd. |
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| AbstractList | This paper addresses the inverse heat conduction problem encountered in thermal manufacturing processes. A numerical control algorithm is developed for distributed parameter conduction systems, based on Galerkin optimization of an energy index employing Green's functions. Various temperature profiles of variable complexity are studied, using the proposed technique, in order to determine the surface heat input distribution necessary to generate the desired temperature field inside a solid body. Furthermore, the effect of altering the iterative time step and duration of processing time, on the convergence of the solution generated by the aforementioned method is investigated. It is proved that despite the variations in numerical processing, the iterative technique is able to solve the problem of inverse heat conduction in the thermal processing of materials. This paper addresses the inverse heat conduction problem encountered in thermal manufacturing processes. A numerical control algorithm is developed for distributed parameter conduction systems, based on Galerkin optimization of an energy index employing Green's functions. Various temperature profiles of variable complexity are studied, using the proposed technique, in order to determine the surface heat input distribution necessary to generate the desired temperature field inside a solid body. Furthermore, the effect of altering the iterative time step and duration of processing time, on the convergence of the solution generated by the aforementioned method is investigated. It is proved that despite the variations in numerical processing, the iterative technique is able to solve the problem of inverse heat conduction in the thermal processing of materials. Copyright © 2004 John Wiley & Sons, Ltd. |
| Author | Alaeddine, Marios Doumanidis, Charalabos C. |
| Author_xml | – sequence: 1 givenname: Marios surname: Alaeddine fullname: Alaeddine, Marios email: alae77@yahoo.com organization: Department of Mechanical Engineering, Tufts University, Medford, MA 02155, U.S.A – sequence: 2 givenname: Charalabos C. surname: Doumanidis fullname: Doumanidis, Charalabos C. organization: Nanomanufacturing Program Director, National Science Foundation, Arlington, VA 22230, U.S.A |
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| Cites_doi | 10.1016/S0009-2509(00)00037-3 10.1006/jmaa.1997.5649 10.1016/S0017-9310(00)00212-X 10.1016/S0009-2509(01)00123-3 10.1115/1.3679871 10.1016/B978-0-08-027624-3.50008-2 10.1007/978-3-642-76436-3 10.1016/0017-9310(96)00034-8 10.1299/kikai1938.44.1633 10.2514/3.50211 10.1016/S0009-2509(02)00419-0 10.1080/002071700219614 10.1115/1.3688700 10.2514/3.49417 10.1299/kikaib.54.2893 10.1201/9780429258367 10.1016/0017-9310(95)00229-4 10.1016/S0017-9310(00)00040-5 |
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| References | Kailath T. Linear Systems. Prentice-Hall: Englewood Cliffs, NJ, 1980. Mills AF. Basic Heat and Mass Transfer. Irwin Professional Publishing: Burr Ridge, 1995. Shumakov NV. A method for the experimental study of the process of heating a solid body. Soviet Physics-Technical Physics 1957; 2:771-781 (Translated by American Institute of Physics). Burggraf OR. An exact solution of the inverse problem in heat conduction theory and application. Journal of Heat Transfer 1964; 86:373-382. Lesnic D, Elliott L, Ingham DB. Application of the boundary element method to inverse heat conduction problems. International Journal of Heat and Mass Transfer 1996; 39(7):1503-1517. Bendersky E, Christofides PD. Optimization of transport-reaction processes using nonlinear model reduction. Chemical Engineering Science 2000; 55:4349-4366. Ray WH, Lainiotis DG. Distributed Parameter Systems: Identification, Estimation, and Control. Marcel Dekker: New York, 1978. Armaou A, Christofides PD. Dynamic optimization of dissipative PDE systems using nonlinear order reduction. Chemical Engineering Science 2002; 57:5083-5114. Baker J, Christofides PD. Finite-dimensional approximation and control of nonlinear parabolic PDE systems. International Journal of Control 2000; 73:439-456. Shoji M, Ono N. Application of the boundary element to the inverse problem of heat conduction. Transactions of the Japan Society of Mechanical Engineers 1988; 54(506B) :2893-2900 (in Japanese). Imber M. Temperature extrapolation mechanism for two-dimensional heat flow. AIAA Journal 1974; 12(8):1089-1093. Sparrow EM, Haji-Sheikh A, Lundgren TS. The inverse problem in transient heat conduction. Journal of Heat Transfer 1964; 86:369-375. Imber M, Khan J. Prediction of transient temperature distributions with embedded thermocouples. AIAA Journal 1972; 10(6):784-789. Beck JV, Blackwell B, St.Clair JrAR. Inverse Heat Conduction: Ill-Posed Problems. Wiley: New York, 1985. Monde M. Analytical method in inverse heat transfer problem using Laplace transform technique. International Journal of Heat and Mass Transfer 2000; 43(21):3965-3975. Carslaw HS, Jaeger JC. Conduction of Heat in Solids (2nd edn). Clarendon Press: Oxford, 1959. Chen HT, Lin SY, Fang LC. Estimation of surface temperature in two-dimensional inverse heat conduction problems. International Journal of Heat and Mass Transfer 2001; 44(8):1455-1463. Christofides PD, Daoutidis P. Finite-dimensional control of parabolic PDE systems using approximate inertial manifolds. Journal of Mathematical Analysis and Applications 1997; 216:398-420. Beck JV, Cole KD, Haji-Sheikh A, Litkouhi B. Heat Conduction Using Green's Functions. Hemisphere Publishing Corporation: London, Washington DC, 1992. Alifanov OM. Inverse Heat Transfer Problems. Springer: Berlin, 1994. Beck JV, Blackwell B, Haji-Sheikh A. Comparison of some inverse heat conduction methods using experimental data. International Journal of Heat and Mass Transfer 1996; 39(17):3649-3657. Antoniades C, Christofides PD. Integrating nonlinear output feedback control and optimal actuator/sensor placement for transport-reaction processes. Chemical Engineering Science 2001; 56:4517-4535. Tzafestas SG (ed). Distributed Parameter Control Systems. Pergamon Press: Oxford, 1982. Shoji M. Study of inverse problems of heat conduction. Transactions of the Japan Society of Mechanical Engineers 1978; 44(381) :1633-1643 (in Japanese). Stolz JrG. Numerical solutions to an inverse problem of heat conduction for simple shapes. Journal of Heat Transfer 1960; 82:20-26. 1960; 82 1997; 216 1974; 12 1996; 39 1978; 44 2000; 43 2002; 57 2000; 55 1988; 54 2000; 73 1996 1985 1964; 86 1995 1994 1972; 10 1982 1992 1980 2001; 44 1957; 2 2001; 56 1959 1978 Carslaw HS (e_1_2_1_26_2) 1959 Shumakov NV (e_1_2_1_4_2) 1957; 2 e_1_2_1_22_2 e_1_2_1_23_2 e_1_2_1_20_2 e_1_2_1_21_2 e_1_2_1_27_2 e_1_2_1_24_2 Beck JV (e_1_2_1_15_2) 1985 e_1_2_1_7_2 e_1_2_1_5_2 e_1_2_1_2_2 e_1_2_1_11_2 e_1_2_1_3_2 e_1_2_1_12_2 Kailath T (e_1_2_1_6_2) 1980 e_1_2_1_10_2 Sparrow EM (e_1_2_1_17_2) 1964; 86 e_1_2_1_16_2 e_1_2_1_13_2 Mills AF (e_1_2_1_25_2) 1995 e_1_2_1_19_2 e_1_2_1_8_2 e_1_2_1_9_2 Ray WH (e_1_2_1_14_2) 1978 e_1_2_1_18_2 |
| References_xml | – reference: Alifanov OM. Inverse Heat Transfer Problems. Springer: Berlin, 1994. – reference: Imber M. Temperature extrapolation mechanism for two-dimensional heat flow. AIAA Journal 1974; 12(8):1089-1093. – reference: Antoniades C, Christofides PD. Integrating nonlinear output feedback control and optimal actuator/sensor placement for transport-reaction processes. Chemical Engineering Science 2001; 56:4517-4535. – reference: Lesnic D, Elliott L, Ingham DB. Application of the boundary element method to inverse heat conduction problems. International Journal of Heat and Mass Transfer 1996; 39(7):1503-1517. – reference: Imber M, Khan J. Prediction of transient temperature distributions with embedded thermocouples. AIAA Journal 1972; 10(6):784-789. – reference: Tzafestas SG (ed). Distributed Parameter Control Systems. Pergamon Press: Oxford, 1982. – reference: Carslaw HS, Jaeger JC. Conduction of Heat in Solids (2nd edn). Clarendon Press: Oxford, 1959. – reference: Shoji M. Study of inverse problems of heat conduction. Transactions of the Japan Society of Mechanical Engineers 1978; 44(381) :1633-1643 (in Japanese). – reference: Beck JV, Blackwell B, Haji-Sheikh A. Comparison of some inverse heat conduction methods using experimental data. International Journal of Heat and Mass Transfer 1996; 39(17):3649-3657. – reference: Bendersky E, Christofides PD. Optimization of transport-reaction processes using nonlinear model reduction. Chemical Engineering Science 2000; 55:4349-4366. – reference: Mills AF. Basic Heat and Mass Transfer. Irwin Professional Publishing: Burr Ridge, 1995. – reference: Shoji M, Ono N. Application of the boundary element to the inverse problem of heat conduction. Transactions of the Japan Society of Mechanical Engineers 1988; 54(506B) :2893-2900 (in Japanese). – reference: Beck JV, Blackwell B, St.Clair JrAR. Inverse Heat Conduction: Ill-Posed Problems. Wiley: New York, 1985. – reference: Armaou A, Christofides PD. Dynamic optimization of dissipative PDE systems using nonlinear order reduction. Chemical Engineering Science 2002; 57:5083-5114. – reference: Kailath T. Linear Systems. Prentice-Hall: Englewood Cliffs, NJ, 1980. – reference: Burggraf OR. An exact solution of the inverse problem in heat conduction theory and application. Journal of Heat Transfer 1964; 86:373-382. – reference: Baker J, Christofides PD. Finite-dimensional approximation and control of nonlinear parabolic PDE systems. International Journal of Control 2000; 73:439-456. – reference: Beck JV, Cole KD, Haji-Sheikh A, Litkouhi B. Heat Conduction Using Green's Functions. Hemisphere Publishing Corporation: London, Washington DC, 1992. – reference: Christofides PD, Daoutidis P. Finite-dimensional control of parabolic PDE systems using approximate inertial manifolds. Journal of Mathematical Analysis and Applications 1997; 216:398-420. – reference: Chen HT, Lin SY, Fang LC. Estimation of surface temperature in two-dimensional inverse heat conduction problems. International Journal of Heat and Mass Transfer 2001; 44(8):1455-1463. – reference: Ray WH, Lainiotis DG. Distributed Parameter Systems: Identification, Estimation, and Control. Marcel Dekker: New York, 1978. – reference: Monde M. Analytical method in inverse heat transfer problem using Laplace transform technique. International Journal of Heat and Mass Transfer 2000; 43(21):3965-3975. – reference: Shumakov NV. A method for the experimental study of the process of heating a solid body. Soviet Physics-Technical Physics 1957; 2:771-781 (Translated by American Institute of Physics). – reference: Stolz JrG. Numerical solutions to an inverse problem of heat conduction for simple shapes. Journal of Heat Transfer 1960; 82:20-26. – reference: Sparrow EM, Haji-Sheikh A, Lundgren TS. The inverse problem in transient heat conduction. Journal of Heat Transfer 1964; 86:369-375. – year: 1985 – volume: 43 start-page: 3965 issue: 21 year: 2000 end-page: 3975 article-title: Analytical method in inverse heat transfer problem using Laplace transform technique publication-title: International Journal of Heat and Mass Transfer – volume: 57 start-page: 5083 year: 2002 end-page: 5114 article-title: Dynamic optimization of dissipative PDE systems using nonlinear order reduction publication-title: Chemical Engineering Science – volume: 2 start-page: 771 year: 1957 end-page: 781 article-title: A method for the experimental study of the process of heating a solid body publication-title: Soviet Physics—Technical Physics – volume: 54 start-page: 2893 issue: 506B year: 1988 end-page: 2900 article-title: Application of the boundary element to the inverse problem of heat conduction publication-title: Transactions of the Japan Society of Mechanical Engineers – year: 1992 – year: 1994 – volume: 12 start-page: 1089 issue: 8 year: 1974 end-page: 1093 article-title: Temperature extrapolation mechanism for two‐dimensional heat flow publication-title: AIAA Journal – start-page: 632 year: 1996 end-page: 636 – volume: 55 start-page: 4349 year: 2000 end-page: 4366 article-title: Optimization of transport‐reaction processes using nonlinear model reduction publication-title: Chemical Engineering Science – volume: 82 start-page: 20 year: 1960 end-page: 26 article-title: Numerical solutions to an inverse problem of heat conduction for simple shapes publication-title: Journal of Heat Transfer – year: 1959 – year: 1982 – volume: 44 start-page: 1633 issue: 381 year: 1978 end-page: 1643 article-title: Study of inverse problems of heat conduction publication-title: Transactions of the Japan Society of Mechanical Engineers – volume: 73 start-page: 439 year: 2000 end-page: 456 article-title: Finite‐dimensional approximation and control of nonlinear parabolic PDE systems publication-title: International Journal of Control – volume: 39 start-page: 3649 issue: 17 year: 1996 end-page: 3657 article-title: Comparison of some inverse heat conduction methods using experimental data publication-title: International Journal of Heat and Mass Transfer – year: 1980 – volume: 10 start-page: 784 issue: 6 year: 1972 end-page: 789 article-title: Prediction of transient temperature distributions with embedded thermocouples publication-title: AIAA Journal – volume: 39 start-page: 1503 issue: 7 year: 1996 end-page: 1517 article-title: Application of the boundary element method to inverse heat conduction problems publication-title: International Journal of Heat and Mass Transfer – volume: 86 start-page: 373 year: 1964 end-page: 382 article-title: An exact solution of the inverse problem in heat conduction theory and application publication-title: Journal of Heat Transfer – volume: 86 start-page: 369 year: 1964 end-page: 375 article-title: The inverse problem in transient heat conduction publication-title: Journal of Heat Transfer – year: 1995 – volume: 44 start-page: 1455 issue: 8 year: 2001 end-page: 1463 article-title: Estimation of surface temperature in two‐dimensional inverse heat conduction problems publication-title: International Journal of Heat and Mass Transfer – year: 1978 – volume: 216 start-page: 398 year: 1997 end-page: 420 article-title: Finite‐dimensional control of parabolic PDE systems using approximate inertial manifolds publication-title: Journal of Mathematical Analysis and Applications – volume: 56 start-page: 4517 year: 2001 end-page: 4535 article-title: Integrating nonlinear output feedback control and optimal actuator/sensor placement for transport‐reaction processes publication-title: Chemical Engineering Science – volume: 86 start-page: 369 year: 1964 ident: e_1_2_1_17_2 article-title: The inverse problem in transient heat conduction publication-title: Journal of Heat Transfer – ident: e_1_2_1_9_2 doi: 10.1016/S0009-2509(00)00037-3 – ident: e_1_2_1_11_2 doi: 10.1006/jmaa.1997.5649 – volume-title: Linear Systems year: 1980 ident: e_1_2_1_6_2 – ident: e_1_2_1_24_2 doi: 10.1016/S0017-9310(00)00212-X – ident: e_1_2_1_12_2 doi: 10.1016/S0009-2509(01)00123-3 – ident: e_1_2_1_5_2 doi: 10.1115/1.3679871 – ident: e_1_2_1_13_2 doi: 10.1016/B978-0-08-027624-3.50008-2 – ident: e_1_2_1_2_2 doi: 10.1007/978-3-642-76436-3 – volume: 2 start-page: 771 year: 1957 ident: e_1_2_1_4_2 article-title: A method for the experimental study of the process of heating a solid body publication-title: Soviet Physics—Technical Physics – ident: e_1_2_1_3_2 doi: 10.1016/0017-9310(96)00034-8 – ident: e_1_2_1_18_2 doi: 10.1299/kikai1938.44.1633 – ident: e_1_2_1_20_2 doi: 10.2514/3.50211 – volume-title: Basic Heat and Mass Transfer year: 1995 ident: e_1_2_1_25_2 – ident: e_1_2_1_7_2 – ident: e_1_2_1_8_2 doi: 10.1016/S0009-2509(02)00419-0 – ident: e_1_2_1_10_2 doi: 10.1080/002071700219614 – volume-title: Conduction of Heat in Solids year: 1959 ident: e_1_2_1_26_2 – ident: e_1_2_1_16_2 doi: 10.1115/1.3688700 – ident: e_1_2_1_21_2 doi: 10.2514/3.49417 – volume-title: Distributed Parameter Systems: Identification, Estimation, and Control year: 1978 ident: e_1_2_1_14_2 – ident: e_1_2_1_19_2 doi: 10.1299/kikaib.54.2893 – ident: e_1_2_1_27_2 doi: 10.1201/9780429258367 – ident: e_1_2_1_23_2 doi: 10.1016/0017-9310(95)00229-4 – volume-title: Inverse Heat Conduction: Ill‐Posed Problems year: 1985 ident: e_1_2_1_15_2 – ident: e_1_2_1_22_2 doi: 10.1016/S0017-9310(00)00040-5 |
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| Title | Distributed parameter thermal controllability: a numerical method for solving the inverse heat conduction problem |
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