A method of optimal control is presented as a numerical tool for solving the sea ice heat transfer problem governed by a parabolic partial differential equation. Taken the deviation between the calculated ice temperature and the measurements as the performance criterion, an optimal control model of distributed parameter systems with specific constraints of thermal properties of sea ice was proposed to determine the thermal diffusivity of sea ice. Based on sea ice physical processes, the parameterization of the thermal diffusivity was derived through field data. The simulation results illustrated that the identified parameterization of the thermal diffusivity is reasonably effective in sea ice thermodynamics. The direct relation between the thermal diffusivity of sea ice and ice porosity is physically significant and can considerably reduce the computational errors. The successful application of this method also explained that the optimal control model of distributed parameter systems in conjunction with the engineering background has great potential in dealing with practical problems.
J. Appl. Math.
2013(SI17):
1-8
(2013).
DOI: 10.1155/2013/760378
Y. Gao, “Demyanov difference of two sets and optimality conditions of Lagrange multiplier type for constrained quasidifferentiable optimization,” Journal of Optimization Theory and Applications, vol. 104, no. 2, pp. 377–394, 2000. MR1752323 10.1023/A:1004613814084 Y. Gao, “Demyanov difference of two sets and optimality conditions of Lagrange multiplier type for constrained quasidifferentiable optimization,” Journal of Optimization Theory and Applications, vol. 104, no. 2, pp. 377–394, 2000. MR1752323 10.1023/A:1004613814084
J. Chen and M. Gerdts, “Smoothing technique of nonsmooth Newton methods for control-state constrained optimal control problems,” SIAM Journal on Numerical Analysis, vol. 50, no. 4, pp. 1982–2011, 2012. MR3022207 1252.49045 10.1137/110822177 J. Chen and M. Gerdts, “Smoothing technique of nonsmooth Newton methods for control-state constrained optimal control problems,” SIAM Journal on Numerical Analysis, vol. 50, no. 4, pp. 1982–2011, 2012. MR3022207 1252.49045 10.1137/110822177
K. A. Hoo and D. Zheng, “Low-order control-relevant models for a class of distributed parameter systems,” Chemical Engineering Science, vol. 56, no. 23, pp. 6683–6710, 2001. K. A. Hoo and D. Zheng, “Low-order control-relevant models for a class of distributed parameter systems,” Chemical Engineering Science, vol. 56, no. 23, pp. 6683–6710, 2001.
N. Beniich, A. El Bouhtouri, and D. Dochain, “Input constrained adaptive tracking for a nonlinear distributed parameter tubular reactor,” International Journal of Adaptive Control and Signal Processing, vol. 24, no. 4, pp. 249–260, 2010. MR2656218 1193.93117 10.1002/acs.1127 N. Beniich, A. El Bouhtouri, and D. Dochain, “Input constrained adaptive tracking for a nonlinear distributed parameter tubular reactor,” International Journal of Adaptive Control and Signal Processing, vol. 24, no. 4, pp. 249–260, 2010. MR2656218 1193.93117 10.1002/acs.1127
H. Li, J. Liu, C. P. Chen, and H. Deng, “A simple model-based approach for fluid dispensing analysis and control,” IEEE/ASME Transactions on Mechatronics, vol. 12, no. 4, pp. 491–503, 2007. H. Li, J. Liu, C. P. Chen, and H. Deng, “A simple model-based approach for fluid dispensing analysis and control,” IEEE/ASME Transactions on Mechatronics, vol. 12, no. 4, pp. 491–503, 2007.
M. A. Demetriou, “Integrated actuator-sensor placement and hybrid controller design of flexible structures under worst case spatiotemporal disturbance variations,” Journal of Intelligent Material Systems and Structures, vol. 15, no. 12, pp. 901–921, 2004. M. A. Demetriou, “Integrated actuator-sensor placement and hybrid controller design of flexible structures under worst case spatiotemporal disturbance variations,” Journal of Intelligent Material Systems and Structures, vol. 15, no. 12, pp. 901–921, 2004.
Y. Sakawa, “Solution of an optimal control problem in a distributed-parameter system,” IEEE Transactions on Automatic Control, vol. 9, pp. 420–426, 1964. MR0179022 10.1109/TAC.1964.1105753 Y. Sakawa, “Solution of an optimal control problem in a distributed-parameter system,” IEEE Transactions on Automatic Control, vol. 9, pp. 420–426, 1964. MR0179022 10.1109/TAC.1964.1105753
G. M. Bahaa, “Optimality conditions for infinite order distributed parabolic systems with multiple time delays given in integral form,” Journal of Applied Mathematics, vol. 2012, Article ID 672947, 25 pages, 2012. MR2959976 1252.49031 G. M. Bahaa, “Optimality conditions for infinite order distributed parabolic systems with multiple time delays given in integral form,” Journal of Applied Mathematics, vol. 2012, Article ID 672947, 25 pages, 2012. MR2959976 1252.49031
G. Wang, “Optimal control of parabolic differential equations with two point boundary state constraints,” SIAM Journal on Control and Optimization, vol. 38, no. 5, pp. 1639–1654, 2000. MR1766434 0963.49018 10.1137/S0363012998338132 G. Wang, “Optimal control of parabolic differential equations with two point boundary state constraints,” SIAM Journal on Control and Optimization, vol. 38, no. 5, pp. 1639–1654, 2000. MR1766434 0963.49018 10.1137/S0363012998338132
H. Li and C. Qi, “Modeling of distributed parameter systems for applications–-a synthesized review from time-space separation,” Journal of Process Control, vol. 20, no. 8, pp. 891–901, 2010. H. Li and C. Qi, “Modeling of distributed parameter systems for applications–-a synthesized review from time-space separation,” Journal of Process Control, vol. 20, no. 8, pp. 891–901, 2010.
W. Lv, E. Feng, and Z. Li, “A coupled thermodynamic system of sea ice and its parameter identification,” Applied Mathematical Modelling, vol. 32, no. 7, pp. 1198–1207, 2008. MR2409376 1187.80024 10.1016/j.apm.2007.03.006 W. Lv, E. Feng, and Z. Li, “A coupled thermodynamic system of sea ice and its parameter identification,” Applied Mathematical Modelling, vol. 32, no. 7, pp. 1198–1207, 2008. MR2409376 1187.80024 10.1016/j.apm.2007.03.006
H. Fang, J. Wang, E. Feng, and Z. Li, “Parameter identification and application of a distributed parameter coupled system with a movable inner boundary,” Computers & Mathematics with Applications, vol. 62, no. 11, pp. 4015–4020, 2011. MR2859957 1236.86003 H. Fang, J. Wang, E. Feng, and Z. Li, “Parameter identification and application of a distributed parameter coupled system with a movable inner boundary,” Computers & Mathematics with Applications, vol. 62, no. 11, pp. 4015–4020, 2011. MR2859957 1236.86003
B. Tan, P. Lu, Z. Li, E. Feng, and J. Wang, “Simulation and domain identification of sea ice thermodynamic system,” Journal of Applied Mathematics, vol. 2012, Article ID 532870, 16 pages, 2012. B. Tan, P. Lu, Z. Li, E. Feng, and J. Wang, “Simulation and domain identification of sea ice thermodynamic system,” Journal of Applied Mathematics, vol. 2012, Article ID 532870, 16 pages, 2012.
Y. Bai, Z. Li, P. Lu, L. Zhang, and E. Feng, “Optimization and identification on current drag coefficients of isolated ice floe based on experimental data,” Journal of Dalian University of Technology, vol. 47, no. 6, pp. 885–889, 2007 (Chinese). Y. Bai, Z. Li, P. Lu, L. Zhang, and E. Feng, “Optimization and identification on current drag coefficients of isolated ice floe based on experimental data,” Journal of Dalian University of Technology, vol. 47, no. 6, pp. 885–889, 2007 (Chinese).
Y. Bai, Z. Li, M. Han, and P. Lu, “Approaches to revise ice thermal diffusivity from measured time series of temperature in a river,” in Proceedings of the 6th International Symposium on Test and Measurement (ISTM '05), pp. 2980–2986, Dalian, China, 2005. Y. Bai, Z. Li, M. Han, and P. Lu, “Approaches to revise ice thermal diffusivity from measured time series of temperature in a river,” in Proceedings of the 6th International Symposium on Test and Measurement (ISTM '05), pp. 2980–2986, Dalian, China, 2005.
D. J. Pringle, H. Eicken, H. J. Trodahl, and L. G. E. Backstrom, “Thermal conductivity of landfast Antarctic and Arctic sea ice,” Journal of Geophysical Research C, vol. 112, no. 4, Article ID C04017, 2007. D. J. Pringle, H. Eicken, H. J. Trodahl, and L. G. E. Backstrom, “Thermal conductivity of landfast Antarctic and Arctic sea ice,” Journal of Geophysical Research C, vol. 112, no. 4, Article ID C04017, 2007.
M. Leppäranta and T. Manninen, “The brine and gas content of sea ice with attention to low salinities and high temperatures,” Internal Report, Finnish Institute of Marine Research, Helsinki, Finland, 1988. M. Leppäranta and T. Manninen, “The brine and gas content of sea ice with attention to low salinities and high temperatures,” Internal Report, Finnish Institute of Marine Research, Helsinki, Finland, 1988.
Z. Li, D. S. Sodhi, and P. Lu, “Distribution of ice engineering design criteria of Bohai,” Engineering Mechanics, vol. 23, no. 6, pp. 167–172, 2006 (Chinese). Z. Li, D. S. Sodhi, and P. Lu, “Distribution of ice engineering design criteria of Bohai,” Engineering Mechanics, vol. 23, no. 6, pp. 167–172, 2006 (Chinese).
Z. Li, W. Peng, T. Zhang, M. Lu, and S. Gao, “Effect of porosity on the uniaxial compressive strength of ice in Liaodong Gulf,” in Proceedings of the 10th International Symposium on Okhotsk Sea, Sea Ice and People, Abstracts, pp. 358–363, Hokkaido, Japan, 1995. Z. Li, W. Peng, T. Zhang, M. Lu, and S. Gao, “Effect of porosity on the uniaxial compressive strength of ice in Liaodong Gulf,” in Proceedings of the 10th International Symposium on Okhotsk Sea, Sea Ice and People, Abstracts, pp. 358–363, Hokkaido, Japan, 1995.
C. Petrich, P. J. Langhorne, and Z. F. Sun, “Modelling the interrelationships between permeability, effective porosity and total porosity in sea ice,” Cold Regions Science and Technology, vol. 44, no. 2, pp. 131–144, 2006. C. Petrich, P. J. Langhorne, and Z. F. Sun, “Modelling the interrelationships between permeability, effective porosity and total porosity in sea ice,” Cold Regions Science and Technology, vol. 44, no. 2, pp. 131–144, 2006.
D. M. Cole, “The microstructure of ice and its influence on mechanical properties,” Engineering Fracture Mechanics, vol. 68, no. 17-18, pp. 1797–1822, 2001. D. M. Cole, “The microstructure of ice and its influence on mechanical properties,” Engineering Fracture Mechanics, vol. 68, no. 17-18, pp. 1797–1822, 2001.
A. J. Semtner, “A model for the thermodynamic growth of sea ice in numerical investigations of climate,” Journal of Physical Oceanography, vol. 6, no. 3, pp. 379–389, 1976. A. J. Semtner, “A model for the thermodynamic growth of sea ice in numerical investigations of climate,” Journal of Physical Oceanography, vol. 6, no. 3, pp. 379–389, 1976.
G. F. N. Cox and W. F. Weeks, “Equations for determining the gas and brine volumes in sea-ice samples,” Journal of Glaciology, vol. 29, no. 102, pp. 306–316, 1983. G. F. N. Cox and W. F. Weeks, “Equations for determining the gas and brine volumes in sea-ice samples,” Journal of Glaciology, vol. 29, no. 102, pp. 306–316, 1983.
D. L. Anderson and W. F. Weeks, “A theoretical analysis of sea ice strength,” Transactions, American Geophysical Union, vol. 39, no. 4, pp. 632–640, 1958. D. L. Anderson and W. F. Weeks, “A theoretical analysis of sea ice strength,” Transactions, American Geophysical Union, vol. 39, no. 4, pp. 632–640, 1958.
W. F. Weeks and A. Assur, The Mechanical Properties of Sea Ice, Part II, Section C3, V.S. Cold Regions Research and Engineering Laboratory, Cold Regions Science and Engineering, Hanover, Germany, 1967. W. F. Weeks and A. Assur, The Mechanical Properties of Sea Ice, Part II, Section C3, V.S. Cold Regions Research and Engineering Laboratory, Cold Regions Science and Engineering, Hanover, Germany, 1967.
H. D. Gans and P. R. Woodmansee, “An improved method of modeling porous materials using finite elements,” Computers and Structures, vol. 44, no. 5, pp. 1055–1063, 1992. H. D. Gans and P. R. Woodmansee, “An improved method of modeling porous materials using finite elements,” Computers and Structures, vol. 44, no. 5, pp. 1055–1063, 1992.
P. O. Moslet, “Field testing of uniaxial compression strength of columnar sea ice,” Cold Regions Science and Technology, vol. 48, no. 1, pp. 1–14, 2007. P. O. Moslet, “Field testing of uniaxial compression strength of columnar sea ice,” Cold Regions Science and Technology, vol. 48, no. 1, pp. 1–14, 2007.
L. Shi, Y. Bai, Z. Li, B. Cheng, and M. Leppäranta, “Preliminary results on relationship between thermal diffusivity and porosity of sea ice in the Antarctic,” Chinese Journal of Polar Science, vol. 20, no. 1, pp. 72–80, 2009. L. Shi, Y. Bai, Z. Li, B. Cheng, and M. Leppäranta, “Preliminary results on relationship between thermal diffusivity and porosity of sea ice in the Antarctic,” Chinese Journal of Polar Science, vol. 20, no. 1, pp. 72–80, 2009.
J. Nocedal and S. J. Wright, Numerical Optimization, Springer Series in Operations Research and Financial Engineering, Springer, New York, NY, USA, 1999. MR1713114 J. Nocedal and S. J. Wright, Numerical Optimization, Springer Series in Operations Research and Financial Engineering, Springer, New York, NY, USA, 1999. MR1713114
Y. J. Cao and Q. H. Wu, “Convergence analysis of adaptive genetic algorithms,” in Proceedings of the 2nd International Conference on Genetic Algorithms in Engineering Systems: Innovations and Applications (GALESIA '97), No. 446, pp. 85–89, September 1997. Y. J. Cao and Q. H. Wu, “Convergence analysis of adaptive genetic algorithms,” in Proceedings of the 2nd International Conference on Genetic Algorithms in Engineering Systems: Innovations and Applications (GALESIA '97), No. 446, pp. 85–89, September 1997.
M. Srinivas and L. M. Patnaik, “Adaptive probabilities of crossover and mutation in genetic algorithms,” IEEE Transactions on Systems, Man and Cybernetics, vol. 24, no. 4, pp. 656–667, 1994. M. Srinivas and L. M. Patnaik, “Adaptive probabilities of crossover and mutation in genetic algorithms,” IEEE Transactions on Systems, Man and Cybernetics, vol. 24, no. 4, pp. 656–667, 1994.