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Application of the inverse method to estimate the time-varying heat sources on infinite cylinders.
|關鍵字:||inverse problem;逆問題;heat conduction;infinite cylinders;heat source;熱傳導;細長圓柱;熱源||出版社:||應用數學系所||引用:|| 吳信毅，以混合逆算法配合實驗數據探討逆向熱傳導問題，博士論文，成功大學，2006。  張浮明，應用微分值積法(DQM)探討逆向熱傳導問題，博士論文，中興大學，2008。 英文參考文獻  O.M. Alifanov, E.A. Artyukhin and S.V. Rumyantsev, Extreme Method for Ill-posed Problems with Applications to Inverse Heat Transfer Problem, Begell house, Inc NY, 1995.  A.N. Tikhnov and V.Y. Arsenin, Solution of Ill-posed Problems, Winston and Washington DC, 1977.  J. V. Beck, B Blackwell and C.R. St. Clair, Inverse Heat Conduction ill-posed Problems, Wiley, NY, 1985.  M. N. Ozisik, Boundary Value Problem of Heat Conduction, Dover Publication, INC, NY, 1989.  M. N. Ozisik, Heat conduction, , John Wiley, and Sons, NY, 1993.  D. N. Ghosh Roy, Methods of Inverse Problems in Physics, CRC press, 1991.  C.R. Vogel, Computational Methods for Inverse Problems, SIAM, 2002.  O.M. Alifanov, Inverse Heat Transfer Problems, Springer – Verlag, NY, 1994.  E. Hensel, Inverse Theory and Applications for Engineers, Prentice-Hall, Englewood Cliffs. NJ. 1991.  V. A. Morozov and M stessin, Regularization Methods for Ill-Posed Problems, CRC Press, Boca Raton, FL. 1993.  O. M. Alifanov and V. V. Mikhailov, Solution of the Non-linear Inverse Thermal Conductivity Problem by the Iteration Method, J. of Engineering Physics, 35, pp. 1501-1506, 1978.  O. M. Alifanov, Solution of an Inverse Problem of Heat Conduction by Iteration Method, Journal of Engineering Physics, 25, pp. 471-476, 1974.  R. G. Arledge and A. Haji-Sheikh, An Iterative Approach to The Solution of Inverse Heat Conduction Problems, Numerical Transfer, 1, pp . 365-376, 1978.  N. D’Souza, Numerical Solution of One-dimensional Inverse Transient Heat Conduction by Finite Difference Method, ASME J. Heat Transfer, No. 75-WA/HT-81, 1975.  J. V. Beck and H. Wolf, The Non-linear Inverse Heat Conduction Problem, ASME J. Heat Transfer, No. 65-HT-40 , 1965.  T. T. Lam and W. K. Yeung, Inverse Determination of Thermal Conductivity for One-dimensional Problems, J. Thermophys. Heat Transf. 9 (2), pp. 335-344, 1995.  B. R. Bass, Application of the Finite Element Method to the Nonlinear Inverse Heat Conduction Problem Using Beck’s Second Method, ASME J.of Engineering for Industry, 102, pp. 168-176, 1980.  T. Yoshimura and K. Ikuta, Inverse Heat-Conduction Problem by Finite Element Formulation, Int. J. System Sci. 16 , pp. 1365-1376, 1985.  C. K. Hsieh and Alain J. Kassab, A General Method for the Solution of Inverse Heat Conduction Problems with Partially Unknown System Geometries, Int. J. Heat Mass Transfer, 29, No 1, pp 47-58, 1986.  C. L. Chang and M. Chang, Non-iteration Estimation of Thermal Conductivity Using Finite Volume Method, Int. Commun. Heat Mass Transf. 33, pp. 1013-1020, 2006.  C. Y. Yang, Non-iterative Solution of Inverse Heat Conduction Problems in One Dimension. Commun. Number. Methods Eng. 13, p419-429, 1997.  C. Y. Yang, A Linear Inverse Model for the Temperature- Dependent Thermal Conductivity determination in one –dimensional Problems. App1. Math. Modeling, 22, p p.1-9, 1998.  C. Y. Yang, Estimation of the Temperature – Dependent Thermal Conductivity in Inverse Heat Conduction Problems, App1. Math. Modeling, 23, pp. 469-478, 1999.  C. H. Huang and M. N. Ozisik, A Direct Integration Approach for Simultaneously Estimating Spatially Varying Thermal Conductivity and Heat Capacity, J. Heat Fluid Flow 11, pp.262-268,1990.  H. C. Huang and M. N. Ozisik, Conjugate Gradient Method for Determining Unknown Contact Conductance During Metal Casting, Int. J. Heat Mass Transfer 21, pp. 55-70, 1992.  H. T. Chen and S. M. Chang, Application of the Hybird Method to Inverse Heat Conduction Problems, Int. J. Heat Mass Transfer, 33, No.4, pp. 621-628, 1990.  H. T. Chen and J. Y. Lin, Hybird Laplace Transfer technique for non-linear transient thermal problems, Int. J. Heat mass Transfer, 34(4/5), pp. 1301-1308.  R. Bellman, J. Casti, Differential Quadrature and Long Term Integration. J. Math. Anal. Appl. 34, pp. 235-238, 1971.  R. Bellman, and B. G. Kashef and J. Casti, Differential Quadrature：A Technique for the Rapid Solution of Nonlinear Partial Differential Equations, J. Comput. Phys. 10, pp. 40-52, 1972.  M. I. Char, F. P. Chang and B. C. Tai, Inverse Determination of thermal Conductivity by Differential Quadrature Method, Int. Commun. Heat Mass Transfer 35, pp.113-119, 2008.  I. Frank, An Application of Least Squares Method to the Solution of The Inverse Problem of Heat Conduction, J. Heat Transfer, 85C, pp. 378-379, 1963.  N. Al-Khalidy, On The Solution of Parabolic and Hyperbolic Inverse Heat Conduction Problems, Int. J. Heat Mass Transfer, 41, N pp. 3731-3740, 1998.||摘要:||
This paper estimates the heat source of infinite cylinders at different time and in various locations by inverse method. First, equally divide time and space in the formula and discretize them using the second-order finite difference method. Next, substitute into the quantities of the known thermal conductivity, boundary conditions, initial conditions, as well as the measured temperatures of every location and time. However, there are errors in the measured temperatures of every location and time. In this paper, the measured temperatures are the results of adding random errors of 1%, 3%, and 5% to the analytical solution, respectively. Using these noisy measured temperatures to inverse calculate the heat sources results in more error in the estimated values than the exact heat source solution. Consequently, Savitzky-Gollay digital filter method is adopted to smooth the measured temperatures by weight averaging seven or eleven measured points, respectively, prior to substituting the values into the discretized equations for the inverse calculation. The result demonstrates that comparing to the unsmoothed estimated heat source value, the smoothed heat source value is much more precise.
In addition, the research finds that Savitzky-Gollay digital filter method is more effective to estimate the heat source when there are multiple measured points. In the digital filter method, inverse calculation gives more accuracy after smoothing eleven measured points than seven measured points. Nevertheless, if the amount of measured points is not large, smoothing seven measured points would be accurate enough.
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