Comparison of Three Versions of Conjugate Gradient Method in Predicting an Unknown Irregular Boundary Profile
An inverse geometry problem is solved to predict an
unknown irregular boundary profile. The aim is to minimize the
objective function, which is the difference between real and
computed temperatures, using three different versions of Conjugate
Gradient Method. The gradient of the objective function, considered
necessary in this method, obtained as a result of solving the adjoint
equation. The abilities of three versions of Conjugate Gradient
Method in predicting the boundary profile are compared using a
numerical algorithm based on the method. The predicted shapes show
that due to its convergence rate and accuracy of predicted values, the
Powell-Beale version of the method is more effective than the
Fletcher-Reeves and Polak –Ribiere versions.
[1] C. H. Huang, S.P. Wang, "A three-dimensional Inverse heat conduction
problem in estimating surface heat flux by conjugate gradient method",
International Journal of Heat and Mass Transfer, vol. 42, pp. 3387-
3403, 1999.
[2] F. Y. Zhao, D. Liu, G. F. Tang, "Numerical determination of boundary
heat fluxes in an enclosure dynamically with natural convection through
Fletcher-Reeves gradient method", Computer and Fluids, vol. 38, pp.
797-809, 2009.
[3] T. P. Lin, "Inverse heat conduction problem of simultaneously
determining thermal conductivity, heat capacity and heat transfer
coefficient", Master thesis, Department of Mechanical Engineering,
Tatung Institute of Technology, Taipei, Taiwan, 1998.
[4] H. R. B. Orlande, G. S. Dulikravich, "Inverse heat transfer problems and
their solutions within the Bayesian framework", in 2012 Numerical Heat
Transfer, ECCOMAS Special Interest Conference
[5] H. M. Park, O. Y. Chung, "An inverse natural convection problem of
estimating the strength of a heat source", International Journal of Heat
and Mass Transfer, vol. 42, pp.4259-4273, 1999.
[6] A. Ellabib and A. Nachaoui, "On the numerical solution of a free
boundary identification problem", Inverse Problems Eng. 9(3), pp.235-
260, 2001.
[7] C. R. Su and C. K. Chen, "Geometry estimation of the furnace inner
wall by an inverse approach", International Journal of Heat and Mass
Transfer, vol. 50, pp. 3767-3773, 2007.
[8] D. Lesnic, L. Elliott, B. Ingham, "Application of boundary element
method to inverse heat conduction problems", International Journal of
Heat and Mass Transfer, vol. 39, No. 7, pp. 1503-1517, 1996.
[9] A.N. Tikhonov and V.Y. Arsenin, Solution of Ill-Posed Problems,
Winston and Sons, Washington, DC (1977).
[10] J.V. Beck, B. Blackwell, C.St. Clair, "Inverse Heat Conduction: Illposed
Problems", Wiley, New York, 1985
[11] M. J. Colaco, H. R. B. Orlande, "Comparison of different versions of the
Conjugate Gradient Method of Function Estimation", Numerical Heat
Transfer, Part A, 36, pp. 229-249, 1999.
[12] H. M. Park, O. Y. Chung, "On the solution of an inverse natural
convection problem using various conjugate gradient methods",
International Journal for Numerical Methods In Engineering, vol. 47,
pp. 821-842, 2000.
[13] C.H. Huang, B.H. Chao, "An inverse geometry problem in identifying
irregular boundary configurations", Int. Journal of Heat Mass Transfer,
vol. 40, pp. 2045-2053, 1997.
[14] M. Ozisik, H.R.B. Orlande, "Inverse Heat Transfer", Taylor & Francis,
2000.
[1] C. H. Huang, S.P. Wang, "A three-dimensional Inverse heat conduction
problem in estimating surface heat flux by conjugate gradient method",
International Journal of Heat and Mass Transfer, vol. 42, pp. 3387-
3403, 1999.
[2] F. Y. Zhao, D. Liu, G. F. Tang, "Numerical determination of boundary
heat fluxes in an enclosure dynamically with natural convection through
Fletcher-Reeves gradient method", Computer and Fluids, vol. 38, pp.
797-809, 2009.
[3] T. P. Lin, "Inverse heat conduction problem of simultaneously
determining thermal conductivity, heat capacity and heat transfer
coefficient", Master thesis, Department of Mechanical Engineering,
Tatung Institute of Technology, Taipei, Taiwan, 1998.
[4] H. R. B. Orlande, G. S. Dulikravich, "Inverse heat transfer problems and
their solutions within the Bayesian framework", in 2012 Numerical Heat
Transfer, ECCOMAS Special Interest Conference
[5] H. M. Park, O. Y. Chung, "An inverse natural convection problem of
estimating the strength of a heat source", International Journal of Heat
and Mass Transfer, vol. 42, pp.4259-4273, 1999.
[6] A. Ellabib and A. Nachaoui, "On the numerical solution of a free
boundary identification problem", Inverse Problems Eng. 9(3), pp.235-
260, 2001.
[7] C. R. Su and C. K. Chen, "Geometry estimation of the furnace inner
wall by an inverse approach", International Journal of Heat and Mass
Transfer, vol. 50, pp. 3767-3773, 2007.
[8] D. Lesnic, L. Elliott, B. Ingham, "Application of boundary element
method to inverse heat conduction problems", International Journal of
Heat and Mass Transfer, vol. 39, No. 7, pp. 1503-1517, 1996.
[9] A.N. Tikhonov and V.Y. Arsenin, Solution of Ill-Posed Problems,
Winston and Sons, Washington, DC (1977).
[10] J.V. Beck, B. Blackwell, C.St. Clair, "Inverse Heat Conduction: Illposed
Problems", Wiley, New York, 1985
[11] M. J. Colaco, H. R. B. Orlande, "Comparison of different versions of the
Conjugate Gradient Method of Function Estimation", Numerical Heat
Transfer, Part A, 36, pp. 229-249, 1999.
[12] H. M. Park, O. Y. Chung, "On the solution of an inverse natural
convection problem using various conjugate gradient methods",
International Journal for Numerical Methods In Engineering, vol. 47,
pp. 821-842, 2000.
[13] C.H. Huang, B.H. Chao, "An inverse geometry problem in identifying
irregular boundary configurations", Int. Journal of Heat Mass Transfer,
vol. 40, pp. 2045-2053, 1997.
[14] M. Ozisik, H.R.B. Orlande, "Inverse Heat Transfer", Taylor & Francis,
2000.
@article{"International Journal of Mechanical, Industrial and Aerospace Sciences:60300", author = "V. Ghadamyari and F. Samadi and F. Kowsary", title = "Comparison of Three Versions of Conjugate Gradient Method in Predicting an Unknown Irregular Boundary Profile", abstract = "An inverse geometry problem is solved to predict an
unknown irregular boundary profile. The aim is to minimize the
objective function, which is the difference between real and
computed temperatures, using three different versions of Conjugate
Gradient Method. The gradient of the objective function, considered
necessary in this method, obtained as a result of solving the adjoint
equation. The abilities of three versions of Conjugate Gradient
Method in predicting the boundary profile are compared using a
numerical algorithm based on the method. The predicted shapes show
that due to its convergence rate and accuracy of predicted values, the
Powell-Beale version of the method is more effective than the
Fletcher-Reeves and Polak –Ribiere versions.", keywords = "Boundary elements, Conjugate Gradient Method,
Inverse Geometry Problem, Sensitivity equation.", volume = "7", number = "1", pages = "94-6", }