### A comprehensive study on the efficiency of three different types of the variable metric method in determining the unknown Rocket inner heat flux

#### Abstract

**In this paper, the unknown heat flux is estimated with Davidon-Fletcher-Powell (DFP), Broydon–Fletcher–Goldfarb–Shanno (BFGS) and Symmetric Rank-one (SR1) version of variable metric method (VMM). The numerical techniques used in this study solved the inverse problems with various boundary and environmental conditions very efficiently. The results shows the sensitivity of measurement errors and different parameter including changes of slope and angle which can be functions of an unknown parameter. Further, the speed of convergence is assessed and the convergence behavior is found. The accuracy of results show that this study is a powerful reference for comparing results obtained based on the three proposed techniques. The solution procedure introduced a general cheap and fast method which can be used for the inverse heat conduction problem in rocket nuzzle and same heat conduction, radiation and convection problems.**

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Tsung-Chien Chen, Chiun-Chien Liu, Inverse estimation of heat flux and temperature on nozzle throat-insert inner contour, International Journal of Heat and Mass Transfer, 51 (2008) 3571–3581.

H.N. Wang, J.H. Wang, A numerical investigation of ablation and transpiration cooling using the local thermal non-equilibrium model [R], in: Proceeding ofthe 42nd AIAA/ASME/SAE/ASEE Joint Propulsion Conference & Exhibit, Sacramento, California, July 9–12, AIAA-(2006)-5264.

J.H. Wang, H.N. Wang, J.G. Sun, J. Wang, Numerical simulation of control ablation by transpiration cooling, Heat Mass Transfer 43 (2007) 471–478.

Junxiang Shi, Jianhua Wang, Inverse problem of transpiration cooling for estimating wall heat flux by LTNE model and CGM method, International Journal of Heat and Mass Transfer 52 (2009) 2714–2720

G. Stolz, Jr., Numerical Solutions to an Inverse Problem of Heat Condu-ction for Simpl Shapes, ASME J. Heat Transfer, vol.82, (1960) pp.20-26.

E. M. Sparrow, A. Haji-Sheikh and T.S. Lundgren, The Inverse Problem in Transient Heat Conduction, ASME J. Appl. Mech, vol. 86,(1964) pp. 369-375.

A.N. Tikhonov and V.Y. Arsenin, Solutions of Ill-Posed Problems, Winston, Washington, (1977).

J. V. Beck, B. Blackwell and C.R. St. Clair, Inverse Heat Conduction-Ill Posed Problem, Wiley, New York, 1985.

Ch. H. H. and H. H. Wu, An Inverse Hyperbolic .Heat Conduction Problem in Estimating Surface Heat Flux by the Conjugate Gradient Method, J. Phys. D:. Appl. Phys., vol. 39, (2006) pp.4087-4096.

Junxiang Shi, Jianhua Wang, Inverse problem of transpiration cooling for estimating wall heat flux by LTNE model and CGM method, International Journal of Heat and Mass Transfer 52 (2009) 2714–2720.

Yun Ky Hong, Seung Wook Baek, Inverse analysis for estimating the unsteadyinlet temperature distribution for two-phase laminar flow in a channel, Int. J.Heat Mass Transfer 49 (2006) 1137–1147.

J.H. Bae, J.M. Hyun, H.S. Kwak, Mixed convection from a multiblock heater in a channel with imposed thermal modulation, Number. Heat Transfer Part A 45 (2004) 329–345.

J.G. Bauzin, N. Laraqi, Simultaneous estimation of frictional heat flux and two thermal contact parameters for sliding contacts, Numer. Heat Transfer 45 (4) (2004) 313–328.

S. Abboudi, A. Artioukhine, Two dimensional computational estima-tion of transient boundary conditions for a flat specimen. In: Proceedings of the fourth International Conference on Inverse Problems Engineering: Theory and Practice, June 13–18, 2002, Rio, Brasil, ASME 2003.

Linhua L, Heping T, Qizheng Y. Inverse radiation problem of temperature filed in three dimensional rectangular furnaces. Int Commun Heat Mass Transfer (1999) ;26: 239–48.

Daun KJ, Morton DP, Howell JR. Geometric optimization of radiant enclosures containing specular surfaces. ASME J Heat Transfer (2003) ;125:845–51.

Franc-a FR, Howell J, Ezekoye OA, Morales JC. Inverse design of thermal systems. In: Hartnett JP, Irvine TF, editors. Advances in heat transfer, vol. 36. New York: Elsevier; (2002). p. 1–110.

S.S Rao Optimization; theory and applications, 2nd edn (9th reprint). New Age International (P) Limited Publishers, New Delhi (1995).

Luks ̌an L, Spedicato E Variable metric methods for unconstrained optimization and nonlinear least squares. J Compute Appl Math (2000) 124: 61–95.

M. Prud'homme, S. Jasmin, Determination of a heat source in porous medium with convective mass diffusion by an inverse method, Interna-tional Journal of Heat and Mass Transfer 46 (2003) 2065–2075.

A. Pourshaghagh, F. Kowsary, A. Behbahaninia, Comparison of four different versions of the variable metric method for solving inverse heat conduction problems, Heat Mass Transfer (2007) 43: 285–294

S.V. Patankar, Numerical Heat Transfer and Fluid Flow McGrawHill New York , (1980).

Z.Z. Zhang, D.H. Cao, J.P. Zeng, Property of a class of variable metric methods, Applied Mathematics Letters 17 (2004) 437–442.

F. Kowsary, A. Behbahaninia, A. Pourshaghaghy, Transient heat flux function estimation utilizing the variable metric method, International Communications in Heat and Mass Transfer 33 (2006) 800–810.

Huang C H and Chen A three-dimensional inverse forced convection problem in estimating surface heat flux by conjugate gradient method Int. J. Heat Mass Transfer 43 (2000) 3171–81.

H.khoshkam, M.Alizadeh, Inverse problem of rocket nozzle throat for estimating inner wall heat flux by Broydon–Fletcher–Goldfarb–Shanno & conjugate gradient method, International Review of Mechanical Engineering (I.RE.M.E), Vol.5,N.5 July 2011.

Pandey, S. K. and Tripathi, D. Unsteady peristaltic transport of Maxwell fluid through ¯finite length tube: application to esophageal swallowing. Appl. Math. Mech. -Engl. Ed., 33(1), 15{24 (2012) DOI 10.1007/s10483-012-1530-9

GB4161-84. Standard Test Method for Plane-Strain Fracture Toughness (KIC) of Metallic Materials (in Chinese), Bureau in Chinese National Standard, Beijing (1984)

De Souza, J. and Silveria-Neto, A. Preliminary results of large eddy simulations of a hydro cyclone. Thermal Engineering, 3(2), 168{173 (2004)

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