0
Research Papers: Conduction

Fuzzy Adaptive Regularization Method for Inverse Steady-State Heat Transfer Problem

[+] Author and Article Information
Kun Wang

School of Energy and Power Engineering,
Chongqing University,
Chongqing 400044, China

Guangjun Wang

School of Energy and Power Engineering,
Chongqing University,
Chongqing 400044, China;
Key Laboratory of Low-Grade Energy Utilization
Technologies and Systems,
Chongqing University,
Ministry of Education,
Chongqing 400044, China
e-mail: wangguangjun@cqu.edu.cn

1Corresponding author.

Contributed by the Heat Transfer Division of ASME for publication in the JOURNAL OF HEAT TRANSFER. Manuscript received June 16, 2018; final manuscript received December 18, 2018; published online January 30, 2019. Assoc. Editor: George S. Dulikravich.

J. Heat Transfer 141(3), 031301 (Jan 30, 2019) (8 pages) Paper No: HT-18-1395; doi: 10.1115/1.4042462 History: Received June 16, 2018; Revised December 18, 2018

For the steady-state heat transfer process, a fuzzy adaptive regularization method (FARM) is proposed to estimate the distributed thermal boundary condition in heat transfer system. First, the relationship model between temperatures at measurement points and parameters to be estimated is established based on sensitivity matrix. The regularization term is introduced into the least-squares objective function, and then the distributed thermal boundary condition is estimated by optimizing the new objective function. A fuzzy inference mechanism is developed to ensure the adaptive ability of FARM in which the regularization parameter is updated based on the residual norm between calculated and measured temperatures at measurement points and the norm of inversion parameters. Taking the plate heat conduction system and fluid–solid conjugate heat transfer system as research objects, the effects of the parameter distribution, the number of measurement points, and measurement errors on the inversion results are discussed by numerical experiments, and comparison with the classical regularization method is also conducted. Results indicate that FARM exhibits a good adaptive ability.

Copyright © 2019 by ASME
Your Session has timed out. Please sign back in to continue.

References

Wang, G. , Luo, Z. , Zhu, L. , Chen, H. , and Zhang, L. , 2012, “ Fuzzy Estimation for Temperature Distribution of Furnace Inner Surface,” Int. J. Therm. Sci., 51, pp. 84–90. [CrossRef]
Huang, C. , Jan, U. , Li, R. , and Shih, A. J. , 2007, “ A Three-Dimensional Inverse Problem in Estimating the Applied Heat Flux of a Titanium Drilling—Theoretical and Experimental Studies,” Int. J. Heat Mass Transfer, 50(17–18), pp. 3265–3277. [CrossRef]
Zhang, L. , Tai, B. L. , Wang, G. , Zhang, K. , Sullivan, S. , and Shih, A. J. , 2013, “ Thermal Model to Investigate the Temperature in Bone Grinding for Skull Base Neurosurgery,” Med. Eng. Phys., 35(10), pp. 1391–1398. [CrossRef] [PubMed]
Beck, J. V. , Blackwell, B. , and St. Clair, J. C. R. , 1985, Inverse Heat Conduction: Ill-Posed Problems, Wiley, New York.
Huang, C. , and Huang, C. , 2007, “ An Inverse Problem in Estimating Simultaneously the Effective Thermal Conductivity and Volumetric Heat Capacity of Biological Tissue,” Appl. Math. Model., 31(9), pp. 1785–1797. [CrossRef]
Watson, G. A. , 2007, “ A Levenberg-Marquardt Method for Estimating Polygonal Regions,” J. Comput. Appl. Math., 208(2), pp. 331–340. [CrossRef]
Zhu, L. , Wang, G. , and Chen, H. , 2011, “ Estimating Steady Multi-Variables Inverse Heat Conduction Problem by Using Conjugate Gradient Method,” Proc. CSEE, 31(8), pp. 58–61 (in Chinese).
Parwani, A. K. , Talukdar, P. , and Subbarao, P. M. V. , 2014, “ Estimation of Transient Boundary Flux for a Developing Flow in a Parallel Plate Channel,” Int. J. Numer. Method Heat Fluid Flow, 24, pp. 522–544. [CrossRef]
Tikhonov, A. N. , 1963, Regularition of Incorrectly Posed Problems, Doklady Akademii Nauk Sssr, Moskva, Russia.
Woodbury, K. A. , and Beck, J. V. , 2013, “ Estimation Metrics and Optimal Regularization in a Tikhonov Digital Filter for the Inverse Heat Conduction Problem,” Int. J. Heat Mass Transfer, 62, pp. 31–39. [CrossRef]
Hong, Y. K. , and Baek, S. W. , 2006, “ Inverse Analysis for Estimating the Unsteady Inlet Temperature Distribution for Two-Phase Laminar Flow in a Channel,” Int. J. Heat Mass Transfer, 49(5–6), pp. 1137–1147. [CrossRef]
Calvetti, D. , Morigi, S. , Reichel, L. , and Sgallari, F. , 2000, “ Tikhonov Regularization and the L-Curve for Large Discrete Ill-Posed Problems,” J. Comput. Appl. Math., 123(1–2), pp. 423–446. [CrossRef]
Engl, H. W. , and Grever, W. , 1994, “ Using the L-Curve for Determining Optimal Regularization Parameters,” Numer. Math., 69(1), pp. 25–31. [CrossRef]
Morozov, V. A. , 1984, Methods for Solving Incorrectly Posed Problems, Springer, New York.
Bakushinsky, A. , and Goncharsky, A. , 1994, Ill-Posed Problems: Theory and Applications, Kluwer Academic Publishers, Dororecht, The Netherlands.
Garshasbi, M. , and Hassani, F. , 2016, “ Boundary Temperature Reconstruction in an Inverse Heat Conduction Problem Using Boundary Integral Equation Method,” Bull. Iran. Math. Soc., 42(5), pp. 1039–1056. https://www.researchgate.net/publication/310615704_Boundary_temperature_reconstruction_in_an_inverse_heat_conduction_problem_using_boundary_integral_equation_method
Mobtil, M. , Bougeard, D. , and Solliec, C. , 2014, “ Inverse Determination of Convective Heat Transfer Between an Impinging Jet and a Continuously Moving Flat Surface,” Int. J. Heat Fluid Fl., 50, pp. 83–94. [CrossRef]
Reichel, L. , and Sadok, H. , 2008, “ A New L-Curve for Ill-Posed Problems,” J. Comput. Appl. Math., 219(2), pp. 493–508. [CrossRef]
Hanke, M. , 1996, “ Limitations of the L-Curve Method in Ill-Posed Problems,” BIT, 36(2), pp. 287–301. [CrossRef]
Vogel, C. R. , 1996, “ Non-Convergence of the L-Curve Regularization Parameter Selection Method,” Inverse Probl., 12(4), pp. 535–547. [CrossRef]
Klir, G. J. , and Yuan, B. , 1995, Fuzzy Sets and Fuzzy Logic: Theory and Applications, Prentice Hall, Upper Saddle River, NJ.
Sargolzaei, J. , Khoshnoodi, M. , Saghatoleslami, N. , and Mousavi, M. , 2008, “ Fuzzy Inference System to Modeling of Crossflow Milk Ultrafiltration,” Appl. Soft Comput., 8(1), pp. 456–465. [CrossRef]
Van Broekhoven, E. , and De Baets, B. , 2006, “ Fast and Accurate Center of Gravity Defuzzification of Fuzzy System Outputs Defined on Trapezoidal Fuzzy Partitions,” Fuzzy Set. Syst., 157(7), pp. 904–918. [CrossRef]
MathWorks, 2018, “Fuzzy Logic Toolboxm” MathWorks, Inc., Natick, MA, accessed Oct. 15, 2018, https://www.mathworks.com/products/fuzzy-logic.html

Figures

Grahic Jump Location
Fig. 1

Physical model of inverse heat transfer problem

Grahic Jump Location
Fig. 2

System of fuzzy adaptive regularization

Grahic Jump Location
Fig. 3

Membership function of ‖F‖ to fuzzy sets A

Grahic Jump Location
Fig. 4

Membership function of ‖E‖ to fuzzy sets B

Grahic Jump Location
Fig. 5

Membership function of Δα to fuzzy sets C

Grahic Jump Location
Fig. 6

Steady-state heat conduction system

Grahic Jump Location
Fig. 7

L-curve for steady-state heat conduction system

Grahic Jump Location
Fig. 8

The regularization parameters fuzzy adaptive process for heat conduction system

Grahic Jump Location
Fig. 9

Inversion results using σ = 0 °C for heat conduction system

Grahic Jump Location
Fig. 10

Inversion results using σ = 0.2 °C for heat conduction system

Grahic Jump Location
Fig. 11

Inversion results using σ = 0.3 °C for heat conduction system

Grahic Jump Location
Fig. 12

Inversion results for triangular temperature distribution using σ = 0 °C for heat conduction system

Grahic Jump Location
Fig. 13

Inversion results for triangular temperature distribution using σ = 0.2 °C for heat conduction system

Grahic Jump Location
Fig. 14

Inversion results using G =10 and σ = 0 °C for heat conduction system

Grahic Jump Location
Fig. 15

Inversion results using G =10 and σ = 0.2 °C for heat conduction system

Grahic Jump Location
Fig. 16

Inversion results using G =6 and σ = 0 °C for heat conduction system

Grahic Jump Location
Fig. 17

Inversion results using G =6 and σ = 0.2 °C for heat conduction system

Tables

Errata

Some tools below are only available to our subscribers or users with an online account.

Related Content

Customize your page view by dragging and repositioning the boxes below.

Related Journal Articles
Related eBook Content
Topic Collections

Sorry! You do not have access to this content. For assistance or to subscribe, please contact us:

  • TELEPHONE: 1-800-843-2763 (Toll-free in the USA)
  • EMAIL: asmedigitalcollection@asme.org
Sign In