Inverse Identification of Temperature-Dependent Thermal Properties Using Improved Krill Herd Algorithm

  • S. C. Sun
  • H. QiEmail author
  • X. Y. Yu
  • Y. T. Ren
  • L. M. Ruan


A novel intelligent algorithm, krill herd (KH), is firstly introduced to solve the inverse identification of temperature-dependent thermal properties of materials. To promote the searching ability and accelerate the convergence velocity, three improved KH (IKH) algorithms are proposed and developed for solving the optimization tasks. The temperature-dependent thermal conductivity and specific heat of a building material are estimated by using the KH algorithms, and the IKHs achieve better performance than the original KHs. Moreover, the functional forms of thermal conductivity of insulating and refractory materials are also reconstructed. The IKH algorithm is proved to be more accurate than other algorithms. Finally, a two-dimensional nonhomogeneous heat conduction model is investigated and the thermal conductivities of materials at specified temperatures are reconstructed, in which no prior information is needed for the expressions of the thermal conductivity to be identified. All the retrieval results show that IKH algorithm is robust and effective for solving the inverse heat conduction problems.


Improved KH algorithm Inverse identification Temperature-dependent thermal property Inverse heat conduction Thermal conductivity 

List of symbols


Coefficient of thermal conductivity of steel


Constant term of energy equation or coefficient of specific heat of steel


Coefficient of thermal conductivity of slag wool, foam brick and silica brick


Specific heat, J/(kg·°C)


Volume specific heat, J/(m3·°C)


Effect coefficient of the best krill individual


Crossover probability


Coefficient of search step


Random movement


Foraging movement


Fitness function


Iteration number


Coefficient of thermal conductivity of carbon steel and aluminum


Fitness value


Length of medium


Lower boundary


Measured signal


Population size


Mutation probability


The number of boundary


Induced movement


A uniformly distributed random number


Heat flux, W/m2


A uniformly distributed random number


Temperature, °C


Upper boundary


Total speed


Foraging speed


x-coordinate, m


Position of krill individual


Value of estimation parameters


y-coordinate, m

Greeks symbols


Local or target effect


Effect provided by the food or the individual best position


A fluctuation


Search step size


Computational accuracy


Relative error


Measurement error


Thermal conductivity, W/(m·°C)


Control number


Density, kg/m3


Standard deviation


Time, s


Inertia weight


A normally distributed random number


Sensitivity coefficient



The best value


Estimated parameter


Exact parameter


Foraging motion


Imaginary food position


Incident value


The maximum value


Measurement value


The minimum value


Induced movement


The current iteration


The last iteration


Outgoing value


Random number



The supports of this work by the National Natural Science Foundation of China (No. 51576053) and the Major National Scientific Instruments and Equipment Development Special Foundation of China (No. 51327803) are gratefully acknowledged. A very special acknowledgment is made to the editors and referees who make important comments to improve this paper.


  1. 1.
    J.B. Beck, B. Blackwell, C.R. St. Clair Jr., Inverse Heat Conduction: Ill-Posed Problems (Wiley-Interscience, New York, 1985)zbMATHGoogle Scholar
  2. 2.
    O.M. Alifanov, Inverse Heat Transfer Problems (Springer, Berlin, 1994)CrossRefGoogle Scholar
  3. 3.
    G. Milano, F. Scarpa, F. Righni, G.C. Bussolino, Ten years of parameter estimation applied to dynamic thermophysical property measurements. Int. J. Thermophys. 22(4), 1227–1240 (2001)CrossRefGoogle Scholar
  4. 4.
    B. Sawaf, M.N. Özisik, An inverse analysis to estimate linearly temperature dependent thermal conductivity components and heat capacity of an orthotropic medium. Int. J. Heat Mass Transf. 38(16), 3005–3010 (1995)CrossRefGoogle Scholar
  5. 5.
    D.W. Tang, N. Araki, An inverse analysis to estimate relaxation parameters and thermal diffusivity with a universal heat conduction equation. Int. J. Thermophys. 21(2), 553–561 (2000)CrossRefGoogle Scholar
  6. 6.
    M. Cui, K. Yang, X.L. Xu, S.D. Wang, X.W. Gao, A modified Levenberg–Marquardt algorithm for simultaneous estimation of multi-parameters of boundary heat flux by solving transient nonlinear inverse heat conduction problems. Int. J. Heat Mass Transf. 97, 908–916 (2016)CrossRefGoogle Scholar
  7. 7.
    G.H. Tang, C.L. Xu, L.T. Shao, B. Zhou, D.Y. Yang et al., Improve algorithms of differential optical absorption spectroscopy for monitoring SO2, NO2 from flue gas. Meas. Sci. Technol. 20(1), 015601 (2009)CrossRefGoogle Scholar
  8. 8.
    J. Zmywaczyk, P. Koniorczyk, Numerical solution of inverse radiative–conductive transient heat transfer problem in a grey participating medium. Int. J. Thermophys. 30(4), 1438–1451 (2009)ADSCrossRefGoogle Scholar
  9. 9.
    M. Cui, N. Li, Y.F. Liu, X.W. Gao, Robust inverse approach for two-dimensional transient nonlinear heat conduction problems. J. Thermophys. Heat Transf. 29(2), 253–262 (2015)CrossRefGoogle Scholar
  10. 10.
    N. Ukrainczyk, Thermal diffusivity estimation using numerical inverse solution for 1D heat conduction. Int. J. Heat Mass Transf. 52(25–26), 5675–5681 (2009)CrossRefGoogle Scholar
  11. 11.
    F.Q. Wang, J.Y. Tan, Z.Q. Wang, Heat transfer analysis of porous media receiver with different transport and thermophysical models using mixture as feeding gas. Energy Conv. Manag. 83, 159–166 (2014)CrossRefGoogle Scholar
  12. 12.
    J. Jiao, Z.X. Guo, Thermal interaction of short-pulsed laser focused beams with skin tissues. Phys. Med. Biol. 54, 4225–4241 (2009)CrossRefGoogle Scholar
  13. 13.
    M. Kosaka, M. Monde, Simultaneous measurement of thermal diffusivity and thermal conductivity by means of inverse solution for one-dimensional heat conduction (anisotropic thermal properties of CFRP for FCEV). Int. J. Thermophys. 36(10–11), 2590–2598 (2015)ADSCrossRefGoogle Scholar
  14. 14.
    M. Cui, W.W. Duan, X.W. Gao, A new inverse analysis method based on a relaxation factor optimization technique for solving transient nonlinear inverse heat conduction problems. Int. J. Heat Mass Transf. 90, 491–498 (2015)CrossRefGoogle Scholar
  15. 15.
    Y.S. Sun, J. Ma, B.W. Li, Z.X. Guo, Predication of nonlinear heat transfer in a convective-radiative fin with temperature-dependent properties by the collocation spectral method. Numer. Heat Transf. B-Fundam 69(1), 68–83 (2016)ADSCrossRefGoogle Scholar
  16. 16.
    S.Y. Zhao, B.M. Zhang, S.Y. Du, X.D. He, Inverse identification of thermal properties of fibrous insulation from transient temperature measurements. Int. J. Thermophys. 30(6), 2020–2035 (2009)ADSCrossRefGoogle Scholar
  17. 17.
    M. Cui, X.W. Gao, J.B. Zhang, A new approach for the estimation of temperature-dependent thermal properties by solving transient inverse heat conduction problems. Int. J. Therm. Sci. 58, 113–119 (2012)CrossRefGoogle Scholar
  18. 18.
    B. Zhang, C.L. Xu, S.M. Wang, An inverse method for flue gas shielded metal surface temperature measurement based on infrared radiation. Meas. Sci. Technol. 27(7), 74002–74012 (2016)CrossRefGoogle Scholar
  19. 19.
    X.C. Yu, Y.C. Bai, M. Cui, X.W. Gao, Inverse analysis of thermal conductivities in transient non-homogeneous and non-linear heat conductions using BEM based on complex variable differentiation method. Sci. China-Phys. Mech. Astron. 56(5), 966–973 (2013)ADSCrossRefGoogle Scholar
  20. 20.
    C.H. Huang, J.Y. Yan, An inverse problem in simultaneously measuring temperature-dependent thermal conductivity and heat capacity. Int. J. Heat Mass Transf. 38(18), 3433–3441 (1995)CrossRefGoogle Scholar
  21. 21.
    C.H. Huang, J.Y. Yan, H.T. Chen, Function estimation in predicting temperature-dependent thermal conductivity without internal measurements. J. Thermophys. Heat Transf. 9(4), 667–673 (1995)CrossRefGoogle Scholar
  22. 22.
    C.H. Huang, J.Y. Yan, An inverse problem in predicting temperature dependent heat capacity per unit volume without internal measurements. Int. J. Numer. Methods Eng. 39(4), 605–618 (1996)CrossRefGoogle Scholar
  23. 23.
    C.H. Huang, S.C. Chin, A two-dimensional inverse problem in imaging the thermal conductivity of a non-homogeneous medium. Int. J. Heat Mass Transf. 43(22), 4061–4071 (2000)CrossRefGoogle Scholar
  24. 24.
    M. Cui, Q.H. Zhu, X.W. Gao, A modified conjugate gradient method for transient nonlinear inverse heat conduction problems: a case study for identifying temperature-dependent thermal conductivities. J. Heat Transf.-Trans. ASME 136(9), 091301 (2014)CrossRefGoogle Scholar
  25. 25.
    W.K. Yeung, T.T. Lam, Second-order finite difference approximation for inverse determination of thermal conductivity. Int. J. Heat Mass Transf. 39(17), 3685–3693 (1996)CrossRefGoogle Scholar
  26. 26.
    C.L. Chang, M. Chang, Inverse estimation of the thermal conductivity in a one-dimensional domain by Taylor series approach. Heat Transf. Eng. 29(9), 830–838 (2008)ADSCrossRefGoogle Scholar
  27. 27.
    C.L. Chang, M. Chang, Inverse determination of thermal conductivity using semi-discretization method. Appl. Math. Model. 33(3), 1644–1655 (2009)CrossRefGoogle Scholar
  28. 28.
    M. Raudenský, J. Horský, J. Krejsa, L. Sláma, Usage of artificial intelligence methods in inverse problems for estimation of material parameters. Int. J. Numer. Methods Heat Fluid Flow 6(8), 19–29 (1996)CrossRefGoogle Scholar
  29. 29.
    B. Czél, K.A. Woodbury, G. Gróf, Inverse identification of temperature-dependent volumetric heat capacity by neural networks. Int. J. Thermophys. 34(2), 284–305 (2013)ADSCrossRefGoogle Scholar
  30. 30.
    B. Czél, G. Gróf, Genetic algorithm-based method for determination of temperature-dependent thermophysical properties. Int. J. Thermophys. 30, 1975–1991 (2009)ADSCrossRefGoogle Scholar
  31. 31.
    Z.H. Ruan, Y. Yuan, Q.X. Chen, C.X. Zhang, Y. Shuai et al., A new multi-function global particle swarm optimization. Appl. Soft Comput. 49, 279–291 (2016)CrossRefGoogle Scholar
  32. 32.
    M.D. Ardakani, M. Khodadad, Identification of thermal conductivity and the shape of an inclusion using the boundary elements method and the particle swarm optimization algorithm. Inverse Probl. Sci. Eng. 17(7), 855–870 (2009)MathSciNetCrossRefGoogle Scholar
  33. 33.
    S. Vakili, M.S. Gadala, Effectiveness and efficiency of particle swarm optimization technique in inverse heat conduction analysis. Numer. Heat Tranf. B-Fundam. 56(2), 119–141 (2009)ADSCrossRefGoogle Scholar
  34. 34.
    N. Tian, J. Sun, W.B. Xu, C.H. Lai, Quantum-behaved particle swarm optimization with ring topology and its application in estimating temperature-dependent thermal conductivity. Numer. Heat Tranf. B-Fundam. 60(2), 73–95 (2011)ADSCrossRefGoogle Scholar
  35. 35.
    Z.Z. He, J.K. Mao, X.S. Han, Non-parametric estimation of particle size distribution from spectral extinction data with PCA approach. Powder Technol. 325(1), 510–518 (2018)CrossRefGoogle Scholar
  36. 36.
    A.H. Gandomi, A.H. Alavi, Krill herd: a new bio-inspired optimization algorithm. Commun. Nonlinear Sci. Numer. Simul. 17, 4831–4854 (2012)ADSMathSciNetCrossRefGoogle Scholar
  37. 37.
    Y.T. Ren, H. Qi, X. Huang, W. Wang, L.M. Ruan, H.P. Tan, Application of improved krill herd algorithms to inverse radiation problems. Int. J. Therm. Sci. 103, 24–34 (2016)CrossRefGoogle Scholar
  38. 38.
    S.C. Sun, H. Qi, F.Z. Zhao, L.M. Ruan, B.X. Li, Inverse geometry design of two-dimensional complex radiative enclosures using krill herd optimization algorithm. Appl. Therm. Eng. 98, 1104–1115 (2016)CrossRefGoogle Scholar
  39. 39.
    H.A. Van den Vorst, BI-CGSTAB: a fast and smoothly converging variant of BI-CG for the solution of nonsymmetric linear systems. SIAM J. Sci. Stat. Comput. 13(2), 631–644 (1992)MathSciNetCrossRefGoogle Scholar
  40. 40.
    E.E. Hofmann, A.G.E. Haskell, J.M. Klinck, C.M. Lascara, Lagrangian modelling studies of Antarctic krill (Euphasia superba) swarm formation. ICES J. Mar. Sci. 61(4), 617–631 (2004)CrossRefGoogle Scholar
  41. 41.
    H.J. Price, Swimming behavior of krill in response to algal patches: a mesocosm study. Limnol. Oceanogr. 34, 649–659 (1989)ADSCrossRefGoogle Scholar
  42. 42.
    G. Boeing, Visual analysis of nonlinear dynamical systems: chaos, fractals, self-similarity and the limits of prediction. Systems 4, 37 (2016)CrossRefGoogle Scholar
  43. 43.
    S.M. Yang, W.Q. Tao, Heat Transfer (in Chinese) (Higher Education Press, Beijing, 2006)Google Scholar

Copyright information

© Springer Science+Business Media, LLC, part of Springer Nature 2018

Authors and Affiliations

  • S. C. Sun
    • 1
  • H. Qi
    • 1
    Email author
  • X. Y. Yu
    • 1
  • Y. T. Ren
    • 1
  • L. M. Ruan
    • 1
  1. 1.School of Energy Science and EngineeringHarbin Institute of TechnologyHarbinPeople’s Republic of China

Personalised recommendations