Abstract
To solve nonlinear equations by an optimization method, scaling is very important. Two types of poor scaling where: (a) the variables differ greatly in magnitude; (b) the merit function of system is highly sensitive to small changes in certain variables and relatively insensitive to changes in other variables. If poor scaling is ignored, the algorithm may produce solutions with poor quality. To solve (a), we can change units of variables. A numerical solution of the nonlinear equations produced by the finite volume method in the forced convective heat transfer of a nanofluid, as a case study, indicates that the poor scaling (b) is solved by using the Euclidean norm of columns of the Jacobian matrix as scaling data, while some researchers proposed diagonal elements of the Hessian matrix as scaling data.
Similar content being viewed by others
References
Nocedal, J., Wright, S.J.: Numerical Optimization. Springer, New York (1999)
Dennis, J.E., Schnabel, R.B.: Numerical Methods for Unconstrained Optimization and Nonlinear Equations. Classics in Applied Mathematics. SIAM, Philadelphia (1987)
Özerinç, S., Kakaç, S., Yazıcıoğlu, A.G.: Enhanced thermal conductivity of nanofluids: a state-of-the-art review. Microfluid. Nanofluid. 8, 145–170 (2010)
Wang, X., Mujumdar, A.S.: Heat transfer characteristics of nanofluids: a review. Int. J. Therm. Sci. 46, 1–19 (2007)
Kakaç, S., Pramuanjaroenkij, A.: Review of convective heat transfer enhancement with nanofluids. Int. J. Heat Mass Transf. 52, 3187–3196 (2009)
Ferziger, J.H., Peric, M.: Computational Methods for Fluid Dynamics. Springer, Berlin (2002)
Versteeg, H.K., Malalasekera, W.: An Introduction to Computational Fluid Dynamics the Finite Volume Method. Longman Scientific & Technical, Harlow (1995)
Bird, R.B., Stewart, W.E., Lightfoot, E.N.: Transport Phenomena, Wiley, New York (2007)
Brady, J.F., Khair, A.S., Swaroop, M.: On the bulk viscosity of suspensions. J. Fluid Mech. 554, 109–123 (2006)
Zhou, S.Q., Ni, R.: Measurement of the specific heat capacity of water-based Al2O3 nanofluid. Appl. Phys. Lett. 92(9), 0931231 (2008)
Kanzow, C., Yamashita, N., Fukushima, M.: Levenberg–Marquardt methods for constrained nonlinear equations with strong local convergence properties. J. Comput. Appl. Math. 172, 375–397 (2004)
Nguyen, C.T., Desgranges, F., Galanis, N., Roya, G., Maréd, T., Boucher, S., Angue Mintsa, H.: Viscosity data for Al2O3–water nanofluid—hysteresis: is heat transfer enhancement using nanofluids reliable? Int. J. Therm. Sci. 47, 103–111 (2008)
Li Calvin, H., Peterson, G.P.: Experimental investigation of temperature and volume fraction variations on the effective thermal conductivity of nanoparticle suspensions (nanofluids). J. Appl. Phys. 99, 084314 (2006)
Pathipakka, G., Sivashanmugam, P.: Heat transfer behaviour of nanofluids in a uniformly heated circular tube fitted with helical inserts in laminar flow. Superlattices Microstruct. 47, 349–360 (2010)
Kays, W.M., Crawford, M.E.: Convective Heat and Mass Transfer. McGraw-Hill, New York (1993)
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Azimi, S.S., Kalbasi, M. & Sadeghifar, H. Sensitivity Analysis of Merit Function in Solving Nonlinear Equations by Optimization. J Optim Theory Appl 162, 191–201 (2014). https://doi.org/10.1007/s10957-013-0439-9
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10957-013-0439-9