A proper orthogonal decomposition analysis method for transient nonlinear heat conduction problems. Part 1: Basic algorithm

References

• J. Yang and X. L. Huang, “Nonlinear transient response of functionally graded plates with general imperfections in thermal environments,” Comput. Methods. Appl. Mech. Eng., vol. 196, no. 25–28, pp. 2619–2630, 2007. DOI: 10.1016/j.cma.2007.01.012.
   Web of Science ®Google Scholar
• A. Korzen and D. Taler, “Modeling of transient response of a plate fin and tube heat exchanger,” Int. J. Therm. Sci, vol. 92, pp. 188–198, 2015. DOI: 10.1016/j.ijthermalsci.2015.01.036.
   Web of Science ®Google Scholar
• A. Riccio et al., “Optimum design of ablative thermal protection systems for atmospheric entry vehicles,” Appl. Therm. Eng., vol. 119, pp. 541–552, 2017. DOI: 10.1016/j.applthermaleng.2017.03.053.
   Web of Science ®Google Scholar
• W. L. Dai and P. R. Woodward, “Numerical simulations for nonlinear heat transfer in a system of multimaterials,” J. Comput. Phys., vol. 139, no. 1, pp. 58–78, 1998. DOI: 10.1006/jcph.1997.5863.
   Web of Science ®Google Scholar
• T.-M. Shih, C.-H. Sung, and B. Yang, “A numerical method for solving nonlinear heat transfer equations,” Numer. Heat Transf., Part B, vol. 54, no. 4, pp. 338–353, 2008. DOI: 10.1080/10407790802182687.
   Web of Science ®Google Scholar
• J. Kujawski, “Analysis of the collocation time finite element method for the nonlinear heat transfer equation,” Commun. Appl. Numer. Methods, vol. 3, no. 2, pp. 103–107, 1987. DOI: 10.1002/cnm.1630030205.
   Google Scholar
• E. Li et al., “Smoothed finite element method with exact solutions in heat transfer problems,” Int. J. Heat Mass Transf., vol. 78, pp. 1219–1231, 2014. DOI: 10.1016/j.ijheatmasstransfer.2014.07.078.
   Web of Science ®Google Scholar
• R. V. Mohan and K. K. Tamma, “Finite-element finite-volume approaches with adaptive time-stepping strategies for transient thermal problems,” Sadhana-Acad. Proc. Eng. Sci., vol. 19, pp. 765–783, 1994. DOI: 10.1007/BF02744404.
   Web of Science ®Google Scholar
• M. Copur and M. N. Eruslu, “Finite volume modeling of the solidification of an axial steel cast impeller,” Metalurgija, vol. 53, no. 2, pp. 149–154, 2014.
   Web of Science ®Google Scholar
• T. Goto and M. Suzuki, “A boundary integral equation method for nonlinear heat conduction problems with temperature-dependent material properties,” Int. J. Heat Mass Transf., vol. 39, no. 4, pp. 823–830, 1996. DOI: 10.1016/0017-9310(95)00167-0.
   Web of Science ®Google Scholar
• M. Mohammadi, M. R. Hematiyan, and L. Marin, “Boundary element analysis of nonlinear transient heat conduction problems involving non-homogenous and nonlinear heat sources using time-dependent fundamental solutions,” Eng. Anal. Bound. Elem., vol. 34, no. 7, pp. 655–665, 2010. DOI: 10.1016/j.enganabound.2010.02.004.
   Web of Science ®Google Scholar
• M. Cui, B. B. Xu, W. Z. Feng, Y. W. Zhang, X. W. Gao, and H. F. Peng, “A radial integration boundary element method for solving transient heat conduction problems with heat sources and variable thermal conductivity,” Numer. Heat Transf., Part B, vol. 73, no. 1, pp. 1–18, 2018. DOI: 10.1080/10407790.2017.1420319.
   Web of Science ®Google Scholar
• A. Singh, I. V. Singh, and R. Prakash, “Meshless element free Galerkin method for unsteady nonlinear heat transfer problems,” Int. J. Heat Mass Transf., vol. 50, no. 5-6, pp. 1212–1219, 2007. DOI: 10.1016/j.ijheatmasstransfer.2006.08.039.
   Web of Science ®Google Scholar
• A. Khosravifard, M. R. Hematiyan, and L. Marin, “Nonlinear transient heat conduction analysis of functionally graded materials in the presence of heat sources using an improved meshless radial point interpolation method,” Appl. Math. Model, vol. 35, no. 9, pp. 4157–4174, 2011. DOI: 10.1016/j.apm.2011.02.039.
   Web of Science ®Google Scholar
• H. M. Zhou, G. Qin, and Z. Y. Wang, “Heat conduction analysis for irregular functionally graded material geometries using the meshless weighted least square method with temperature-dependent material properties,” Numer. Heat Transf., Part B, vol. 75, no. 5, pp. 312–324, 2019. DOI: 10.1080/10407790.2019.1627814.
   Web of Science ®Google Scholar
• J. N. Tang et al., “A new procedure for solving steady-state and transient-state nonlinear radial conduction problems of nuclear fuel rods,” Anna. Nucl. Energy, vol. 110, pp. 492–500, 2017. DOI: 10.1016/j.anucene.2017.05.061.
   Web of Science ®Google Scholar
• K. Yang, H. F. Peng, J. Wang, C. H. Xing, and X. W. Gao, “Radial integration BEM for solving transient nonlinear heat conduction with temperature-dependent conductivity,” Int. J. Heat Mass Transf., vol. 108, pp. 1551–1559, 2017. DOI: 10.1016/j.ijheatmasstransfer.2017.01.030.
   Web of Science ®Google Scholar
• K. Yang, W. Z. Feng, J. Wang, and X. W. Gao, “RIBEM for 2D and 3D nonlinear heat conduction with temperature dependent conductivity,” Eng. Anal. Bound. Elem., vol. 87, pp. 1–8, 2018. DOI: 10.1016/j.enganabound.2017.11.001.
   Web of Science ®Google Scholar
• M. Cui, B. B. Xu, J. Lv, X. W. Gao, and Y. W. Zhang, “Numerical solution of multi-dimensional transient nonlinear heat conduction problems with heat sources by an extended element differential method,” Int. J. Heat Mass Transf., vol. 126, pp. 1111–1119, 2018. DOI: 10.1016/j.ijheatmasstransfer.2018.05.100.
   Web of Science ®Google Scholar
• X. W. Gao et al., “Element differential method for solving general heat conduction problems,” Int. J. Heat Mass Transf., vol. 115, pp. 882–894, 2017. DOI: 10.1016/j.ijheatmasstransfer.2017.08.039.
   Web of Science ®Google Scholar
• X. W. Gao, H. Y. Liu, B. B. Xu, M. Cui, and J. Lv, “Element differential method with the simplest quadrilateral and hexahedron quadratic elements for solving heat conduction problems,” Numer. Heat Transf., Part B, vol. 73, no. 4, pp. 206–224, 2018. DOI: 10.1080/10407790.2018.1461491.
   Web of Science ®Google Scholar
• M. Girault and D. Petit, “Identification methods in nonlinear heat conduction. Part I: Odel reduction,” Int. J. Heat Mass Transf., vol. 48, no. 1, pp. 105–118, 2005. DOI: 10.1016/j.ijheatmasstransfer.2004.06.032.
   Web of Science ®Google Scholar
• G. Kerschen, J. C. Golinval, A. F. Vakakis, and L. A. Bergman, “The method of proper orthogonal decomposition for dynamical characterization and order reduction of mechanical systems: An overview,” Nonlin. Dynam., vol. 41, no. 1–3, pp. 147–169, 2005. DOI: 10.1007/s11071-005-2803-2.
   Web of Science ®Google Scholar
• A. Fic, R. A. Biaecki, and A. J. Kassab, “Solving transient nonlinear heat conduction problems by proper orthogonal decomposition and the finite-element method,” Numer. Heat Transf., Part B, vol. 48, no. 2, pp. 103–124, 2005. DOI: 10.1080/10407790590935920.
   Web of Science ®Google Scholar
• E. Monteiro, J. Yvonnet, and Q. C. He, “Computational homogenization for nonlinear conduction in heterogeneous materials using model reduction,” Comput. Mater. Sci., vol. 42, no. 4, pp. 704–712, 2008. DOI: 10.1016/j.commatsci.2007.11.001.
   Web of Science ®Google Scholar
• D. X. Han, B. Yu, and X. Y. Zhang, “Study on a BFC-based POD-Galerkin reduced-order model for the unsteady-state variable-property heat transfer problem,” Numer. Heat Transf., Part B, vol. 65, no. 3, pp. 256–281, 2014. DOI: 10.1080/10407790.2013.849989.
   Web of Science ®Google Scholar
• A. K. Gaonkar and S. S. Kulkarni, “Application of multilevel scheme and two level discretization for POD based model order reduction of nonlinear transient heat transfer problems,” Comput. Mech., vol. 55, no. 1, pp. 179–191, 2015. DOI: 10.1007/s00466-014-1089-y.
   Web of Science ®Google Scholar
• J. X. Hu, B. J. Zheng, and X. W. Gao, “Reduced order model analysis method via proper orthogonal decomposition for transient heat conduction,” Sci. Sin. Phys., Mech. Astron., vol. 45, no. 1, pp. 014602, 2015. DOI: 10.1360/SSPMA2013-00041.
   Google Scholar
• X. W. Gao, J. X. Hu, and S. Z. Huang, “A proper orthogonal decomposition analysis method for multimedia heat conduction problems,” ASME J. Heat Transf., vol. 138, no. 7, pp. 071301, 2016.
   Web of Science ®Google Scholar
• R. A. Biaecki, A. J. Kassab, and A. Fic, “Proper orthogonal decomposition and modal analysis for acceleration of transient FEM thermal analysis,” Int. J. Numer. Methods Eng., vol. 62, no. 6, pp. 774–797, 2005. DOI: 10.1002/nme.1205.
   Web of Science ®Google Scholar
• L. Sirovich, “Turbulence and the dynamics of coherent structures. Part I: coherent structures,” Quart. Appl. Math., vol. 45, no. 3, pp. 561–571, 1987. DOI: 10.1090/qam/910462.
   Web of Science ®Google Scholar
• R. Ghosh and Y. Joshi, “Error estimation in POD-based dynamic reduced-order thermal modeling of data centers,” Int. J. Heat Mass Transf., vol. 57, no. 2, pp. 698–707, 2013. DOI: 10.1016/j.ijheatmasstransfer.2012.10.013.
   Web of Science ®Google Scholar
• W. L. Wood and R. W. Lewis, “A comparison of time marching schemes for the transient heat conduction equation,” Int. J. Numer. Methods Eng., vol. 9, no. 3, pp. 679–689, 1975. DOI: 10.1002/nme.1620090314.
   Google Scholar
• T. J. R. Hughes, “Unconditionally stable algorithms for nonlinear heat conduction,” Comput. Methods Appl. Mech. Eng., vol. 10, no. 2, pp. 135–139, 1977. DOI: 10.1016/0045-7825(77)90001-9.
   Google Scholar
• M. Cui, K. Yang, X. Xu, S. Wang, and 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., vol. 97, pp. 908–916, 2016. DOI: 10.1016/j.ijheatmasstransfer.2016.02.085.
   Web of Science ®Google Scholar
• K. E. Ragab and L. El-Gabry, “Heat transfer analysis of the surface of a nozzle guide vane in a transonic annular cascade,” J. Therm. Sci. Eng. Appl., vol. 11, no. 1, pp. 011019, 2019.
   Web of Science ®Google Scholar