The adjoint method coupled with the modal identification method for nonlinear model reduction

References

• Videcoq, E, and Petit, D, 2001. Model reduction for the resolution of multidimensional inverse heat conduction problems, International Journal of Heat and Mass Transfer 44 (2001), pp. 1899–1911.
   Web of Science ®Google Scholar
• Girault, M, Petit, D, and Videcoq, E, 2003. The use of model reduction and function decomposition for identifying boundary conditions of a linear thermal system, Inverse Problems in Engineering 11 (2003), pp. 425–455.
   Google Scholar
• Bay, F, Labbé, V, Favennec, Y, and Chenot, JL, 2003. A numerical Model for Induction Heating Processes Coupling Electromagnetism and Thermomechanics, International Journal of Numerical Methods in Engineering 58 (6) (2003), pp. 839–867.
   Web of Science ®Google Scholar
• Sophy, T, and Sadat, H, 2002. A three dimensional formulation of a meshless method for solving fluid flow and heat transfer problems, Numerical Heat Transfer, Part B: Fundamentals 41 (5) (2002), pp. 433–445.
   Web of Science ®Google Scholar
• Patankar, SV, 1980. "Numerical heat transfer and fluid flow". In: Series in Computational Methods in Mechanics and Thermal Sciences. New York: Hemisphere; 1980.
   Google Scholar
• Favennec, Y, The comparison between the discretized continuous gradient and the discrete gradient when dealing with nonlinear parabolic problems, Numerical Heat Transfer, Part B: Fundamentals, (Accepted, to appear in 2005).
   Web of Science ®Google Scholar
• Girault, M, and Petit, D, 2005. Identification methods in nonlinear heat conduction – Part 1: model reduction, International Journal of Heat and Mass Transfer 48 (1) (2005), pp. 105–118.
   Web of Science ®Google Scholar
• Girault, M, and Petit, D, 2005. Identification methods in nonlinear heat conduction – Part 2: inverse problem using a reduced problem, International Journal of Heat and Mass Transfer 48 (1) (2005), pp. 119–133.
   Web of Science ®Google Scholar
• Marshall, SA, 1966. An approximate method for reducing the order of a linear system, Control (1966), pp. 642–643.
   Google Scholar
• Aoki, M, 1968. Control of large-scale dynamic systems by aggregation, IEEE Transactions on Automatic Control 13 (3) (1968), pp. 246–253.
   Web of Science ®Google Scholar
• Litz, L, 1981. Order reduction of linear state space models via optimal approximation of the nondominant modes, North-Holland Publishing Company Large Scale System 2 (1981), pp. 171–184.
   Google Scholar
• Newman, AJ, Model reduction via the Karhunen–Loève expansion, Part I: an exposition. Presented at Technical Research Report of the Institute for Systems Research. 1996, (http://techreports.isr.umd.edu/TechReports/ISR/1996/TR_96-32/TR_96-32.phtml).
   Google Scholar
• Neveu, A, El-Khoury, K, and Flament, B, 1999. Simulation de la conduction non linéaire en régime variable: décomposition sur les modes de branche. Simulation of nonlinear non-steady-state conduction: expansion on the branch modes, International Journal of Thermal Sciences 38 (4) (1999), pp. 289–304.
   Web of Science ®Google Scholar
• Girault, M, and Petit, D, 2004. Resolution of linear inverse forced convection problems using model reduction by the Modal Identification Method: application to turbulent flow in parallel-plate duct, International Journal of Heat and Mass Transfer 47 (17–18) (2004), pp. 3909–3925.
   Web of Science ®Google Scholar
• Gill, PE, Murray, W, and Wright, MH, 1992. Practical Optimization. London, UK: Academic; 1992.
   Google Scholar
• Michaleris, P, Tortorelli, DA, and Vidal, C, 1994. Tangent operator and design sensitivity formulations for transient non-linear coupled problems with applications to elastoplasticity, International Journal of Numerical Methods in Engineering 37 (1994), pp. 2471–2499.
   Web of Science ®Google Scholar
• Tortorelli, DA, and Michaleris, P, 1994. Design sensitivity analysis: overview and review, Inverse Problems in Engineering 1 (1994), pp. 71–105.
   Google Scholar
• Woodbury, KA, 2002. Inverse Analysis Handbook. Boca Raton, FL: CRC Press; 2002.
   Google Scholar
• Favennec, Y, Labbé, V, and Bay, F, 2003. Induction heating processes optimization – a general optimal control approach, Journal of Computational Physics 187 (2003), pp. 68–94.
   Web of Science ®Google Scholar
• Favennec, Y, Labbé, V, and Bay, F, 2004. The ultraweak time coupling in nonlinear multiphysics modelling and related optimization problems, International Journal for Numerical Methods in Engineering 60 (2004), pp. 803–831.
   Web of Science ®Google Scholar
• Céa, J, 1971. Optimisation, théorie et algorithmes. Paris: Dunod; 1971.
   Google Scholar
• Larrouturou, B, and Lions, PL, 1994. Méthodes mathématiques pour les sciences de l’ingénieur: optimisation et analyse numérique. Cours de l’Ecole Polytechnique; 1994.
   Google Scholar
• Jarny, Y, Ozisik, MN, and Bardon, JP, 1991. A general optimization method using adjoint equation for solving multidimensional inverse heat conduction, International Journal of Heat and Mass Transfer 34 (11) (1991), pp. 2911–2919.
   Web of Science ®Google Scholar
• Minoux, M, 1986. Mathematical Programming – Theory and Algorithms. Chichester: John Wiley and Sons; 1986.
   Google Scholar