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

In order to obtain reduced models (RMs) from original detailed ones, the Modal Identification Method (MIM) has been developed for several years for linear and then for nonlinear systems. This identification is performed through the resolution of an inverse problem of parameter estimation. So far, the MIM was used with the Finite Difference Method (FDM) for the computation of the gradient of the functional to be minimized. This leads to important computation times. In order to save it up, the Adjoint Method (AM) has been coupled with the MIM to compute the gradient. All the equations related to the reduced model, the adjoint equations, the gradient and the optimization algorithm are clearly expressed. A test case involving a 3D nonlinear transient heat conduction problem is proposed. The accuracy of the identified RM is shown and the comparison between the proposed AM and the classical FDM shows the drastic reduction of computation time.

[1]  Daniel Petit,et al.  Identification methods in nonlinear heat conduction. Part II: inverse problem using a reduced model , 2005 .

[2]  E. Videcoq,et al.  The Use of Model Reduction and Function Decomposition for Identifying Boundary Conditions of A Linear Thermal System , 2003 .

[3]  Jean-Loup Chenot,et al.  A numerical model for induction heating processes coupling electromagnetism and thermomechanics , 2003 .

[4]  Andrew J. Newman,et al.  Model Reduction via the Karhunen-Loeve Expansion Part I: An Exposition , 1996 .

[5]  Daniel Petit,et al.  Identification methods in nonlinear heat conduction. Part I: Model reduction , 2005 .

[6]  D. Tortorelli,et al.  Tangent operators and design sensitivity formulations for transient non‐linear coupled problems with applications to elastoplasticity , 1994 .

[7]  Y. Favennec.  Comparison between the Discretized Continuous Gradient and the Discrete Gradient when Dealing with Nonlinear Parabolic Problems , 2005 .

[8]  D. Tortorelli,et al.  Design sensitivity analysis: Overview and review , 1994 .

[9]  J. Cea.  Optimisation : théorie et algorithmes , 1971 .

[10]  L. Litz,et al.  Order Reduction of Linear State-Space Models Via Optimal Approximation of the Nondominant Modes , 1980 .

[11]  François Bay,et al.  The ultraweak time coupling in non-linear multiphysics modelling and related optimization problems , 2004 .

[12]  Hamou Sadat,et al.  A MESHLESS FORMULATION FOR THREE-DIMENSIONAL LAMINAR NATURAL CONVECTION , 2002 .

[13]  M. Girault,et al.  Resolution of linear inverse forced convection problems using model reduction by the Modal Identification Method: application to turbulent flow in parallel-plate duct , 2004 .

[14]  Keith A. Woodbury,et al.  Inverse Engineering Handbook , 2002 .

[15]  M. Aoki.  Control of large-scale dynamic systems by aggregation , 1968 .

[16]  Philip E. Gill,et al.  Practical optimization , 1981 .

[17]  Daniel Petit,et al.  Model reduction for the resolution of multidimensional inverse heat conduction problems , 2001 .

[18]  David K. Smith,et al.  Mathematical Programming: Theory and Algorithms , 1986 .

[19]  Bernard Flament,et al.  Simulation de la conduction non linéaire en régime variable: décomposition sur les modes de branche , 1999 .

[20]  Yvon Jarny,et al.  A General Optimization Method using Adjoint Equation for Solving Multidimensional Inverse Heat Conduction , 1991 .

[21]  François Bay,et al.  Induction heating processes optimization a general optimal control approach , 2003 .