A Modified Conjugate Gradient Method for Transient Nonlinear Inverse Heat Conduction Problems: A Case Study for Identifying Temperature-Dependent Thermal Conductivities

Abstract

A M o d ifie d C o n ju g a te G ra d ie n t M e th o d fo r T ra n s ie n t N o n lin e a rIn v e rs e H e a t C o nduction P ro b le m s : A C ase S tudy fo r Id e n tify in g T e m p e ra tu re -D e p e n d e n t T h e rm a l C o n d u c tiv itie s Despite numerous studies of conjugate gradient methods (CGMs), the "sensitivity prob lem" and the "adjoint problem" are inevitable for nonlinear inverse heat conduction problems (IHCPs), which are accompanied by some assumptions and complicated differ entiating processes. In this paper, a modified CGM (MCGM) is presented for the solution o f a specified transient nonlinear IHCP, to recover temperature-dependent thermal con ductivities for a case study. By introducing the complex-variable-differentiation method (CVDM) for sensitivity analysis, the sensitivity problem and the adjoint problem are cir cumvented. Five test examples are given to validate and assess the performance of the MCGM. Model DescriptionTo illustrate the methodology of the inverse algorithm proposed in this work, a case study for identifying temperature-dependent thermal conductivities is presented below.

A new inverse analysis method based on a relaxation factor optimization technique for solving transient nonlinear inverse heat conduction problems

“…In recent years, the first author and the co-authors have been focused on the accurate calculation of sensitivity coefficients in gradient-based methods [3,[23][24][25][26][27]. We introduced the com plex-variable-differentiation method (CVDM) [28] into inverse heat transfer problems, and have successfully overcome the difficulty in accurately calculating sensitivity coefficients.…”

“…Generally, inversion methods can be classified into two categories: one is the gradient-based method and the other is the stochastic method [3,22]. The advantage of the stochastic methods is their capability in searching for the global optimum.…”

“…The advantages of the gradient-based methods are the fast convergence speed and the high accuracy. In gradient-based methods, the determination of sensitivity coefficients [3,23] is the main work. However, sensitivity coefficients are difficult to be precisely calculated if multi-dimensions, transient states and nonlinearities are involved, and this is the main reason why so many stochastic methods have been developed.…”

“…Consequently, the solution of a transient nonlinear inverse heat conduction problem is difficult, especially if multi-dimensions and measurement errors are concerned. IHCPs have been extensively studied [3][4][5][6][7][8][9][10][11][12][13][14] and many methods have been proposed [15][16][17][18][19][20][21].…”

“…IHCPs are ill-posed/conditioned, which are very challenging, because small disturbances of input conditions or measurement errors can cause large solution errors and oscillations. The objective of IHCPs is to identify the boundary or initial conditions, thermo-physical properties or geometric parameters by using additional information such as temperature measurements within the domain or on the boundary [3]. Transient states and temperature-dependent thermo-physical properties are usually involved in IHCPs.…”