| The optimization methods for the inverse heat conduction and solidification problems are discussed. Three different methods, the Tikhonov regularization method, the singular value decomposition (SVD) method, and the Levenberg-Marquardt method, are discussed and their performance is assessed comparatively in the inverse heat conduction problems. Several schemes for choosing the optimal regularization parameters are also discussed. These schemes include the maximum likelihood method (ML), the ordinary cross-validation method (OCV), the generalized cross-validation method (GCV), the discrepancy principle (DP), and the L-curve method. 2-D steady-state heat conduction problems are used for the case studies. Parameter estimation and function estimation for the optimal solution are also discussed and compared using 1-D transient heat conduction problems. In the inverse design solidification problems, on the other hand, the regularization method along with the L-curve method is discussed. The design algorithm is applied to determine the appropriate boundary heat flux distribution to obtain prescribed solid-liquid interfaces in a 2-D cavity. A new finite difference scheme for determining the sensitivity coefficients is proposed in the inverse steady-state solidification problems. A sequential method and a whole time-domain method are used and evaluated in the inverse design of solidification processes. |
