A Modified Levenberg-Marquardt Algorithm For Solving Transient Non-linear Inverse Heat Conduction Problems

The inverse heat conduction problem (IHCP) has a wide range of significant applications in many practical projects such as metallurgy, chemical, material and aerospace engineering. It's very difficult to solve IHCPs due to the ill-posedness. In order to overcome the ill-posedness, a variety of methods are proposed by researchers to find the solutions of IHCPs. In general, these methods can be divided into two categories, the stochastic and the gradient-based. The stochastic methods can achieve global convergence, but such methods need large computational amount with low computational efficiency. Compared with the stochastic methods, gradient-based methods can ensure not only high computational efficiency but also high computational accuracy. However, sensitivity matrix coefficients need to be correctly calculated in gradient-based methods.Levenberg-Marquardt algorithm is a typical gradient-based algorithm, which is often used to solve IHCPs in engineering.In this thesis, a modified Levenberg-Marquardt algorithm using complex variable derivation method to calculate sensitivity coefficients is firstly constructed, and two examples are given to show the advantage of the complex-variable-differentiation-method over the finite difference method at calculating the first order derivatives. Then, a numerical example is given to validate the good performance of the modified Levenberg-Marquardt algorithm. A two-dimensional transient nonlinear inverse heat conduction problem is taken as an example to identify the boundary heat flux with a functional form. The sensitivity analysis, accuracy, efficiency, stability and robustness of the modified method are discussed and compared with the results of the traditional LM algorithm. Finally, the conclusion is given.The present work provides a theoretical basis for the identification of the boundary heat flux for an aircraft. The method can be also used for the identification of temperature-dependent thermal physical properties for thermal protection system (TPS) materials.