二阶非定常多宗量热传导反问题的正则解

引入Bregman距离函数及其加权函数作为正则项，应用Tikhonov正则 化方法，对二阶非定常多宗量热传导反问题进行求解. 利用测量信息和计算信息构造最小二 乘函数，将多宗量反演识别问题转化为一个优化问题. 空间上采用8节点等参元进行离散， 时域上采用时域精细算法进行离散，建立了二阶非定常多宗量热传导问题的有限元正/反演数 值模型. 该模型不仅考虑了非均质和参数分布的影响，而且也便于正反演问题的敏度分析， 可对导热系数和边界条件等宗量进行有效的单一和组合识别. 给出了相关的数值验证，对信 息测量误差以及不同正则项的计算效率作了探讨. 数值结果表明，该方法能够对二阶非定常 多宗量热传导反问题进行有效的求解，并具有较高的计算精度.

A REGULARIZED SOLUTION OF INVERSE SECOND-ORDER TRANSIENT HEAT CONDUCTION PROBLEMS WITH MULTI-VARIABLES

In the present work, Tikhonov's regularization approach has been used to solve inverse two-order transient heat conduction problems with multi-variables, using Bregman distances and weighted Bregman distances in the construction of regularization terms for the Tikhonov's function. The inverse problem is formulated implicitly as an optimization problem with the cost functional of squared residues between calculated and measured quantities.The eight-point finite element is used for the discretization in the space system and a time stepping scheme is used for transient analysis. A finite element model is given, facilitating to sensitivity analysis for direct and inverse problems, and taking account of inhomogeneity and parameters distribution. Combined identifications can be carried out for thermal parameters and boundary conditions etc. Satisfactory numerical validation is given including a preliminary investigation of effect of noise data on the results and the computational efficiency for different regularization terms. Results show that the proposed method can identify single and combined thermal parameters and boundary conditions for two-order transient heat conduction problems with precision.