反演二维瞬态热传导问题随温度变化的导热系数

Identification of Temperature-Dependent Thermal Conductivity for 2-D Transient Heat Conduction Problems

基于边界元法反演二维瞬态热传导问题随温度变化的导热系数.采用Kirchhoff变换将非线性的控制方程转变为线性方程.边界元法用于构建二维瞬态热传导问题的数值分析模型.将反演参数作为优化变量,测点温度计算值与测量值之间的残差平方和作为优化目标函数.引入复变量求导法求解目标函数的梯度矩阵,梯度正则化法用于优化目标函数获得反演结果.探讨时间步长、测点数量和随机偏差对反演结果的影响.减小步长、增加测点数量收敛速度加快.降低了随机偏差,计算结果更精确.算例证明了算法的有效性与稳定性.

The temperature-dependent thermal conductivity was identified for 2-D transient heat conduction problems with the boundary element method （BEM）. The nonlinear governing equations were transformed into linear ones through the Kirchhoff transformation. The BEM was used to build the numerical model for the 2-D transient heat conduction problems. The in- version parameters were defined as the opthulzation variables. The quadratic sum of residual errors between the calculated temperature values and the measured temperature values at the measuring points was regarded as the objective function. The complex variable differentiation method was employed to compute the gradient matrix of the objective function. The gradientregulation method was developed to optimize the objective function. Effects of the time step length, the number of measuring points and the random noise on the inverse results were discussed. With decrease of the time step length and increase of the number of the measuring points, the converging rate quickens. With decrease of the random noise, the results grow accurate. Numerical tests show the effectiveness and stability of the presented method.