Conjugate gradient method (CGM) is developed for the estimation of surface heat flux in two-dimensional variable geometry heat conduction problem from temperature measurements.In typical exemplified case,it is shown that CGM is feasible and robust.Further studies reveal that estimated results are influenced by number of measurement points and initial estimation.The number of measurement points are related to intensity of heat flux spatial variation.And the final heat flux should be chosen as the initial estimation.

Temperature-dependent thermal conductivity k(T)and specific heat C(T)are estimated simultaneously.At first,k(T) and C(T)are rewritten as functions of position and time,k(x,t)and C(x,t),and the gradient of cost function is deduced with adjoint equation method(AEM).Temperature is divided into segments and thermal conductivity and specific heat are expressed as constant in each segment.Gradients of cost function with respect to k(T)and C(<T)are further deduced from gradients of cost function with respect to k(x,t)and C(x,t).With a typical inversion problem,it is shown that the inversion method is feasible,effective and not too sensitive to measurement noise.

A typical inverse heat conduction problem(IHCP) is converted to an optimization problem.An inversion algorithm of adjoint equation method(AEM) is developed.In order to treat ill-posedness of IHCP,a stop criteria used in iterative regularization method is employed.In a typical case,the inversion method of AEM is feasible and not very sensitive to measurement noise as noise level is relatively low.As measurement noise presents,difference between estimated thermal conductivities and exact values at boundaries x=0 and x=L is significant,which is due to the fact that gradient of objective function with respect to the change of thermal conductivity at boundaries is relatively small.Moreover,further research shows that as noise level rises,significant difference between estimated and exact values of thermal conductivity presents and initial value has great influence on estimated values.