Inverse problem of controlling the interface velocity in Stefan problems by conjugate gradient method

Abstract

The conjugate gradient method of minimization with adjoint equation is used successfully to solve the inverse problem in estimating an appropriate boundary control function such that the phase front moves at a desired velocity in the Stefan problem.

It is assumed that no prior information is available on the functional form of the unknown control function, therefore, it is classified as the function estimation in inverse calculation. The stability and accuracy of the inverse analysis using present algorithm are examined by comparing the results of the previous work by Voller [12].

Results show that the estimated control function by using conjugate gradient method did not exhibit oscillatory behavior in the inverse calculations for a broad range of front velocity while in [12] the inverse solutions are very sensitive to phase front velocity, therefore the application of future time stepping [2] is necessary in [12].

The advantage of applying this algorithm in inverse analysis lies in its stability as compared to the conventional minimization process [12]. Artificial future time stepping is unnecessary during the inverse calculation, since it is still an uncertainty in the inverse analysis. Furthermore, the inverse solutions obtained by the present method are found to be more accurate than the solutions obtained by the conventional minimization process.