A new approach to speed up the conjugate gradient method by applying the complex variable differentiation method

Abstract

ABSTRACTThis study represents a novel approach to speed up the solution of nonlinear inverse heat conduction problems (IHCPs) by the implementation of the complex variable differentiation method (CVDM). A difficulty appeared in the solution of nonlinear problems is the great amount of computational time. To handle this problem, a new scheme is introduced to improve the conjugate gradient method (CGM). The main contribution in the improvement of conventional CGM is the simultaneous solution of direct and sensitivity problems by applying the complex variable method. The derivation of sensitivity problem by utilizing the analytic derivative is difficult or impossible, since the direct problem is complicated mathematically in many cases. By this approach, the analytic derivation of sensitivity problem is circumvented, while the Jacobian matrix components are obtained accompanied by the solution of the direct problem. Therefore, the developed scheme results in the reduction of mathematical manipulations. Due to the high nonlinearity of moving boundary inverse problems, the ablation problem is considered as the benchmark to examine the accuracy and effectiveness of the proposed scheme. The simulated results illustrate that the developed scheme has the potential to significantly reducing of the computational expenses while maintaining the quality of numerical solution.