Calculation of sensitivity derivatives in thermal problems by finite differences

The transient thermal analysis of large aerospace vehicles such as the space shuttle requires models with thousands of degrees-of-freedom. The design of thermal protection systems for such vehicles requires the repetitive thermal analysis of the vehicle and the calculation of sensitivities—derivatives of the transient response with respect to design parameters. When the number of design parameters is large, the cost of the sensitivity calculations may become prohibitive. The present work is concerned with reducing the computational cost and errors in sensitivity calculations with one of the simplest available methods—the finite difference approach. A technique for calculating simultaneously the nominal and perturbed solutions required for the finite difference derivative is proposed. An expression for the optimum value of the perturbation parameter is obtained which permits reducing the computational error. An insulated cylinder is used as an example which demonstrates that significant improvements in accuracy and efficiency may be obtained.