Computation of Second-order Design Sensitivities for Steady State Incompressible Laminar Flows Using the Extended Complex Variables Method

In the current paper, the general procedure of the first and second-order sensitivity analysis is presented using the extended complex variables method (ECVM). In the traditional complex variables method, only the imaginary step is used for sensitivity analysis. However, in the ECVM, both of the real and imaginary parts are employed to improve the efficiency of the method. To show this, the ECVM is applied to the steady state incompressible laminar flow around a cylinder. The governing Navier-Stokes equations are solved by the finite element method and then the ECVM is employed. The results are validated through comparing with those of obtained by an analytical as well as the finite difference methods and the convergence rate is investigated. It is illustrated that the first-order sensitivity analysis is not influenced by the change of the step length for both of the traditional and extended complex variables methods. However, it is shown that unlike the traditional complex variables method, the ECVM is less dependent on the step size for calculating the second-order sensitivity. This can be considered as an enhancement in the efficiency of this method. Hence, the ECVM is suggested as an appropriate technique for calculating simultaneously the first and second-order sensitivities with high accuracy as well as low computational cost. The proposed method is applicable to a wide range of problems having simple or complex parameters.