Design sensitivities analysis for transient incompressible laminar flows using extended complex variable method

In the present study, the extended standard complex variable method (ESCVM) was proposed as a robust method for calculating the analysis of design sensitivities, including the first and second order. The standard complex variables method (SCVM) uses only the imaginary step for sensitivity analysis. In contrast, the presented method applies both the imaginary and the real part to enhance the effectiveness of the procedure. To illustrate this, the ESCVM is employed for the transited laminar incompressible flow. The Navier-Stokes equations are solved using finite element analysis and the developed SCVM was then applied to them. It has been shown that the first order (FO) sensitivity analysis is less susceptible to variations in the step size for both the standard and extended SCVM. However, it is evident that, unlike the SCVM, the extended SCVM was less dependent on the step size in estimating the second-order sensitivity (SO). This ability can be seen as an improvement in the efficiency and robustness of the extended standard method for complex variables.