Computation of First and Second-Order Sensitivities for Steady State Incompressible Laminar Flow Using Extended Complex Variables Method

In this paper, extended complex variables method (ECVM) is presented in fluid flow problems for the first and second-order sensitivity analysis. The finite element method is used to solve the Navier-Stokes equations, and the complex variables method is implemented to it. In the complex variables method, a complex step that only includes the imaginary part is used, but in its development, it uses a complex step that includes both the imaginary part and the real part to achieve higher performance. In the first-order sensitivity calculation, the results are not dependent on the step size, but in the second-order sensitivity, the results of the sensitivity depending on the step size and inevitably the developed formulas should be used to obtain higher accuracy. The proposed method is first validated for a problem with a closed-form solution, and the convergence rate is investigated and, then, applied to a uniform flow past a cylindrical cylinder and, finally, the results are compared by finite difference method. The results show that the range of accuracy for second-order sensitivity in the extended complex variable method is doubled compared to the complex variable method and it can be reduced to 10-12. It means that the effectiveness of the proposed method has increased. The introduced method is applicable to a wide range of problems with simple and complex parameters.