Computation of sensitivity derivatives of Navier-Stokes equations using complex variables

Accurate computation of sensitivity derivatives is becoming an important item in Computational Fluid Dynamics (CFD) because of recent emphasis on using nonlinear CFD methods in aerodynamic design, optimization, stability and control related problems. Several techniques are available to compute gradients or sensitivity derivatives of desired flow quantities or cost functions with respect to selected independent (design) variables. Perhaps the most common and oldest method is to use straightforward finite-differences for the evaluation of sensitivity derivatives. Although very simple, this method is prone to errors associated with choice of step sizes and can be cumbersome for geometric variables. The cost per design variable for computing sensitivity derivatives with central differencing is at least equal to the cost of three full analyses, but is usually much larger in practice due to difficulty in choosing step sizes. Another approach gaining popularity is the use of Automatic differentiation software (such as ADIFOR) to process the source code, which in turn can be used to evaluate the sensitivity derivatives of preselected functions with respect to chosen design variables. In principle, this approach is also very straightforward and quite promising. The main drawback is the large memory requirement because memory use increases linearly with the number of design variables. ADIFOR software can also be cumbersome for large CFD codes and has not yet reached a full maturity level for production codes, especially in parallel computing environments. Another viable methodology for computing sensitivity derivatives is based on the adjount approach. This method has received a lot of publicity in recent years due to the pioneering work of Jameson and colleagues, who had focused their attention primarily on inviscid flows. A few applications of this approach to Navier-Stokes equtions have also become available in the last couple of years. The adjoint approach offers great savings and lower memory penalty for a large number of design variables. The major disadvantage of this method is that a different set of adjoint equations must be derived and solved for each different cost function. The overall process of deriving the adjoint equations to achieve compatibility with the finite-difference equations and the corresponding boundary conditions is tedious and time consuming. An alternate approach based on the use of complex variable expansions for computing sensitivity derivatives, was suggested by Lyness and Lyness and Moler. For some unknown reasons, such as the inability of compilers to deal effectively with complex arithmetic, this technique has not been exploited much until Squire and Trapp revived it. Anderson and colleagues have recently made use of complex variables to evaluate sensitivity derivatives with an unstructured grid Navier-Stokes flow solver. This technique is quite straightforward to apply and produces accurate sensitivity derivatives witho