INTRODUCTION
The general optimization problem can be stated as the problem of minimizing a scalar function
subject to some constrains g(x,u). Note that g(x,u) is a vector function. In the above x represents the state variable vector and u the design variable vector. It is
and
where 
and 
.
In the airfoil optimization or in general in the aerodynamic design of flying bodies usually the f(x,u) function represents scalar functions such as the Drag, the Drag to Lift ratio, the negative of the Lift etc. The constraint function g(x,u) represents the governing equations of the flow. Usually they are the Euler equations for inviscid flow or the Navier-Stokes equation for viscous flow. Research at ICAM (Interdisciplinary Center for Applied Mathematics) focuses at solving the optimization problem using the Sensitivity Equation Method. We refer to the sensitivity as the partial derivatives of the conserved or primitive variables of the flow (pressure, density, velocity, energy etc.) with respect to some parameter of the flow such as angle of attack, airfoil thickness, Mach-number etc. It was shown numerically that when we have a stationary flow around an airfoil with constant conditions then the flow resulting from a small perturbation in the parameters of the flow can be approximated by a Taylor series. That means that the perturbed flow solution equals the basic flow plus the product of the sensitivity of the flow times the perturbation of the parameter. Mathematically the above can be written as
It is clear that the above is an efficient
and very fast method on getting perturbed flow (when the sensitivities
are known) solution because all it requires is a matrix multiplication
and an addition. A designer can use the above method to visualize a flow
for conditions other than the design conditions. This allows him to early
identify regions where the design has limited performance and make the
appropriate changes.
Objective of the project was to develop a tool that will allow a designer to peturb some of the parameters of the flow and by using the above mentioned Taylor decomposition to obtain very fast the new flow. Emphasis was given to the design of a user friendly graphical interface.
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