Robust Shape Optimization Problem
The usual single setpoint aerodynamic shape optimization problem can be described in the following rather abstract form
min f(y, p) (14)
y, p
s. t. c(y, p) = 0 (15)
h(y, p) > 0 (16)
We think of equation (15) as the discretized outer flow equation around, e. g., an airfoil described by geometry parameter p e R”p. The vector y is the state vector (velocities, pressure,…) of the flow model (15) and we assume that (15) can be solved uniquely for y for all reasonable geometries p. The objective in (14) f : (y, p) ^ f (y, p) e К typically is the drag to be minimized. The restriction (16) typically denotes lift or pitching moment requirements. To make the discussion here simpler, we assume a scalar valued restriction, i. e., h(y, p) e R. The generalization of the discussions below to more than one restriction is straight forward. In contrast to previous papers on robust aerodynamic optimization, we treat the angle of attack as an fixed parameter which is not adjusted to reach the required lift (cf. e. g. Ref.[29], Ref.[21], Ref.[36] ).
The general deterministic problem formulation (1416) is influenced by stochastic perturbations. We assume that there are uncertain disturbances involved in the form of realvalued random variables з : Q ^ R (or random vectors) associated with a probability measure P with Lebesgue density ф : R ^ R+ such that the expected value of з can be written as
E (s) = s(Z)dP (Z) = хф(х)<іх
Q R
and the expected value of any function g : R ^ R is written as
E(g(s)) = / g(s(Z))dP(Z) = / g(xMx)dx
Q R
The dependence can arise in all aspects, i. e., a naive stochastic variant might be rewritten as
min f (y, p, s) (17)
y, p
s. t. c(y, p, s)= 0 (18)
h(y, p, s) > 0 (19)
This formulation still treats the uncertain parameter as an additional fixed parameter. The optimal solution should be stable with respect to stochastic variations in s. The literature can be classified in the following ideal classes: minmax formulation, semiinfinite formulation and chance constraints.
1.2 MinMax Formulations
The minmax formulation aims at the worstcase scenario.
minmax f (y, p, s(Z)) 
(20) 
y, p Zєn 

s. t. c(y, p,s(Z)) = 0, yZ є n 
(21) 
h(y, p,s(Z)) > 0, yZ є П 
(22) 
The minmax formulation is obviously independent of the stochastic measure P and thus needs only the realizations of the random variable з as input. If the probability density function of the uncertain parameter is not available, this approach could potentially be an attractive strategy. Otherwise, this formulaion will ignore problem specific information, if it is at hand and will lead to overly conservative designs. We do not treat this formulation furthermore in this paper.