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3 Mathematical Aspects of CFD-based Optimization 63
rameter vector p, f : R n × R np → R denotes the objective function,
Eq. (3.2), c : R n × R np → R n the discretized CFD-model, Eq. (3.1) and
h : R n × R np → R m formulates the restrictions, Eq. (3.3). We arrive at the
same formulation also by use of a full space-time discretization of the unsteady
CFD-model, Eq. (3.1). Therefore, we use the formulation Eqs. (3.4)-
(3.6) for general discussions on CFD-model based optimization in Sect. 3.2
with emphasis on stationary optimization problems and discuss special issues
exploiting the structure of unsteady problems in Sect. 3.3.
Again, we assume that the states y are uniquely determined by the state
equation (3.5) if the influence vector p is given, which means that the Jacobian
∂c/∂y is nonsingular. The implicit function theorem ensures the existence of a
function φ such that y = φ(p) and one can reformulate the problem described
by Eqs. (3.4)-(3.6) in a so-called black-box fashion
min
p f(φ(p),p) (3.7)
s.t. h(φ(p),p) ≥ 0 . (3.8)
Formulation (3.7, 3.8) is chosen in straightforward implementations of standard
optimization techniques like genetic algorithms, the Nelder-Mead-Simplex
or simulated annealing. These techniques require a relatively small
amount of implementation effort, lead to a modular coupling of the optimization
task with the simulation task and are often robust with respect to
roughness in the objective function. However, since each evaluation of the
implicit function φ requires a full CFD simulation, these methods lead to an
overall computational effort which is several orders of magnitude higher than
a forward simulation of the model (3.1). Furthermore, the inequality conditions
(3.8), which are an integral and essential part of all realistic problem
formulations, still pose challenges for these techniques. Therefore, we discuss
simultaneous techniques for the direct solution of the constrained optimization
problem (3.4-3.6), which have the potential for high efficiency, leading
to an overall effort of only a small multiple of the forward simulation effort.
3.2 Simultaneous Model-based Optimization
3.2.1 Sequential Quadratic Programming (SQP)
For ease of presentation, we drop the inequalities in problem (3.4-3.6) and
lump together the variables as z := (y,p) in order to investigate the generic
problem