Optimization and Computational Fluid Dynamics - Department of ...

62 H.G. Bock and V. Schulz
3.1 Introduction
Computational fluid dynamics (CFD) models can be written in the general
abstract form
F(˙y(t),y(t),u(t),p,t)=0, t ∈ [0,T] (3.1)
where we use the following nomenclature:
t time within horizon [0,T];
y(t) state vector (dependent variables);
˙y(t) velocity (time derivative) of state vector (dependent variables);
u(t) control vector (independent variables);
p ∈ R np finite-dimensional parameter vector (independent variables).
The variables y(t) and ˙y are functions defined on a spatial domain, the control
vector u(t) are usually functions on (a subset of) the spatial domain
or its boundary. F is a suitable differential operator that includes transport
and diffusion as well as source terms and boundary conditions. We assume
that Eq. (3.1) is an initial-boundary value problem that defines the states
y(t) uniquely, if u(t), p and initial data y(0) are given. In CFD-model based
optimization (CFD-O), we want to determine p and the control function u(t)
in such a way that a scalar objective function such as
J(y,u,p):=
� T
0
j(y(t),u(t),p,t)dt (3.2)
is minimized. The objective function can have different forms in process control,
shape optimization, inverse modeling/parameter estimation or optimum
experimental design. Usually, there arise additional constraints for the state
and control functions in terms of inequalities
r(y(t),u(t),p,t) ≥ 0 (3.3)
modeling technical restrictions. For a sizeable part of the discussion in this
paper, it is convenient to choose an even more abstract formulation of the
optimization problem above.
If we consider stationary problems (i.e., ∂F/∂ ˙y = 0 in Eq. (3.1)) and
choose a spatial discretization for the steady state y and the steady control
u, we can reformulate the problem as
min f(y,p) (3.4)
y,p
s.t. c(y,p) = 0 (3.5)
h(y,p) ≥ 0 . (3.6)
Here, y ∈ R n is the discretized state vector, the influence vector p ∈ R np
now summarizes the discretized control u and the formerly denoted pa-