Optimization and Computational Fluid Dynamics - Department of ...

268 J. Hämäläinen, T. Hämäläinen, E. Madetoja, H. Ruotsalainen
shape of a domain is not known a-priori. Instead, a shape is sought such that
the cost function is minimized (or maximized), and the state equation and
possible constraints are fulfilled. The cost function, also known as the objective
function, measures the quality of the shape. Typically, it is formulated
such that its minimum (or maximum) value corresponds to the best possible
shape (optimal shape). The state equation is a model describing the physical
phenomenon to be studied, as in the case of the Navier-Stokes equations,
for example. The constraints ensure that the optimal shape is reasonable,
for instance, by setting limits to total material usage in solid mechanical
problems, minimum and maximum material thicknesses. Optimal control is
a special case of shape optimization. Instead of the shape of the domain, the
boundary data, for example, are now a-priori unknown. A typical optimal
control problem related to CFD is the search for an optimal velocity profile
at an inlet or optimal heat flux through a wall.
Optimal shape design is said to have originated with Hadamard in 1910
[9], who first supplied a formula for a partial differential equation in order
to evaluate the change due to a boundary modification of the domain. The
first engineering applications were in solid mechanics. CFD emerged through
potential flows, Euler equations, and finally viscous Navier-Stokes equations
including turbulence models. For further information, see [18, 29, 32] and the
bibliographical studies cited there.
In traditional optimal shape design or optimal control problems, there is
only one objective (also known as cost function) to be optimized. However,
everyday engineering problems typically involve several conflicting objectives
that should be achieved simultaneously. A single objective function can be
derived from multiple functions, as a weighted sum, for example, but then the
practical relevance of the cost function values may be lost. Instead, it is more
reasonable to handle multiple objectives as they arise naturally from engineering
problems, by using methods of multi-objective optimization [26, 34]. The
importance of multi-objective optimization becomes even more significant
when dealing with large industrial processes and taking into account consecutive
unit-processes, raw material, energy consumption, and end-product
quality and price, each of which have their own objectives. Applications of
multi-objective optimization are not only necessary to handle engineering
problems, but will play an important role also in decision support systems in
the future.
Computing capacity is always limited which leads to compromises between
accuracy, scope and computing time as illustrated in Fig. 9.1. Naturally, there
is a need for detailed small-scale modeling such as Direct Numerical Simulation
(DNS) which requires high performance computing even without any
optimization. But the detailed models cannot be coupled with optimization
if total computational (CPU) time is to be kept reasonable. When a new
product is being designed, a relatively long CPU time, days or even weeks,
is considered acceptable. Therefore, the optimal shape design can be used
based on a complex CFD model. But when a fast response, in seconds or