Optimization and Computational Fluid Dynamics - Department of ...

150 René A. Van den Braembussche
attention in recent years. In what follows one will describe the fast, multipoint
and multidisciplinary optimization method, developed at the von Kármán
Institute, and its application to different turbomachinery designs.
6.2 Optimization Methods
Optimization methods attempt to determine the design variables Xi(i =1,n)
that minimize an objective function OF(U(Xi),Xi)whereU(Xi)isthesolution
of the flow equations R(U(Xi),Xi) = 0 and subject to nA performance
constraints Aj(U(Xi),Xi) ≤ 0(j =1,nA) andnG geometrical constraints
Gk(Xi) ≤ 0(k =1,nG).
Specifying as Objective Function (OF) the difference between a prescribed
and calculated pressure distribution results in an inverse design method. Aiming
for the improvement of the overall performance leads to a global optimization
technique.
Numerical optimization procedures consist of the following components,
described in more detail in the following paragraphs:
• a choice of independent design parameters and the definition of the addressable
part of the design space.
• definition of an OF quantifying the performance. Any standard analysis
tool can be used to calculate the components such as lift, drag, efficiency,
mass flow, manufacturing cost or a combination of all of them.
• a search mechanism to find the optimum combination of the design parameters,
i.e., the one corresponding to the minimum of the OF.
6.2.1 Search Mechanisms
There are two main groups of search mechanisms:
• Analytical ones calculate the required geometry changes in a deterministic
way from the output of the performance evaluation. A common one is
the steepest descent method approaching the area of minimum OF by
following the path with the largest negative gradient on the OF surface
(Fig. 6.2). This approach requires the calculation of the direction of the
largest gradient of the OF and the step length. A comprehensive overview
of gradient based optimization techniques is given by Vanderplaats [17].
• Zero-order or stochastic procedures require only function evaluations.
They make a random or systematic sweep of the design space or use evolutionary
theories such as Genetic Algorithms (GA) or Simulated Annealing
(SA).