Optimization Methods
Summary
- Overview
- Aerodynamic Optimization
- Compexity of Real-Life Problems
- Splitting up the Problem
- Nature of the Search Space
- State-of-the-Art Methods
- Non-gradient Methods
In the last decade or so methods have been developed for treating intrinsically difficult minimization problems, including dis-joint and non convex search spaces. These methods, based on non-analytical approaches, have found their way in the design of aerodynamic components and systems.
When the solution is known to be in a neighborood of an optimum, gradient methods are widely used.
Several methods and computer codes are available, that come as black
boxes to be used in different problems. This class of methods clearly fails when a
radical new solution is sought. Hence the interest in this new class of methods.
The idea behind a `design' procedure is to locate a satisfactory solution in
the design space. A procedure is said to be efficient if the number of
solutions evaluated is small as compared to the size of the solution space.
The same procedure is said to be robust if it finds the minimum under
search spaces that feature a complex topology.
With the increasing complexity of engineering problems the job of optimization
becomes more challenging. Rather that designing the best component possible,
the quest is now for the best configuration that can operate efficiently over
a wide range of operating conditions.
In aerospace and turbomachinery design it is required to produce the best
configuration that satisfies at the same time aero- thermodynamic, aeroelastic,
mechanic, manifacturing, time and budget requirements.
In general, each engineering problem can be described by different subsets of
problems. Ideally, the subproblems are disjoint, e.g. they depend on free parameters
that do not appear in the larger problem. When this not occur, a decision has to be
made as to how to choose the parameters and their constraints. For the method to
produce good results the choice of the free parameters and of the objective function
is of the utmost importance.
When the search space is known to be non-convex, the optimum cannot be approached by
traditional gradient-based methods. Instead, the search may start from different
initial points, and proceed according to some heuristic procedures. Even then, no
guarantee exists of obtaining the global optimum.
Complex solution topologies (disjoint and non-convex) emerge when a problem is
defined by a very large number of parameters (typically, several dozens to several
hundreds, sometimes even more), with constraints on at least some of them.
In the attempt to overcome prohibitive complexity of large-scale problems,
some heuristic search mechanisms have been developed. These include:
Special pre-conditioning techniques may be used to smooth the objective function, so
that the search of a global minimum becomes more tractable.
One approach, called global continuation method, is based on a smoothing
technique, which transforms the objective function into a smoother function with
fewer local minima.
There is a staggering number of publications concerned with numerical optimization
methods. This is a field of research on its own.
Aerodynamic Optimization
Complexity of Real-Life Problems
Splitting up the Problem
Nature of the Search Space
State-of-the-Art Methods