Optimal solutions are fragile and
should be generally avoided. This unpopular statement enjoys substantial
practical and philosophical argumentation and now, thanks to
complexity, we can be even more persuasive. However, this short note is
about making optimisation a bit more easy. If you really insist on
pursuing optimality, there is an important point to keep in mind.

Let us examine the case illustrated below: designing a
composite conical structure in which the goal is to keep mass under
control, as well as the fluxes and axial and lateral frequencies. The
System Map shown below reflects which (red) design variables (ply
thicknesses) influence the performance of the structure in the nominal
(initial) configuration, prior to optimisation. In addition, the map
also illustrates how the various outputs (blue nodes) relate to each
other.

In fact, one may conclude that, for example, the following relationships exist:

- t_20013 - Weight
- Weight - Axial frequency
- Min Flux - Max Flux
- t_20013 controls the Lateral Frequency
- t_20013 also controls the Axial Frequency
- Lateral Frequency and Axial Frequency are related to each other
- etc.

As one may conclude, the outputs are tightly coupled:
if you change one you cannot avoid changing the others. Let's first see
how optimisation is handled when one faces multiple - often conflicting
- objectives:

minimise y = COST (y_1, y_2, ..., y_n) where y_k
stands for the k-th performance descriptor (e.g. mass, stiffness, etc.).
In many cases weights are introduced as follows:

minimise y = COST (w_1 * y_1, w_2 * y_2, ..., w_n * y_n).

The fundamental problem with such a formulation (and
all similar MDO-type formulations) is that the various performance
descriptors are often dependent (just like the example above indicates)
and the analysts doesn't know. The cost function indicated above is a
mathematical statement of a conflict, whereby the y's compete for
protagonism. This competition is driven by an optimisation algorithm
which knows nothing of the structure of the corresponding System Map and
of the existence of the relationships contained therein. Imagine, for
example that you are trying to reduce a variable (e.g. mass) and
increase, at the same time another (e.g. frequency). Suppose also that
you don't know that these two variables are strongly related to
each other: the relationship looks typically like this: f = SQRT(k/m).
Here, f and m, outputs of the problem, are related - changing one
modifies the other. This is inevitable. In a more intricate situation,
in which hundreds of design variables are involved, along with tens or
hundreds of performance descriptors, the problem really becomes
numerically tough. The optimisation algorithm has a very hard time. What
is the solution?

If you cannot avoid optimisation, the we suggest the following approach:

- Define your baseline design.
- Run a Monte Carlo Simulation, in which you randomly perturb the design variables (inputs).
- Process the results using OntoSpace, obtaining the System Map.
- Find the INDEPENDENT outputs (performance descriptors), or, in the case the aren't any, those outputs which have the lowest degree in the System Map. There are tools in OntoSpace that actually help to do this.
- Build your cost function using only those variables, leaving the others out.

This approach "softens" the problem from a numerical
point of view and reduces the mentioned conflicts between output
variables. Attempting to formulate a multi-disciplinary problem without
knowing a-priori how the various disciplines interact (i.e. without the
System Map) is risky, to say the least.

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