  Parametric shape optimization aims at minimizing an objective function f(x)
where x are CAD parameters. This task is difficult when f is the output of an
expensive-to-evaluate numerical simulator and the number of CAD parameters is
large. Most often, the set of all considered CAD shapes resides in a manifold
of lower effective dimension in which it is preferable to build the surrogate
model and perform the optimization. In this work, we uncover the manifold
through a high-dimensional shape mapping and build a new coordinate system made
of eigenshapes. The surrogate model is learned in the space of eigenshapes: a
regularized likelihood maximization provides the most relevant dimensions for
the output. The final surrogate model is detailed (anisotropic) with respect to
the most sensitive eigenshapes and rough (isotropic) in the remaining
dimensions. Last, the optimization is carried out with a focus on the critical
dimensions, the remaining ones being coarsely optimized through a random
embedding and the manifold being accounted for through a replication strategy.
At low budgets, the methodology leads to a more accurate model and a faster
optimization than the classical approach of directly working with the CAD
parameters.
