Robust shape optimization with small uncertaintie.

Abstract: In this talk, we propose two approaches for dealing with small uncertainties
in geometry and topology optimization of structures. Uncertainties occur in
the loadings, the material properties, the geometry or the imposed vibration
frequency. A first approach, in a worst-case scenario, amounts to linearize
the considered cost function with respect to the uncertain parameters, then
to consider the supremum function of the obtained linear approximation, which
can be rewritten as a more `classical’ function of the design, owing to standard
adjoint techniques from optimal control theory.
The resulting `linearized worst-case’ objective function turns out to be the sum of the initial cost functionand of a norm of an adjoint state function, which is dual with respect to the considered norm over perturbations.

A second approach considers objective functions which are mean values, variances or
failure probabilities of standard cost functions under random uncertainties.
By assuming that the uncertainties are small and generated by a finite number
$N$ of random variables, and using first- or second-order Taylor expansions,
we propose a deterministic approach to optimize approximate objective functions.
The computational cost is similar to that of a multiple load problems where the
number of loads is $N$.

We demonstrate the effectiveness of both approaches on various parametric and geometric optimization problems for elastic structures in two space dimensions.

Posted on Dec 16, 2022 in Optimization and Equilibrium, Seminars