Robust shape optimisation with small uncertainties

Abstract:

In this talk we propose two approaches for dealing with small uncertainties in geometry and topology optimisation 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 linearise the considered cost function with respect to the uncertain parameters, then to consider the supremum function of the obtained linear approximation, whichcan be rewritten as a more `classical’ function of the design, owing to standardadjoint techniques from optimal control theory.

The resulting `linearised worst-case’ objective function turns out to be the sum of the initial cost and 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 optimise 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 optimisation problems for elastic structures in two space dimensions.

The talk is based on joint work with Charles Dapogny (LJK, Grenoble).

Posted on Jun 20, 2016 in Mathematical Mechanics, Seminars