We review the optimal design of an arterial bypass graft following either a (i) boundaryoptimal control approach, or a (ii) shape optimization formulation. The main focus isquantifying and treating the uncertainty in the residual flow when the hosting artery isnot completely occluded, for which the worst-case in terms of recirculation effects isinferred to correspond to a strong orifice flow through near-complete occlusion.Aworst-case optimal control approach is applied to the steady Navier-Stokes equations in 2Dto identify an anastomosis angle and a cuffed shape that are robust with respect to apossible range of residual flows. We also consider a reduced order modelling frameworkbased on reduced basis methods in order to make the robust design problem computationallyfeasible. The results obtained in 2D are compared with simulations in a 3D geometry butwithout model reduction or the robust framework.

The goal of this paper is to study the so-called worst-case or robustoptimal design problem for minimal compliance. In the context of linear elasticity we seek an optimal shape which minimizes the largest, or worst,compliance when the loads are subject to some unknown perturbations.We first prove that, for a fixed shape, there exists indeed a worst perturbation (possibly non unique) that we characterize as the maximizer of a nonlinear energy. We also propose a stable algorithm tocompute it. Then, in the framework of Hadamard method, wecompute the directional shape derivative of this criterion which isused in a numerical algorithm, based on the level set method,to find optimal shapes that minimize the worst-case compliance.Since this criterion is usually merely directionally differentiable,we introduce a semidefinite programming approach toselect the best descent direction at each step of agradient method. Numerical examples are given in 2-d and 3-d.