Boundary control and shape optimization for the robust design of bypass anastomoses under uncertainty
1 Modelling and Scientific Computing,
Mathematics Institute of Computational Science and Engineering, École Polytechnique
Fédérale de Lausanne, Station 8, EPFL, 1015
2 MOX, Modellistica e Calcolo Scientifico, Dipartimento di Matematica F. Brioschi, Politecnico di Milano, Piazza Leonardo da Vinci 32, 20133 Milano, Italy.
3 Now at SISSA MathLab, International School for Advanced Studies, Via Bonomea 265, 34136 Trieste, Italy.
We review the optimal design of an arterial bypass graft following either a (i) boundary optimal control approach, or a (ii) shape optimization formulation. The main focus is quantifying and treating the uncertainty in the residual flow when the hosting artery is not completely occluded, for which the worst-case in terms of recirculation effects is inferred to correspond to a strong orifice flow through near-complete occlusion.A worst-case optimal control approach is applied to the steady Navier-Stokes equations in 2D to identify an anastomosis angle and a cuffed shape that are robust with respect to a possible range of residual flows. We also consider a reduced order modelling framework based on reduced basis methods in order to make the robust design problem computationally feasible. The results obtained in 2D are compared with simulations in a 3D geometry but without model reduction or the robust framework.
Mathematics Subject Classification: 35Q93 / 49Q10 / 76D05
Key words: Optimal control / shape optimization / arterial bypass grafts / uncertainty / worst-case design / reduced order modelling / Navier-Stokes equations
© EDP Sciences, SMAI, 2013