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 Lausanne, Switzerland.
toni.lassila@epfl.ch; alfio.quarteroni@epfl.ch
² MOX, Modellistica e Calcolo Scientifico, Dipartimento di Matematica F. Brioschi, Politecnico di Milano, Piazza Leonardo da Vinci 32, 20133 Milano, Italy.
alfio.quarteroni@polimi.it
³ Now at SISSA MathLab, International School for Advanced Studies, Via Bonomea 265, 34136 Trieste, Italy.
andrea.manzoni@sissa.it; gianluigi.rozza@sissa.it

Received: 1 February 2012

Abstract

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.