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.

Classification: 35Q93, 49Q10, 76D05

Keywords: optimal control, shape optimization, arterial bypass grafts, uncertainty, worst-case design, reduced order modelling, Navier-Stokes equations

@article{M2AN_2013__47_4_1107_0, author = {Lassila, Toni and Manzoni, Andrea and Quarteroni, Alfio and Rozza, Gianluigi}, title = {Boundary control and shape optimization for the robust design of bypass anastomoses under uncertainty}, journal = {ESAIM: Mathematical Modelling and Numerical Analysis - Mod\'elisation Math\'ematique et Analyse Num\'erique}, publisher = {EDP-Sciences}, volume = {47}, number = {4}, year = {2013}, pages = {1107-1131}, doi = {10.1051/m2an/2012059}, mrnumber = {3082291}, language = {en}, url = {http://www.numdam.org/item/M2AN_2013__47_4_1107_0} }

Lassila, Toni; Manzoni, Andrea; Quarteroni, Alfio; Rozza, Gianluigi. Boundary control and shape optimization for the robust design of bypass anastomoses under uncertainty. ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique, Volume 47 (2013) no. 4, pp. 1107-1131. doi : 10.1051/m2an/2012059. http://www.numdam.org/item/M2AN_2013__47_4_1107_0/

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