Adjoint shape sensitivities of blood flows considering non-Newtonian properties

This article discusses the derivation and numerical implementation of an adjoint system, to the primal Navier–Stokes equations, for the computation of shape sensitivities of ducted blood flows considering non-Newtonian fluid properties. The ever-growing advancements in blood flow simulations are, naturally, accompanied by an increased interest in the optimization of related medical devices. In the majority of the computational studies, the Newtonian assumption is used to describe the rheology of blood. While this assumption has been shown to satisfactorily capture the flow when it is governed by high shear rates, it falls short at low shear rates. A rich variety of viscosity models has been proposed to tackle this shortcoming. In this article we show how such models can be incorporated into an adjoint system targeting to produce the shape sensitivity which can be used by a gradient-based optimization method for the minimization of an objective functional. A general formulation of the adjoint equations is proposed, in which contributions of the non-Newtonian properties explicitly occur. The numerical implementation is discussed and the validity of the method is assessed by means of numerical experiments of steady blood flows in a 2D stenosed duct, where results are compared against second-order finite-difference (FD) studies. The proposed methodology is then applied to CAD-free, gradient-based shape optimizations of an idealized 3D arterial bypass-graft operating at three relevant Reynolds numbers. It is observed that the impact of the adjoint viscosity treatment is amplified in low shear-rate flow regimes while fades for higher shear-rates, analogous to its primal counterpart.