Shape optimization in steady blood flow: A numerical study of non-Newtonian effects

Abstract

We investigate the influence of the fluid constitutive model on the outcome of shape optimization tasks, motivated by optimal design problems in biomedical engineering. Our computations are based on the Navier-Stokes equations generalized to non-Newtonian fluid, with the modified Cross model employed to account for the shear-thinning behavior of blood. The generalized Newtonian treatment exhibits striking differences in the velocity field for smaller shear rates. We apply sensitivity-based optimization procedure to a flow through an idealized arterial graft. For this problem we study the influence of the inflow velocity, and thus the shear rate. Furthermore, we introduce an additional factor in the form of a geometric parameter, and study its effect on the optimal shape obtained.

Effect of fluid dynamics and device mechanism on biofluid behaviour in microchannel systems: Modelling biofluids in a microchannel biochip separator

“…Parameter values are as follows [25,26] The Carreau-Yasuda model shows that non-Newtonian blood flow is a shear-thin flow.…”

Parallel One-Shot Lagrange--Newton--Krylov--Schwarz Algorithms for Shape Optimization of Steady Incompressible Flows

“…In [6,7,43], a Newtonian flow modeled by the Stokes equations is used to study the design of the incoming branch of the bypass (the toe) into the coronary. A non-Newtonian flow described by the Navier-Stokes equations is used to study the design of the entire bypass in [2,3,40]. In our numerical experiments, a Newtonian flow governed by the Navier-Stokes equations is used to study the design of the entire bypass.…”

“…(3) Solve the design equations to update the shape; return to (1) unless a stopping condition is met. The three steps have to be performed sequentially [2,3,6,7,26,32,34,35,43]. These algorithms are often called nested analysis and design (NAND) and they involve an iterative algorithm and need to solve the state equations repeatedly, which make this computing extremely time consuming and in some cases not practical.…”

“…On the theoretical side there are several publications dealing with the existence of solution and the sensitivity analysis to the problems; see e.g., [19,20,21,27,39,46] and references therein. On the practical side, optimal shape design has played an important role in many industrial applications, for example, aerodynamic shape design [25,33,34,35,44], artery bypass design [2,3,6,7,40,43], microfluidic biochip design [4,24,38] and so on. Most of these optimization problems have constraints imposed by partial differential equations (PDE) or other physical and geometrical conditions.…”

“…The last term of (2.2) is a regularization term which provides the regularity of the boundary of the domain Ω α . In some approaches [2,3], some restrictions of the geometry are included in the constraints, e.g., certain thickness or volume, instead of a regularization term in the objective function. In this paper, we use a regularization term as in [19].…”