Mathematical and numerical modeling of solute dynamics in blood flow and arterial walls

The numerical modeling of solutes absorption processes by the arterial wall is of paramount interest for the understanding of the relationships between the local features of blood flow, the nourishing of the inner arterial wall by the blood solutes, and the pathologies that can appear when this process is for some reason perturbed. In the present work, two models for the solutes dynamics are investigated. In the first model, which is essentially based on the one introduced by Rappitsch and Perktold (1996) and Rappitsch, Perktold, and Pernkopf (1997), the Navier-Stokes equations for an incompressible fluid, describing the blood velocity and pressure fields, are coupled with an advection-diffusion equation for the solute concentration. The wellposedness of this model is discussed. The second model considers also the solutes dynamics "inside" the arterial wall, described by a pure diffusion equation. Actually, this is a heterogeneous model, coupling different equations in different parts of the domain at hand. Its wellposedness is proven. Moreover, in view of the numerical study, an iterative finite element method by subdomains is proposed and its convergence properties are analyzed. Finally, several numerical results comparing the different models in situations of physiologic interest are illustrated

Numerical simulation of sediment dynamics with free surface flows

Arwa Mrad

We present a numerical model for the simulation of 3D mono-dispersed sediment dynamics in a Newtonian flow with free surfaces. The physical model is a macroscopic model for the transport of sediment based on a sediment concentration with a single momentum balance equation for the mixture (fluid and sediments). The model proposed here couples the Navier-Stokes equations, with a volume-of-fluid (VOF) approach for the tracking of the free surfaces between the liquid and the air, plus a nonlinear advection equation for the sediments (for the transport, deposition, and resuspension of sediments). The numerical algorithm relies on a splitting approach to decouple diffusion and advection phenomena such that we are left with a Stokes operator, an advection operator, and deposition/resuspension operators. For the space discretization, a two-grid method couples a finite element discretization for the resolution of the Stokes problem, and a finer structured grid of small cells for the discretization of the advection operator and the sediment deposition/resuspension operator. SLIC, redistribution, and decompression algorithms are used for post-processing to limit numerical diffusion and correct the numerical compression of the volume fraction of liquid. The numerical model is validated through numerical experiments. We validate and benchmark the model with deposition effects only for some specific experiments, in particular erosion experiments. Then, we validate and benchmark the model in which we introduce resuspension effects. After that, we discuss the limitations of the underlying physical models. Finally, we consider a one-dimensional diffusion-convection equation and study an error indicator for the design of adaptive algorithms. First, we consider a finite element backward scheme, and then, a splitting scheme that separates the diffusion and the convection parts of the equation.

EPFL2021

Numerical simulation of free surface flows

Vincent Maronnier, Marco Picasso, Jacques Rappaz

A numerical model is presented for the simulation of complex fluid flows with free surfaces. The unknowns are the velocity and pressure fields in the liquid region, together with a function defining the volume fraction of liquid. Although the mathematical formulation of the model is similar to the volume of fluid (VOF) method, the numerical schemes used to solve the problem are different. A splitting method is used for the time discretization. At each time step, two advection problems and a generalized Stokes problem are to be solved. Two different grids are used for the space discretization. The two advection problems are solved on a fixed, structured grid made out of small rectangular cells, using a forward characteristic method. The generalized Stokes problem is solved using a finite element method on a fixed, unstructured mesh. Numerical results are presented for several test cases: the filling of an S-shaped channel, the filling of a disk with core, the broken dam in a confined domain. (C) 1999 Academic Press.

1999

A saddle point approach to an optimal boundary control problem for steady Navier-Stokes equations

Andrea Manzoni, Alfio Quarteroni

In this paper we propose a saddle point approach to solve boundary control problems for the steady Navier-Stokes equations with mixed Dirichlet-Neumann boundary conditions, both in two and three dimensions. We provide a comprehensive theoretical framework to address (i) the well posedness analysis for the optimal control problem related to this system and (ii) the derivation of a system of first-order optimality conditions. We take advantage of a suitable treatment of boundary Dirichlet controls (and data) realized by means of Lagrange multipliers. In spite of the fact that this approach is rather common, a detailed analysis is still missing for mixed boundary conditions. We consider the minimization of quadratic cost (e.g., tracking-type or vorticity) functionals of the velocity. A descent method is then applied for numerical optimization, exploiting the Galerkin finite element method for the discretization of the state equations, the adjoint (Oseen) equations and the optimality equation. Numerical results are shown for simplified two-dimensional fluid flows in a tract of blood vessel where a bypass is inserted; to avoid to simulate the whole bypass configuration, we represent its action by a boundary velocity control.

AMER INST MATHEMATICAL SCIENCES-AIMS2019