| Blood flow simulation is a very challenging topic in Applied Mathematics. It involves state of the art techniques of image segmentation, computational fluid dynamics as well as fluid/structure interaction. The potential application for vascular diseases would be a computer-assisted diagnostic: blood flow characteristics such as blood velocity, pressure or artery wall shear stress are very important to know but difficult to measure in vivo.; Here, we focus on the methods for fluid/flexible-structure interaction and more specifically on the Immersed Boundary Method (IBM) of C.S. Peskin. The IBM combines Eulerian and Lagrangian descriptions of flow and moving elastic boundaries using the Dirac delta function as an interpolation tool. Incompressible Navier-Stokes (NS) and elasticity theory can be unified by the same set of equations to get a combined model of the interaction. This method is very easy to implement and versatile, so that it has numerous applications in bio-engineering or in more general computational fluid dynamics. For blood flow simulation, the IBM can be used to compute the motion of large blood cells, on a small scale, and the interaction between the artery wall and the pulsating flow, on a large scale.; We present a numerical study of the accuracy, the stability and the efficiency of the IBM based on the implementation of several mathematical tools. These implementations are being made on test cases that are relevant for the IBM applications, keeping in mind that we do not want to increase its computational cost and keep its simplicity for three-dimensional coding.; Finally, we introduce an integrated approach to quickly compute an incompressible NS flow in a section of a large blood vessel using medical imaging data, in order to provide a first order approximation of the shear stress and the pressure on the artery wall. |
