The numerical simulation of hemodynamics is increasingly recognized as a valuable analysis tool for the bioengineering laboratories who design implantable vascular grafts, and it is thought to become in the near future a complement to the medical doctors’ analysis for the choice of interventional methods to restore a proper irrigation in diseased arteries. The computation of non-newtonian blood flows in arteries with compliant walls was made possible by tightly integrating inside the inner iterations of an implicit time scheme :  A Navier-Stokes laminar fluid flow solver with an ALE formulation,  a structural model of membrane elastic deformation,  a dynamic volume remeshing module. The complex nature of the mathematical model formed by the coupled fluid and structural mechanics equations, completed by highly unsteady boundary conditions, led to a choice of fully time-implicit algorithms with optimized sub-iterations. In each physical domain (fluid, structure) the inner iterations use a local dual time step for converging the balance equations to n equibrium, with time derivatives at 2nd order. At each inner exchange iteration, the whole fluid grid is recomputed from the new position of the structural nodes. These algorithms were successfully applied to different 3D, unsteady hemodynamics problems in a coronary artery. The complex, patients’dependent geometry of a stenosed artery was reconstructed in CAD format from medical imaging. A wide number of physical parameters can be represented in these simulations : - the non-newtonian nature of the rheology, with the apparent viscosity varying by a factor 7 between the high and low shear regions, - the different elasticity of the vessels in the azimuth and the longitudinal directions, - a structural damping coefficient to represent the artery volume and its motion into surrounding tissues (this last parameter would require a more in-depth analysis). A project of pre-industrial importance could be conducted for a bio-engineering laboratory that designs a wide range of vascular grafts. It permitted to evaluate the improved geometry of the graft at the place where it is fixed in the receiving artery, through a computation of the stability of the flow at the walls and of the fluctuation of the wall shear stress with time at different positions. The time spectrum of the surface integral of the wall shear stress at this distal location, with Fourier coefficients peaks, led to valuable quality criteria