Читать книгу Computational Modeling and Simulation Examples in Bioengineering - Группа авторов - Страница 20

Blood flow in aorta has a time‐dependent 3D flow, so the time‐dependent and full three‐dimensional Navier–Stokes equations were solved. The laminar flow condition appropriate for this type of analysis [107] was used. The finite element code was validated using the analytical solution for shear stress and velocities through the curved tube [108]. A penalty formulation will be used [109]. The incremental–iterative form of the equations for time step and equilibrium iteration “i” is:

(1.1)

The left upper index “t + Δt” denotes that the quantities are evaluated at the end of time step. The matrix M_v is mass matrix, K_vv and J_vv are convective matrices, K_μv is the viscous matrix, K_vp is the pressure matrix, and F_v and F_p are forcing vectors. The pressure is eliminated at the element level through the static condensation. For the penalty formulation, the incompressibility constraint is defined in the following manner:

(1.2)

where λ is a relatively large positive scalar so that p/λ is a small number (practically zero).