`Stent Design Blood Flow SimulationKirill Pichon GostafEric Bonnetier, Didier Bresch, Vuk Milisic, Amélie Rambaud 15 September 2009PeopleThis work was done in collaboration with: Eric Bonnetier Prof Didier Bresch DR CNRS Vuk Milisic CR1 CNRS Amélie Rambaud ENS LyonObjective: Asymptotic modeling and numerical simulations of blood flows in arteries with wired multi-layer stents. Mathematical modeling of multi-layered stents inserted in the human cardio-vascular network. Investigation of realistic 3D geometries. Collaboration with an industrial partner: Cardiatis, BelgiumKirill GostafFreefem++ workshop 20092AneurysmAn aneurysm is an abnormal bulge in the wall of an artery. Often, no symptoms... becomes large, begins to leak blood, ruptures.National Heart, Lung and Blood Institute, USA.Kirill GostafFreefem++ workshop 20093Stenting Saccular AneurysmThe multilayer stent design reduces flow velocity within the aneurysm, improves laminar flow in the main artery and surrounding branches, pressure and tension reduction within the aneurismal sac, Cardiatis. Frontal flow Tangential flowKirill GostafFreefem++ workshop 20094Stenting Saccular AneurysmThe multilayer stent design reduces flow velocity within the aneurysm, improves laminar flow in the main artery and surrounding branches, pressure and tension reduction within the aneurismal sac, Cardiatis. Frontal flow Tangential flowKirill GostafFreefem++ workshop 20095Direct Simulation 2D: Pressure Driven FlowBlood flow impinges directly upon the struts Stokes problemNnodes = 55k-u +p=0 u=0in  in  on lf ,rt ,strut on in ,outdiv u = 0 u· =0p = pin on in p = pout on out Pressure imposed = Dirichlet velocity Direct finite element simulation  to be a reference solution Same number of elements on a stent strutKirill Gostaf Freefem++ workshop 2009 Nnodes = 104k6Direct Simulation 3D: Two VesselsStokes direct simulation mesh Catia v5 Nnodes = 280K, Ntet =1.5M P2/P1 FE discretization solver FreeFem++ CG symmetric matrix 4.8GbKirill GostafFreefem++ workshop 20097Direct Simulation 3D: Two VesselsStokes direct simulation mesh Catia v5 Nnodes = 280K, Ntet =1.5M P2/P1 FE discretization solver FreeFem++ CG symmetric matrix 4.8GbEXTREMELY EXPENSIVE!Kirill GostafFreefem++ workshop 20098Direct Simulation 3D: Two VesselsStokes direct simulation mesh Catia v5 Nnodes = 280K, Ntet =1.5M P2/P1 FE discretization solver FreeFem++ CG symmetric matrix 4.8GbParallel solution EXTREMELY EXPENSIVE!one time step: 120 hoursHomogenization theoryKirill GostafFreefem++ workshop 20099HomogenizationWe replace struts by the homogenization wall law Bi-domain formulation  -u     ,j+p ,j = 0in jdiv u ,j = 0 in j R u ,j on  [u ,p ] · n =Discontinuous pressure Mono-domain numerical treatment with continuous pressure on velocity Kirill Gostafpressure Freefem++ workshop 2009 10Domain DecompositionDirichlet-Neumann method [Quarteroni et.al.'88] o Consider an initial guess u on the interface :(1) k k  L (u1 , p1 ) = 0 in 1 uk = g1 on 1 \  1 k k u1 = u on   k k  L (u2 , p2 ) = 0 in 2  k u2 = g2 on 2 \  k  [u ,p ] · n = R u k on  2(2)Correct the solution u until convergence, relaxation parameter  :(3)Kirill Gostafk k k u +1 = (1 - ) u +  u2Freefem++ workshop 200911Velocity Profile: Flow Rate0 -0.005 -0.01 Velocity Profile -0.015 -0.02 -0.025 -0.03solid lines = Homogenized symbols = Direct solution6 struts 10 struts 16 struts 20 struts 0.4 0.6 Artery Cross-Section 0.8 1-0.03500.2Kirill GostafFreefem++ workshop 200912Velocity Profile: Near the Homogenized Border0 Direct -0.005 Homogenized (FEM) Theory Velocity Profile -0.01-0.015-0.02-0.025-0.0300.20.4 0.6 Artery Cross-Section0.81Kirill GostafFreefem++ workshop 200913Homogenization Error100L2 H1 3/2 10-11Error10-210-310-410-210 -1100Kirill GostafFreefem++ workshop 200914Aneurysm Side 3D: FreeFem++ codeSolid geometry + mesh created with CATIA v5 CATIA v5 to FreeFem++ convert tool Load mesh, create finite element spacesmesh3 Th(&quot;aneurysm_side_h30.mesh&quot;); fespace Xh(Th,P23d); Xh u1,v1, u2,v2, u3,v3; fespace Mh(Th,P13d); Mh p,q;//velocity FE space//pressure FE spaceKirill GostafFreefem++ workshop 200915Aneurysm Side 3D: FreeFem++ codeVariational form: Stokes equationproblem Stokes([u1,u2,u3,p],[v1,v2,v3,q],solver=UMFPACK) = int3d(Th)( dx(u1)*dx(v1) + dy(u1)*dy(v1) + dz(u1)*dz(v1) + dx(u2)*dx(v2) + dy(u2)*dy(v2) + dz(u2)*dz(v2) + dx(u3)*dx(v3) + dy(u3)*dy(v3) + dz(u3)*dz(v3) - p*dx(v1) - p*dy(v2) - p*dz(v3) - dx(u1)*q - dy(u2)*q - dz(u3)*q - epsPenalty*p*q - int2d(Th,1)( pin*v1) + int2d(Th,2)(pout*v2) //p*div(w) //div(u)*q //penalization //penalization//negative 0X //positive 0Y+ on(1,u2=0,u3=0) //Gamma_in, u2=u3=0 + on(2,u1=0,u3=0) //Gamma_out, u1=u3=0 + on(20,u1=0,u2=0,u3=0);Kirill GostafFreefem++ workshop 200916Aneurysm Side 3D: Velocity FieldKirill GostafFreefem++ workshop 200917Aneurysm Side 3D: Pulsatile flowPressure solutionKirill GostafFreefem++ workshop 200918Aneurysm Side 3D: Pulsatile flowVelocity magnitudeKirill GostafFreefem++ workshop 200919`

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