cemracs_vivabrain_slides

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Feel++ Projects : VivaBrain
Céline Caldini-Queiros, Vincent Chabannes,
Tarik Madini, Jussara Marandola, Christophe
Prud’homme, Marcela Szopos, Ranine
Tarabay
CEMRACS’12
Marseille, August 23, 2012
C. Caldini-Queiros, V. Chabannes (CEMRACS’12) VIVABRAIN August 28 2012 1 / 23

Introduction
Objectives
Describe the problem :
complex reality
(multi-physics,
multi-scale).
Simplify.
Develop a mathematical
model.
Solve the mathematical
equations.
Display results.

Introduction
Physical models and boundary conditions
Which forces induces blood flow in the cerebral arteries ?
pressure drop
gravity
others?
Related questions
Fluid dynamic model : Newtonian vs Non-Newtonian?
Fluid or Fluid-Structure model?
Inflow : pressure, velocity profile ?
Outflow : do nothing strategy ?
How does surrounding tissues affect the blood flow ?

Introduction
Table Of Contents
1 Mathematical models and methods
2 HPC with feel++
3 Numerical results
4 Conclusion

Mathematical models and methods
2 HPC with feel++
3 Numerical results
4 Conclusion

Navier-Stokes
Modelling blood : Newtonian ﬂuid with incompressible behaviour.
We use Stokes or Navier-Stokes equations



ρ∂u
∂t + ρ (u · ) u − · σ = f
· u = 0
+ Boundary conditions
with :
stress tensor : σ = −pI + τ
deviatoric stress tensor : τ = 2µD
strain rate : D = 1
2 u + ( u)
T

Pressure drop
Impose the inlet pressure
Variational formulation with weak Dirichlet condition :
Find (u, p) ∈ H1
(Ω) × L2
(Ω) such as ∀(v, q) ∈ H1
(Ω) × L2
(Ω) :
Ω
( u: v − p · v + · uq) +
Γwall
(− u + pI)n · v
+
Γwall
(− v + qI)n · u +
Γwall
γ
h
u · v =
Ω
f · v −
Γinlet
(pinletn) · v,
Variational formulation with strong Dirichlet condition :
Find (u, p) ∈ H1
Γwall
(Ω) × L2
(Ω) such as ∀(v, q) ∈ H1
Γwall
(Ω) × L2
(Ω) :
Ω
( u: v − p · v + · uq) =
Ω
f · v −
Γinlet
(pinletn) · v,

Inﬂow Dirichlet conditions
Create a 2D submesh with the
inlet faces
Solve a Laplacian on the
submesh
Interpolate the Laplacian
solution on the 3D mesh and
ﬁnally the inlet velocity is :
u = ˜un on Γinlet

Inﬂow Dirichlet conditions
∆ ˜u = f in Γinlet
˜u = O on ∂Γinlet
Create a 2D submesh with the
inlet faces
Solve a Laplacian on the
submesh
Interpolate the Laplacian
solution on the 3D mesh and
ﬁnally the inlet velocity is :
u = ˜un on Γinlet

HPC with feel++
2 HPC with feel++
3 Numerical results
4 Conclusion

HPC with feel++
Feel++ [et al., ]
Finite Element Embedded Library in C++ : a C++ library
Features
Generalized Galerkin (cG, dG) methods in 1D, 2D and 3D with meshes
from simplices and hypercubes
Domain speciﬁc language embedded in C++ for variational formulation
Seamless interpolation tool (space, toppological dimension and mesh)
integrated in the variational language
Concepts/Classes mimic closely the mathematics which allows for
genericity and to handle complexity (Operators, function spaces,
elements of function spaces ...)
Seamless transition from 1D to 2D to 3D thanks to the variational
language (e.g. same code used either for 2D or 3D computations)
http://www.feelpp.org

HPC with feel++
Example : Stokes
ﬁnd (u, p) ∈ Xh = Xu
h × Xp
h such that :
Ω
u : v −
Ω
p · v =
Ω
fv ∀v ∈ Xu
h
Ω
q · u = 0 ∀q ∈ Xp
h
form2( _test=Xh, _trial=Xh, _matrix=D ) =
integrate( _range=elements(mesh),
_expr=trace(trans(deft)*grad(v)) ) // Ω
u : v
-integrate( _range=elements(mesh),
_expr=idt(p)*div(v) ) //− Ω
p · v
+integrate( _range=elements(mesh),
_expr=id(q)*divt(u) ); // Ω
q · u
form1( _test=Xh, _vector=F ) =
integrate( _range=elements(mesh),
_expr=f*id(v) ); // Ω
fv

HPC with feel++
Feel++ //
Contruction of partitioned mesh : use Gmsh
[Geuzaine and Remacle, 2009] with chaco or metis partitioners.
Degrees of freedom distribution :
Step 1 : build the local dof table with
ghost dofs
Step 2 : build the global dof table
without ghost dofs
the single global dof belongs to the process
of smallest rank
communication : update the id of
interprocess dofs on the global table
Algebraic parallel solver : Petsc libraries
[Balay et al., 2011, Balay et al., 2010, Balay et al., 1997, Smith, 2006]

Numerical results
2 HPC with feel++
3 Numerical results
4 Conclusion

Numerical results
Aorta
Physical model : Stokes
Boundary conditions :
inlets : Dirichlet condition (u = g) computed with Laplacian subproblem
outlets : do nothing ( σn = 0 )
wall : no slip (u = 0)
Solutions strategies : Additive Schwarz Method with LU Preconditioner in
each block
(a) pressure (b) streamlines colored with velocity magni-
tude

Numerical results
Aorta
P2P1
nDof : 322966
nProc : 48
Linear solve converged in 474 iterations (tolerance 1e−14
)
inflow : 0.0813213
outflow : 0.0813301
flow difference : 8.81439e − 06
P3P2
nDof : 1101523
nProc : 48
Linear solve converged in 477 iterations (tolerance 1e−14
)
inflow : 0.0813643
outflow : 0.0813649
flow difference : 5.79816e − 07

Numerical results
Aneurysm on artery
Physical model : Navier-Stokes
inlets : Pressure during 3ms
wall : no slip (u = 0)
Solutions strategies : Additive Schwarz Method with LU Preconditioner in
each block and Newton method for nonlinear term.
Statistic : nProc = 48, nDof : 1171496, number of iteration with linear
solver ≈ 50, time elapsed for each time step : 450 s
(c) pressure at t=0.0008 (d) pressure at t=0.0015

Numerical results
Cerebrovenous system
Figure : 1. Superior sagittal sinus ; 2.
Right sinus ; 3. Conﬂuence of sinuses ;
4. Transverse sinus ; 5. Sigmoid sinus.
Cerebrovenous system : not well
examined, only partially
understood.
Anatomy : complex
three-dimensional structure, often
asymetric, presenting considerably
more variable pattern than the
arterial system.

Numerical results
Cerebrovenous system : Mesh generation
MRA image segmentation (N.
Passat[Passat, 2005], LSIIT)
Final mesh (O. Génevaux,
LSIIT, S. Salmon, EA4535)

Numerical results
Physical model : Stokes
inlets : pressure ( σn = gn )
Solutions strategies : Additive Schwarz Method with ILU Preconditioner
in each block
nDof : 1173594, number of iterations : 1184 (tolerance 1e−14
)
(a) mesh partitioning (b) pressure
Figure : Partionining and pressure solution with 48 proc

Numerical results
Figure : Streamlines with 48 proc

Conclusion
2 HPC with feel++
3 Numerical results
4 Conclusion

Conclusion
Conclusion
Achievement
Simple simulations in complex geometries
Parallel computing experimentation with Feel++ on large cluster (thanks
to MesoCentre Marseille)
Inlet Dirichlet velocity (Laplacian subproblem)
Perspective
Vessels are endowed with valves (e.g. leg veins). What about cerebral
veins ?
One step further : modeling the whole circuit (coupling arterial and
venous system)
What are the main differences between arteries and veins from a
modeling standpoint ?
Boundary conditions : essential step to get realistic simulations

Conclusion
References I
Balay, S., Brown, J., , Buschelman, K., Eijkhout, V., Gropp, W. D., Kaushik, D., Knepley, M. G., McInnes, L. C., Smith, B. F., and Zhang, H. (2010).
PETSc users manual.
Technical Report ANL-95/11 - Revision 3.1, Argonne National Laboratory.
Balay, S., Brown, J., Buschelman, K., Gropp, W. D., Kaushik, D., Knepley, M. G., McInnes, L. C., Smith, B. F., and Zhang, H. (2011).
PETSc Web page.
http://www.mcs.anl.gov/petsc.
Balay, S., Gropp, W. D., McInnes, L. C., and Smith, B. F. (1997).
Efficient management of parallelism in object oriented numerical software libraries.
In Arge, E., Bruaset, A. M., and Langtangen, H. P., editors, Modern Software Tools in Scientific Computing, pages 163–202. Birkhäuser Press.
et al., C. P.
Feel++: The finite element embedded language (and library) in c++.
http://www.feelpp.org.
Geuzaine, C. and Remacle, J.-F. (2009).
Gmsh: a three-dimensional finite element mesh generator with built-in pre-and post-processing facilities.
International Journal for Numerical Methods in Engineering, 79(11):1309–1331.
Passat, N. (2005).
Contributiona la segmentation des réseaux vasculaires cérébraux obtenus en IRM. Intégration de connaissance anatomique pour le guidage
d’outils de morphologie mathématique.
PhD thesis, PhD thesis, Université Louis Pasteur de Strasbourg.
Smith, B. (2006).
Petsc introductory tutorial.

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