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Strongly Scalable Parallel Simulations of High-resolution Modelsin Computational Cardiology
Christoph Augustin, Gernot Plank
in coop. G. Haase, M. Liebmann, O. Steinbach, G. Holzapfel, A. Neic, A. Prassl, T. Fastl, T. Eriksson, A. Crozier
Medical University of Graz
SFB Mathematical Optimization andApplications in Biomedical Science
Modeling and Simulations in Biomechanics, September 15th, 2014
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Outline
Electromechanical Modeling
Parallel Strategies
Configuration
Numerical Examples
Open Tasks and Perspectives
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Outline
Electromechanical Modeling
Parallel Strategies
Configuration
Numerical Examples
Open Tasks and Perspectives
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Mechanics of Cardiovascular Tissues
matrixmuscle fiber
collagen fiber
n0sheet-normal
axis
s0
f0
sheet axis
fiber axis
Find the displacement u such that
− Div FS(u, x) = 0 for x ∈ Ω ,
u(x) = uD(x) for x ∈ ΓD ,
FS(u, x)n(x) = tN(x) for x ∈ ΓN.
S = Sp(u, x) + Sa(Vm, η, u, x)
• F = I + Grad u the deformation gradient
• Sp the passive 2nd Piola-Kirchhoff stress tensor1,2
• Sa the active 2nd Piola-Kirchhoff stress tensor2,3
• uD the prescribed displacement
• tN the prescribed traction
• n the normal vector
• Vm the transmembrane voltage and
• η state variables
1 Holzapfel and Ogden 2009. Philos. Trans. R. Soc. Lond. Ser. A, pp. 3445–3475.2 Eriksson et al. 2013. Mathematics and Mechanics of Solids, pp. 592–606.3 Smith et al. 2004. Acta Numer., pp. 371–431.
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Passive Stress Model
Constitutive equation using the free-energy function Ψ
Sp = 2∂Ψ(C)
∂C, Ψ(C) = U(F) + Ψiso(C) + Ψaniso(C), Ψ (locally) convex
nearly incompressible: penalty with κ
e.g., U(J) =κ
2(J − 1)2
, J = det F
isotropic components: ground matrix, elastin
e.g., Ψiso(C) =c
2(J−2/3 tr(C) − 3)
anisotropic components4,5: fibers, sheets
Ψaniso(C) =a
2b
exp[b(J−2/3If − 1)2] − 1
invariant If = Ff 0 · Ff 0: stretch in fiber direction
Loading
Unloading
0 10 20 30 40 50 60
100
200
300
400
500
600
Stress S (F/A), kPa
Sti
ffnes
sd
S/
dλ
,kP
a
4 Fung 1967. American Journal of Physiology, pp. 1532–1544.5 Eriksson, Gasser, Holzapfel, Ogden, 2000–2014
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Electical Activation in the Myocardium
The Bidomain equations6,7 describe the spread of cardiac electrical activity
Figure : Bidomain representation ofcardiac tissue in 2D
find (Vm, φe, η) such that
∇ · (σi + σe)C−1
∇φe = −∇ · σiC−1
∇Vm,
∇ · σiC−1
∇Vm = −∇ · σiC−1
∇φe + β Im,
Im = Cm
∂Vm
∂t+ Iion(Vm, η, u),
∂η
∂t= f (Vm, η)
• Vm = φi − φe the transmembrane voltage
• φi, φe intra- and extracellular potential
• η a vector of state variables
• σi, σe conductivity tensors
• Im(Vm, η) transmembrane current flow
• Ii, Ie, Iion current densities
• β surface to volume ratio of cardiac cells
Simplification: in our experiments we replace C−1
by the identity matrix
6 Tung 1978. PhD thesis,7 Vigmond et al. 2007. Prog. Biophys. Mol. Biol., pp. 3–18.
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Active Stress Models
Relaxed Contracted
Actin
Myosin+ ATP
+ Ca2+
(+)
(+)
(+)
(+)
(-)
(-)
Z disk
• Active stress is generated by electricalactivation in the myocardium
Sa = Sa(Vm, η, u) I−sf (f 0 ⊗ f 0)
with f 0 the myocyte fiber orientation
→ s = 12
mathematical, s = 1 mechanical choice
• Cell models to compute scalar-values stressterm Sa
→ Weakly coupled electromechanicse.g. NPStress9: Sa = ε(Vm)(kSa
Vm − Sa)
→ Strongly Coupled Electromechanicse.g. Rice10: h(Sa, Sa, Vm, η, λ, λ) = 0
8 Ambrosi and Pezzuto 2012. J. Elast., pp. 199–212.9 Nash and Panfilov 2004. Progress in Biophysics and Molecular Biology, pp. 501 –522.
10 Rice et al. 2008. Biophysical Journal, pp. 2368 –2390.
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Outline
Electromechanical Modeling
Parallel Strategies
Configuration
Numerical Examples
Open Tasks and Perspectives
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Motivation
High-resolution geometries
• meshes with up to N = O(109) degrees of freedom
• direct solvers: solving time O(N2), very high memory consumption
• iterative solvers: solving time O(N), lower memory consumption
→ we require strongly scalable parallel algorithms using iterative solvers
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Why do we need HR geometries?
Easy to motivate in electrics:
• very small features influence wavepropagation
⇒ resolutions below 100 µm would beworthwhile
• i.e. cellular level, O(109) cells
Harder to do so in mechanics:• use same mesh as in electics
no data mapping or mesh coarsening needed
• some phenomena of interest involve small-scale features infarcts and ischemic regions multiple tissue layers and thin structures as papillary muscles, heart strings, valves
• parallel framework is available improves computational time for smaller meshes as well
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Global Problem
Ω
FEM + Newton yields linearized system:
K′(uk)∆u = F − K(uk)
uk+1 = u
k +∆u.
Decomposition:p
∑
i=1
A⊤i K
′i (u
ki )Ai∆ui
solve with algebraic multigrid method (AMG11,12)
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Algebraic Multigrid Method
Ω3
Ω5
Ω6
Ω7
Ω4
Ω2Ω1
FEM + Newton yields linearized system:
K′(uk)∆u = F − K(uk)
uk+1 = u
k +∆u.
Decomposition:p
∑
i=1
A⊤i K
′i (u
ki )Ai∆ui
solve with algebraic multigrid method (AMG11,12)
restriction prol
onga
tion
pre-smoothing
post-smoothing
fine grid
base level
11 Plank et al. 2007. Biomedical Engineering, IEEE Transactions on, pp. 585–596.12 Neic et al. 2012. IEEE Trans. Biomed. Eng., pp. 2281–2290.
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Finite Element Tearing and Interconnecting
Ω1
Ω2
Ω3
Ω4
Ω5
Ω6
Ω7
FETI: Finite element tearing and interconnectingTearing:
K′1,k
. . .
K′p,k
∆u1
...∆up
= −
K 1,k
...K p,k
generally ∆ui 6= ∆uj on the interface ΓC
13 Farhat and Roux 1991. Int. J. Numer. Methods Engrg., pp. 1205–1227.14 Klawonn and Rheinbach 2010. ZAMM Z. Angew. Math. Mech., pp. 5–32.15 Augustin, Holzapfel, and Steinbach 2014. Int. J. Numer. Meth. Engrg., pp. 290–312.
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Finite Element Tearing and Interconnecting
Ω1
Ω2
Ω3
Ω4
Ω5
Ω6
Ω7
FETI: Finite element tearing and interconnectingInterconnecting: with Lagrange multipliers λ andboolean matrices Bi
K′1,k B
⊤1
. . ....
K′p,k B
⊤p
B1 · · · Bp 0
∆u1
...∆up
λ
= −
K 1,k
...K p,k
0
K†
i,k a generalized inverse this yields
P⊤
p∑
i=1
Bi K†i,kB
⊤i λ = P
⊤
p∑
i=1
BiK†i,k f i .
13 Farhat and Roux 1991. Int. J. Numer. Methods Engrg., pp. 1205–1227.14 Klawonn and Rheinbach 2010. ZAMM Z. Angew. Math. Mech., pp. 5–32.15 Augustin, Holzapfel, and Steinbach 2014. Int. J. Numer. Meth. Engrg., pp. 290–312.
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Software
• FETI: C. Pechstein, C. A.
• Cardiac electromechanics: CARP: G. Plank, E. Vigmond, C. A. PETSc: Krylov Solver Package with BoomerAMG (hypre)
→ general AMG for various problems Parallel Toolbox (PT): M. Liebmann, A. Neic, G. Haase
→ special AMG for electrics and mechanics
• Tarantula, ParaView, ParMetis, PaStiX, MUMPS, FEAP
Start electrics Solver mechanics
TIME++ S = Sp + Sa(Vm, η, u, f )
u
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Outline
Electromechanical Modeling
Parallel Strategies
Configuration
Numerical Examples
Open Tasks and Perspectives
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Geometry - MR Images
16 from the Visible Heart Lab (www.vhlab.umn.edu/)
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Geometry - Smoothing
17 with K. Bredis and M. Holler (KFU Graz)
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Myocardium Model
• Tetrahedral or hybrid meshes18
• Parameters19, fiber directions20
• Decomposition with ParMETIS
• Computations on SuperMUC Leibniz Rechenzentrum Munich Nr 12 in Top500 List - June 2014 147 456 cores
18 Prassl et al. 2009. IEEE Trans. Biomed. Engineering, pp. 1318–1330.19 Eriksson et al. 2013. Mathematics and Mechanics of Solids, pp. 592–606.20 Bayer et al. 2012. Ann. Biomed. Eng., pp. 2243–2254.
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Circulatory System - Pressure BC
• A to B: loading
• B to C: isovolumetric contraction
• C to D: ventricular ejection
• D to E: isovolumetric relaxation
• E to B: refilling
C
B
A
E
D
70 140
0
3
16
LV volume (ml)
LV p
ressure
(kPA
)
Challenges:
• MRI images usually taken at point B→ unloading of the geometry needed
• in isovolumetric phases the cavityvolumes have to stay constant
• pressure volume realtions in ejectionand filling phase
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Fixed Location Boundary Conditions
• heart covered by double layered membrane(pericard)
• space between layers is filled with fluid
• attached to diaphragm and pleura
• fix base (not physiological)
• contact problem21
• use bath from bid. model⇒ soft elastic material⇒ apply D-BC to bath
21 Fritz et al. 2013. Biomech Model Mechanobiol,
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Fixed Location Boundary Conditions
• heart covered by double layered membrane(pericard)
• space between layers is filled with fluid
• attached to diaphragm and pleura
• fix base (not physiological)
• contact problem21
• use bath from bid. model⇒ soft elastic material⇒ apply D-BC to bath
21 Fritz et al. 2013. Biomech Model Mechanobiol,
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Fixed Location Boundary Conditions
• heart covered by double layered membrane(pericard)
• space between layers is filled with fluid
• attached to diaphragm and pleura
• fix vessel in- and outlets
21 Fritz et al. 2013. Biomech Model Mechanobiol,
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Outline
Electromechanical Modeling
Parallel Strategies
Configuration
Numerical Examples
Open Tasks and Perspectives
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10
100
1000
10000
64 128 256 512 1024 2048 4096
Number of processes [-]
AssemblingPC Setup
SolverMonodomain
Total
Scaling for Electromechanics - AMG
• 3 686 631 nodes
• 11 059 893 DOF
• 20 524 957 tets
• 100 timesteps
• ≈ 5 NS each
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GPU
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Outline
Electromechanical Modeling
Parallel Strategies
Configuration
Numerical Examples
Open Tasks and Perspectives
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Open Tasks and Perspectives
Ongoing work• finish GPU assembling for mechanics• TH-elements, dynamic model, adaptive timestepping, . . .• block system solvers and preconditioning• projections between fine and coarse mesh• fitting parameters to experiments (Cardioproof project)
Wish list• contact problems, hemodynamics and FSI
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