simulation du système cardiovasculaire et modélisation réduite

Jean-Frédéric GerbeauProject-team REO

INRIA Paris-Rocquencourt & Laboratoire J-L. LionsFrance

Simulation du système cardiovasculaire

et modélisation réduite6 février 2013

Université de Rouen

2

Cardiovascular system

Aneurisms

Atherosclerosis

Biomedical devices

Electrophysiology(Arythmia, drug effect,...)

Vascular diseases

Outline

• Computational Hemodynamics- Motivations- Inverse problems in fluid-structure

• Electrocardiograms (ECGs)• Reduced order modeling - Motivations- The Approximated Lax Pair method

3

Outline



4

Computational hemodynamics:Hierarchy of models

• 3D Model (Navier-Stokes + Nonlinear elasticity)

Variables : velocity, pressure, displacement

• 1D Model (Euler equation, 1775)

Variables : flow rate, mean pressure, section area

• 0D Model (Ordinary Differential Equations)

Variables : flow rate, pressure drop

Γw0

Γ

ΩS S

w

1 2

z

A(z,t)Q(z,t)

Rp

Rd

C

(compartment modeling or boundary conditions)

Three-dimensional modelsBefore 2000 : • Pioneer works: Navier-Stokes equations, simple boundary

conditions, rarely on real anatomies2000 - 2010 :• Patient-specific anatomies: homemade solutions; free (VMTK) or

commercial software (Amira, Mimics,...)• Fluid-solid interaction: tremendous progress, now present in

commercial software (not very efficient though!)• Boundary conditions: many workarounds... still an open topic!• Main applications: stents, aneurysms, ...

Today (and tomorrow) :• Diagnosis: assimilate medical data in simulations• Prognosis & planning: long term evolution, adaption, remodeling,...• Find problems where all these works can be really useful!

6

Progress in FSI

7

1999 2007

2003

Computational time2001 2003 2007 2012100 20 4 1

Only due to progress in algorithms !

1st example: surgical planning

8

Glenn-Fontan surgery• congenital heart disease• multi-step complex

procedure

Glenn FontanNumerical simulations• Patient-specific geometries• Forecast pressure drop, flow split, wall-shear stress

Troianowski, Taylor, Feinstein, Vignon-ClementelStanford/INRIA

http://www.americanheart.org

I. Vignon-Clementel, A. Birolleau, G. Troianowski (INRIA & Stanford)

Geometry from 5 patients MRI data

10

Y-graft in general improves:• Energy efficiency• Flow distribution• SVC pressure under rest & exercise

Troianowski, Taylor, Feinstein, Vignon-Clementel

Challenge:

• About 20 “outlets”• 3 parameters per

outlets....

Rp

Rd

C

2d example: avoid clinical exams ?

11

Source: O. Peruta

• Example: aortic coarctation • After surgical repair, patients must

be followed on a regular basis• Exercise test is often necessary to

assess the patient condition• Question: With computer simulations, can we extrapolate

the rest test to avoid the stress test ?• Maybe... if we are able to evaluate the artery wall stiffness

Collaboration with R. Hose, I. Valverde, P. Beerbaum (euHeart project)

Aortic coarctation

12

Data: I. Valverde, P. Beerbaum (KCL). Segmentation and processing: R. Hose et al. (Sheffield), C. Bertoglio (INRIA)

(euHeart project)

Medical data assimilation

13

Estimate parameters- artery wall stiffness- boundary condition- ...

Measurements- medical imaging (CT, MRI, ...)- blood flow (US, PC-MRI,...)- pressure (catheter)- ...

Access to hidden quantities

- pressure- wall stress

Models- Navier-Stokes equations- Solid mechanics- ...

Improve measures- regularization- interpolation- ...

Data assimilation

14

• Partial observations of X: Z = H(X)

• Uncertainties on the initial condition X0 and the parameters ✓

• FSI dynamical system:⇢

BX = A(X, �) +RX(0) = X0

• Time discretization: Xn+1 = Fn+1(Xn, �)

• State variable: X=[u, p,df ,d,v]

• Parameters: ✓ = [Young modulus, viscosity, boundary conditions, . . . ]

Data assimilation

15

⇢X0 = X0 + ⇣X✓ = ✓ + ⇣✓

• Variational approach: - Optimization algorithms- Usually based on gradient (adjoint equations)

• Filtering approach:- Sequential correction of the state and the parameters

• Uncertainties: ⇣ = [⇣X , ⇣✓]

• Minimize

J(�) =1

2

Z T

0⇥Z �H(X)⇥2W dt+

1

2⇥�⇥2P

where X = X(�) is solution to the problem.

16Simulation : C.Bertoglio

Rp

Rd

• Parameter estimation: Young modulus Ei = 2�i in 5 regions

• Observations: wall displacement on fluid-solid interface

Example: Compliance estimationParameter estimation

17

0 0.2 0.4 0.6 0.8−0.5

0

0.5

1

1.5

2

Time [s]

Dis

plac

emen

t [m

m]

Region 1Region 2Region 3Region 4Region 5Noise

Region SNR

1 0.43

2 1.25

3 8.8

4 1.8

5 0.75

• Gaussian noise (10%) and time resampling �tobs

= 10�t

• Synthetic data

• Two tuning coeffecients:

– A priori covariance on parameters: ↵I– Gain: �


18

0 0.2 0.4 0.6 0.80

0.5

1

1.5

2

2.5

3

3.5

Time [s]

Youn

gs m

odul

us [

MPa

]

� = 9,⇥ = 1


Simulation : C.Bertoglio

19

0 0.2 0.4 0.6 0.8−4

−2

0

2

4

Time [s]

θ

α = 4 − β = 100

0 0.2 0.4 0.6 0.8−4

−2

0

2

4

Time [s]

θ

α = 9 − β = 20

• We can plot �n ±p

diag(P �n)

• The filtering algorithm provides an a posteriori covariance P �n


Simulation : C.Bertoglio

Experimental validation

0 0.1 0.2 0.3 0.4 0.5 0.6−1.5

−1

−0.5

0

0.5

1

Time [s]

log 2(E

)

FSI + data assmilationMechanical test

• Silicon rubber aortic phantom

Data: Gaddum, Rutten, Beerbaum (KCL). Segmentation and processing: Hose, Barber (Sheffield)

Simulation: Bertoglio, JFG (INRIA)(euHeart project)

• Many works about fluid boundary conditions... but not that many about wall !

C. Taylor et al. , StanfordAbdominal aorta

Blood flow in aortaExternal tissue support

• Typical b.c. on the external part of the vessel :

�sn = p0n

• A simple and affordable way to model the external tissues :

�sn = �ksd� cs�d

�t

Moireau, Xiao, Astorino, Figueroa, Chapelle, Taylor, JFG, (Biomech Model Mech 2011)

DataContours without supportContour with support

Legend:Solid

Fluid Pressure with supportPressure without supportFlow with supportFlow without support

DiastoleSystole

spinespine

spinesp

ine

spinesp

ine

π

π

π

1

2

3

π1

π2

π3

60

80

100

120

140

0.0 0.3 0.90.6

60

80

100

120

140

0.0 0.3 0.90.60

10

20

30

40

0

40

80

120

160

60

80

100

120

140

0.0 0.3 0.90.60

90

180

270

360

π0

π0

(mmHg)

(cc/s)

(mmHg)

(cc/s)

Example 3: External tissue supportModeling

Moireau, Xiao, Astorino, Figueroa, Chapelle, Taylor, JFG, (Biomech Model Mech 2011)

Outline



23

e.g.: Pullan et al. 05, Sundes et al. 06,...

8<

:

div(�TruT) = 0+ transmission condition

on the epicardium

Torso:Poisson equation

Electrocardiograms

24

ui

ue

APD�⌅⇤

⌅⇥

CmdVm

dt+ Iion(Vm, g) = 0

dg

dt+ G(Vm, g) = 0

Cell scale: Hodgkin-Huxley-like models

8>>>><

>>>>:

Am

✓C

m

�Vm

�t+ I

ion

(Vm

, g)

◆� div(�

i

rui

) = Am

Iapp

,

div(�e

rue

) + div(�i

rui

) = 0

�g

�t+G(V

m

, g) = 0,

Myocardium: bidomain models

12-lead ECG

25

Boulakia, Cazeau, Fernández, JFG, Zemzemi, Annals Biomed Engng 2010

• Simulated ECG:

• Real ECG:

-2

-1

0

1

2

0 200 400 600 800

I

-2

-1

0

1

2

0 200 400 600 800

II

-2

-1

0

1

2

0 200 400 600 800

III

-2

-1

0

1

2

0 200 400 600 800

aVR

-2

-1

0

1

2

0 200 400 600 800

aVL

-2

-1

0

1

2

0 200 400 600 800

aVF

-2

-1

0

1

2

0 200 400 600 800

V1

-2

-1

0

1

2

0 200 400 600 800

V2

-2

-1

0

1

2

0 200 400 600 800

V3

-2

-1

0

1

2

0 200 400 600 800

V4

-2

-1

0

1

2

0 200 400 600 800

V5

-2

-1

0

1

2

0 200 400 600 800

V6

-2

-1

0

1

2

0 200 400 600 800

I

-2

-1

0

1

2

0 200 400 600 800

II

-2

-1

0

1

2

0 200 400 600 800

III

-2

-1

0

1

2

0 200 400 600 800

aVR

-2

-1

0

1

2

0 200 400 600 800

aVL

-2

-1

0

1

2

0 200 400 600 800

aVF

-2

-1

0

1

2

0 200 400 600 800

V1

-2

-1

0

1

2

0 200 400 600 800

V2

-2

-1

0

1

2

0 200 400 600 800

V3

-2

-1

0

1

2

0 200 400 600 800

V4

-2

-1

0

1

2

0 200 400 600 800

V5

-2

-1

0

1

2

0 200 400 600 800

V6

-2

-1

0

1

2

0 200 400 600 800

I

-2

-1

0

1

2

0 200 400 600 800

II

-2

-1

0

1

2

0 200 400 600 800

III

-2

-1

0

1

2

0 200 400 600 800

aVR

-2

-1

0

1

2

0 200 400 600 800

aVL

-2

-1

0

1

2

0 200 400 600 800

aVF

-2

-1

0

1

2

0 200 400 600 800

V1

-2

-1

0

1

2

0 200 400 600 800

V2

-2

-1

0

1

2

0 200 400 600 800

V3

-2

-1

0

1

2

0 200 400 600 800

V4

-2

-1

0

1

2

0 200 400 600 800

V5

-2

-1

0

1

2

0 200 400 600 800

V6

-2

-1

0

1

2

0 200 400 600 800

I

-2

-1

0

1

2

0 200 400 600 800

II

-2

-1

0

1

2

0 200 400 600 800

III

-2

-1

0

1

2

0 200 400 600 800

aVR

-2

-1

0

1

2

0 200 400 600 800

aVL

-2

-1

0

1

2

0 200 400 600 800

aVF

-2

-1

0

1

2

0 200 400 600 800

V1

-2

-1

0

1

2

0 200 400 600 800

V2

-2

-1

0

1

2

0 200 400 600 800

V3

-2

-1

0

1

2

0 200 400 600 800

V4

-2

-1

0

1

2

0 200 400 600 800

V5

-2

-1

0

1

2

0 200 400 600 800

V6

-2

-1

0

1

2

0 200 400 600 800

I

-2

-1

0

1

2

0 200 400 600 800

II

-2

-1

0

1

2

0 200 400 600 800

III

-2

-1

0

1

2

0 200 400 600 800

aVR

-2

-1

0

1

2

0 200 400 600 800

aVL

-2

-1

0

1

2

0 200 400 600 800

aVF

-2

-1

0

1

2

0 200 400 600 800

V1

-2

-1

0

1

2

0 200 400 600 800

V2

-2

-1

0

1

2

0 200 400 600 800

V3

-2

-1

0

1

2

0 200 400 600 800

V4

-2

-1

0

1

2

0 200 400 600 800

V5

-2

-1

0

1

2

0 200 400 600 800

V6

-2

-1

0

1

2

0 200 400 600 800

I

-2

-1

0

1

2

0 200 400 600 800

II

-2

-1

0

1

2

0 200 400 600 800

III

-2

-1

0

1

2

0 200 400 600 800

aVR

-2

-1

0

1

2

0 200 400 600 800

aVL

-2

-1

0

1

2

0 200 400 600 800

aVF

-2

-1

0

1

2

0 200 400 600 800

V1

-2

-1

0

1

2

0 200 400 600 800

V2

-2

-1

0

1

2

0 200 400 600 800

V3

-2

-1

0

1

2

0 200 400 600 800

V4

-2

-1

0

1

2

0 200 400 600 800

V5

-2

-1

0

1

2

0 200 400 600 800

V6

-2

-1

0

1

2

0 200 400 600 800

I

-2

-1

0

1

2

0 200 400 600 800

II

-2

-1

0

1

2

0 200 400 600 800

III

-2

-1

0

1

2

0 200 400 600 800

aVR

-2

-1

0

1

2

0 200 400 600 800

aVL

-2

-1

0

1

2

0 200 400 600 800

aVF

-2

-1

0

1

2

0 200 400 600 800

V1

-2

-1

0

1

2

0 200 400 600 800

V2

-2

-1

0

1

2

0 200 400 600 800

V3

-2

-1

0

1

2

0 200 400 600 800

V4

-2

-1

0

1

2

0 200 400 600 800

V5

-2

-1

0

1

2

0 200 400 600 800

V6

-2

-1

0

1

2

0 200 400 600 800

I

-2

-1

0

1

2

0 200 400 600 800

II

-2

-1

0

1

2

0 200 400 600 800

III

-2

-1

0

1

2

0 200 400 600 800

aVR

-2

-1

0

1

2

0 200 400 600 800

aVL

-2

-1

0

1

2

0 200 400 600 800

aVF

-2

-1

0

1

2

0 200 400 600 800

V1

-2

-1

0

1

2

0 200 400 600 800

V2

-2

-1

0

1

2

0 200 400 600 800

V3

-2

-1

0

1

2

0 200 400 600 800

V4

-2

-1

0

1

2

0 200 400 600 800

V5

-2

-1

0

1

2

0 200 400 600 800

V6

-2

-1

0

1

2

0 200 400 600 800

I

-2

-1

0

1

2

0 200 400 600 800

II

-2

-1

0

1

2

0 200 400 600 800

III

-2

-1

0

1

2

0 200 400 600 800

aVR

-2

-1

0

1

2

0 200 400 600 800

aVL

-2

-1

0

1

2

0 200 400 600 800

aVF

-2

-1

0

1

2

0 200 400 600 800

V1

-2

-1

0

1

2

0 200 400 600 800

V2

-2

-1

0

1

2

0 200 400 600 800

V3

-2

-1

0

1

2

0 200 400 600 800

V4

-2

-1

0

1

2

0 200 400 600 800

V5

-2

-1

0

1

2

0 200 400 600 800

V6

-2

-1

0

1

2

0 200 400 600 800

I

-2

-1

0

1

2

0 200 400 600 800

II

-2

-1

0

1

2

0 200 400 600 800

III

-2

-1

0

1

2

0 200 400 600 800

aVR

-2

-1

0

1

2

0 200 400 600 800

aVL

-2

-1

0

1

2

0 200 400 600 800

aVF

-2

-1

0

1

2

0 200 400 600 800

V1

-2

-1

0

1

2

0 200 400 600 800

V2

-2

-1

0

1

2

0 200 400 600 800

V3

-2

-1

0

1

2

0 200 400 600 800

V4

-2

-1

0

1

2

0 200 400 600 800

V5

-2

-1

0

1

2

0 200 400 600 800

V6

-2

-1

0

1

2

0 200 400 600 800

I

-2

-1

0

1

2

0 200 400 600 800

II

-2

-1

0

1

2

0 200 400 600 800

III

-2

-1

0

1

2

0 200 400 600 800

aVR

-2

-1

0

1

2

0 200 400 600 800

aVL

-2

-1

0

1

2

0 200 400 600 800

aVF

-2

-1

0

1

2

0 200 400 600 800

V1

-2

-1

0

1

2

0 200 400 600 800

V2

-2

-1

0

1

2

0 200 400 600 800

V3

-2

-1

0

1

2

0 200 400 600 800

V4

-2

-1

0

1

2

0 200 400 600 800

V5

-2

-1

0

1

2

0 200 400 600 800

V6

-2

-1

0

1

2

0 200 400 600 800

I

-2

-1

0

1

2

0 200 400 600 800

II

-2

-1

0

1

2

0 200 400 600 800

III

-2

-1

0

1

2

0 200 400 600 800

aVR

-2

-1

0

1

2

0 200 400 600 800

aVL

-2

-1

0

1

2

0 200 400 600 800

aVF

-2

-1

0

1

2

0 200 400 600 800

V1

-2

-1

0

1

2

0 200 400 600 800

V2

-2

-1

0

1

2

0 200 400 600 800

V3

-2

-1

0

1

2

0 200 400 600 800

V4

-2

-1

0

1

2

0 200 400 600 800

V5

-2

-1

0

1

2

0 200 400 600 800

V6

V5

V6

V1

V2

V3

V4

D1

aVR

aVL

aVFD2

D3

(from www.wikipedia.org)

http://www.wikipedia.org

http://www.wikipedia.org

Bundle branch blocks

Healthy case Right bundle branch block

Chapelle, Fernández, JFG, Moireau, Sainte-Marie, Zemzemi, FIMH 2009

26

Outline


• Electrocardiograms (ECGs)• Reduced order modeling

- Motivations- The Approximated Lax Pair method

27

Motivation: Blood flows

• Output of interest Ex: Pressure drop

28

• Rapid prototypingBoundary conditions (Windkessel), constitutive law coefficients,...

• Moderate variabilityExamples: pulmonary arteries

Astorino, JFG, Pantz, Traoré, CMAME 2009

• OptimizationEx: arterial by-pass optimization (Quarteroni-Rozza, Sankaran-Marsden)

Aortic coarctation

Motivation: electrophysiology

29

• OptimizationEx: optimize multi-sites pacing

Forward

Inverse

• Inverse problemsElectrocardiology

Potential in heart and the torso Electrocardiogram(ECG)

Reduced Order Modeling

Many options:• Simplify the geometry and/or the physics

Examples:- 1D model for blood flows (Formaggia, Peiró, ...)

- Eikonal model in electrophysiology (Franzone, Sermesant, Frangi,...)

• Keep the equations and the geometry, but reduce the approximation spaceExamples:- Reduced Basis Method (Patera, Maday, Quarteroni, Rozza,...)

- Proper Orthogonal Decomposition (POD), or Karhunen-Loève expansion (Iollo, Farhat, Karniadakis, Kunisch, Gunzburger, Volkwein,...)

30

POD in a nutshell

31

• Let ('i)i=1..n be a finite element basis

• Singular Value Decomposition: S = ⇥�⇤T, with � = diag(�1, . . . ,�p)

• Reduced Order Model (ROM): search for Uh =PN

j=1 Uj�j such that:d

dt(Uh,�i) + a(Uh,�i) = (f,�i), �i = 1..N

• (�1, . . . ,�N ): N columns of � corresponding to the N largest �i, N � n

• Let S be the matrix whose columns are the Si, i = 1..p.

• Full Order Model (FOM): search for uh =Pn

j=1 uj⇥j such that:d

dt(uh,⇥i) + a(�;uh,⇥i) = (f,⇥i), �i = 1..n

• Compute p “snapshots” (i.e. solutions at p time instants, or parameters �):

S1(u11, . . . , u

1n), . . . , S

p(up1, . . . , u

pn)

32

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1−0.2

0

0.2

0.4

0.6

0.8

1

1.2

x

u

t=0exact POD(10)

0 5 10 15 20 25 30 35 40 45 5010−18

10−16

10−14

10−12

10−10

10−8

10−6

10−4

10−2

100

n

eigenvalues

POD with 10 modes

Simulation : D. Lombardi

Example 1:

@u

@t

� @

2u

@x

2= 0

Eigenvalues

33

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

x

u

t=0 t=1

0 5 10 15 20 25 30 35 40 45 5010−8

10−7

10−6

10−5

10−4

10−3

10−2

10−1

100

n

eigenvalues


0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1−0.2

0

0.2

0.4

0.6

0.8

1

1.2

x

u

exactPOD(10)

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1−0.4

−0.2

0

0.2

0.4

0.6

0.8

1

x

u

exactPOD(50)

POD with 10 modes POD with 50 modes

Example 2:

@u

@t

+ c

@u

@x

= 0

Eigenvalues

Basic idea

34

Uh =NX

j=1

Uj�j(x, t)

Uh =NX

j=1

Uj(t)�j(x)

• Instead of

• Try to work with something like :

• Question : how to propagate the modes ?

Step 1: Semi Classical Signal Analysis (SCSA)

• Let u(x) be a non-negative signal and the “scattering” operator

35

L�(u)� = �� u�

• And approximate u(x) by the Deift-Trubowitz formula

u(x) = �

�1

N�X

m=1

m�

2m

• Solve the eigenvalue problem

L�(u)� = ��

where �m are the negative eigenvalueswith m =p��m

Laleg, Crépeau, Sorine (2007 & 2012)

• Choose � > 0 to achieve a given accuracy

• Remark: can be exact for “reflectionless” potentials

36

Reconstruction of aortic pressureFrom: Laleg, Médigue, Papelier, Crépeau, et al. (2010)

Lax pair in a nutshell

37

• If an operator M(t) is such that

• If (�k(t),�k(t)) is an eigenpair of L(t) then

- @t�k = 0

- @t�k(t) = M(t)�k(t)

@tL+ LM�ML = 0

• Let L(t) be a self-adjoint operator in a Hilbert space

Lax pair in a nutshell

• Example: Korteweg de Vries equation:

38

@t

u+ 6u@x

u+ @3x

u = 0

• Lax pair:

L(u)v = �@2x

v � uv

M(u)v = 4@3x

v + 6u@x

v + 3v@x

u

• Example: 1-soliton: if

then:

with: �1(x) = sech(x)

u0(x) =

21

2�

21(1x)

u(x, t) =

21

2�

21(1(x�

21t))

Goal: • Consider a general nonlinear evolution PDE :

39

⇢@tu = F (u)u(0) = u0

• For example: F (u) = �u+ u(1� u)

• Difficulty: the Lax pair are not known in general

• Use the SCSA and the Lax pair to define a reduced order model

40

@t�m = ��hF (u) m, mi

• The evolution of �m is governed by

• The evolution of m satifies, for p 2 {1, . . . , NM},

h@t m, pi = Mmp(u)

(Mmp(u) =

�

�p � �mhF (u) m, pi, if p 6= m and �p 6= �m,

Mmp(u) = 0, if p = m or �p = �m.with

⇢@tu = F (u)u(0) = u0

• For m 2 {1, . . . , NM}, let �m(t) be an eigenvalue of L�(u(x, t))and m(x, t) an associated eigenfunction, normalized in L

2(⌦)

• Assume there exists an operator M(u) such that

@t m = M(u) m.

Proposition: let u be solution to Step 2: Approximation of Lax Pair (ALP)

Step 3: Approximated “inverse scattering”

41

• (�m,�m)m=1..N� : eigenfunctions/values of L�(u)

• We look for an approximation of u:

u =

N�X

k=1

↵k�2k

• Inserting this expression in

• We findN�X

k=1

↵kh�2k,�

2mi = � 1

�(�m + h��m,�mi) .

which gives the ↵k.

hL�(u)�m,�pi = �mh�m,�pi

42

Summary: the ALP algorithmInitialisation (SCSA of u0): compute NM eigenmodes m.

The N� first are denoted �m and

u0(x) ⇡ �

�1

N�X

m=0

m�m(x)

1. Compute the matrix M(u

n) approximating the operator M(u(t

n))

2. Compute the eigenvalues �

n+1m : �

n+1m = �

nm � �thF (u

n)

nm,

nmi

3. Compute the eigenfunctions

n+1m :

n+1m =

PNM

p=1

hI + �tM(u

n) +

�t2

2 M(u

n)

2i

mp

np

4. Solve for ↵

n+1p :

PN�p=1 ↵

n+1p h�2p,�2mi = � 1

�

��

n+1p + h��n+1

m ,�

n+1m i

�.

5. Compute u

n+1with u

n+1=

PN�p=1 ↵

n+1p

��

n+1p

�2

Numerical test cases• Korteweg-de Vries:

★ one-soliton (analytically known)★ only one negative eigenvalue

43

−15 −10 −5 0 5 10 150

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2

x

u

t=0 t=T

−15 −10 −5 0 5 10 15−0.5

0

0.5

1

1.5

2

x

u

exactROM

−15 −10 −5 0 5 10 150

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2

x

u

exactROM

10 modes 25 modes


44

• Korteweg-de Vries: ★ three-soliton★ three negative eigenvalues

−15 −10 −5 0 5 10 150

0.5

1

1.5

2

2.5

3

3.5

4

4.5

x

u

t=0

−15 −10 −5 0 5 10 150

0.5

1

1.5

2

2.5

3

3.5

4

x

u

t=T


• Comparison of:★ projection of the analytic solution on 20 POD

modes built of snapshots obtained from the half first time history

★ ALP integration with 20 modes

−15 −10 −5 0 5 10 15−0.5

0

0.5

1

1.5

2

2.5

3

3.5

4

x

u

exactSTMPOD

• Fisher-Kolmogorov-Petrovski-Piskunov:

45

@u

@t��u = u(1� u)

46

• P1 Finite-element:★ 9000 dof ★ Time step 2.5e-4

• Challenges:★ Sharp front, splitting in two directions★ No precomputed solutions : POD cannot work


47

• ALP★ 6 modes for the solution, 25 modes for the operator★ Time step 2.5e-3 (10 times larger than FEM)


48

55 main arteries

Γw0

Γ

ΩS S

w

1 2

z

A(z,t)Q(z,t)

velocity and pressure

flow rate and mean pressure(D. Lombardi)

In progress....

Perspectives

• Many things to do★ Adaptation of the number of modes★ Different operators for the scattering step★ Role of positive eigenvalues★ Error estimator★ Analysis★ ...

• Applications (in progress)★ systemic network of arteries (Damiano Lombardi)★ cardiac electrophysiology (Elisa Schenone)

49

Inria Research Report 8137 available on HALhttp://hal.inria.fr/hal-00752810

http://hal.inria.fr/hal-00752810

http://hal.inria.fr/hal-00752810

• Remark:

50

Z

⌦(@t m)2 ⇡

NMX

n,l=1

Mml(u)Mmn(u)h n, li =NMX

n=1

Mmn(u)2

• Thus, the Frobenius norm of M(u) can be used as an estimator of the approximation of the dynamics

simulation du système cardiovasculaire et modélisation réduite

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