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Model reduction of steady fluid-structure interaction problems with
reduced basis methods and free-form deformations
Toni Lassila∗, Gianluigi Rozza†
∗Institute of Mathematics
†Modelling and Scientific Computing
Helsinki University of Technology Institute of Analysis and Scientific Computing
Ecole Polytechnique Federale de Lausanne
10th Finnish Mechanics Days, Jyvaskyla, Finland
December 3-4, 2009
Toni Lassila Model reduction of steady fluid-structure interaction problems with reduced basis methods 1 / 19
Outline
Fluid-structure interaction problems in cardiovascular modelling
Model reduction for fluid-structure interaction problemsI Step #1: Free-form deformations for parametric shape deformationI Step #2: Reduced basis method for efficient fluid solution
Test problem for fluid-structure interactionI Stokes fluid + 1-d elliptic wallI Parametric coupling of fluid and structureI Fixed-point algorithm for the reduced system
Conclusions and future work
Toni Lassila Model reduction of steady fluid-structure interaction problems with reduced basis methods 2 / 19
Motivation for FSI: Cardiovascular Modelling
Images courtesy of A.Quarteroni (EPFL)
Patient physiology → medical imaging → computational model → simulation
Interests:I Modelling the onset of pathologies (aneurysms, atherosclerosis)I Simulation and planning of surgeriesI Modelling of drug release and transfer in blood flow
Cardiovascular system is a complex flow network with different time and spatial scales
Arterial walls flexible with relatively large displacements
Toni Lassila Model reduction of steady fluid-structure interaction problems with reduced basis methods 3 / 19
Fluid-Structure Interaction in Cardiovascular Modelling
Fluid-structure interaction = coupled Navier-Stokes + nonlinear
elasticity ⇒ high computational cost
Arterial wall moves ⇒ free boundary problem
Full 3-d solution only in small sections of cardiovascular system
Even highly parallel codes take days of computing time
Modelling of full cardiovascular system (heart, main arteries,
peripheral circulation) requires multiscale approach and reduced
order models
Need way to reduce both complexity of state equations and geometric
complexity of free boundary problem
Simulation by G.Fourestey (EPFL)
Toni Lassila Model reduction of steady fluid-structure interaction problems with reduced basis methods 4 / 19
What Is (Parametric) Model Order Reduction?
“Model Order Reduction (MOR) is a cross-disciplinary field that strives to
systematically reduce a complicated ODE or PDE model to a simpler and
computationally more tractable model, while preserving the most important
dynamics of the model.”
In our case the objective is: given a finite-element discretized parametric PDE
problem to find the fluid solution uh such that
A(µ)uh = fh, (dim uh = N )
where µ ∈ D is a low-dimensional parameter vector, find reduced basis matrix
Z ∈ RN ×N and reduced state uhN (µ) = Z T uh(µ) with N N s.t. solution of
the reduced system
Z T A(µ)ZuhN = Z T fh
gives an approximate solution within some acceptable tolerance for a specific
range of parameters
||ZuhN (µ)−uh(µ)||< ε ∀µ ∈D
Toni Lassila Model reduction of steady fluid-structure interaction problems with reduced basis methods 5 / 19
Test Problem of Steady Fluid-Structure Interaction
η displacement of wall from rest
position Σ0
K spring constant to displacements in
the normal direction
τ interface traction caused by the fluid
Stokes fluid + 1-d membrane [G98,M05]
Fluid:ν
∫Ω(η)
∇u ·∇w dΩ−∫
Ω(η)p∇ ·w dΩ =
∫Ω(η)
fF ·wdΩ ∀w ∈ (H10,Σ(η)(Ω(η)))2,∫
Ω(η)q∇ ·u dΩ = 0 ∀q ∈ L2(Ω(η)),
u = u0 on ∂Ω(η)\Σ(η), u = 0 on Σ(η)
Wall: ∫Σ0
K(x)η′φ′ dΓ =
∫Σ0
τ(p,u)φ dΓ ∀φ ∈H10 (Σ0)
Coupling:
τ(p,u) =[pn−ν
(∂u∂n + ∂u
∂n
T)]T
[0
1
]on Σ(η)
Toni Lassila Model reduction of steady fluid-structure interaction problems with reduced basis methods 6 / 19
Model Reduction Strategy for Fluid-Structure Interaction
Standard Fluid-Structure Interaction Reduced Fluid-Structure Interaction
Toni Lassila Model reduction of steady fluid-structure interaction problems with reduced basis methods 7 / 19
Reduction Step #1: Geometry Parameterization
T (·; µ ′)
Ω(µ)
Ω(µ ′)
Ω0
T (·; µ)
Assume there exists a reference configuration for the fluid domain Ω0
Choose parameter space D and parametric map T (x; µ) : Ω0×D → Rd
For each parameter vector µ ∈D we get parametric domain Ω(µ) = T (Ω0; µ)
Many parameterization choices: boundary splines, free-form deformations,
point-based T-splines, transfinite mapping, . . .
Toni Lassila Model reduction of steady fluid-structure interaction problems with reduced basis methods 8 / 19
Free-Form Deformations for Shape Parameterization [SP86]
T (x; µ)Ω(0)
P0i ,j
Pi ,j
Ω(µ)
D
Choose a L×M lattice of control points P0i ,j around the reference shape
Introduce parameters µ`m as displacements of each control point
Perturbed control points Pi ,j = P0i ,j + µ i ,j define a parametric domain map
T (x; µ) =L−1
∑`=0
M−1
∑m=0
(P0
i ,j + µ i ,j
)b`,m(x)
Tensor product Bernstein basis polynomials form a partition of unity
b`,m(x1,x2) =
(L−1
`
)(M−1
m
)(1−x1)L−`−1x `
1(1−x2)M−m−1xm2
Toni Lassila Model reduction of steady fluid-structure interaction problems with reduced basis methods 9 / 19
Parametric Stokes Equations in Fixed Domain
Parametric domain Ω(µ) obtained as the image T (Ω0; µ) of fixed reference domain
Transform the PDEs on Ω(µ) to parametric PDEs on Ω0
Parametric transformation tensors for the viscous term νT (x; µ) := J−TT J−1
T det(JT )
and the pressure-divergence term χT (x; µ) := J−1T det(JT )
Weak form of the parametric Stokes equations on a fixed reference domain:
Find u(µ) ∈H10 (Ω0)×H1
0 (Ω0)×D and p(µ) ∈ L2(Ω0)×D s.t.
∫Ω0
(ν
∂uk
∂xi[νT ]i ,j
∂vk
∂xj+ p [χT ]k,j
∂vk
∂xj
)dΩ0 =
∫Ω0
det(JT )[fF ]k dΩ0,
∀v ∈H10 (Ω0)×H1
0 (Ω0)∫Ω0
q [χT ]k,j∂uk
∂xjdΩ0 = 0,
∀q ∈ L2(Ω0)
Free-boundary problem is now reduced to low-dimensional parameter
space D .
Toni Lassila Model reduction of steady fluid-structure interaction problems with reduced basis methods 10 / 19
Parametric Coupling of Fluid and Structure
Fixed-point algorithm
1 [Fluid substep] For a given µk solve the fluid problem in Ω(µk ) to obtain (u(µk ),p(µk ))
2 Compute assumed interface traction τ =[p(µk )n−ν
(∂u(µk )
∂n + ∂u(µk )T
∂n
)n]T[
0
1
]3 [Structure substep] Solve for assumed wall displacement η ∈H1
0 (Σ0) using assumed traction∫Σ0
K(x)η′φ′ dΓ =
∫Σ0
τφ dΓ ∀φ ∈H10 (Σ0)
4 [Parametric projection substep] Solve minimization problem
µk+1 := argmin
µ
∫Σ|η(µ)− η(µ
k )|2 dΓ
to obtain next parameter value. Displacement η(µ) is obtained using T (x; µ).
5 Iterate until ||µk+1−µk ||< εtol .
Toni Lassila Model reduction of steady fluid-structure interaction problems with reduced basis methods 11 / 19
Reduction Step #2: Reduced Basis Methods for Parametric PDEs
Problem: FE solution (uh(µ),ph(µ)) ∈ X h×Qh too expensive
to compute for many different values of µ.
Observation: Dependence of the bilinear forms A (·, ·; µ) and
B(·, ·; µ) on µ is smooth ⇒ parametric manifold of solutions in
X h×Qh is smooth
Solution: Choose a representative set of parameter values
µ1, . . . ,µN with N N
Snapshot solutions uh(µ1), . . . ,uh(µN ) span a subspace X hN for
the velocity and p(µ1), . . . ,p(µN ) span a subspace QhN for the
pressure
Galerkin reduced basis formulation
For any parameter vector µ ∈D find reduced solution uhN (µ) ∈ X h
N
and phN (µ) ∈Qh
N such that
A (uhN (µ),v; µ) +B(ph
N (µ),v; µ) = 〈F h(µ),v〉 for all v ∈ X hN
B(q,uhN (µ); µ) = 〈Gh(µ),q〉 for all q ∈Qh
N
Toni Lassila Model reduction of steady fluid-structure interaction problems with reduced basis methods 12 / 19
Comparison Between Finite Element and Reduced Basis Methods
FE basis functions
Locally supported
Generic, work for many problems
A priori estimates readily available
RB basis functions
Globally supported
Constructed for specific problem
A posteriori estimates required to
guarantee approximation stability
Toni Lassila Model reduction of steady fluid-structure interaction problems with reduced basis methods 13 / 19
How to Choose Parameter Snapshots µ1, . . . ,µN?
Greedy Algorithm [GP05,RHP08,RV07]
1 Large (but finite) training set of parameters Ξtrain ⊂D
2 Choose first snapshot µ1 and obtain first approximation space for velocity X h1 = span(uh(µ1))
and pressure Qh1 = span(ph(µ1)) and
3 Next snapshot is chosen as
µn = argmax
µ∈Ξtrain
∆(uhn−1(µ)),
where ∆(uhn−1(µ)) is an efficiently computable upper bound for the error
εn(µ) := infuh
n−1(µ)∈X hn−1
||uh(µ)−uhn−1(µ)||1
4 Construct next spaces X hn = span(uh(µ1), . . . ,uh(µn)) and Qh
n = span(ph(µ1), . . . ,ph(µn)).
Repeat from until upper bound of error ∆ sufficiently small
Finally we perform Gram-Schmidt to obtain a basis ξ vnN
n=1 for the velocity
space X hN and a basis ξ p
n Nn=1 for the pressure space Qh
N . To stabilize the
reduced velocity-pressure pair it is necessary to add the so called “supremizer”
solutions to the velocity space [RV07]. Total RB dimension is therefore 3N.
Toni Lassila Model reduction of steady fluid-structure interaction problems with reduced basis methods 14 / 19
Are There Computational Savings in Practice?
Assembly of RB system can depend on N ⇒ no computational savings are realized
Assumption of affine parameterization
A (v ,w ; µ) =Ma
∑m=1
Θma (µ)A m(v ,w), B(p,w ; µ) =
Mb
∑q=1
Θmb (µ)Bm(p,w)
leads to a split
A (ξvn ,ξ v
n′ ; µ) =Ma
∑M=1
Θma (µ)A m(ξ
vn ,ξ v
n′ ), B(ξpn ,ξ v
n′ ; µ) =Mb
∑m=1
Θmb (µ)Bm(ξ
pn ,ξ v
n′ )
so that the matrices Am and Bm do not depend on µ and can be precomputed (offline stage)
After precomputation, RB system assembly and solution independent from N (online stage)
When parameterization nonaffine, use Empirical Interpolation Method [BMNP04]
For any µ ∈D find reduced velocity uN (µ) and reduced pressure pN (µ)
s.t. (Ma
∑m=1
Θma (µ)Am
)uN +
(Mb
∑m=1
Θmb (µ)Bm
)pN = F(µ)
Mb
∑m=1
Θmb (µ)[Bm]T uN = G(µ).
Toni Lassila Model reduction of steady fluid-structure interaction problems with reduced basis methods 15 / 19
Results of Model Reduction for the Test Problem
Iteration Coupling Step size Cumulative
step # error ||µk+1−µk || time
1 1.664e-6 4.187e-4 3 s
50 9.981e-8 3.661e-5 120 s
100 7.835e-8 1.488e-5 239 s
150 6.861e-8 1.323e-5 359 s
200 6.073e-8 1.193e-5 478 s
250 5.433e-8 1.076e-5 597 s
286 5.047e-8 9.987e-6 683 s
Table: Fixed-point iteration convergence
0 0.5 1 1.5 2 2.5 3−1.5
−1
−0.5
0
0.5
1
1.5x 10
−3
Channel length x1
Dis
plac
emen
t
DisplacementAssumed Displacement
Figure: Displacement vs. assumed displacement at the
end of the fixed-point iteration
Inflow velocity v0 = 30 cm/s, blood viscosity ν = 0.035 g/cm·s, spring constant K = 62.5 g/s2
Snapshot solutions: Taylor-Hood P2/P1 finite elements with N = 18 423 degrees of freedom
Free-form deformations: 6 parameters to deform channel wall
Reduced basis dimension: N = 16
Reduction in fluid system size: 383 : 1
Reduction in geometric complexity (compared to nodal deformation): 27 : 1
Toni Lassila Model reduction of steady fluid-structure interaction problems with reduced basis methods 16 / 19
Summary
Model reduction applied to steady fluid-structure interaction problemI Reduction #1: Geometry parameterization with free-form deformationsI Reduction #2: Reduced basis methods for fluid solution in parametric domainI Parametric coupling via fixed-point algorithm
Future workI Implementation of a posteriori error estimates for reduced Stokes equationsI Proof of fixed-point iteration convergenceI Elasticity equation for the wallI Navier-Stokes equations for the fluidI Unsteady problems
Toni Lassila Model reduction of steady fluid-structure interaction problems with reduced basis methods 17 / 19
Thank you for your attention.
Toni Lassila Model reduction of steady fluid-structure interaction problems with reduced basis methods 18 / 19
Bibliography
BMNP04 M. Barrault, Y. Maday, N.C. Nguyen, and A.T. Patera. An ‘empirical interpolation’ method: application to efficient
reduced-basis discretization of partial differential equations. C. R. Math. Acad. Sci. Paris, 339(9):667–672, 2004.
G98 C. Grandmont. Existence et unicite de solutions d’un probleme de couplage fluide-structure bidimensionnel stationnaire.
C. R. Math. Acad. Sci. Paris, 326:651–656, 1998.
GP05 M.A. Grepl and A.T. Patera. A posteriori error bounds for reduced-basis approximations of parametrized parabolic partial
differential equations. ESAIM Math. Modelling Numer. Anal., 39(1):157–181, 2005.
M05 C.M. Murea. The BFGS algorithm for a nonlinear least squares problem arising from blood flow in arteries. Comput.
Math. Appl., 49:171–186, 2005.
RHP08 G. Rozza, D.B.P. Huynh, and A.T. Patera. Reduced basis approximation and a posteriori error estimation for affinely
parametrized elliptic coercive partial differential equations. Arch. Comput. Methods Engrg., 15:229–275, 2008.
RV07 G. Rozza and K. Veroy. On the stability of the reduced basis method for Stokes equations in parametrized domains.
Comput. Methods Appl. Mech. Engrg., 196(7):1244–1260, 2007.
SP86 T.W. Sederberg and S.R. Parry. Free-form deformation of solid geometric models. Comput. Graph., 20(4), 1986.
Toni Lassila Model reduction of steady fluid-structure interaction problems with reduced basis methods 19 / 19