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 F´ ed´ erale de Lausanne 10th Finnish Mechanics Days, Jyv¨ askyl¨ a, Finland December 3-4, 2009 Toni Lassila Model reduction of steady fluid-structure interaction problems with reduced basis methods 1 / 19
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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
]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