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