Reduced Formulation of Steady Fluid-structure Interaction with - - PowerPoint PPT Presentation

▶
reduced formulation of steady fluid structure interaction
SMART_READER_LITE
LIVE PREVIEW

Reduced Formulation of Steady Fluid-structure Interaction with - - PowerPoint PPT Presentation

Jun 29, 2023 270 likes •499 views

Reduced Formulation of Steady Fluid-structure Interaction with Parametric Coupling Toni Lassila , , Alfio Quarteroni , , Gianluigi Rozza Department of Mathematics and Systems Analysis MOX - Modellistica e Calcolo

slide-1
SLIDE 1

Reduced Formulation of Steady Fluid-structure Interaction with Parametric Coupling

Toni Lassila∗,◦, Alfio Quarteroni†,×, Gianluigi Rozza†

∗Department of Mathematics and Systems Analysis × MOX - Modellistica e Calcolo Scientifico

School of Science and Technology Dipartimento di Matematica “F . Brioschi” Aalto University Politecnico di Milano

†Chair of Modelling and Scientific Computing

Mathematics Institute of Computational Science and Engineering ´ Ecole Polytechnique F´ ed´ erale de Lausanne

  • Supported by the Emil Aaltonen Foundation

IV International Symposium on Modelling of Physiological Flows, Chia Laguna, Sardinia, June 2-5, 2010

Fluid-structure Interaction with Parametric Coupling 1 / 21

slide-2
SLIDE 2

Outline

“Towards reducing the geometric complexity of FSI problems” Motivation for fluid-structure interaction

Previous three days of this conference...

Steady fluid-structure interaction problem

Incompressible Stokes equations for fluid Generalized 1-d string model for the structure Coupling between traction applied by fluid and structural displacement

Parametric flow geometry

Parametric free-form deformation of geometry Fluid equations on fixed domain with parametric coefficients

Model reduction

Fluid-structure coupling variables are the geometric parameters Iterative scheme in parameter space Reduced basis method for approximation of parametric Stokes

Fluid-structure Interaction with Parametric Coupling 2 / 21

slide-3
SLIDE 3

Approaches to Reduced Modelling of Fluid-Structure Interaction

Classical model reduction applied to linear systems of ODEs and PDEs ⇒ not very useful for FSI with strong geometric nonlinearity. Some new approaches have been proposed: Proper Orthogonal Decomposition (review in [DH01]) Eigendecomposition-based method for approximating an ensemble of trajectories of a given dynamical system Widely used in aeroelasticity simulations (not so much in hemodynamics) Cons: Computationally expensive, error of reduced model difficult to estimate Geometrical multiscale (review in [FQV09]) Different fidelity models (0D vs. 1D vs. 3D-models) used in different parts of the cardiovascular system, coupled together with suitable boundary conditions Combines modelling scales ranging from peripheral circulation all the way to the major arteries Cons: Physically meaningful boundary conditions between 0D-1D-3D models are challenging (talk of C. Malossi) Reduced basis element method [LMR06] Decomposition of complex flow network to a small collection of “simple elements” like T-junctions and straight pipes, combined with reduced basis method (talk by L. Iapichino)

Fluid-structure Interaction with Parametric Coupling 3 / 21

slide-4
SLIDE 4

Our Model Reduction Strategy for Fluid-Structure Interaction

Standard Fluid-Structure Interaction Reduced Fluid-Structure Interaction

Fluid-structure Interaction with Parametric Coupling 4 / 21

slide-5
SLIDE 5

Steady Fluid-Structure Interaction Model Problem

Γin Γout Ω(η) −△u +∇p = 0 ∇·u = 0 Σ(η) η(x1) L

Fluid:              −ν△u +∇p = f, in Ω(η) ∇·u = 0, in Ω(η) u = u0,

  • n Γin ∪Γout

u = 0,

  • n Σ(η)

Structure:      ε d4η dx4

1

−K d2η dx2

1

+η = τΣ, for x1 ∈ (0,L) η(0) = η′(0) = η(L) = η′(L) = 0 Coupling condition: τΣ =

  • pn −ν
  • ∇u +∇ut

n

  • ·
  • 1
  • ,
  • n Σ(η)

Existence proved in [G98] with fixed point argument + some additional regularity assumptions.

Fluid-structure Interaction with Parametric Coupling 5 / 21

slide-6
SLIDE 6

Choice of Structural Model and Treatment of Boundary Conditions

Generalized 1-d string model for arterial wall [QTV00] in the steady case: −kGh ∂ 2η ∂x2

1

+ Eh 1−ν2

P

η R0(x1)2 = τΣ,

  • n x1 ∈ (0,L),

where h = wall thickness, k = Timoshenko shear correction factor, G = shear modulus, E = Young modulus, νP = Poisson ratio, and R0(x1) = radius of the reference configuration at distance x1 from inflow. We choose to include a fourth order singular perturbation term ε d4η

dx4 1

, which after nondimensionalizing the equations gives ε d4η dx4

1

−K d2η dx2

1

+η = τΣ, for x1 ∈ (0,L) with the boundary conditions η(0) = η′(0) = η(L) = η′(L) = 0.

Fluid-structure Interaction with Parametric Coupling 6 / 21

slide-7
SLIDE 7

Steady Fluid-Structure Interaction Problem (weak form)

Incompressible Stokes fluid + 1-d membrane structure Find (u,p,η) ∈ V (Ω(η))×Q(Ω(η))×S (0,L) s.t.      A(u,v)+B(p,v) = F,v for all v ∈ V (Ω(η)) B(q,u) = 0 for all q ∈ Q(Ω(η)) C(η,φ) = R(u,p),φ for all φ ∈ S (0,L). , where we have the bilinear forms for the incompressible Stokes equations A(u,v) = ν

  • Ω(η) ∇u ·∇v dΩ,

B(q,v) = −

  • Ω(η) q∇·v dΩ,

and the linear form F,v =

  • Ω(η) f ·v dΩ

and the structural bilinear form C(η,φ) = ε

  • Σ0

d2η dx2

1

d2φ dx2

1

dx1 +K

  • Σ0

dη dx1 dφ dx1 dx1 +

  • Σ0

ηφ dx1. The fluid residual is the normal component of the normal Cauchy stress of the fluid.

Fluid-structure Interaction with Parametric Coupling 7 / 21

slide-8
SLIDE 8

Reduction Step #1: Free-form Deformations of the Fluid Domain

fixed reference domain deformed parametric domain

  • f control points

parameters = displacements FFD control points

P0

ℓ,m

P0

ℓ,m + µℓ,m

FFD map T(·,µ)

parameter

  • T(·,µ)

matrix µ

Ψ−1 affine map Ψ Ω0 Ω(µ)

Recalling from the talk of A. Manzoni...

Fluid-structure Interaction with Parametric Coupling 8 / 21

slide-9
SLIDE 9

Parametric Fluid Equations on the Fixed Reference Domain

Parametric FFD deformation map T(·;µ) : Ω0 → Ω(µ) defined as T = Ψ−1 ◦ T ◦Ψ where

  • T(

x;µ) =

L

∑

ℓ=0 M

∑

m=0

bL,M

ℓ,m (

x)

  • P0

ℓ,m + µℓ,m

  • and its Jacobian matrix JT (x;µ) := ∇xT define the transformation tensors [RV07]

νT (x;µ) := J−t

T J−1 T det(JT ),

χT (x;µ) := J−1

T det(JT )

used to map fluid problem back to reference domain: find ( u, p) ∈ V (Ω0)×Q(Ω0) s.t.                     

  • Ω0
  • ν ∂

uk ∂ xi [νTη ]i,j ∂ vk ∂ xj + p [χTη ]k,j ∂ vk ∂ xj

  • dΩ0 =
  • Ω0

det(JTη )[f F +flift]k dΩ0, for all v ∈ V (Ω0)

  • Ω0
  • q [χTη ]k,j

∂ uk ∂ xj dΩ0 = 0, for all q ∈ Q(Ω0) Recalling from the talk of A. Manzoni...

Fluid-structure Interaction with Parametric Coupling 9 / 21

slide-10
SLIDE 10

Parametric Coupling of Fluid and Structure

Standard iterative scheme for fluid-structure coupling

Ω(ηk) − → (uk,pk) Stokes update ↑ ↓ fluid residual geometry ηk+1 ← − R(uk,pk) structural equation

Our parametric coupling approach

µk − → Ω(µk) − → ( uk, pk) parametric domain Stokes update ↑ ↓ parameters µk+1 ← −

  • η

← − R(µk) least squares fit structure

Fluid-structure Interaction with Parametric Coupling 10 / 21

slide-11
SLIDE 11

Parametric Coupling Algorithm

Fixed-point algorithm for weak parametric coupling Start with initial guess for the parameter µ0 and set k = 0.

1

[Fluid substep] Solve the discretized fluid problem in Ω0 to obtain ( uh(µk), ph(µk))

2

Form the discrete fluid residual R(µk) := G

  • F −A

uk

h −B

pk

h

  • where G(

vh) takes the boundary normal trace of any vh on Σ0

3

[Structure substep] Solve for assumed structural displacement ˆ η(µk) ∈ S s.t. C(ˆ ηh,φ) = R(µk),φ for all φ ∈ S

4

[Parametric projection substep] Solve “inverse problem” of finding parameter value that gives best fit to the assumed displacement ˆ η(µk) µk+1 := argmin

¯ µ

  • Σ |ηh(¯

µ)− ˆ ηh(µk)|2 dΓ to obtain next parameter value. Displacement ηh(x; ¯ µ) = T(x; ¯ µ)−T(x;0) is given by the FFD and requires no structural equation solutions.

5

Iterate until |µk+1 − µk| < TOL.

Fluid-structure Interaction with Parametric Coupling 11 / 21

slide-12
SLIDE 12

Reduction Step #2: Reduced Basis Methods for Parametric PDEs

Problem: FE solution (uh(µ),ph(µ)) ∈ Vh ×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 Vh ×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 V N

h for

the velocity and p(µ1),...,p(µN) span a subspace QN

h for the

pressure Galerkin reduced basis formulation For given parameter vector µ ∈ D find approximate solution uN

h (µ) ∈ V N h and pN h (µ) ∈ QN h in reduced spaces such that

A(uN

h (µ),v;µ)+B(pN h (µ),v;µ) = F h(µ),v

for all v ∈ V N

h

B(q,uN

h (µ);µ) = 0

for all q ∈ QN

h

Recalling from the talk of A. Manzoni...

Fluid-structure Interaction with Parametric Coupling 12 / 21

slide-13
SLIDE 13

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 to guarantee reliability and accuracy

Fluid-structure Interaction with Parametric Coupling 13 / 21

slide-14
SLIDE 14

Algorithm for Building the Reduced Basis

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 V 1

h = span(uh(µ1))

and pressure Q1

h = span(ph(µ1)) 3

Next snapshot is chosen as µn = argmax

µ∈Ξtrain

∆n−1(µ), where ∆n(µ) is an efficiently computable upper bound for the error εn(µ) := inf

un h(µ)∈V n h

||uh(µ)−un

h(µ)||1 ≤ ∆n(µ) 4

Construct next spaces V n

h = span(uh(µ1),...,uh(µn)) and Qn h = span(ph(µ1),...,ph(µn)).

Repeat from until upper bound of error ∆ sufficiently small. Finally we perform Gram-Schmidt to obtain a basis {ξ v

n}N n=1 for the velocity space V N h and a basis

{ξ p

n }N n=1 for the pressure space QN h . 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.

Fluid-structure Interaction with Parametric Coupling 14 / 21

slide-15
SLIDE 15

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

Θm

a (µ)Am(v,w),

B(p,w;µ) =

Mb

∑

q=1

Θm

b (µ)Bm(p,w)

leads to a split A(ξ v

n,ξ v n′;µ) = Ma

∑

M=1

Θm

a (µ)Am(ξ v n,ξ v n′),

B(ξ p

n ,ξ v n′;µ) = Mb

∑

m=1

Θm

b (µ)Bm(ξ p n ,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 is 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

Θm

a (µ)Am

  • uN +

Mb

∑

m=1

Θm

b (µ)Bm

  • pN = F(µ)

Mb

∑

m=1

Θm

b (µ)[Bm]T uN = 0. Fluid-structure Interaction with Parametric Coupling 15 / 21

slide-16
SLIDE 16

Free-form Deformation Setup for the Model Problem

14×2 grid of control points (not all shown in figure) Ten control points in the top row move in x2-direction, others fixed P = 10 parameters Each parameter varies in range µi ∈ [−0.25,0.25] (small deformations)

Fluid-structure Interaction with Parametric Coupling 16 / 21

slide-17
SLIDE 17

Results of Model Reduction for the Model Problem

Parametric coupling Free-form deformation with P = 10 parameters Fixed point iteration until |µk+1 − µk| < 1e-3 Coupling accuracy measured with Jk =

  • Σ |η(µk+1)− ˆ

η(µk)|2 dΓ Nodes on the fluid-structure boundary NB = 41 Reduction in number of coupling variables 4:1 Reduced basis approximation Greedy algorithm picks N = 20 velocity basis functions ⇒ total size of reduced basis 3N = 60 Basis functions computed using P2/P1 FEM with N = 22227 DOFs Reduction in size of Stokes system (FEM vs. RB) is 370:1. Iteration Coupling Step size step # error |µk+1 − µk| 3.77e-3 2.22e-1 1 4.39e-3 1.72e-1 5 2.28e-4 3.57e-2 10 4.88e-5 1.56e-2 14 3.08e-6 6.99e-4

0.5 1 1.5 2 2.5 3 −0.04 −0.03 −0.02 −0.01 0.01 0.02 0.03 0.04 Length z Displacement Assumed displacement Displacement η(µk)

Fluid-structure Interaction with Parametric Coupling 17 / 21

slide-18
SLIDE 18

How Many Free-form Deformation Parameters Are Needed?

Coupled reduced problem solved with P = 3,...,9 parameters free, others fixed Coupling accuracy measured with Jk =

  • Σ |η(µk+1)− ˆ

η(µk)|2 dΓ until convergence achieve to tolerance |µk+1 − µk| <1e-4 Logarithmic best-fit line indicates convergence like P−α with α ≈ 0.495 FFDs are spline-based and possess good general approximation properties Using P = 20 parameters could obtain coupling up to 10−9 accuracy 5 10 15 20 10

−9

10

−8

10

−7

10

−6

10

−5

10

−4

10

−3

# of FFD parameters P Final coupling accuracy (displacement)

Fluid-structure Interaction with Parametric Coupling 18 / 21

slide-19
SLIDE 19

Conclusion

New model reduction approach to steady fluid-structure interaction

Parameterization of fluid domain with free-form deformations Parametric coupling of applied traction and structural displacement Reduced basis method for efficient approximation of parametric Stokes solutions

Fluid system dimension reduction 370:1 compared to FEM Coupling variables reduction 4:1 compared to explicit nodal deformations Coupling error behaves like P−1/2 as number of FFD parameters P increases Future work

Navier-Stokes equations for the fluid Coupling in stress space instead of displacement space Unsteady problems, ALE formulation with parametric maps Alternatives to free-form deformation parameterization

Fluid-structure Interaction with Parametric Coupling 19 / 21

slide-20
SLIDE 20

Thank you for your attention.

Fluid-structure Interaction with Parametric Coupling 20 / 21

slide-21
SLIDE 21

References

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. FQV09 L. Formaggia, A. Quarteroni and A. Veneziani. Multiscale models of the vascular system. In: L. Formaggia,

  • A. Quarteroni, A. Veneziani (Eds.). Cardiovascular Mathematics, Springer, 2009.

G98 C. Grandmont. Existence et unicit´ e de solutions d’un probl` eme de couplage fluide-structure bidimensionnel

  • stationnaire. C. R. Math. Acad. Sci. Paris, 326:651–656, 1998.

DH01 E. Dowell and K. Hall. Modeling of fluid-structure interaction. Annu. Rev. Fluid Mech. 33:445–490, 2001. 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. LR09 T. L. and G. Rozza. Model reduction of steady fluid-structure interaction problems with free-form deformations and reduced basis methods. Proc. 10th Finnish Mech. Days, Jyv¨ askyl¨ a, Finland, December 2009, pages 454–465, 2009. LMR06 A. E. Løvgren, Y. Maday and E. M. Rønquist. A reduced basis element method for the steady Stokes problem. ESAIM Math. Modelling Numer. Anal., 40(3):529–552, 2006. QTV00 A. Quarteroni, M. Tuveri and A. Veneziani. Computational vascular fluid dynamics: problems, models and methods.

  • Comp. Vis. Sci. 2:163–197, 2000.

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.

Fluid-structure Interaction with Parametric Coupling 21 / 21