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Adaptive Time-Step Control for Nonlinear Fluid-Structure Interaction Lukas Failer and Thomas Wick Lehrstuhl M17, Fakultt fr Mathematik Technische Universitt Mnchen Centre de Mathmatiques Appliques (CMAP) cole Polytechnique

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Adaptive Time-Step Control for Nonlinear Fluid-Structure Interaction

Lukas Failer and Thomas Wick

Lehrstuhl M17, Fakultät für Mathematik Technische Universität München Centre de Mathématiques Appliquées (CMAP) École Polytechnique Université Paris-Saclay

Nov 10, 2016 Workshop 2 of the RICAM special semester 2016

Thomas Wick (Ecole Polytechnique) Time step control for FSI 1

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Overview

1 Motivation 2 Modeling FSI 3 Discretization 4 Dual-weighted residual time-step control 5 Truncation-based time step control 6 Conclusions

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Motivation1

Goal of this work: Investigate adaptive time step control for fluid-structure interaction

  • using dual-weighted residual error estimation;
  • heuristic time step control based on the truncation error.

1Main work carried out during a 3-months stay in 2016 of Lukas Failer at RICAM

Linz

Thomas Wick (Ecole Polytechnique) Time step control for FSI 3

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Fluids and solids in their standard systems

Equations for fluid flows (Navier Stokes) - Eulerian

∂tv + (v · ∇v) − ∇ · σ(v, p) = 0, ∇ · v = 0, in Ωf × I, +bc and initial conditions with Cauchy stress tensor σ(v, p) = −pI + ρf νf (∇v + ∇vT).

Equations for (nonlinear) elasticity - Lagrangian

∂2

t ˆ

u − ˆ ∇ · (ˆ F Σ(ˆ u)) = 0 in Ωs × I, +bc and initial conditions with the stress ˆ F Σ(ˆ u) = 2µs ˆ E + λstrace( ˆ E)I, the strain ˆ E = ( F FT − I) and F = I + ∇ˆ u.

Coupling conditions on Γi, ˆ Γi

vf = ˆ vs and σ(v, p)nf = F Σ(ˆ u)ˆ ns.

  • Ωs

Ωf Γi

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1 Motivation 2 Modeling FSI 3 Discretization 4 Dual-weighted residual time-step control 5 Truncation-based time step control 6 Conclusions

Thomas Wick (Ecole Polytechnique) Time step control for FSI 5

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Challenges in FSI modeling

  • Dealing and coupling of different classes of partial differential equations

(PDEs): elliptic, parabolic, hyperbolic that require different mathematical analysis and numerical tools;

  • Nonlinearities in various equations and nonlinear coupling terms;
  • Multidomain character and interface coupling conditions;
  • Combining different coordinate systems: Eulerian and Lagrangian;
  • Moving boundaries, i.e., moving interfaces;
  • Designing robust and efficient numerical methods.

Thomas Wick (Ecole Polytechnique) Time step control for FSI 6

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Variational-monolithic ALE fluid-structure interaction

Formulation

Find vector-valued velocities, vector-valued displacements and a scalar-valued fluid pressure, i.e., {ˆ vf , ˆ vs, ˆ uf , ˆ us, ˆ pf } ∈ {ˆ vD

f + ˆ

V0

f,ˆ v} × ˆ

Ls × {ˆ uD

f + ˆ

V0

f,ˆ u} × {ˆ

uD

s + ˆ

V0

s } × ˆ

L0

f , such that ˆ

vf (0) = ˆ v0

f ,

ˆ vs(0) = ˆ v0

s, ˆ

uf (0) = ˆ u0

f , and ˆ

us(0) = ˆ u0

s are satisfied, and for almost all times t ∈ I holds:

Fluid momentum    (ˆ J ˆ ̺f ∂tˆ vf , ˆ ψv)

Ωf + ( ˆ

̺f ˆ J( F−1(ˆ vf − ˆ w) · ∇)ˆ vf ), ˆ ψv)

Ωf + (ˆ

Jˆ σf F−T, ∇ ˆ ψv)

Ωf

+ ˆ ̺f νf ˆ J( F−T ∇ˆ vT

f ˆ

nf ) F−T, ˆ ψvˆ

Γout = 0

∀ ˆ ψv ∈ ˆ V0

f,ˆ Γi,

Solid momentum, 1st eq.

  • ( ˆ

̺s∂tˆ vs, ˆ ψv)

Ωs + (

F Σ, ∇ ˆ ψv)

Ωs + (

Σv(ˆ vs), ∇ ˆ ψv)

Ωs

= 0 ∀ ˆ ψv ∈ ˆ V0

s ,

Fluid mesh motion

  • (ˆ

σmesh, ∇ ˆ ψu)

Ωf

= 0 ∀ ˆ ψu ∈ ˆ V0

f,ˆ u,ˆ Γi,

Solid momentum, 2nd eq.

  • ˆ

̺s(∂t ˆ us − ˆ vs, ˆ ψu)

Ωs

= 0 ∀ ˆ ψu ∈ ˆ Ls, Fluid mass conservation

  • (

div (ˆ J F−1ˆ vf ), ˆ ψp)

Ωf

= 0 ∀ ˆ ψp ∈ ˆ L0

f .

Thomas Wick (Ecole Polytechnique) Time step control for FSI 7

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A compact formulation

Formulation (Compact FSI in space-time formulation)

Find a velocity ˆ v ∈ ˆ vD + Wv, a displacement ˆ u ∈ ˆ uD + Wu and a pressure ˆ p ∈ L2(I; Lf,0) with the initial conditions ˆ v(0) = ˆ v0 and ˆ u(0) = ˆ u0 fulfilling the weak formulation: ( (ˆ Jρ0

f ∂tˆ

v, ϕ) )f + ( (ˆ Jρ0

f (

F−1(ˆ v − ∂t ˆ u) · ∇)ˆ v, ϕ) )f + ( (ˆ Jˆ σf F−T, ∇ϕ) )f +( (ρ0

s ∂tˆ

v, ϕ) )s + ( ( F Σs, ∇ϕ) )s − ρf νf F−T∇ˆ vT, ϕ

  • ut

−( (ˆ Jρ0

f ˆ

f, ϕ) )f + ( (ρ0

s ˆ

f, ϕ) )s = 0 ∀ϕ ∈ L2(I; Vv) ( (ˆ σmesh, ∇ψ) )f + ( (∂t ˆ u − ˆ v, ψ) )s = 0 ∀ψ ∈ L2(I; Vu) ( (div(ˆ J F−1ˆ v), ξ) )f = 0 ∀ξ ∈ L2(I; Lf) where ( (u, v) )f = T

0 (u, v)Ω dt.

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1 Motivation 2 Modeling FSI 3 Discretization 4 Dual-weighted residual time-step control 5 Truncation-based time step control 6 Conclusions

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Standard discretization so far

  • Temporal discretization based on finite differences
  • Spatial discretization based on inf-sup stable finite elements
  • Nonlinear solution using Newton’s method with quasi-Newton steps

and line-search backtracking

  • Solution of linear equation systems (direct solver - unfortunately) 2

2Iterative FSI solvers at RICAM, e.g., Langer/Yang; 2015, 2016; Jodlbauer; 2016.

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Numerical observations: FSI benchmark3

Benchmark configuration:

Figure: Flow around cylinder with elastic beam with circle-center C = (0.2, 0.2) and radius r = 0.05. Figure: Velocity field and mesh deformation at two snapshots using the harmonic

  • MMPDE. The displacement extremum is displayed at left and a small deformation is

shown at right.

3Hron/Turek; 2006

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Long time FSI computations

  • 200

200 400 600 800 2 4 6 8 10 12 Drag Time Secant CN (v) Tangent CN

  • 200

200 400 600 800 2 4 6 8 10 12 Drag Time Tangent CN (vw) Secant CN shifted (v)

Figure: Blow-up (using the time step k = 0.01) of the un-stabilized Crank-Nicolson schemes (secant and tangent) whereas the shifted Crank-Nicolson schemes is stable

  • ver the whole time interval. Moreover the secant Crank-Nicolson scheme exhibits

the instabilities earlier than the tangent version.

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Long time FSI computations (cont’d)

50 100 150 200 250 300 350 400 2 4 6 8 10 12 Drag Time Secant CN shifted (v) FS 50 100 150 200 250 300 350 400 2 4 6 8 10 12 Drag Time Tangent CN Secant CN shifted (v) FS

Figure: Top: stable solution (using the large time step k = 0.01) computed by the shifted Crank-Nicolson and the Fractional-Step-θ scheme. Recall the blow-up of the un-stabilized Crank-Nicolson scheme in this case. Bottom: using the smaller time step k = 0.001 yields stable solutions for any time stepping scheme.

  • A smaller time step works, but of course takes more computational

power and the interest in implicit time-stepping is to get rid of any time step restrictions.

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Choice of time-stepping scheme

  • Motivated by the previous results, we are interested in a space-time

formulation and the shifted Crank-Nicolson (where θ = 0.5 + δt) and Fractional-Step-θ schemes;

  • Shifted Crank-Nicolson: 2nd order, A-stable (but not strongly), depends
  • n characteristic time step size δt
  • Fractional-Step-θ: 2nd order, strongly A-stable
  • Recently a Galerkin interpretation of the Fractional-Step-θ scheme has

been presented by Richter/Meidner; 2014, 2015;

  • Based on their results, we further extend to fluid-structure interaction

(current work Failer/Wick; 2016).

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1 Motivation 2 Modeling FSI 3 Discretization 4 Dual-weighted residual time-step control 5 Truncation-based time step control 6 Conclusions

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Prerequisites

  • Use a Petrov-Galerkin formulation in time;
  • Apply DWR (dual-weighted residual) estimator from

Becker/Rannacher; 1996,2001;

  • Adjoint time-stepping is tricky;

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Petrov-Galerkin discretization in time

Definition of function spaces: Xv,u

k

= {vk ∈ C(¯ I, V)| vk|Im ∈ P1(Im, V), m = 1, 2, . . . , M} Xp

k = {pk ∈ L2(I, Lf,0| pk|Im ∈ P0(Im, Lf,0, m = 1, 2, . . . , M}

Test space for the momentum equation and mesh motion4: Xv,u

k,θ = {ϕk ∈ L2(I, V)| ϕk|Im ∈ Pθ(Im, V), m = 1, 2, 3, . . . , M}

with Pθ(Im, V) := {Φm ¯ ϕk,m| ¯ ϕk,m ∈ V} where Φm(t) = 1 + 6(θm − 0.5)2t − tm−1 − tm ∆tm

  • Only the test functions are θ-dependent

⇒ Method easily applicable to nonlinear equations

4Meidner/Richter; 2014

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Petrov-Galerkin discretization in time

Formulation

Find Uk := (vk, uk, pk) ∈ UD + Xk := Xv,u

k

× Xv,u

k

× Xp

k such that

AF(Uk)(ϕk) + As(Uk)(ϕk) + AΓ(Uk)(ϕk) = F(Uk)(ϕk) + IV(Uk)(ϕk) ∀ϕk ∈ Xv

k,θ,

AM(Uk)(ψk) + AV(Uk)(ψk) = IU(Uk)(ψk) ∀ψk ∈ Xu

k,θ,

AD(Uk)(ξk) = 0 ∀ξk ∈ Xp

k.

where IV and IU contain the initial conditions.

Formulation

Find Uk := (vk, uk, pk) ∈ UD + Xk := Xv,u

k

× Xv,u

k

× Xp

k such that

A(U)(Ψk) = F(Ψk) + I(Uk)(Ψk) ∀Ψk ∈ Xk with A(Uk)(Ψ) := AF(Uk)(ϕk) + . . . + AD(Uk)(ξk).

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DWR estimator5

  • Let J(U) be a quantity of interest, i.e., a goal functional.

Lagrange functional: L : X × X → R, L(U, Z) := J(U) − A(U)(Z) + F(Z) Error in the functional of interest: J(U) − J(Uk) = L(U, Z) − L(Uk, Zk)

5Becker/Rannacher; 2001

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DWR estimator (cont’d)

  • U: exact solution
  • Uk: solution of Petrov-Galerkin discretization
  • ˜

Uk: solution of the time-stepping scheme

  • L(·)(·): continuous Lagrange functional
  • ˜

L(·)(·): Lagrange functional of the time-stepping scheme Remarks:

  • We can only compute ˜

Uk

  • Quadrature error of same order as time-stepping scheme

For the error holds then: J(U) − J( ˜ Uk) = L(U, Z) − ˜ L( ˜ Uk, ˜ Zk) (1) = L(U, Z) − L( ˜ Uk, ˜ Zk) + L( ˜ Uk, ˜ Zk) − ˜ L( ˜ Uk, ˜ Zk) (2) = L(U, Z) − L( ˜ Uk, ˜ Zk) + F(Zk) − A( ˜ Uk)( ˜ Zk) (3)

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Adjoint equations

  • Running backward in time;
  • Need to store primal solution;
  • Monolithic space-time formulation allows to write down consistent

adjoint formulation. Find Z := (zv, zp, zu) ∈ X such that A′(U)(Φ, Z) = J′(U)(Φ) Φ ∈ X Petrov-Galerkin semi-discretized adjoint equation:

Formulation

Find Zk := (zv

k, zp k, zu k) ∈ Xv k,θ × Xp k × Xu k,θ such that

A′(Uk)(Φk, Zk) = J′(Uk)(Φk) Φ ∈ Xk.

  • Lengthy equations!

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Example

The θ-dependent quadrature rule leads to the following terms in the adjoint time-stepping scheme:

  • I(ρf Jk(F−1

k (vk − ∂tuk) · ∇ ˜

Φm(t)ϕm), zv

k) dt

(4) ≈ kmθm(ρf Jk,m(F−1

k,m(vk,m − uk,m − uk,m−1

km ) · ∇ ˜ ϕm(t)ϕm), zv

k,m)

(5) + km+1(1 − θm+1)(ρf Jk,m(F−1

k,m(vk,m − uk,m+1 − uk,m

km ) · ∇ ˜ ϕm(t)ϕm), zv

k,m+1)

(6)

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Numerical test: FSI 2 benchmark

  • Goal functional:

J( ˆ U) =

I ˆ

uy(A)2 dt.

Figure: Flow around cylinder with elastic beam with circle-center C = (0.2, 0.2) and radius r = 0.05. Figure: FSI2: comparison of uniform refinement of the time grid and adaptive refinement based on DWR.

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Numerical test: FSI 3 benchmark with new inflow profile

  • Goal functional (drag): J( ˆ

U) =

I

  • ˆ

ΓI ˆ

σf ˆ nf e1 ds dt.

  • Inflow profile:

Figure: FSI3: Drag (left) and at right comparison of uniform refinement of the time grid and adaptive refinement based on DWR.

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1 Motivation 2 Modeling FSI 3 Discretization 4 Dual-weighted residual time-step control 5 Truncation-based time step control 6 Conclusions

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An heuristic (truncation-based) estimator in time 6

Proposition

Per time step, we compute once the problem with 2k, compute the goal functional J( ˆ U), and then compute two times with time step size k again. After we compare J( ˆ Uk) and J( ˆ U2k) by evaluating the absolute error abserr := |J( ˆ U2k) − J( ˆ Uk)|. Let the error tolerance TOLTS be given. We evaluate: abserr := |J( ˆ U2k) − J( ˆ Uk)|, θ = γ ∗ TOLTS abserr K , (7) where γ ≈ 1 is a safety factor, γ = 0.9), K =

1 15 . We then compute the new time step size as

knew =

  • kold

1 ≤ θ ≤ 1.2, kold ∗ θ

  • therwise.

(8) Moreover, we check whether kmin ≤ knew ≤ kmax. If knew < 0.5kold, i.e., the new time step would be much smaller than the old

  • ne, we redo the current time step.
  • The check (8) prevents high time step oscillations and just keeps the old time step if the

difference is not too large.

  • The last check knew < 0.5kold is expensive since the entire calculation of the solution of

this time step has to be repeated. In practice this happens rarely, but is necessary since a dramatic decrease of the time step size indicates that the current solution is by far not accurate enough.

6based on Turek; 1999; Gustafsson/Lundh/Soederlind; 1988

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An heuristic (truncation-based) estimator in time (cont’d)

Algorithm

The adaptive time step control for One-Step-θ schemes reads at tn:

1

Set kold := kn−1

2

Perform

  • one computation with 2kold, evaluate J(ˆ

u2k) at tn + 2kold;

  • compute two steps with kold, evaluate at the end of the 2nd step J(ˆ

uk) at tn + kold + kold;

3

Evaluate abserr and compute θ;

4

Compute the new knew;

5

If knew < 0.5kold, set kold := knew and go to Step (ii);

6

Otherwise accept knew and set kn = knew;

7

Increment n → n + 1 and go to Step (i).

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Application to a FSI phase-field fracture system 8

Figure: Display of the velocity field at two different times in which the highest forward flow and the highest backward flow (negative x-velocity) are obtained.

  • A fracture is prescribed in the solid part7;
  • Parabolic time-dependent sinus-type inflow velocity profile on ˆ

Γin.

7Wick; 2016, JCP 8Wick; 2016 to appear in Radon RICAM FSI book (collection)

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Numerical results

  • Based on the heuristic (truncation-based) error-estimator
  • 0.02

0.02 0.04 0.06 10 20 30 40 50 ux [m] Time [s] TOL = 0.01 TOL = 0.001 TOL = 0.0001 0.2 0.4 0.6 0.8 1 10 20 30 40 50 k[s] Time [s] TOL = 0.01 TOL = 0.001 TOL = 0.0001

Figure: Display of the goal functional (left) and the adaptively obtained time step

  • sizes. We observe significant reduced time step sizes for each tolerance.

0.01 0.02 0.03 0.04 0.05 10 20 30 40 50 ABS error Time [s] TOL = 0.01 TOL = 0.001 TOL = 0.0001 0.005 0.01 0.015 0.02 10 20 30 40 50 ABS error Time [s] 0.0005 0.001 0.0015 0.002 10 20 30 40 50 ABS error Time [s] 5e-05 0.0001 0.00015 0.0002 10 20 30 40 50 ABS error Time [s]

Figure: The evolution of the absolute error is shown. The entire evolution is shown in the top left figure. Then, in the other three figures, in each of them, the respective tolerance is shown as a black dotted line. We observe very well that the absolute error evolution centers around the given tolerance. Thus, the error estimator chooses the time steps in such a way that indeed (as expected) the absolute error is reduced according to the prescribed tolerance.

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1 Motivation 2 Modeling FSI 3 Discretization 4 Dual-weighted residual time-step control 5 Truncation-based time step control 6 Conclusions

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Conclusions

  • Development of adaptive time step control for multiphysics (i.e.,

fluid-structure interaction) problems;

  • Development of dual-weighted residual and heuristic error estimators;
  • Results are good but for the chosen numerical examples less impressive

than we thought!

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Q.E.D. Thanks for your valuable time!

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Variational-monolithic coupling of ALE-FSI with PPF

Formulation (Coupling of ALE-FSI with PFF)

Let ˆ pF ∈ H1( Ωs ∪ C) be given. Find vector-valued velocities, vector-valued displacements, a scalar-valued fluid pressure, and a scalar-valued phase-field function, that is to say that {ˆ vf , ˆ vs, ˆ uf , ˆ us, ˆ pf , ˆ ϕs} ∈ {ˆ vD

f + ˆ

V0

f,ˆ v} × ˆ

Ls × {ˆ uD

f + ˆ

V0

f,ˆ u} × {ˆ

uD

s + ˆ

V0

s } × ˆ

L0

f × H1(

Ωs ∪ C), such that ˆ vf (0) = ˆ v0

f ,

ˆ vs(0) = ˆ v0

s , ˆ

uf (0) = ˆ u0

f , ˆ

us(0) = ˆ u0

s and ˆ

ϕs(0) = ˆ ϕ0

s are satisfied, and for almost all times t ∈ I holds:

Fluid momentum    (ˆ J ˆ ̺f ∂t ˆ vf , ˆ ψv)

Ωf + ( ˆ

̺f ˆ J( F−1(ˆ vf − ˆ w) · ∇)ˆ vf ), ˆ ψv)

Ωf + (ˆ

Jˆ σf F−T, ∇ ˆ ψv)

Ωf

+ρf νf ˆ J( F−T ∇ˆ vT

f ˆ

nf ) F−T, ˆ ψvˆ

Γout = 0

∀ ˆ ψv ∈ ˆ V0

f,ˆ Γi ,

Solid momentum, 1st eq.    ( ˆ ̺s∂t ˆ vs, ˆ ψv)

Ωs +

  • E( ˆ

ϕs) F Σ, ∇ ˆ ψv

  • Ωs

+

  • E( ˆ

ϕs) Σv(ˆ vs), ∇ ˆ ψv

  • Ωs

+( ˆ ϕ2

s ˆ

pF, ∇ · ˆ ψv) = 0 ∀ ˆ ψv ∈ ˆ V0

s ,

Fluid mesh motion

  • (ˆ

σmesh, ∇ ˆ ψu)

Ωf

= 0 ∀ ˆ ψu ∈ ˆ V0

f,ˆ u,ˆ Γi,

Solid momentum, 2nd eq.

  • ˆ

̺s(∂t ˆ us − ˆ vs, ˆ ψu)

Ωs

= 0 ∀ ˆ ψu ∈ ˆ Ls, Fluid mass conservation

  • (

div (ˆ J F−1 ˆ vf ), ˆ ψp)

Ωf

= 0 ∀ ˆ ψp ∈ ˆ L0

f ,

Phase-field        (1 − κ)(ˆ J ˆ ϕs ( Σ + Σv) : ˆ E, ˆ ψϕ)

Ωs + 2(ˆ

J ˆ ϕs pF ∇ · ˆ us, ˆ ψϕ)

Ωs

+Gc

  • − 1

ε (ˆ

J(1 − ˆ ϕs), ˆ ψϕ) + ε(ˆ J( ∇ ˆ ϕs F−1) F−T, ∇ ˆ ψϕ)

  • Ωs

+ ˆ J[Ξ + γ( ˆ ϕs − ˆ ϕold

s )]+, ˆ

ψϕ

  • Ωs

= 0 ∀ ˆ ψϕ ∈ H1( Ωs ∪ C).

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