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
Overview 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 2
Motivation 1 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. 1 Main 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
Fluids and solids in their standard systems Equations for fluid flows (Navier Stokes) - Eulerian ∂ t v + ( v · ∇ v ) − ∇ · σ ( v , p ) = 0, ∇ · v = 0, in Ω f × I , + bc and initial conditions with Cauchy stress tensor σ ( v , p ) = − pI + ρ f ν f ( ∇ v + ∇ v T ) . Equations for (nonlinear) elasticity - Lagrangian in � ∂ 2 u − ˆ ∇ · ( ˆ F � Σ ( ˆ u )) = 0 Ω s × I , t ˆ + bc and initial conditions F T − I ) and � with the stress ˆ F � u ) = 2 µ s ˆ E + λ s trace ( ˆ E ) I , the strain ˆ E = ( � F � F = I + � Σ ( ˆ ∇ ˆ u . Coupling conditions on Γ i , ˆ Γ i Ω f Γ i σ ( v , p ) n f = � F � � v f = ˆ v s and Σ ( ˆ u ) ˆ n s . Ω s Thomas Wick (Ecole Polytechnique) Time step control for FSI 4
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
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
Variational-monolithic ALE fluid-structure interaction Formulation Find vector-valued velocities, vector-valued displacements and a scalar-valued fluid pressure, i.e., v D f + ˆ V 0 v } × ˆ u D f + ˆ V 0 u D s + ˆ V 0 s } × ˆ L 0 v 0 { ˆ p f } ∈ { ˆ L s × { ˆ u } × { ˆ v f ( 0 ) = ˆ v f , ˆ v s , ˆ u f , ˆ u s , ˆ f , such that ˆ f , f ,ˆ f ,ˆ v 0 u 0 u 0 v s ( 0 ) = ˆ ˆ s , ˆ u f ( 0 ) = ˆ f , and ˆ u s ( 0 ) = ˆ s are satisfied, and for almost all times t ∈ I holds: ( ˆ ̺ f ˆ J ( � F − 1 ( ˆ w ) · � Ω f + ( ˆ σ f � F − T , � v f , ˆ ψ v ) � v f ) , ˆ ψ v ) � ∇ ˆ ψ v ) � J ˆ ̺ f ∂ t ˆ Ω f + ( ˆ v f − ˆ ∇ ) ˆ J ˆ Ω f Fluid momentum F − T � ψ v ∈ ˆ J ( � n f ) � ̺ f ν f ˆ v T F − T , ˆ ψ v � ˆ ∀ ˆ V 0 + � ˆ ∇ ˆ f ˆ Γ out = 0 Γ i , f ,ˆ � ψ v ∈ ˆ Ω s + ( � F � Σ , � Ω s + ( � v s ) , � v s , ˆ ψ v ) � ∇ ˆ ψ v ) � ∇ ˆ ψ v ) � ∀ ˆ V 0 Solid momentum, 1st eq. ( ˆ ̺ s ∂ t ˆ Σ v ( ˆ = 0 s , Ω s � ψ u ∈ ˆ σ mesh , � ∇ ˆ ψ u ) � ∀ ˆ V 0 ( ˆ = 0 Fluid mesh motion Γ i , Ω f u ,ˆ f ,ˆ � ψ u ∈ ˆ v s , ˆ ψ u ) � ∀ ˆ Solid momentum, 2nd eq. ̺ s ( ∂ t ˆ ˆ u s − ˆ = 0 L s , Ω s � ψ p ∈ ˆ ( � J � F − 1 ˆ div ( ˆ v f ) , ˆ ψ p ) � ∀ ˆ L 0 = 0 f . Fluid mass conservation Ω f Thomas Wick (Ecole Polytechnique) Time step control for FSI 7
A compact formulation Formulation (Compact FSI in space-time formulation) v D + W v , a displacement ˆ u D + W u and a pressure ˆ p ∈ L 2 ( I ; L f,0 ) with the Find a velocity ˆ v ∈ ˆ u ∈ ˆ v 0 and ˆ u 0 fulfilling the weak formulation: initial conditions ˆ v ( 0 ) = ˆ u ( 0 ) = ˆ ( ˆ J ρ 0 ( ˆ J ρ 0 f ( � F − 1 ( ˆ ( ˆ σ f � F − T , ∇ ϕ ) ( v , ϕ ) ) f + ( v − ∂ t ˆ u ) · ∇ ) ˆ v , ϕ ) ) f + ( ) f f ∂ t ˆ J ˆ ( ρ 0 ( � F � � ρ f ν f � F − T ∇ ˆ v T , ϕ � +( s ∂ t ˆ v , ϕ ) ) s + ( Σ s , ∇ ϕ ) ) s − � � out ( ˆ J ρ 0 f ˆ ( ρ 0 s ˆ ∀ ϕ ∈ L 2 ( I ; V v ) − ( f , ϕ ) ) f + ( f , ϕ ) ) s = 0 ∀ ψ ∈ L 2 ( I ; V u ) ( ( ˆ σ mesh , ∇ ψ ) ) f + ( ( ∂ t ˆ u − ˆ v , ψ ) ) s = 0 ( div ( ˆ J � F − 1 ˆ ∀ ξ ∈ L 2 ( I ; L f ) ( v ) , ξ ) ) f = 0 ) f = � T where ( ( u , v ) 0 ( u , v ) Ω dt. Thomas Wick (Ecole Polytechnique) Time step control for FSI 8
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 9
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 2 Iterative FSI solvers at RICAM, e.g., Langer/Yang; 2015, 2016; Jodlbauer; 2016. Thomas Wick (Ecole Polytechnique) Time step control for FSI 10
Numerical observations: FSI benchmark 3 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. 3 Hron/Turek; 2006 Thomas Wick (Ecole Polytechnique) Time step control for FSI 11
Long time FSI computations Secant CN (v) Tangent CN (vw) 800 800 Tangent CN Secant CN shifted (v) 600 600 400 400 Drag Drag 200 200 0 0 -200 -200 2 4 6 8 10 12 2 4 6 8 10 12 Time Time 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 over the whole time interval. Moreover the secant Crank-Nicolson scheme exhibits the instabilities earlier than the tangent version. Thomas Wick (Ecole Polytechnique) Time step control for FSI 12
Long time FSI computations (cont’d) 400 400 Secant CN shifted (v) Tangent CN FS Secant CN shifted (v) 350 350 FS 300 300 250 250 Drag Drag 200 200 150 150 100 100 50 50 0 0 2 4 6 8 10 12 2 4 6 8 10 12 Time Time 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. Thomas Wick (Ecole Polytechnique) Time step control for FSI 13
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 + δ t ) and Fractional-Step- θ schemes; • Shifted Crank-Nicolson: 2nd order, A-stable (but not strongly), depends on characteristic time step size δ t • 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). Thomas Wick (Ecole Polytechnique) Time step control for FSI 14
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 15
Prerequisites • Use a Petrov-Galerkin formulation in time; • Apply DWR (dual-weighted residual) estimator from Becker/Rannacher; 1996,2001; • Adjoint time-stepping is tricky; Thomas Wick (Ecole Polytechnique) Time step control for FSI 16
Petrov-Galerkin discretization in time Definition of function spaces: X v , u = { v k ∈ C ( ¯ I , V ) | v k | I m ∈ P 1 ( I m , V ) , m = 1, 2, . . . , M } k X p k = { p k ∈ L 2 ( I , L f ,0 | p k | I m ∈ P 0 ( I m , L f ,0 , m = 1, 2, . . . , M } Test space for the momentum equation and mesh motion 4 : X v , u k , θ = { ϕ k ∈ L 2 ( I , V ) | ϕ k | I m ∈ P θ ( I m , V ) , m = 1, 2, 3, . . . , M } with P θ ( I m , V ) : = { Φ m ϕ k , m | ϕ k , m ∈ V } ¯ ¯ where Φ m ( t ) = 1 + 6 ( θ m − 0.5 ) 2 t − t m − 1 − t m ∆ t m • Only the test functions are θ -dependent ⇒ Method easily applicable to nonlinear equations 4 Meidner/Richter; 2014 Thomas Wick (Ecole Polytechnique) Time step control for FSI 17
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