An all-speed relaxation scheme for the simulation of multi-material flows Emanuela Abbate Joint work with G. Puppo and A. Iollo SHARK-FV May 2018, Portugal Universit` a degli Studi dell’Insubria Memphis team, Inria Bordeaux Universit´ e Bordeaux, Institut de Math´ ematiques

Table of contents

1. Eulerian model for compressible materials 2. Low Mach limits 3. Relaxation all-speed scheme 4. Numerical results 5. Numerical model for multi-material flows 6. Preliminary 2D multi-material results 7. Open work and questions

Abbate

All-speed scheme

1

Eulerian model for compressible materials

Eulerian framework 2D system of conservation equations  ∂t ρ + ∇x · (ρu) = 0      ∂t (ρu) + ∇x · (ρu ⊗ u − σ) = 0  ∂t ([∇x Y ]) + ∇x (u · [∇x Y ]) = 0     ∂t (ρe) + ∇x · (ρeu − σ | u) = 0 Equations for a general medium in the deformed configuration • ρ density • u Eulerian velocity field • σ = σ (ρ, s, [∇x Y ]) Cauchy stress tensor • e = 21 u 2 +  total energy per unit mass ( internal energy) Standard Euler equations for gas-dynamics are in this class Abbate

All-speed scheme

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Eulerian framework: deformation • Ω0 ∈ R2 initial configuration of a continuous medium • Ωt ∈ R2 deformed configuration at time t

Abbate

All-speed scheme

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Eulerian framework: deformation • Ω0 ∈ R2 initial configuration of a continuous medium • Ωt ∈ R2 deformed configuration at time t

• Forward characteristics: X (ξ, t) : Ω0 × [0, T ] → Ωt • Eulerian velocity u : Ωt × [0, T ] → R2 with ∂t X (ξ, t) = u (X (ξ, t) , t)

Abbate

All-speed scheme

3

Eulerian framework: deformation • Ω0 ∈ R2 initial configuration of a continuous medium • Ωt ∈ R2 deformed configuration at time t

• Forward characteristics: X (ξ, t) : Ω0 × [0, T ] → Ωt • Eulerian velocity u : Ωt × [0, T ] → R2 with ∂t X (ξ, t) = u (X (ξ, t) , t) Backward characteristics Y (x, t) : Ωt × [0, T ] → Ω0 • for (x, t) in Ωt recover ξ in Ω0 • from Y (X (ξ, t) , t) = ξ: transport equation ∂t Y + u · ∇x Y = 0 • gradient of the deformation [∇x Y (x, t)] Abbate

All-speed scheme

3

Eulerian framework: state law Constitutive law (internal energy) neo-hookean elastic solid

z }| {  1−γ
p∞ χ κ(s) 1 ε(ρ, s, ∇x Y ) = −b − aρ + + (Tr B([∇x Y ]) − 2) γ−1 ρ ρ ρ0 | {z } van der Waals gas

|

{z

}

stiffened gas

⇒ Unique model to treat gas, fluids and elastic solids ⇒ Euler system is easily recovered • a = b = 0 for a perfect gas • p∞ = 0 (no intermolecular forces) • χ = 0 (no elastic deformation) De Brauer, Iollo, Milcent. A Cartesian scheme for compressible multimaterial models in 3D. Journal of Computational Physics, 2016. Godunov. Elements of continuum mechanics. 1978. Abbate

All-speed scheme

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Considering only direction x1 "

Y,11 • [∇x Y ] = Y,12

0 1

# ⇒ equation on Y11 is redundant: Y11 =

ρ ρ0

• ρ = det ([∇x Y ]) ρ0 1D system 



ρ ρu   1   ψ = ρu2  ,  2  Y,1  ρe

Abbate

∂t ψ + ∂x1 F (ψ) = 0     F (ψ) =   

 ρu1  ρu12 − σ 11   ρu1 u2 − σ 21    u1 Y,12 + u2
ρe − σ 11 u1 − σ 21 u2

All-speed scheme

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Considering only direction x1 "

Y,11 • [∇x Y ] = Y,12

0 1

# ⇒ equation on Y11 is redundant: Y11 =

ρ ρ0

• ρ = det ([∇x Y ]) ρ0 1D system 



ρ ρu   1   ψ = ρu2  ,  2  Y,1  ρe

∂t ψ + ∂x1 F (ψ) = 0     F (ψ) =   

 ρu1  ρu12 − σ 11   ρu1 u2 − σ 21    u1 Y,12 + u2
ρe − σ 11 u1 − σ 21 u2

Non-null components of the stress tensor:      σ 11 = −p (ρ, s) + 2χJ −1 B 11 − Tr B = −p (ρ, s) + χ 1 − Y 2
2 − (ρ/ρ )2 0 ,1 2  21 −1 21 2 σ

Abbate

= 2χJ

B

= −2χY,1 .

All-speed scheme

5

Wave speeds For this formulation: 5 waves 1. two longitudinal waves (relative to the normal stress) with speed r q 2 λ1,5 = u1 ± c 2 /2 + χ/ρ (α + β) + 1/ρ (ρc 2 /2 + χ (α − β)) + 4χ2 δ 2

α, β and δ: functions of [∇x Y ]. Abbate

All-speed scheme

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Wave speeds For this formulation: 5 waves 1. two longitudinal waves (relative to the normal stress) with speed r q 2 λ1,5 = u1 ± c 2 /2 + χ/ρ (α + β) + 1/ρ (ρc 2 /2 + χ (α − β)) + 4χ2 δ 2

2. two shear waves (relative to the tangential stress) with speed r q 2 λ2,4 = u1 ± c 2 /2 + χ/ρ (α + β) − 1/ρ (ρc 2 /2 + χ (α − β)) + 4χ2 δ 2

α, β and δ: functions of [∇x Y ]. Abbate

All-speed scheme

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Wave speeds For this formulation: 5 waves 1. two longitudinal waves (relative to the normal stress) with speed r q 2 λ1,5 = u1 ± c 2 /2 + χ/ρ (α + β) + 1/ρ (ρc 2 /2 + χ (α − β)) + 4χ2 δ 2

2. two shear waves (relative to the tangential stress) with speed r q 2 λ2,4 = u1 ± c 2 /2 + χ/ρ (α + β) − 1/ρ (ρc 2 /2 + χ (α − β)) + 4χ2 δ 2

3. one material wave with speed λ3 = u1

α, β and δ: functions of [∇x Y ]. Abbate

All-speed scheme

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Low Mach limits

Non-dimensionalization ⇒ Introduction of two speeds: 1. Speed of sound s c (ρ, s, ∇x Y ) =

r p ∂p γ γ−1 (p + p∞ ) = γk (s) ρ = ∂ρ s=const ρ r

2. “Elastic speed”: uel =

Abbate

2χ ρ

All-speed scheme

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Non-dimensionalization ⇒ Introduction of two speeds: 1. Speed of sound s c (ρ, s, ∇x Y ) =

r p ∂p γ γ−1 (p + p∞ ) = γk (s) ρ = ∂ρ s=const ρ r

2. “Elastic speed”: uel =

2χ ρ

⇒ Introduction of two “Mach numbers”: u1 1. the standard acoustic Mach number: M = sc u1 ρu12 2. an “elastic Mach number”: Mχ = = uel 2χ Abbate

All-speed scheme

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Non-dimensionalization • advective scale • acoustic scale • elastic scale   ∂t ρ + ∂x (ρu1 ) = 0    
2    ∂x 1 − Y,12 − (ρ/ρ0 )2
 χ ∂ p x   =0 ∂t (ρu1 ) + ∂x ρu12 + 2 −   M 2 Mχ2   2  ∂x Y   ∂t (ρu2 ) + ∂x (ρu1 u2 ) + χ 2,1 = 0 Mχ

 ∂t Y,12 + ∂x u1 Y,12 + u2 = 0  
!      χ tr B − 2 γ (p + γp∞ ) 1 2 p + γp∞ 1 3   ∂ ρu + + ρ|u| + u + + ∂ t 1 x   2 M 2 (γ − 1) 2Mχ2 2 M 2 (γ − 1)          2    χ 2 ∂x tr B − 2 − χ 1 − Y,12 − (ρ/ρ0 )2 u1 + 2χY,12 u2 = 0.





2Mχ

Abbate, Iollo, Puppo. An all-speed relaxation scheme for gases and compressible materials. Journal of Computational Physics, 2017. Abbate

All-speed scheme

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Low Mach limits Small deformations: [∇x Y ] ' I ⇒ Y,12 ' 0 Characteristic speeds reduce to: p • longitudinal waves: λ1,5 ' u1 ± c 2 + 2χ/ρ p • shear waves: λ2,4 ' u1 ± 2χ/ρ

First limit: “Acoustic and shear low Mach regime” ( M1 Mχ  1 • stiffness from both M and Mχ • O (M) ' O (Mχ ) ⇒ c ' uel and p + p∞ ' χ • longitudinal and shear waves are all consistently faster than the material wave Abbate

All-speed scheme

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Low Mach limits Small deformations: [∇x Y ] ' I ⇒ Y,12 ' 0 Characteristic speeds reduce to: p • longitudinal waves: λ1,5 ' u1 ± c 2 + 2χ/ρ p • shear waves: λ2,4 ' u1 ± 2χ/ρ

Second limit: “Acoustic only low Mach regime” ( M1 M  Mχ • stiffness from M only (pressure gradients only) • c  |u1 | and c  uel ⇒ p + p∞  χ • only longitudinal waves are all consistently faster than the material wave Abbate

All-speed scheme

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Relaxation all-speed scheme

Relaxation method Relaxation system   ∂t ψ + ∂x1 v + ∂x2 w = 0     1 ∂t v + A1 ∂x1 ψ = (F (ψ) − v) η    1  ∂t w + A2 ∂x2 ψ = (G (ψ) − w) η • η > 0 relaxation rate (η → 0) • A1 = diag {ai1 } and A2 = diag {ai2 } relaxation matrices • subcharacteristic condition: 2 2 A1 − F0 (ψ) ≥ 0 and A2 − G0 (ψ) ≥ 0

∀ψ

⇒ linearity of the advective operator ⇒ direction by direction relaxation Jin, Xin. The relaxation schemes for systems of conservation laws in arbitrary space dimensions. Comm. in Pure and Applied Math., 1995. Abbate

All-speed scheme

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Implicit scheme For simplicity of notation: scheme derivation in 1D Implicit time discretization  n+1 ψ − ψn    + ∂x vn+1 = 0 ∆t

 vn+1 − vn 1   + A∂x ψ n+1 = F ψ n+1 − vn+1 ∆t η

Abbate

All-speed scheme

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Implicit scheme For simplicity of notation: scheme derivation in 1D Implicit time discretization  n+1 ψ − ψn    + ∂x vn+1 = 0 ∆t

 vn+1 − vn 1   + A∂x ψ n+1 = F ψ n+1 − vn+1 ∆t η Stiff source: Taylor expansion (one iteration of Newton method)

F ψ n+1 = F (ψ n ) + F0 (ψ n ) ψ n+1 − ψ n ⇒ F0 (ψ) computed analytically ⇒ ( MΨn+1 + NVn+1 = r PΨn+1 + QVn+1 = s

Abbate

All-speed scheme

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Implicit scheme For simplicity of notation: scheme derivation in 1D Implicit time discretization  n+1 ψ − ψn    + ∂x vn+1 = 0 ∆t

 vn+1 − vn 1   + A∂x ψ n+1 = F ψ n+1 − vn+1 ∆t η Stiff source: Taylor expansion (one iteration of Newton method)

F ψ n+1 = F (ψ n ) + F0 (ψ n ) ψ n+1 − ψ n ⇒ F0 (ψ) computed analytically ⇒ ( MΨn+1 + NVn+1 = r PΨn+1 + QVn+1 = s Unconditional stability (∆t dictated by accuracy not by stability) Abbate

All-speed scheme

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Spatial discretization

Finite volumes spatial discretization  vi+1/2 − vi−1/2  ∂t ψ i + =0 ∆x ψ − ψ i−1/2 1  ∂t vi + A i+1/2 = (F (ψ i ) − vi ) ∆x η

Abbate

All-speed scheme

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Spatial discretization

Finite volumes spatial discretization  vi+1/2 − vi−1/2  ∂t ψ i + =0 ∆x ψ − ψ i−1/2 1  ∂t vi + A i+1/2 = (F (ψ i ) − vi ) ∆x η Centered 1 (vi+1 − vi−1 ) 2∆x
1 = ψ i+1 − ψ i−1 2∆x

 D (∂x v)cent = D (∂ ψ) x cent

Abbate

All-speed scheme

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Spatial discretization Finite volumes spatial discretization  vi+1/2 − vi−1/2  ∂t ψ i + =0 ∆x ψ − ψ i−1/2 1  ∂t vi + A i+1/2 = (F (ψ i ) − vi ) ∆x η Centered 1 (vi+1 − vi−1 ) 2∆x
1 = ψ i+1 − ψ i−1 2∆x

 D (∂x v)cent = D (∂ ψ) x cent

Upwind (along characteristic var. v ± A1/2 ψ)      v + A1/2 ψ = v + A1/2 ψ 

 v − A1/2 ψ

Abbate

i+1/2

 i = v − A1/2 ψ

i+1/2

i+1

All-speed scheme

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Spatial discretization Finite volumes spatial discretization  vi+1/2 − vi−1/2  ∂t ψ i + =0 ∆x ψ − ψ 1 i−1/2  ∂t vi + A i+1/2 = (F (ψ i ) − vi ) ∆x η Centered 1 (vi+1 − vi−1 ) 2∆x
1 = ψ i+1 − ψ i−1 2∆x

 D (∂x v)cent = D (∂ ψ) x cent

Upwind (along characteristic var. v ± A1/2 ψ)  1/2
 D (∂x v) = 1 (vi+1 − vi−1 ) − A ψ i+1 − 2ψ i + ψ i−1 upw  D (∂x ψ)upw =

Abbate

2∆x 2∆x
A−1/2 1 ψ i+1 − ψ i−1 − (vi+1 − 2vj + vi−1 ) 2∆x 2∆x

All-speed scheme

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All-speed scheme Hybrid scheme Convex combination on the local Mach number: D (∂x u)hyb =f (Mloc )D (∂x u)upw + (1 − f (Mloc ))D (∂x u)cent with f (Mloc ) = min{1, Mloc } or f (Mloc ) =

Abbate

All-speed scheme

arctan(Mloc ) π/2

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All-speed scheme Hybrid scheme Convex combination on the local Mach number: D (∂x u)hyb =f (Mloc )D (∂x u)upw + (1 − f (Mloc ))D (∂x u)cent with f (Mloc ) = min{1, Mloc } or f (Mloc ) =

arctan(Mloc ) π/2

All-speed spatial discretization 
f (Mloc ) 1/2 1  D (∂x v)hyb = (vi+1 − vi−1 ) − A ψ i+1 − 2ψ i + ψ i−1 2∆x 2∆x
 f (Mloc ) −1/2 1 D (∂ ψ) = ψ i+1 − ψ i−1 − A (vi+1 − 2vi + vi−1 ) . x hyb 2∆x 2∆x

Abbate, Iollo, Puppo. An all-speed relaxation scheme for gases and compressible materials. Journal of Computational Physics, 2017. Abbate

All-speed scheme

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CFL condition

⇒ unconditional stability (implicit time discretization) ⇒ constraint on ∆t dictated only by accuracy reasons

Abbate

All-speed scheme

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CFL condition

⇒ unconditional stability (implicit time discretization) ⇒ constraint on ∆t dictated only by accuracy reasons

• material constraint: ∆t ≤ ∆x|u1 | → flow speed → accurate only on material waves → less demanding (much larger ∆t when M → 0)

Abbate

All-speed scheme

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CFL condition

⇒ unconditional stability (implicit time discretization) ⇒ constraint on ∆t dictated only by accuracy reasons

• material constraint: ∆t ≤ ∆x|u1 | → flow speed → accurate only on material waves → less demanding (much larger ∆t when M → 0)

• acoustic constraint: ∆t ≤ ∆xλmax → fastest characteristic (speed of sound) → accurate on all waves → computationally demanding when M → 0

Abbate

All-speed scheme

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Numerical results

Gresho vortex Mmax = 0.1 Initial condition Rotating vortex centered at (x, y ) = (0.5, 0.5). Angular velocity distribution:

uφ (x, y , 0) =

   5r

0 ≤ r ≤ 0.2

2 − 5r 0.2 ≤ r ≤ 0.4   0 r ≥ 0.4

Pressure gradient built to balance the centrifugal
2 force, background pressure: p0 = ρ/ γMmax

All-speed implicit scheme (t = 1)

Abbate

Explicit-upwind scheme (t = 1)

All-speed scheme

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Material waves: perfect gas Sod shock tube (t = 0.1644s) On the contact wave: M ' 0.9 CFL conditions: • explicit relaxation scheme: νac = 0.4 • implicit relaxation scheme: νmat = 0.4 (corresponding to νac = 0.9)

Abbate

All-speed scheme

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Material waves: perfect gas Sod shock tube (t = 0.1644s) On the contact wave: M ' 0.9 CFL conditions: • explicit relaxation scheme: νac = 0.4 • implicit relaxation scheme: νmat = 0.4 (corresponding to νac = 0.9)

Low Mach tube: M ' 6 · 10−3 (t = 0.25s and t = 150s)

• expl: νac = 0.9 (∆t = 1.2·10−3 s) • impl: νmat = 0.4 (∆t = 4.4·10−2 s)

Abbate

All-speed scheme

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Material waves: copper “Acoustic and shear low Mach limit”
• p∞ ' χ ' O 1010 • on the contact wave: M ' 2.6 · 10−3 and Mχ ' 3.15 · 10−3 Tube filled with copper: short times (t = 6 · 10−5 s)

• explicit relaxation scheme: νac = 0.4 (∆t = 8.7 · 10−8 ) • implicit relaxation scheme: • νmat = 0.3 (∆t = 1.7 · 10−5 ) and νmat = 0.15 (∆t = 8.3 · 10−6 ) • νac = 0.9 (∆t = 1.7 · 10−7 ) Abbate

All-speed scheme

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Material waves: copper “Acoustic and shear low Mach limit”
• p∞ ' χ ' O 1010 • on the contact wave: M ' 2.6 · 10−3 and Mχ ' 3.15 · 10−3 Tube filled with copper: long times (t = 0.04s)

Contact wave: from x0 = 250m to x0 = 250.7m Abbate

All-speed scheme

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Computational costs

High computational costs of the all-speed scheme: • additional unknowns (relaxation variables direction by direction) • fully implicit scheme (big matrices to invert)

Abbate

All-speed scheme

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Computational costs

High computational costs of the all-speed scheme: • additional unknowns (relaxation variables direction by direction) • fully implicit scheme (big matrices to invert) ⇒ To reduce the computational effort: 1. parallelization of the code (MPI-C++ code) 2. adaptive mesh refinement (AMR) on octree grids • number of dofs consistently reduced where smooth behaviour is expected • local increase of accuracy in specific areas of interest

Abbate

All-speed scheme

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Computational costs

High computational costs of the all-speed scheme: • additional unknowns (relaxation variables direction by direction) • fully implicit scheme (big matrices to invert) ⇒ To reduce the computational effort: 1. parallelization of the code (MPI-C++ code) 2. adaptive mesh refinement (AMR) on octree grids • number of dofs consistently reduced where smooth behaviour is expected • local increase of accuracy in specific areas of interest

Bitpit library for mesh generation Optimad Engineering. Bitpit Web page. http://optimad.github.io/bitpit/modules/, 2017.

Abbate

All-speed scheme

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Adaptive mesh refinement Hierarchical data structure: recursive decomposition of space. Linear Quadtree data structure: • defined in a square • each internal node has exactly 4 children • Z-order (Morton) index to each cell • only the leafs of the tree structure are stored • parallel communications: first layer of neighbouring cells

Raeli, Bergmann, Iollo. A finite-difference method for the variable coefficient Poisson equation on hierarchical Cartesian meshes, JCP, 2018. Abbate

All-speed scheme

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AMR: numerical entropy production

AMR pivoted by the numerical entropy production of the scheme • entropy pair (η, ζ) • entropy inequality: ∂t η + ∇x · ζ ≤ 0

Numerical entropy Numerical entropy Sij for each cell Cij of the domain:  ∆t   ∆t  n+1 n
n+1 n+1 n+1 Sijn+1 = η(ψ n+1 ) − η ψ + ζ − ζ + ζ − ζ ij ij i−1/2,j i,j−1/2 ∆x1 i+1/2,j ∆x2 i,j+1/2

ζi+1/2,j : “all-speed” numerical entropy flux  √a f (M )   1  n+1 max loc n+1 n+1 n+1 n+1 ζi+1,j + ζi,j − ηi+1,j − ηi,j ζi+1/2,j = 2 2 Puppo, Semplice. Numerical entropy and adaptivity for finite volume schemes, Comm. in Comp. Physics, 2011.

Abbate

All-speed scheme

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AMR: numerical entropy production AMR algorithm at every time step: 1. compute Sijn+1 2. if Sijn+1 > Sref mark Cij for refinement 3. if Sijn+1 < Scoa mark Cij for coarsening 4. ∆t computed with the chosen CFL constraint using the smallest cells size ∆x of the grid. Level of refinement L = d: 2d × 2d grid 2D Riemann problem: adaptive grids

2 levels of refinement: Lmin = 5, Lmax = 7 Abbate

4 levels of refinement: Lmin = 5, Lmax = 9 All-speed scheme

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AMR: numerical entropy production 2D Riemann problem ← − → − Two contact and two acoustic waves: C21 , S 32 , C34 , R 41 p2 = 1 u2 = 0

ρ2 = 2

p1 = 1.

ρ1 = 1

v2 = −0.3

u1 = 0

v1 = −0.4

p3 = 0.4

ρ3 = 1.0625

p4 = 0.4

u3 = 0

v3 = 0.2145

u4 = 0

Uniform grid (Lmin = Lmax = 10): 1048576 cells

Abbate

ρ4 = 0.5197 v4 = −1.1259

AMR (Lmin = 5, Lmax = 10): 146578 cells

All-speed scheme

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Numerical model for multi-material flows

General framework for walls and interfaces

General implicit scheme  n+1 n+1 n+1 vi+1/2 − vi−1/2  − ψ ni ψ   i + =0 ∆t ∆x n+1 n+1 ψ i+1/2 − ψ i−1/2

 v n+1 − v ni 1   i − vin+1 +A = F ψ n+1 i ∆t ∆x η

Abbate

All-speed scheme

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General framework for walls and interfaces

General implicit scheme  n+1 n+1 n+1 vi+1/2 − vi−1/2  − ψ ni ψ   i + =0 ∆t ∆x n+1 n+1 ψ i+1/2 − ψ i−1/2

 v n+1 − v ni 1   i − vin+1 +A = F ψ n+1 i ∆t ∆x η Mono-material scheme: n+1 ∀i = 0, ..N vi+1/2 and ψ n+1 i+1/2 computed with • upwind scheme • centered scheme • all-speed scheme

Abbate

All-speed scheme

21

General framework for walls and interfaces

Introduction of an interface in position xB

material 1

Ck xk

w-k+1/2

material 2

Ck+1 xk+1/2

xk+1

material 1

xB

material 2

xk

xk+1 xk+1/2

interface region

Abbate

w+k+1/2

All-speed scheme

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General framework for walls and interfaces Introduction of an interface in position xB

material 1

Ck xk

w-k+1/2

material 2

Ck+1 xk+1/2

xk+1

w+k+1/2

material 1

xB

material 2

xk

xk+1 xk+1/2

interface region

Wall/interface “artificially” set to coincide with the closest numerical interface xk+1/2 : • cell Ck assigned to material 1 • cell Ck+1 assigned to material 2 ⇒ numerical error of order O (∆x) Abbate

All-speed scheme

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General framework for walls and interfaces Introduction of an interface in position xB

material 1

Ck xk

w-k+1/2

material 2

Ck+1 xk+1/2

w+k+1/2

material 1

xk+1

xB

material 2

xk

xk+1 xk+1/2

interface region

Multi-material scheme: n+1 1. ∀i 6= k vi+1/2 and ψ n+1 i+1/2 computed with

• upwind scheme • centered scheme • all-speed scheme + 2. special care devoted to interface cells: w− k+1/2 and wk+1/2 Abbate

All-speed scheme

22

Interface cells Scheme for the interface cells   − n+1 n+1  n+1 n v − vk−1/2  k+1/2  ψk − ψk   + =0 ∆t

∆x 

−   n+1 n ψ n+1 − ψ n+1

 k+1/2 k−1/2 v − v 1 k  k  F ψ n+1 − vkn+1 +A = k ∆t ∆x η   + n+1 n+1  n+1 n v − v  k+3/2 k+1/2  ψ k+1 − ψ k+1   + =0



∆t

∆x 

n+1  n  ψ n+1  vn+1 − vk+1 k+3/2 − ψ k+1/2   k+1 +A

∆t

+ =

∆x



1 n+1 F ψ n+1 k+1 − vk+1 . η

Distinction between left and right interface values 

ψ k+1/2

−

 +
−
+ 6= ψ k+1/2 and vk+1/2 6= vk+1/2

to be reconstructed with ad hoc conditions Abbate

All-speed scheme

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Moving walls (prescribed velocity) Moving wall conditions −   ρn+1 = 32 ρn+1 − 21 ρn+1 k k−1   k+1/2 

−
n+1 uk+1/2 = u1∗ t n+1    n+1 − 3 n+1 1 n+1 pk+1/2 = 2 pk − 2 pk−1

and

+  1 n+1  ρn+1 = 23 ρn+1 k+1 − 2 ρk+2   k+1/2 

+
n+1 uk+1/2 = u1∗ t n+1    n+1 + 3 n+1 1 n+1 pk+1/2 = 2 pk+1 − 2 pk+2 .

• u1∗ (t): velocity externally imposed • left/right extrapolation for ρ and p • solids: left/right extrapolation of ∇Y • relaxation variables (linearization): ( − 

− = F ψ n+1 k+1/2  +  + n+1 vk+1/2 = F ψ n+1 k+1/2 n+1 vk+1/2

Abbate

All-speed scheme

24

Beam elongations Copper beams surrounded by a perfect gas M − ' 0.14; M + ' 1 u1∗ = 500m/s: on the interface
= 0.65, tend = 10−4

Wall displacement: xB (0) = 0.6, xB tend

600

104

500

normal velocity [m/s]

density [Kg/m3]

103 102 101 100 0.0

exact upwind centered all-speed 0.2

0.4

0.6

0.8

400 300 200

exact upwind centered all-speed

100 0 0.0

1.0

0.2

0.4

0.6

0.8

1.0

u1∗ = 25m/s: on the interface M − ' 5.5 · 10−3 ; M + ' 1
Wall displacement: xB (0) = 1.75, xB tend

= 1.756, tend = 2.5 · 10−4

25

normal velocity [m/s]

density [Kg/m3]

104

103

102 0.00

Abbate

exact upwind centered all-speed 0.25

0.50

0.75

1.00

1.25

1.50

1.75

2.00

20 15 10

exact upwind centered all-speed

5 0 0.00

0.25

All-speed scheme

0.50

0.75

1.00

1.25

1.50

1.75

2.00

25

Multi-material interfaces Sharp interface model: interface treated as a moving boundary 1. everything unknown at the interface (u is not imposed) 2. need to prescribe equilibrium: balancing of the forces

Abbate, Iollo, Puppo. An implicit scheme for moving walls and multi-material interfaces. In preparation Abbate

All-speed scheme

26

Multi-material interfaces Sharp interface model: interface treated as a moving boundary 1. everything unknown at the interface (u is not imposed) 2. need to prescribe equilibrium: balancing of the forces For fluids interactions: • pressure has to balanced: continuity of p and u1 ⇒ mean at xk+1/2 • ρ discontinuous at the interface ⇒ left/right extrap Fluid/fluid interface conditions −   ρn+1 = 32 ρn+1 − 12 ρn+1 k k−1   k+1/2  n+1 uk+1/2

  

n+1 pk+1/2

−

−

=

1 2

n+1 ukn+1 + uk+1




=

1 2

n+1 pkn+1 + pk+1




+  1 n+1  ρn+1 = 32 ρn+1 k+1 − 2 ρk+2   k+1/2  +

and

n+1 uk+1/2 =     n+1 + pk+1/2 =

1 2

n+1 ukn+1 + uk+1

1 2


n+1 pkn+1 + pk+1 .




Abbate, Iollo, Puppo. An implicit scheme for moving walls and multi-material interfaces. In preparation Abbate

All-speed scheme

26

Multi-material interfaces

When solids are involved, in general: • forces have to be balanced: continuity of σ 11 and u1 ⇒ mean at xk+1/2
• ρ and Y,12 discontinuous at the interface ⇒ left/right extrap

Abbate

All-speed scheme

27

Multi-material interfaces

When solids are involved, in general: • forces have to be balanced: continuity of σ 11 and u1 ⇒ mean at xk+1/2
• ρ and Y,12 discontinuous at the interface ⇒ left/right extrap Moreover, we distinguish: 1. solid/solid interactions:
−
+ 21 21 • σk+1/2 = σk+1/2 ⇒ mean at xk+1/2 • continuity of u2 ⇒ mean at xk+1/2

2. solid/fluid interactions:
−
+ 21 21 = σk+1/2 =0 • σk+1/2 • discontinuity of u2 ⇒ left/right extrap

Abbate

All-speed scheme

27

Open work

Numerical oscillations and instabilities arise when solving 1. supersonic flows (M > 1) 2. strong shock waves interacting with interfaces This is due to the balancing of p (or σ in solids) by computing the mean: equivalent to a “centered” discretization ⇒ Need to build an all-speed extension: • add some viscosity if Mach gets high • fully implicit framework (introducing a Riemann solver really complicates things!!)

Abbate

All-speed scheme

28

Fluid/fluid: mono-material Mono-material problems: slow material waves treated as interfaces Gas/gas: M ' 6 · 10−3

Water/water: M ' 2.5 · 10−3 996.3

exact explicit-upwind all-speed mono upwind multi all-speed multi

exact explicit-upwind all-speed mono upwind multi all-speed multi

996.2

density [Kg/m3]

density [Kg/m3]

0.9956

0.9946

996.1 996.0 995.9 995.8 995.7

0.9936

196

198

200

202

204

995.6

196

198

200

202

204

• mono-material explicit-upwind: wrong viscosity on material waves in the low Mach regime • mono-material all-speed: correct viscosity on material waves in the low Mach regime • multi-material schemes: sharp treatment of interfaces ⇒ “wave reconstruction” VS “wave approximation” Abbate

All-speed scheme

29

Fluid/fluid: multi-material Water/gas interfaces: water expansion Multi-regime: on the interface M − ' 0.035; M + ' 0.13


Interface displacement: xB (0) = 0.7, xB tend

= 0.712, tend = 2.4 · 10−4

103

upwind centered all-speed

102 0.0

normal velocity [m/s]

density [Kg/m3]

50

upwind centered all-speed 0.2

0.4

0.6

0.8

40 30 20 10 0 0.0

1.0

0.2

0.4

0.6

0.8

1.0

Low Mach: on the interface M − ' 6.5 · 10−3 ; M + ' 0.027


Interface displacement: xB (0) = 200, xB tend

= 201.103, tend = 0.095

25

102

upwind centered all-speed 0

Abbate

normal velocity [m/s]

density [Kg/m3]

103

50

100

upwind centered all-speed

20 15 10 5 0

150

200

250

300

350

400

0

50

All-speed scheme

100

150

200

250

300

350

400

30

Solid/solid Copper/copper interface with shear (u2 = 100 on the right) On the interface M ' 0.14. Interface displacement: xB (0) = 0.75, xB (tend ) = 0.794, tend = 5.6 · 10−5 upwind centered all-speed

1.0

9500 9000 8500

0.0

0.2

0.4

0.6

0.8

1.0

1.2

0.4

100

transverse velocity [m/s]

700

normal velocity [m/s]

0.6

0.0 0.0

1.4

800 600 500 400 300 200

upwind centered all-speed

100

Abbate

upwind centered all-speed

0.2

8000

0 0.0

1e11

0.8

pressure [Pa]

density [Kg/m3]

10000

0.2

0.4

0.6

0.8

1.0

1.2

1.4

80

0.2

0.4

0.6

0.8

1.0

1.2

1.4

0.6

0.8

1.0

1.2

1.4

upwind centered all-speed

60 40 20 0 0.0

All-speed scheme

0.2

0.4

31

Solid/fluid Copper/gas interface: copper expansion Multi-regime: on the interface M − ' 4.5 · 10−3 ; M + ' 0.15


Interface displacement: xB (0) = 1.1, xB tend

= 1.1023, tend = 1.1 · 10−4

pressure [Pa]

103

102 0.0

300

normal velocity [m/s]

109

250

upwind centered all-speed 0.2

0.4

0.6

0.8

1.0

1.2

1.4

50 0.4

0.4

0.6

0.8

1.0

1.2

1.4

1e9

100

0.2

0.2

0.2

150

Abbate

0.0

upwind centered all-speed

200

0 0.0

upwind centered all-speed

108

normal stress [Pa]

density [Kg/m3]

104

0.6

0.8

1.0

1.2

1.4

0.4 0.6

upwind centered all-speed

0.8 1.0 0.0

All-speed scheme

0.2

0.4

0.6

0.8

1.0

1.2

1.4

32

Preliminary 2D multi-material results

2D multi-material model Level set function for interface tracking ∂t ϕ + u · ∇x ϕ = 0

Abbate

All-speed scheme

33

2D multi-material model Level set function for interface tracking ∂t ϕ + u · ∇x ϕ = 0 Integration with a semi-lagrangian method: • x n+1 any grid node, xd corresponding departure point
• ϕn+1 x n+1 = ϕn (xd )  xˆ = x n+1 − ∆t · u n x n+1
2 x = x n+1 − ∆t · u n+1/2 (ˆ x) d

RK2 time integration and biquadratic interpolation in space

Abbate

All-speed scheme

33

2D multi-material model Level set function for interface tracking ∂t ϕ + u · ∇x ϕ = 0 Integration with a semi-lagrangian method: • x n+1 any grid node, xd corresponding departure point
• ϕn+1 x n+1 = ϕn (xd )  xˆ = x n+1 − ∆t · u n x n+1
2 x = x n+1 − ∆t · u n+1/2 (ˆ x) d

RK2 time integration and biquadratic interpolation in space ⇒ “suitable” for theAMR implementation constraints  3
∆x ⇒ low accuracy: O ∆t 2 + ' O ∆x 2 ∆t Min, Gibou. A second order accurate level set method on non-graded adaptive Cartesian grids. Journal of Computational Physics, 2007. Abbate

All-speed scheme

33

Shock-bubble interactions Helium bubble in air: γHe = 1.648, γair = 1.4 Left moving shock with Mmax ' 1 ρHe = 0.2226, ρair ,L = 1.225, ρair ,R = 1.6861 pHe = pair ,L = 101325, pair ,R = 159059, uair ,R = −113.534 Grid: 1024 × 1024 cells

102µs

260µs

445µs

675µs

Marquina, Moulet. A flux-split algorithm applied to conservative models for multicomponent compressible flows. JCP, 2007. Abbate

All-speed scheme

34

Shock-bubble interactions Air bubble in water: γw = 4.4, γair = 1.4, p∞,w = 6.8 · 108 , p∞,air = 0 Left moving shock with Mmax ' 1 ρair = 100, ρw ,L = 1000, ρw ,R = 1130 pair = pw ,L = 107 , pair ,R = 108 , uair ,R = −122 Grid: 1024 × 1024 cells

600µs

800µs

1500µs

950µs Abbate

All-speed scheme

34

Open work and questions

Open work Conclusions • low Mach limit in Eulerian framework • all-speed scheme for general state laws • simulation of material wave propagation • simulation of multi-material flows: sharp treatment of the interfaces

Ongoing work • increase order of precision mono-material scheme • 2D multi-material solver for solids (ongoing implementation) • multi-material AMR Open questions • increase precision for level-set integration • all-speed multi-material scheme Abbate

All-speed scheme

35

Thanks for your attention

Nozzle flow (quasi-1D approximation) Stationary solution  ∂t ρ + ∂x (ρu) = −ρu ∂Sx S


∂t (ρu) + ∂x ρu 2 + p = −ρu 2 ∂Sx S



Abbate

∂t (ρe) + ∂x (u (ρe + p)) = −u (ρe + p)

All-speed scheme

∂x S S

.

36

Nozzle flow (quasi-1D approximation) Stationary solution  ∂t ρ + ∂x (ρu) = −ρu ∂Sx S


∂t (ρu) + ∂x ρu 2 + p = −ρu 2 ∂Sx S



∂t (ρe) + ∂x (u (ρe + p)) = −u (ρe + p)

∂x S S

.

Three test cases: 1. perfect gas flow in subsonic regime: • γ = 1.4 and p∞ = 0 • pinflow = 1 and poutflow = 0.9

2. perfect gas flow in low Mach regime: • γ = 1.4 and p∞ = 0 • pinflow = 1 and poutflow = 0.99999

3. stiffened gas flow (almost incompressible): • γ = 2.4 and p∞ = 6.8 · 108 Pa • pinflow = 10 and poutflow = 1 Abbate

All-speed scheme

36

Nozzle flow (quasi-1D approximation) Stationary solution  ∂t ρ + ∂x (ρu) = −ρu ∂Sx S


∂t (ρu) + ∂x ρu 2 + p = −ρu 2 ∂Sx S



∂t (ρe) + ∂x (u (ρe + p)) = −u (ρe + p)

∂x S S

.

Subsonic flow of a perfect gas: M ∈ [0.45; 0.7] Explicit-upwind scheme

Abbate

All-speed implicit scheme

All-speed scheme

36

Nozzle flow (quasi-1D approximation) Low Mach flow of a perfect gas: M ∈ [4; 9] · 10−3 Explicit-upwind scheme

Abbate

All-speed implicit scheme

All-speed scheme

37

Nozzle flow (quasi-1D approximation) Low Mach flow of a perfect gas: M ∈ [4; 9] · 10−3 Explicit-upwind scheme

All-speed implicit scheme

Low Mach flow of a stiffened gas: M ∈ [7.26; 8.67] · 10−5 Explicit-upwind scheme

Abbate

All-speed implicit scheme

All-speed scheme

37

Nozzle flow - convergence Gas flow: subsonic

Gas flow: low Mach

Water flow

Abbate

All-speed scheme

38

Material waves: elastic material “Acoustic only low Mach limit”

• p∞ = O 108  χ = O 105 • on the contact wave: M ' 3 · 10−3 and Mχ ' 0.15

Tube filled with hyperelastic material at t = 0.18s

• explicit relaxation scheme: νac = 0.4 (∆t = 6.89 · 10−6 s) • implicit relaxation scheme: νmat = 0.05 (∆t = 2.325 · 10−4 s) Abbate

All-speed scheme

39

Material waves: elastic material “Acoustic only low Mach limit”

• p∞ = O 108  χ = O 105 • on the contact wave: M ' 3 · 10−3 and Mχ ' 0.15

Tube filled with hyperelastic material at t = 0.18s

• explicit relaxation scheme: νac = 0.4 (∆t = 6.89 · 10−6 s) • implicit relaxation scheme: νmat = 0.05 (∆t = 2.325 · 10−4 s) Abbate

All-speed scheme

39