[PPT] - SHARK-FV 2014 The MOOD Ideas MOOD for multi-mat. flows The good PowerPoint Presentation

SLIDE 1

LA-UR 14-22765

Extension of the MOOD Method to the Saurel-Petitpas-Berry Model for Multi-Material Compressible Flows Fourth-order 2D results: The good & the not-yet-good-enough

aSteven Diot, aMarianne François, bEdward Dendy.

aFluid Dynamics and Solid Mechanics (T-3) bComputational Physics and Methods (CCS-2)

Los Alamos National Laboratory, Los Alamos, NM, USA.

SHARK-FV 2014

SLIDE 2

The MOOD Ideas MOOD for multi-mat. flows The good The not-yet-good-enough Conclusion

Outline

Ideas behind the MOOD method Extension to a multi-material compressible flow model Fourth-order 2D results: better accuracy and improved efficiency Fourth-order 2D results: investigating the not-so-good Conclusion and future work

SHARK-FV — diot@lanl.gov — 1/23

SLIDE 3

The MOOD Ideas MOOD for multi-mat. flows The good The not-yet-good-enough Conclusion

Multi-dimensional Optimal Order Detection

Origins

Developed during my Ph.D. in Toulouse (FR) under S. Clain & R. Loubère
Very-high-order Finite Volume method for single-material Euler equations
Alternative to WENO limiting on multidimensional unstructured meshes
Sucessfully tested up to 6th-order of accuracy on 3D polyhedral meshes
Papers: JCP 2011, CAF 2012, IJNMF 2013 =

⇒ public.lanl.gov/diot

Main ideas

Use only one unlimited polynomial reconstruction per cell & per degree
Check after the time update (a posteriori) if the solution is acceptable
If not, locally recompute the solution with a lower-order scheme
In the worst case, use the first-order scheme as parachute

SHARK-FV — diot@lanl.gov — 2/23

SLIDE 4

The MOOD Ideas MOOD for multi-mat. flows The good The not-yet-good-enough Conclusion

Multi-dimensional Optimal Order Detection

Origins

Developed during my Ph.D. in Toulouse (FR) under S. Clain & R. Loubère
Very-high-order Finite Volume method for single-material Euler equations
Alternative to WENO limiting on multidimensional unstructured meshes
Sucessfully tested up to 6th-order of accuracy on 3D polyhedral meshes
Papers: JCP 2011, CAF 2012, IJNMF 2013 =

⇒ public.lanl.gov/diot

Main ideas

Use only one unlimited polynomial reconstruction per cell & per degree
Check after the time update (a posteriori) if the solution is acceptable
If not, locally recompute the solution with a lower-order scheme
In the worst case, use the first-order scheme as parachute

SHARK-FV — diot@lanl.gov — 2/23

SLIDE 5

The MOOD Ideas MOOD for multi-mat. flows The good The not-yet-good-enough Conclusion

Multi-dimensional Optimal Order Detection

Origins

Developed during my Ph.D. in Toulouse (FR) under S. Clain & R. Loubère
Very-high-order Finite Volume method for single-material Euler equations
Alternative to WENO limiting on multidimensional unstructured meshes
Sucessfully tested up to 6th-order of accuracy on 3D polyhedral meshes
Papers: JCP 2011, CAF 2012, IJNMF 2013 =

⇒ public.lanl.gov/diot

Main ideas

Use only one unlimited polynomial reconstruction per cell & per degree
Check after the time update (a posteriori) if the solution is acceptable
If not, locally recompute the solution with a lower-order scheme
In the worst case, use the first-order scheme as parachute

SHARK-FV — diot@lanl.gov — 2/23

SLIDE 6

The MOOD Ideas MOOD for multi-mat. flows The good The not-yet-good-enough Conclusion

Multi-dimensional Optimal Order Detection

Tools

The framework to control the decrementing process developed in JCP 2011
There is a need for Detection Criteria to define an acceptable solution

Detection Criteria

PAD - Physical Admissiblity Detection

⋆ All criteria required to obtain a physical solution ⋆ Typically for Euler with perfect gas EOS — ρ > 0 and p > 0

DMP+u2 - Discrete Maximum Principle + Relaxation

⋆ DMP is used as a relevant detector of numerical oscillations ⋆ u2 developped to overcome the 2nd-order lock at smooth extrema ⋆ Typically for Euler with perfect gas EOS:

→ DMP on ρ — min(un

i , un j ) ≤ u⋆ i ≤ max(un i , un j ) (j looping over neighbors)

→ u2 if DMP violated — comparisons of local curvatures approximations

SHARK-FV — diot@lanl.gov — 3/23

SLIDE 7

The MOOD Ideas MOOD for multi-mat. flows The good The not-yet-good-enough Conclusion

Multi-dimensional Optimal Order Detection

Tools

The framework to control the decrementing process developed in JCP 2011
There is a need for Detection Criteria to define an acceptable solution

Detection Criteria

PAD - Physical Admissiblity Detection

⋆ All criteria required to obtain a physical solution ⋆ Typically for Euler with perfect gas EOS — ρ > 0 and p > 0

DMP+u2 - Discrete Maximum Principle + Relaxation

⋆ DMP is used as a relevant detector of numerical oscillations ⋆ u2 developped to overcome the 2nd-order lock at smooth extrema ⋆ Typically for Euler with perfect gas EOS:

→ DMP on ρ — min(un

i , un j ) ≤ u⋆ i ≤ max(un i , un j ) (j looping over neighbors)

→ u2 if DMP violated — comparisons of local curvatures approximations

SHARK-FV — diot@lanl.gov — 3/23

SLIDE 8

The MOOD Ideas MOOD for multi-mat. flows The good The not-yet-good-enough Conclusion

Multi-dimensional Optimal Order Detection

Tools

The framework to control the decrementing process developed in JCP 2011
There is a need for Detection Criteria to define an acceptable solution

Detection Criteria

PAD - Physical Admissiblity Detection

⋆ All criteria required to obtain a physical solution ⋆ Typically for Euler with perfect gas EOS — ρ > 0 and p > 0

DMP+u2 - Discrete Maximum Principle + Relaxation

⋆ DMP is used as a relevant detector of numerical oscillations ⋆ u2 developped to overcome the 2nd-order lock at smooth extrema ⋆ Typically for Euler with perfect gas EOS:

→ DMP on ρ — min(un

i , un j ) ≤ u⋆ i ≤ max(un i , un j ) (j looping over neighbors)

→ u2 if DMP violated — comparisons of local curvatures approximations

SHARK-FV — diot@lanl.gov — 3/23

SLIDE 9

The MOOD Ideas MOOD for multi-mat. flows The good The not-yet-good-enough Conclusion

The MOOD concepts — Today’s state & future

Already done - papers in press or in preparation

OpenMP parallelization of the 3D Euler code G. Moebs
Application into a VHO ADER scheme M. Dumbser, R. Loubère
Extension to convection-diffusion & shallow-water The Portuguese Team
Used to design entropy-preserving 2nd-order schemes V. Desveaux, C. Berthon
Design of a MOOD method for compr. multi-mat. flows with diffuse interface

On-Going & future

Development of a VHO remapping process for ALE R. Loubère, M. Kucharik
Design of a MOOD method for multi-material flows with sharp interface
Other systems of PDE’s: elastic-plastic, Navier-Stokes, etc.
Any original/useful idea of applications

= ⇒ Supports the fact that the a posteriori approach of MOOD is relevant

SHARK-FV — diot@lanl.gov — 4/23

SLIDE 10

The MOOD Ideas MOOD for multi-mat. flows The good The not-yet-good-enough Conclusion

The MOOD concepts — Today’s state & future

Already done - papers in press or in preparation

OpenMP parallelization of the 3D Euler code G. Moebs
Application into a VHO ADER scheme M. Dumbser, R. Loubère
Extension to convection-diffusion & shallow-water The Portuguese Team
Used to design entropy-preserving 2nd-order schemes V. Desveaux, C. Berthon
Design of a MOOD method for compr. multi-mat. flows with diffuse interface

On-Going & future

Development of a VHO remapping process for ALE R. Loubère, M. Kucharik
Design of a MOOD method for multi-material flows with sharp interface
Other systems of PDE’s: elastic-plastic, Navier-Stokes, etc.
Any original/useful idea of applications

= ⇒ Supports the fact that the a posteriori approach of MOOD is relevant

SHARK-FV — diot@lanl.gov — 4/23

SLIDE 11

The MOOD Ideas MOOD for multi-mat. flows The good The not-yet-good-enough Conclusion

6-eqation model of Saurel-Petitpas-Berry (SPB)

Reference: J. Comput. Phys. 228 (2009) 1678-1712. The non-conservative hyperbolic system is given by              ∂tU + ∇ · F(U) = ∂t(α1) + V · ∇(α1) = µ(p1 − p2) ∂t(α1ρ1e1) + ∇ · (α1ρ1e1V) + α1p1∇ · V = −¯ pIµ(p1 − p2) ∂t(α2ρ2e2) + ∇ · (α2ρ2e2V) + α2p2∇ · V = ¯ pIµ(p1 − p2)

Conservation of material masses, mixture momenta and mixture total energy

U =      α1ρ1 α2ρ2 ρV ρE      and F(U) =      α1ρ1V α2ρ2V ρV ⊗ V + pI (ρE + p)V      Stiffened gas EOS: pk = ρkek(γk − 1) − Πkγk

SHARK-FV — diot@lanl.gov — 5/23

SLIDE 12

The MOOD Ideas MOOD for multi-mat. flows The good The not-yet-good-enough Conclusion

6-eqation model of Saurel-Petitpas-Berry (SPB)

Reference: J. Comput. Phys. 228 (2009) 1678-1712. The non-conservative hyperbolic system is given by              ∂tU + ∇ · F(U) = ∂t(α1) + V · ∇(α1) = µ(p1 − p2) ∂t(α1ρ1e1) + ∇ · (α1ρ1e1V) + α1p1∇ · V = −¯ pIµ(p1 − p2) ∂t(α2ρ2e2) + ∇ · (α2ρ2e2V) + α2p2∇ · V = ¯ pIµ(p1 − p2)

Conservation of material masses, mixture momenta and mixture total energy
Non-conservative volume fraction advection equation + relaxation terms
Non-conservative material internal energies equations + relaxation terms
k

αk =1, ρ=

k

αkρk, p=

k

αkpk, ρc2 =

k

αkρkc2

k and

k

αkρkek

!

=ρe − → Solved in 4 steps: 1) Solve the system w/o relaxation terms 2) Correct the material internal energies (1) 3) Perform the relaxation step 4) Correct the material internal energies (2)

SHARK-FV — diot@lanl.gov — 5/23

SLIDE 13

The MOOD Ideas MOOD for multi-mat. flows The good The not-yet-good-enough Conclusion

6-eqation model of Saurel-Petitpas-Berry (SPB)

Reference: J. Comput. Phys. 228 (2009) 1678-1712. The non-conservative hyperbolic system is given by              ∂tU + ∇ · F(U) = ∂t(α1) + V · ∇(α1) = µ(p1 − p2) ∂t(α1ρ1e1) + ∇ · (α1ρ1e1V) + α1p1∇ · V = −¯ pIµ(p1 − p2) ∂t(α2ρ2e2) + ∇ · (α2ρ2e2V) + α2p2∇ · V = ¯ pIµ(p1 − p2)

Conservation of material masses, mixture momenta and mixture total energy
Non-conservative volume fraction advection equation + relaxation terms
Non-conservative material internal energies equations + relaxation terms
k

αk =1, ρ=

k

αkρk, p=

k

αkpk, ρc2 =

k

αkρkc2

k and

k

αkρkek

!

=ρe − → Solved in 4 steps: 1) Solve the system w/o relaxation terms 2) Correct the material internal energies (1) 3) Perform the relaxation step 4) Correct the material internal energies (2)

SHARK-FV — diot@lanl.gov — 5/23

SLIDE 14

The MOOD Ideas MOOD for multi-mat. flows The good The not-yet-good-enough Conclusion

6-eqation model of Saurel-Petitpas-Berry (SPB)

Reference: J. Comput. Phys. 228 (2009) 1678-1712. The non-conservative hyperbolic system is given by              ∂tU + ∇ · F(U) = ∂t(α1) + V · ∇(α1) = µ(p1 − p2) ∂t(α1ρ1e1) + ∇ · (α1ρ1e1V) + α1p1∇ · V = −¯ pIµ(p1 − p2) ∂t(α2ρ2e2) + ∇ · (α2ρ2e2V) + α2p2∇ · V = ¯ pIµ(p1 − p2)

Conservation of material masses, mixture momenta and mixture total energy
Non-conservative volume fraction advection equation + relaxation terms
Non-conservative material internal energies equations + relaxation terms
k

αk =1, ρ=

k

αkρk, p=

k

αkpk, ρc2 =

k

αkρkc2

k and

k

αkρkek

!

=ρe − → Solved in 4 steps: 1) Solve the system w/o relaxation terms 2) Correct the material internal energies (1) 3) Perform the relaxation step 4) Correct the material internal energies (2)

SHARK-FV — diot@lanl.gov — 5/23

SLIDE 15

The MOOD Ideas MOOD for multi-mat. flows The good The not-yet-good-enough Conclusion

6-eqation model of Saurel-Petitpas-Berry (SPB)

Reference: J. Comput. Phys. 228 (2009) 1678-1712. The non-conservative hyperbolic system is given by              ∂tU + ∇ · F(U) = ∂t(α1) + V · ∇(α1) = µ(p1 − p2) ∂t(α1ρ1e1) + ∇ · (α1ρ1e1V) + α1p1∇ · V = −¯ pIµ(p1 − p2) ∂t(α2ρ2e2) + ∇ · (α2ρ2e2V) + α2p2∇ · V = ¯ pIµ(p1 − p2)

Conservation of material masses, mixture momenta and mixture total energy
Non-conservative volume fraction advection equation + relaxation terms
Non-conservative material internal energies equations + relaxation terms
k

αk =1, ρ=

k

αkρk, p=

k

αkpk, ρc2 =

k

αkρkc2

k and

k

αkρkek

!

=ρe − → Solved in 4 steps: 1) Solve the system w/o relaxation terms 2) Correct the material internal energies (1) 3) Perform the relaxation step 4) Correct the material internal energies (2)

SHARK-FV — diot@lanl.gov — 5/23

SLIDE 16

The MOOD Ideas MOOD for multi-mat. flows The good The not-yet-good-enough Conclusion

First-order discretization of SPB model

Conservative part of the system

Ωi

∂tU(x, t) dx = −

Ωi

∇ · F(U(x, t)) dx, = −

j∈ν(i)
fij

F(U(x, t)) · nij dS, leads to Un+1

i

= Un

i − ∆t

|Ωi|

j∈ν(i)

|fij| F(Ui, Uj, nij), where F is an approximate Riemann Solver in the nij direction. = ⇒ We use HLLC from SPB. (which also provides the normal velocity u⋆

nij = V⋆ · nij)

SHARK-FV — diot@lanl.gov — 6/23

SLIDE 17

The MOOD Ideas MOOD for multi-mat. flows The good The not-yet-good-enough Conclusion

First-order discretization of SPB model

Volume fraction equations

Using that ∂tα1 + V · ∇(α1) = ∂tα1 + ∇ · (α1V) − α1∇ · V, we have

Ωi

∂tα1 dx = −

Ωi
∇ · (α1V) − α1∇ · V
dx,

approx

≈ −

j∈ν(i)
fij

(α1V) · nij dS − α1

fij

V · nij dS

,

and (α1)n+1

i

= (α1)n

i − ∆t

|Ωi|

j∈ν(i)

|fij|

α⋆

1 u⋆ nij − (α1)n i (u⋆ nij)

,

where u⋆

nij is given by the Riemann Solver

α⋆

1 is obtained as constant along fluid trajectories

SHARK-FV — diot@lanl.gov — 6/23

SLIDE 18

The MOOD Ideas MOOD for multi-mat. flows The good The not-yet-good-enough Conclusion

First-order discretization of SPB model

Energies equations

Following the same track, we get

Ωi

∂t(αkρkek) dx = −

Ωi
∇ ·
(αkρkek)V
− (αkpk)∇ · V
dx,

approx.

≈ −

j∈ν(i)
fij

((αkρkek)V) · nij dS − (αkpk)

fij

V · nij dS

,

and (αkρkek)n+1

i

= (αkρkek)n

i − ∆t

|Ωi|

j∈ν(i)

|fij|

(αkρkek)⋆u⋆

nij − (αkpk)n i (u⋆ nij)

,

where u⋆

nij is given by the Riemann Solver

(αkρkek)⋆ is determined by Hugoniot relation proposed in SPB

SHARK-FV — diot@lanl.gov — 6/23

SLIDE 19

The MOOD Ideas MOOD for multi-mat. flows The good The not-yet-good-enough Conclusion

Pressure Relaxation - Energies Correction

Energies correction (1) (only affects αkρkek)

Compute internal energy delta:

δρe = ρe −

k(αkρkek)old

Distribute it according to (αkρk)

ρ : (αkρkek)new =(αkρkek)old + (αkρk) ρ δρe Relaxation step (only affects αk)

After manipulations, the relaxation step consists in solving for p (w/ Newton)
k

(αkρk)νk(p) = 1, νk(p) = ν0

k

p0

k + γkπk + (γk − 1)ˆ

pI p + γkπk + (γk − 1)ˆ pI , (αkρk) constant.

Compute ρk =νk(p)−1 and deduce the corrected vol. fraction αk = αkρk

ρk Energies correction (2) (only affects αkρkek)

Compute the mixture pressure pmix =
ρe −
k

αkγkπk γk − 1

/
k

αk γk − 1

Deduce the corrected material energies from EOS ek =ek(αk, αkρk, pmix)

SHARK-FV — diot@lanl.gov — 7/23

SLIDE 20

The MOOD Ideas MOOD for multi-mat. flows The good The not-yet-good-enough Conclusion

MOOD criteria for the multi-material SPB model

Detection Criteria

PAD: Physical Admissibility Detection

⋆ Positivity of volume fractions: αk > 0, ∀k, ⋆ Positivity of material densities: αkρk > 0, ∀k, ⋆ Positivity of material int. energies: αkρkek > 0, ∀k.

DMP+u2 applied on material densities αkρk, ∀k.

Remarks

PAD seems to ensure robustness, but it is still to be proven
DMP+u2 on αkρk reduces/prevents spurious oscillations
Large number of possible combinations of detection criteria

SHARK-FV — diot@lanl.gov — 8/23

SLIDE 21

The MOOD Ideas MOOD for multi-mat. flows The good The not-yet-good-enough Conclusion

MOOD criteria for the multi-material SPB model

Detection Criteria

PAD: Physical Admissibility Detection

⋆ Positivity of volume fractions: αk > 0, ∀k, ⋆ Positivity of material densities: αkρk > 0, ∀k, ⋆ Positivity of material int. energies: αkρkek > 0, ∀k.

DMP+u2 applied on material densities αkρk, ∀k.

Remarks

PAD seems to ensure robustness, but it is still to be proven
DMP+u2 on αkρk reduces/prevents spurious oscillations
Large number of possible combinations of detection criteria

SHARK-FV — diot@lanl.gov — 8/23

SLIDE 22

The MOOD Ideas MOOD for multi-mat. flows The good The not-yet-good-enough Conclusion

MOOD criteria for the multi-material SPB model

Detection Criteria

PAD: Physical Admissibility Detection

⋆ Positivity of volume fractions: αk > 0, ∀k, ⋆ Positivity of material densities: αkρk > 0, ∀k, ⋆ Positivity of material int. energies: αkρkek > 0, ∀k.

DMP+u2 applied on material densities αkρk, ∀k.

Remarks

PAD seems to ensure robustness, but it is still to be proven
DMP+u2 on αkρk reduces/prevents spurious oscillations
Large number of possible combinations of detection criteria

SHARK-FV — diot@lanl.gov — 8/23

SLIDE 23

The MOOD Ideas MOOD for multi-mat. flows The good The not-yet-good-enough Conclusion

Numerical results - General comments

About the implementation

2D unstructured (polygonal) multi-material research code
Polynomial reconstructions of arbitrary degree, here P1 & P3
The MOOD parameters are always the same:

⋆ Cascade MOOD-P1: P1 → MUSCL → P0 ⋆ Cascade MOOD-P3: P3 → P2 → MUSCL → P0 ⋆ MUSCL is P1 coupled with Barth-Jespersen limiter Numerical parameters

Automatic time-stepping with CFL=0.25
Mixture enforced at initialization with a smallest volume fraction of 10−6

(since materials have to be mixed due to the diffuse interface treatment) → e.g. when simulating two separated pure materials, αk = (1 − 10−6) where the pure material lies and 10−6 elsewhere

SHARK-FV — diot@lanl.gov — 9/23

SLIDE 24

The MOOD Ideas MOOD for multi-mat. flows The good The not-yet-good-enough Conclusion

2-mat. contact disc. transport - 400 cells - Solution

tend = 0.035s ρ u p γ π Boundary Cond. Left State on [0.0; 0.5] 1 10 1 2.0 Inflow at x=0.0 Right State on [0.5; 1.0] 2 10 1 1.4 Outflow at x=1.0

SHARK-FV — diot@lanl.gov — 10/23

SLIDE 25

The MOOD Ideas MOOD for multi-mat. flows The good The not-yet-good-enough Conclusion

2-mat. contact disc. transport - 400 cells - Solution

tend = 0.035s ρ u p γ π Boundary Cond. Left State on [0.0; 0.5] 1 10 1 2.0 Inflow at x=0.0 Right State on [0.5; 1.0] 2 10 1 1.4 Outflow at x=1.0

SHARK-FV — diot@lanl.gov — 10/23

SLIDE 26

The MOOD Ideas MOOD for multi-mat. flows The good The not-yet-good-enough Conclusion

Abgrall–Karni shock tube - 800 cells - Solution

tend = 0.01s ρ u p γ π Boundary Cond. Left State on [0.0; 0.5] 1 500 1.4 Reflective at x=0.0 Right State on [0.5; 1.0] 1 0.2 1.6 Reflective at x=1.0

SHARK-FV — diot@lanl.gov — 11/23

SLIDE 27

The MOOD Ideas MOOD for multi-mat. flows The good The not-yet-good-enough Conclusion

Abgrall–Karni shock tube - 800 cells - Solution

tend = 0.01s ρ u p γ π Boundary Cond. Left State on [0.0; 0.5] 1 500 1.4 Reflective at x=0.0 Right State on [0.5; 1.0] 1 0.2 1.6 Reflective at x=1.0

SHARK-FV — diot@lanl.gov — 11/23

SLIDE 28

The MOOD Ideas MOOD for multi-mat. flows The good The not-yet-good-enough Conclusion

Abgrall–Karni shock tube - 800 cells - Convergence

MOOD-P1 is more accurate than MUSCL while both 2nd-order
4th-order MOOD-P3 is more accurate than 2nd-order MOOD-P1
Convergence rates: MUSCL 0.53 – MOOD-P1 0.63 – MOOD-P3 0.77

SHARK-FV — diot@lanl.gov — 11/23

SLIDE 29

The MOOD Ideas MOOD for multi-mat. flows The good The not-yet-good-enough Conclusion

Abgrall–Karni shock tube - 800 cells - Efficiency

MOOD-P1 is much more efficient than MUSCL while both 2nd-order
The better MOOD-P3 convergence rate implies that

MOOD-P3 is more and more efficient than MOOD-P1

SHARK-FV — diot@lanl.gov — 11/23

SLIDE 30

The MOOD Ideas MOOD for multi-mat. flows The good The not-yet-good-enough Conclusion

Water–Air shock tube - 800 cells - Solution

tend = 220µ ρ u p γ π Boundary Cond. Left State on [0.0; 0.7] 1000 1.109 4.4 6.108 Reflective at x=0.0 Right State on [0.7; 1.0] 50 1.105 1.4 Reflective at x=1.0

SHARK-FV — diot@lanl.gov — 12/23

SLIDE 31

The MOOD Ideas MOOD for multi-mat. flows The good The not-yet-good-enough Conclusion

Water–Air shock tube - 800 cells - Solution

tend = 220µ ρ u p γ π Boundary Cond. Left State on [0.0; 0.7] 1000 1.109 4.4 6.108 Reflective at x=0.0 Right State on [0.7; 1.0] 50 1.105 1.4 Reflective at x=1.0

SHARK-FV — diot@lanl.gov — 12/23

SLIDE 32

The MOOD Ideas MOOD for multi-mat. flows The good The not-yet-good-enough Conclusion

Water–Air shock tube - 800 cells - Convergence

MOOD-P1 is more accurate than MUSCL while both 2nd-order
4th-order MOOD-P3 is more accurate than 2nd-order MOOD-P1
Convergence rates: MUSCL 0.57 – MOOD-P1 0.75 – MOOD-P3 0.82

SHARK-FV — diot@lanl.gov — 12/23

SLIDE 33

The MOOD Ideas MOOD for multi-mat. flows The good The not-yet-good-enough Conclusion

Water–Air shock tube - 800 cells - Efficiency

MOOD-P1 is much more efficient than MUSCL while both 2nd-order
The better MOOD-P3 convergence rate implies that

MOOD-P3 tends to be more and more efficient than MOOD-P1

SHARK-FV — diot@lanl.gov — 12/23

SLIDE 34

The MOOD Ideas MOOD for multi-mat. flows The good The not-yet-good-enough Conclusion

R22 shocked bubble in air - Initialization

Experiment

J. F. Haas, B. Sturtevant. “Interaction of weak shock waves with cylindrical and

spherical gas inhomogeneities” Simulations

J. J. Quirk, S. Karni. “On the dynamics of a shock-bubble interaction”
S. Kokh, F. Lagoutière. “An anti-diffusive numerical scheme for the simulation of

interfaces between compressible fluids by means of a five-equation model”

Final time = 1020.0 µs
Wall boundary conditions on left/top/bottom

SHARK-FV — diot@lanl.gov — 13/23

SLIDE 35

The MOOD Ideas MOOD for multi-mat. flows The good The not-yet-good-enough Conclusion

R22 shocked bubble in air - Resolution Study

Isolines of material volume fraction (t = 767µs) MOOD-P1 MOOD-P3 200 × 40 400 × 80 800 × 160 1600 × 320

→ 4th-order MOOD-P3 method is less diffusive than 2nd-order MOOD-P1 one → The MOOD-P3 method seems to capture the solution features on coarser grids

SHARK-FV — diot@lanl.gov — 14/23

SLIDE 36

The MOOD Ideas MOOD for multi-mat. flows The good The not-yet-good-enough Conclusion

R22 shocked bubble in air - 4th-order vs 2nd-order

Numerical Schlieren images |∇ρ| MOOD-P3 850x170 Schlieren MOOD-P1 1600x320 Schlieren \[0.2cm]

→ 4th-order method computes similar structures on a 3.5× coarser mesh: ∼3× faster

SHARK-FV — diot@lanl.gov — 15/23

SLIDE 37

The MOOD Ideas MOOD for multi-mat. flows The good The not-yet-good-enough Conclusion

R22 shocked bubble in air - 4th-order vs 2nd-order

Map of the CellPD used at the end of the MOOD loop MOOD-P3 850x170 CellPD MOOD-P1 1600x320 CellPD \[0.2cm]

→ CellPD decrementing mostly occurs around discontinuities as expected.

SHARK-FV — diot@lanl.gov — 16/23

SLIDE 38

The MOOD Ideas MOOD for multi-mat. flows The good The not-yet-good-enough Conclusion

R22 shocked bubble in air - 4th-order vs 2nd-order

Isolines of material volume fraction MOOD-P3 850x170

Vol. Frac.

MOOD-P1 1600x320

Vol. Frac.

→ The interface is only slightly more diffused with the MOOD-P3. → MOOD-P3 gives a qualitatively equivalent solution in 3× less CPU time than MOOD-P1.

SHARK-FV — diot@lanl.gov — 17/23

SLIDE 39

The MOOD Ideas MOOD for multi-mat. flows The good The not-yet-good-enough Conclusion

So, what’s not good enough yet?

Main defects of the presented method

1. The MOOD solution is too sensitive to the smallest volume fraction
2. The 1st-order scheme is not mathematically robust and so is MOOD

Possible solutions for 2. (not discussed here)

Derive a robust first-order scheme. Doable?
Develop a positivity-enforcement technique à la MOOD?

(• Remark though, that’s a problem for all(?) multi-mat. diffuse methods!) Main reason for 1.: one MOOD paradigm

We use unlimited polynomial reconstructions and...

...rely on the MOOD process to get clean results

Didn’t seem to generate artifacts for the Euler equations
But clearly does for this model, see next slides

→ Solutions are under investigation and discussions are welcome!

SHARK-FV — diot@lanl.gov — 18/23

SLIDE 40

The MOOD Ideas MOOD for multi-mat. flows The good The not-yet-good-enough Conclusion

The impact of the smallest volume fraction

Test conditions

We consider the two-material Sod shock tube

tend = 0.2s ρ u p γ π Boundary Cond. Left State on [0.0; 0.5] 1 2 2.0 Reflective at x=0.0 Right State on [0.5; 1.0] 0.125 0.1 1.4 Reflective at x=1.0

We use the cascades P3 →P2 →P0 and P3 →P2 →MUSCL→P0
We consider 3 different smallest volume fractions: 10−8, 10−4 and 10−1

What do we look at?

Solution behavior at the interface between the two materials
Improvements when MUSCL is used in the cascade
Solution behavior when using only the PAD

Remarks

The problem we solve is never 2 separated pure mat.
This is particularly visible with a smallest vol. frac. of 10−1
While the approximation with smaller smallest vol. frac. is good

SHARK-FV — diot@lanl.gov — 19/23

SLIDE 41

The MOOD Ideas MOOD for multi-mat. flows The good The not-yet-good-enough Conclusion

Min. Volume Fraction Sensitivity - Sod

MOOD PAD+DMP+u2 Cascade w/o MUSCL Average Percentage of Retained Polynomial Degrees Min αk P3 P2 P0 10−8 93.14% 1.16% 5.25% 10−4 92.97% 1.60% 5.42% 10−1 91.98% 1.67% 6.35%

SHARK-FV — diot@lanl.gov — 20/23

SLIDE 42

The MOOD Ideas MOOD for multi-mat. flows The good The not-yet-good-enough Conclusion

Min. Volume Fraction Sensitivity - Sod

MOOD PAD+DMP+u2 Cascade w MUSCL Average Percentage of Retained Polynomial Degrees Min αk P3 P2 MUSCL P0 10−8 92.69% 0.91% 4.18% 2.23% 10−4 92.16% 1.05% 4.58% 2.20% 10−1 91.50% 0.96% 4.43% 3.10%

SHARK-FV — diot@lanl.gov — 20/23

SLIDE 43

The MOOD Ideas MOOD for multi-mat. flows The good The not-yet-good-enough Conclusion

Min. Volume Fraction Sensitivity - Sod

MOOD PAD Cascade w/o MUSCL Average Percentage of Retained Polynomial Degrees Min αk P3 P2 P0 10−8 97.50% 0.42% 2.07% 10−4 98.92% 0.23% 0.85% 10−1 99.99% 0.00% 0.01%

SHARK-FV — diot@lanl.gov — 20/23

SLIDE 44

The MOOD Ideas MOOD for multi-mat. flows The good The not-yet-good-enough Conclusion

Min. Volume Fraction Sensitivity - Sod

MOOD PAD Cascade w/o MUSCL Average Percentage of Retained Polynomial Degrees Min αk P3 P2 MUSCL P0 10−8 96.99% 0.93% 2.08% 0.00% 10−4 98.54% 0.46% 1.00% 0.00% 10−1 99.99% 0.00% 0.01% 0.00%

SHARK-FV — diot@lanl.gov — 20/23

SLIDE 45

The MOOD Ideas MOOD for multi-mat. flows The good The not-yet-good-enough Conclusion

A simpler problem for a better understanding

Test conditions

We consider the water-air contact disc. transport

tend = 229µs ρ u p γ π Boundary Cond. Left State on [0.0; 0.5] 1000 100 1.105 4.4 6.108 Inflow at x=0.0 Right State on [0.5; 1.0] 50 100 1.105 1.4 Outflow at x=1.0

And the cascades P1 →P0,

P3 →P2 →P0 and P3 →P2 →MUSCL→P0

And 2 different values for the smallest volume fractions: 10−8 and 10−1

What do we look at?

Solution behavior at the interface between the two materials
Pressure solution that should remain constant in time
Solution behavior when using only the PAD

Remarks

It is a sanity check for presure osc. problems
MOOD-P1 solution doesn’t exhibit step-like artifacts...
...but is also affected by the smallest volume fractions value

SHARK-FV — diot@lanl.gov — 21/23

SLIDE 46

The MOOD Ideas MOOD for multi-mat. flows The good The not-yet-good-enough Conclusion

Min. Volume Fraction Sensitivity - Water-Air

MOOD PAD+DMP+u2 Cascade w/o MUSCL Average Percentage of Retained Polynomial Degrees MOOD-P3 MOOD-P1 Min αk P3 P2 P0 P1 P0 10−8 95.87% 0.94% 3.19% 97.09% 2.91% 10−1 95.95% 1.04% 3.00% 97.42% 2.58%

SHARK-FV — diot@lanl.gov — 22/23

SLIDE 47

The MOOD Ideas MOOD for multi-mat. flows The good The not-yet-good-enough Conclusion

Min. Volume Fraction Sensitivity - Water-Air

MOOD PAD+DMP+u2 Cascade w/o MUSCL Average Percentage of Retained Polynomial Degrees MOOD-P3 MOOD-P1 Min αk P3 P2 P0 P1 P0 10−8 95.87% 0.94% 3.19% 97.09% 2.91% 10−1 95.95% 1.04% 3.00% 97.42% 2.58%

SHARK-FV — diot@lanl.gov — 22/23

SLIDE 48

The MOOD Ideas MOOD for multi-mat. flows The good The not-yet-good-enough Conclusion

Min. Volume Fraction Sensitivity - Water-Air

MOOD PAD+DMP+u2 Cascade w MUSCL Average Percentage of Retained Polynomial Degrees MOOD-P3 Min αk P3 P2 MUSCL P0 10−8 93.64% 0.99% 4.94% 0.43% 10−1 93.43% 1.25% 3.32% 2.00%

SHARK-FV — diot@lanl.gov — 22/23

SLIDE 49

The MOOD Ideas MOOD for multi-mat. flows The good The not-yet-good-enough Conclusion

Min. Volume Fraction Sensitivity - Water-Air

MOOD PAD+DMP+u2 Cascade w MUSCL Average Percentage of Retained Polynomial Degrees MOOD-P3 Min αk P3 P2 MUSCL P0 10−8 93.64% 0.99% 4.94% 0.43% 10−1 93.43% 1.25% 3.32% 2.00%

SHARK-FV — diot@lanl.gov — 22/23

SLIDE 50

The MOOD Ideas MOOD for multi-mat. flows The good The not-yet-good-enough Conclusion

Min. Volume Fraction Sensitivity - Water-Air

MOOD PAD Cascade w/o MUSCL Average Percentage of Retained Polynomial Degrees MOOD-P3 MOOD-P1 Min αk P3 P2 P0 P1 P0 10−8 97.50% 0.40% 2.09% 97.38% 2.62% 10−1 99.53% 0.12% 0.35% 99.51% 0.49%

SHARK-FV — diot@lanl.gov — 22/23

SLIDE 51

The MOOD Ideas MOOD for multi-mat. flows The good The not-yet-good-enough Conclusion

Min. Volume Fraction Sensitivity - Water-Air

MOOD PAD Cascade w/o MUSCL Average Percentage of Retained Polynomial Degrees MOOD-P3 MOOD-P1 Min αk P3 P2 P0 P1 P0 10−8 97.50% 0.40% 2.09% 97.38% 2.62% 10−1 99.53% 0.12% 0.35% 99.51% 0.49%

SHARK-FV — diot@lanl.gov — 22/23

SLIDE 52

The MOOD Ideas MOOD for multi-mat. flows The good The not-yet-good-enough Conclusion

Min. Volume Fraction Sensitivity - Water-Air

= ⇒ The less MOOD decrements when using only PAD... ...the better the MOOD solution with DMP+U2 is!

SHARK-FV — diot@lanl.gov — 22/23

SLIDE 53

The MOOD Ideas MOOD for multi-mat. flows The good The not-yet-good-enough Conclusion

Conclusion

The good

First extension of MOOD to a multi-material compressible flows model
The 4th-order scheme improves the solution quality and the efficiency
The MOOD extension is as easy as in the Euler equations case
Adding MUSCL in the cascade improves results...

The not-yet-good-enough

...but may cause loss of equilibrium at interfaces
There are still numerical step-like artifacts close to interfaces
The MOOD scheme is as robust as the 1st-order one

The (possible) future paths

Derive a robust first-order scheme for this model
Find a way to detect & limit oscillations of the reconstructions
Extend MOOD to multi-mat. flows with sharp interfaces treatment

SHARK-FV — diot@lanl.gov — 23/23

SLIDE 54

Thanks for your attention!

aSteven Diot, aMarianne François, bEdward Dendy.

http://public.lanl.gov/diot

Publications (5) S. Diot, M. M. François, E. D. Dendy, A higher-order unsplit 2D direct Eulerian finite volume method for two-material compressible flows based on the MOOD paradigms, Int. J. Numer. Meth. Fl. (2014) under review. (4) M. Dumbser, R. Loubère, S. Diot, A New Family of High Order Unstructured MOOD and ADER Finite Volume Schemes for Multidimensional Systems of Hyperbolic Conservation Laws, Commun. Comp. Phys (2014) accepted. (3) S. Diot, R. Loubère, S. Clain, The MOOD method in the three-dimensional case: Very-High-Order Finite Volume Method for Hyperbolic Systems, Int. J. Numer. Meth. Fl. 73 (2013) 362-392. (2) S. Diot, S. Clain, R. Loubère, Improved detection criteria for the Multi-dimensional Optimal Order Detection (MOOD) on unstructured meshes with very high-order polynomials, Comput. Fluids 64 (2012) 43–63. (1) S. Clain, S. Diot, R. Loubère, A high-order finite volume method for systems of conservation laws — Multi-dimensional Optimal Order Detection (MOOD), J. Comput. Phys. 230 (2011) 4028–4050.

SLIDE 55

Polynomial reconstruction Detection Criteria

Polynomial reconstruction

Form of polynomial reconstruction of degree d

Un

i (x) = Un i +

1≤|α|≤d

Rα

(x − ci)α −

1 |Ωi|

Ωi

(x − ci)αdx

α ∈ Rm multiindex =

⇒ covers all monomials

Rα ∈ R unknown poly. coefficients s.t. #(R) =

m

i=1(d+i)

m!

− 1 Computation of Rα

Approximations of mean values on a local stencil
Local stencil contains more than #(R) cells
Resolution of an overdetermined linear system ≡ Least-Squares

SHARK-FV — diot@lanl.gov — 2/0

SLIDE 56

Polynomial reconstruction Detection Criteria

Polynomial reconstruction

Form of polynomial reconstruction of degree d

Un

i (x) = Un i +

1≤|α|≤d

Rα

(x − ci)α −

1 |Ωi|

Ωi

(x − ci)αdx

α ∈ Rm multiindex =

⇒ covers all monomials

Rα ∈ R unknown poly. coefficients s.t. #(R) =

m

i=1(d+i)

m!

− 1 Computation of Rα

Approximations of mean values on a local stencil
Local stencil contains more than #(R) cells
Resolution of an overdetermined linear system ≡ Least-Squares

SHARK-FV — diot@lanl.gov — 2/0

SLIDE 57

Polynomial reconstruction Detection Criteria

Polynomial reconstruction

Mean value of Un

i on a Ωj

Un

i +

1≤|α|≤d

Rα

1

|Ωj|

Ωj

(x − ci)αdx − 1 |Ωi|

Ωi

(x − ci)αdx

Packing in a Matrix-vector form

      ¯ x(1,0,0)

1

¯ x(0,1,0)

1

¯ x(0,0,1)

1

· · · ¯ x(0,0,d)

1

¯ x(1,0,0)

2

¯ x(0,1,0)

2

¯ x(0,0,1)

2

· · · ¯ x(0,0,d)

2

. . . . . . . . . . . . . . . ¯ x(1,0,0)

N

¯ x(0,1,0)

N

¯ x(0,0,1)

N

· · · ¯ x(0,0,d)

N

             R(1,0,0) R(0,1,0) R(0,0,1) . . . R(0,0,d)        =      Un

1 − Un i

Un

2 − Un i

. . . Un

N − Un i

     AR = ¯ U

SHARK-FV — diot@lanl.gov — 3/0

SLIDE 58

Polynomial reconstruction Detection Criteria

Polynomial reconstruction

Mean value of Un

i on a Ωj

Un

i +

1≤|α|≤d

Rα

¯

xα

j

Packing in a Matrix-vector form

      ¯ x(1,0,0)

1

¯ x(0,1,0)

1

¯ x(0,0,1)

1

· · · ¯ x(0,0,d)

1

¯ x(1,0,0)

2

¯ x(0,1,0)

2

¯ x(0,0,1)

2

· · · ¯ x(0,0,d)

2

. . . . . . . . . . . . . . . ¯ x(1,0,0)

N

¯ x(0,1,0)

N

¯ x(0,0,1)

N

· · · ¯ x(0,0,d)

N

             R(1,0,0) R(0,1,0) R(0,0,1) . . . R(0,0,d)        =      Un

1 − Un i

Un

2 − Un i

. . . Un

N − Un i

     AR = ¯ U

SHARK-FV — diot@lanl.gov — 3/0

SLIDE 59

Polynomial reconstruction Detection Criteria

Polynomial reconstruction

Mean value of Un

i on a Ωj

Un

i +

1≤|α|≤d

Rα

¯

xα

j

Packing in a Matrix-vector form

      ¯ x(1,0,0)

1

¯ x(0,1,0)

1

¯ x(0,0,1)

1

· · · ¯ x(0,0,d)

1

¯ x(1,0,0)

2

¯ x(0,1,0)

2

¯ x(0,0,1)

2

· · · ¯ x(0,0,d)

2

. . . . . . . . . . . . . . . ¯ x(1,0,0)

N

¯ x(0,1,0)

N

¯ x(0,0,1)

N

· · · ¯ x(0,0,d)

N

             R(1,0,0) R(0,1,0) R(0,0,1) . . . R(0,0,d)        =      Un

1 − Un i

Un

2 − Un i

. . . Un

N − Un i

     AR = ¯ U

SHARK-FV — diot@lanl.gov — 3/0

SLIDE 60

Polynomial reconstruction Detection Criteria

Polynomial reconstruction

Idea for resolution AR = ¯ U ⇔ AtAR = At ¯ U ⇔ R = (AtA)−1At ¯ U ⇔ R = A† ¯ U How to get A†

Using a QR decomposition of A:

⋆ A = QR, with Q t = Q−1 and R triangular superior A† = ((QR)t(QR))−1At ⇒ A† = (RtR)−1At

Other possibilities: SVD, etc.
Pre-processing: pseudo-inverse matrices are stored

SHARK-FV — diot@lanl.gov — 4/0

SLIDE 61

Polynomial reconstruction Detection Criteria

Polynomial reconstruction

Idea for resolution AR = ¯ U ⇔ AtAR = At ¯ U ⇔ R = (AtA)−1At ¯ U ⇔ R = A† ¯ U How to get A†

Using a QR decomposition of A:

⋆ A = QR, with Q t = Q−1 and R triangular superior A† = ((QR)t(QR))−1At ⇒ A† = (RtR)−1At

Other possibilities: SVD, etc.
Pre-processing: pseudo-inverse matrices are stored

SHARK-FV — diot@lanl.gov — 4/0

SLIDE 62

Polynomial reconstruction Detection Criteria

Polynomial reconstruction

Idea for resolution AR = ¯ U ⇔ AtAR = At ¯ U ⇔ R = (AtA)−1At ¯ U ⇔ R = A† ¯ U How to get A†

Using a QR decomposition of A:

⋆ A = QR, with Q t = Q−1 and R triangular superior A† = ((QR)t(QR))−1At ⇒ A† = (RtR)−1At

Other possibilities: SVD, etc.
Pre-processing: pseudo-inverse matrices are stored

SHARK-FV — diot@lanl.gov — 4/0

SLIDE 63

Polynomial reconstruction Detection Criteria

Polynomial reconstruction

Idea for resolution AR = ¯ U ⇔ AtAR = At ¯ U ⇔ R = (AtA)−1At ¯ U ⇔ R = A† ¯ U How to get A†

Using a QR decomposition of A:

⋆ A = QR, with Q t = Q−1 and R triangular superior A† = ((QR)t(QR))−1At ⇒ A† = (RtR)−1At

Other possibilities: SVD, etc.
Pre-processing: pseudo-inverse matrices are stored

SHARK-FV — diot@lanl.gov — 4/0

SLIDE 64

Polynomial reconstruction Detection Criteria

Polynomial reconstruction

Idea for resolution AR = ¯ U ⇔ AtAR = At ¯ U ⇔ R = (AtA)−1At ¯ U ⇔ R = A† ¯ U How to get A†

Using a QR decomposition of A:

⋆ A = QR, with Q t = Q−1 and R triangular superior A† = ((QR)t(QR))−1At ⇒ A† = (RtR)−1At

Other possibilities: SVD, etc.
Pre-processing: pseudo-inverse matrices are stored

SHARK-FV — diot@lanl.gov — 4/0

SLIDE 65

Polynomial reconstruction Detection Criteria

MOOD — Detection criteria for transport

Convection equation ∂tu + ∇ ·

Vu
= 0,

with V = V(x) ∈ Rm, u ∈ R

Linear PDE modeling the transport of a scalar quantity
If ∇ · V = 0 then solution fulfills a maximum principle
Simple problem but contains most of problems related to higher-order

Detection Process: DMP as 1st filter Check if u⋆

i fulfills the Discrete Maximum Principle:

min

j∈ν(i)(un i , un j ) ≤ u⋆ i ≤ max j∈ν(i)(un i , un j )

If A = {DMP}, the solution always fulfills a DMP
But 2nd-order error at smooth extrema due to DMP on mean values
So DMP violation must be allowed at smooth extrema to reach VHO

SHARK-FV — diot@lanl.gov — 5/0

SLIDE 66

Polynomial reconstruction Detection Criteria

MOOD — Detection criteria for transport

Detection Process: u2 as 2nd filter

Only checked if the DMP is violated, i.e. 2nd filter
Intends to distinguish discontinuities and smooth extrema
Uses approximations of local curvatures over j ∈ ν(i) ∪ {i}

Xj =∂xx Uj, Yj =∂yy Uj, Zj =∂zz Uj with

Uj ∈ P2

X min

i

= min

j∈ν(i)

Xj, Xi
,

X max

i

=max

j∈ν(i)

Xj, Xi
Ymin

i

= min

j∈ν(i)

Yj, Yi
,

Ymax

i

=max

j∈ν(i)

Yj, Yi
Zmin

i

= min

j∈ν(i)

Zj, Zi
,

Zmax

i

=max

j∈ν(i)

Zj, Zi
Low memory storage and CPU vs dmax polynomial recon.

SHARK-FV — diot@lanl.gov — 6/0

SLIDE 67

Polynomial reconstruction Detection Criteria

MOOD — Detection criteria for transport

Definition (u2 detection criterion) A candidate solution u⋆

i in cell Ωi which violates the DMP is nonetheless

acceptable if X min

i

X max

i

> 0 and

X min

i

X max

i

> 1 − ε

and Ymin

i

Ymax

i

> 0 and

Ymin

i

Ymax

i

> 1 − ε

and Zmin

i

Zmax

i

> 0 and

Zmin

i

Zmax

i

> 1 − ε

where ε is a smoothness parameter.

Very good results obtained by fixing ε = 1/2
Challenging mathematical analysis still to be investigated

SHARK-FV — diot@lanl.gov — 7/0