CCLS
Descriptions
1st order
High-order
CCDG numerical results
Conclusion
Positivity-preserving cell-centered Lagrangian schemes F. Vilar, P.-H. Maire and C.-W. Shu Brown University, Division of Applied Mathematics 182 George Street, Providence, RI 02912
July 21st, 2014
July 21st, 2014
Franc¸ois Vilar
Positivity-preserving cell-centered scheme
1 / 58
CCLS
Descriptions
1st order
High-order
CCDG numerical results
1
Cell-Centered Lagrangian schemes
2
Lagrangian and Eulerian descriptions
3
Compatible first-order positivity-preserving discretization
4
High-order positivity-preserving extension
5
CCDG numerical results
6
Conclusion
July 21st, 2014
Franc¸ois Vilar
Positivity-preserving cell-centered scheme
Conclusion
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CCLS
Descriptions
1st order
High-order
CCDG numerical results
1
Cell-Centered Lagrangian schemes
2
Lagrangian and Eulerian descriptions
3
Compatible first-order positivity-preserving discretization
4
High-order positivity-preserving extension
5
CCDG numerical results
6
Conclusion
July 21st, 2014
Franc¸ois Vilar
Positivity-preserving cell-centered scheme
Conclusion
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CCLS
Descriptions
1st order
High-order
CCDG numerical results
Conclusion
Finite volume schemes on moving mesh J. K. Dukowicz: CAVEAT scheme, 1986 ´ GLACE scheme, 2005 B. Despres: P.-H. Maire: EUCCLHYD scheme, 2007 J. Cheng: High-order ENO conservative Lagrangian scheme, 2007 G. Kluth: Cell-centered Lagrangian scheme for the hyperelasticity, 2010 S. Del Pino: Curvilinear finite-volume Lagrangian scheme, 2010 P. Hoch: Finite volume method on unstructured conical meshes, 2011 A. J. Barlow: Dual grid high-order Godunov scheme, 2012 D. E. Burton: Godunov-like method for solid dynamics, 2013
DG scheme on initial mesh ` R. Loubere: DG scheme for Lagrangian hydrodynamics, 2004 Z. Jia: DG spectral finite element for Lagrangian hydrodynamics, 2010 F. Vilar: High-order DG scheme for Lagrangian hydrodynamics, 2012 July 21st, 2014
Franc¸ois Vilar
Positivity-preserving cell-centered scheme
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CCLS
Descriptions
Flow transformation
1st order
High-order
CCDG numerical results
Governing equations
1
Cell-Centered Lagrangian schemes
2
Lagrangian and Eulerian descriptions
3
Compatible first-order positivity-preserving discretization
4
High-order positivity-preserving extension
5
CCDG numerical results
6
Conclusion
July 21st, 2014
Franc¸ois Vilar
Positivity-preserving cell-centered scheme
Conclusion Equation of state
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CCLS
Descriptions
1st order
Flow transformation
High-order
CCDG numerical results
Governing equations
Conclusion Equation of state
Flow transformation of the fluid The fluid flow is described mathematically by the continuous transformation, Φ, so-called mapping such as Φ : X −→ x = Φ(X , t) n
N Φ x = Φ(X, t)
01
X
0011
Ω
∂ω
∂Ω
ω
Figure: Notation for the flow map.
where X is the Lagrangian (initial) coordinate, x the Eulerian (actual) coordinate, N the Lagrangian normal and n the Eulerian normal
Deformation Jacobian matrix: deformation gradient tensor F = ∇X Φ = July 21st, 2014
∂x ∂X
and
J = det F > 0
Franc¸ois Vilar
Positivity-preserving cell-centered scheme
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CCLS
Descriptions
1st order
Flow transformation
High-order
CCDG numerical results
Governing equations
Conclusion Equation of state
Trajectory equation dx = U(x, t), dt
x(X , 0) = X
Material time derivative d ∂ f (x, t) = f (x, t) + U ∇x f (x, t) dt ∂t
Transformation formulas FdX = dx
Change of shape of infinitesimal vectors
0
ρ = ρJ JdV = dv
Mass conservation Measure of the volume change
JF−t NdS = nds
Nanson formula
Differential operators transformations ∇x P = J1 ∇X (P JF−t ) ∇x U = July 21st, 2014
1 J ∇X
(JF
−1
U)
Franc¸ois Vilar
Gradient operator Divergence operator Positivity-preserving cell-centered scheme
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CCLS
Descriptions
1st order
Flow transformation
High-order
CCDG numerical results
Conclusion
Governing equations
Equation of state
Piola compatibility condition ∇x (JF−t ) = 0
Z =⇒ Ω
∇x (JF−t ) dV =
Z
JF−t N dS =
Z
∂Ω
n ds = 0 ∂ω
Deformation gradient tensor dF − ∇X U = 0 dt
Actual configuration d 1 ( ) − ∇x U = 0 dt ρ dU ρ + ∇x P = 0 dt de ρ + ∇x (PU) = 0 dt ρ
Initial configuration d 1 ( ) − ∇X (JF−1 U) = 0 dt ρ dU ρ0 + ∇X (P JF−t ) = 0 dt de ρ0 + ∇X (JF−1 PU) = 0 dt ρ0
Specific internal energy ε = e − 12 U 2 July 21st, 2014
Franc¸ois Vilar
Positivity-preserving cell-centered scheme
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CCLS
Descriptions
1st order
High-order
Flow transformation
CCDG numerical results
Governing equations
Conclusion Equation of state
Ideal EOS for the perfect gas P = ρ (γ − 1) ε If
ρ>0
then
where
a=
ε>0
q
γP ρ
a2 > 0
⇐⇒
(⇔ P > 0)
Stiffened EOS for water P = ρ (γ − 1) ε − γ P ? If
ρ>0
then
where
ρε > P ?
a=
⇐⇒
q
γ (P+P ? ) ρ
a2 > 0
(⇔ P > −P ? )
Jones-Wilkins-Lee (JWL) EOS for the detonation-products gas P = ρ (γ − 1) ε + f (ρ) If
ρ>0
July 21st, 2014
then
where
ε>0
Franc¸ois Vilar
=⇒
a=
q
γ P−f (ρ)+ρ f 0 (ρ) ρ
a2 > 0
(⇔ P > f (ρ) ≥ 0)
Positivity-preserving cell-centered scheme
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CCLS
Descriptions
Schemes
1st order
High-order
Time step constraint
CCDG numerical results Positivity
1
Cell-Centered Lagrangian schemes
2
Lagrangian and Eulerian descriptions
3
Compatible first-order positivity-preserving discretization
4
High-order positivity-preserving extension
5
CCDG numerical results
6
Conclusion
July 21st, 2014
Franc¸ois Vilar
Positivity-preserving cell-centered scheme
Conclusion Stability
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CCLS
Descriptions
Schemes
1st order
High-order
Time step constraint
CCDG numerical results Positivity
Conclusion Stability
Mass averaged values equations X 1 1 n n mc ( )n+1 = mc ( )nc + ∆t U np lpc npc c ρ ρ p∈Q(∂ωc ) X mc U cn+1 = mc U nc − ∆t F npc p∈Q(∂ωc )
mc ecn+1
=
mc ecn
− ∆t
X
U np F npc
p∈Q(∂ωc )
Definitions Z Z 1 1 ρ0 ψ dV = ρ ψ dv mc Ωc mc ωc = Pc lpc npc − Mpc (U p − U c )
ψc = F pc
mean value subcell forces
Momentum and total energy conservation X c∈C(p) July 21st, 2014
F pc = 0
=⇒
(
X
c∈C(p) Franc¸ois Vilar
Mpc )U p =
X
(Pc lpc npc + Mpc U c )
c∈C(p) Positivity-preserving cell-centered scheme
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CCLS
Descriptions
Schemes
1st order
High-order
Time step constraint
Conclusion
Positivity
GLACE assumptions a) Q(∂ωc ) = P(ωc )
CCDG numerical results
Stability
p+
the node set
npp+ p
lpp+
lpcnpc
− − + + b) lpc npc = lpc npc + lpc npc = 12 lp− p np− p + 12 lpp+ npp+
c) Mpc = Zpc lpc npc ⊗ npc X X d) U p = ( Mpc )−1 (Pc lpc npc + Mpc U c ) c∈C(p)
ωc
l p− p
n p− p p−
c∈C(p)
EUCCLHYD assumptions Same assumptions a), b) and d) as GLACE − − − + + + + c) Mpc = Zpc lpc npc ⊗ n− pc + Zpc lpc npc ⊗ npc
July 21st, 2014
Franc¸ois Vilar
Positivity-preserving cell-centered scheme
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CCLS
Descriptions
Schemes
High-order
CCDG numerical results
Conclusion
Positivity
Stability
p
p+
00 1100 11 1 11 0 00 11 00 00 11 1 0 11 00 00 11 1 0 1 0 1 0 1 0 00 11
1st order Time step constraint
1 0 0 1 00 11 00 11
ωc
1 0 0 1 00 11 11 00
ωc3
p−
00 11 00 11 00 11 0 1 0 1
ωR
ωc4
p+
p
ωc2
m
ωL
ωc5
p
ωc1
Node ell set
Middle point ell set
Cell-centered DG (CCDG) assumptions a) Q(∂ωc ) =
[
(Q(pp+ ) \ {p+ })
p∈P(ωc )
b) For q ∈ Q(pp+ ),
Z lq nq|pp+ =
1
λq (ζ) 0
X k∈Q(pp+ )
∂λk (x k × e z ) dζ ∂ζ
For p ∈ P(ωc ), lpc npc = lp np|p− p + lp np|pp+ For q ∈ Q(pp+ ) \ {p, p+ }, lqc nqc = lq nq|pp+ July 21st, 2014
Franc¸ois Vilar
Positivity-preserving cell-centered scheme
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CCLS
Descriptions
1st order
Schemes
High-order
CCDG numerical results
Time step constraint
Positivity
Conclusion Stability
CCDG assumptions c) For p ∈ P(ωc ),
− − − + + + + Mpc = Zpc lpc npc ⊗ n− pc + Zpc lpc npc ⊗ npc
For q ∈ Q(pp+ ) \ {p, p+ }, d) For p ∈ P(ωc ), U p = (
X
Mpc = Zpc lpc npc ⊗ npc Mpc )−1
c∈C(p)
For q ∈ Q(pp+ ) \ {p, p+ },
July 21st, 2014
Franc¸ois Vilar
Up =
X
(Pc lpc npc + Mpc U c )
c∈C(p)
ZpL U L + ZpR U R PR − PL − npL ZpL + ZpR ZpL + ZpR
Positivity-preserving cell-centered scheme
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CCLS
Descriptions
1st order
Schemes
High-order
CCDG numerical results
Time step constraint
Positivity
Conclusion Stability
CFL condition System eigenvalues:
−a, 0, a ∀c,
∆t ≤ Ce
vcn ac Lc
Volume control Relative volume variation: ∀c,
|vcn+1 − vcn | ≤ Cv vcn
∆t ≤ Cv
vcn |
X
p∈Q(∂ωc )
July 21st, 2014
Franc¸ois Vilar
n n U np lpc npc |
Positivity-preserving cell-centered scheme
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CCLS
Descriptions
1st order
Schemes
High-order
Time step constraint
CCDG numerical results
Conclusion
Positivity
Stability
Solution vector W = ( ρ1 , U, e)t
Admissible convex set G = {W, G = {W,
ρ > 0, ε = e − 12 U 2 > 0} 2
ρ > 0, ε = e − 12 U >
?
P ρ
for ideal and JWL EOS }
for stiffened EOS
First-order positivity-preserving scheme If Wnc = (( ρ1 )nc , U nc , ecn )t ∈ G, then under which constraint Wn+1 ∈G ? c
Positive density If ( ρ1 )nc > 0
then
Thus if Cv < 1 July 21st, 2014
( ρ1 )n+1 > 0 ⇐⇒ ( ρ1 )nc > − c
then
( ρ1 )nc =
Franc¸ois Vilar
vcn mc
∆t mc
X
n n U np lpc npc
p∈Q(∂ωc )
> 0 =⇒ ( ρ1 )n+1 = c
vcn+1 mc
>0
Positivity-preserving cell-centered scheme
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CCLS
Descriptions
Schemes
1st order
High-order
Time step constraint
CCDG numerical results Positivity
Conclusion Stability
Positive internal energy εc = ec − 12 (U c )2 εcn+1
∆t = εnc − mc
X
U np
p
F npc
−
X p
U nc
F npc
∆t X n 2 ( F ) + 2mc p pc
!
Properties F pc = Pc lpc npc − Mpc (U p − U c ) X X lpc npc = lpp+ npp+ = 0 p∈Q(∂ωc )
p∈P(ωc )
Definitions ∆t mc V p = U np − U nc λc =
July 21st, 2014
Franc¸ois Vilar
Positivity-preserving cell-centered scheme
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CCLS
Descriptions
Schemes
1st order
High-order
Time step constraint
CCDG numerical results
Conclusion
Positivity
Stability
Definitions εcn+1 = Ac + λc Bc Pcn vcn+1 − vcn ρnc vcn X λc X Mpc V p )2 Bc = Mpc V p V p − ( 2 p p Ac = εnc −
Ac > 0 for ideal and JWL EOS If Bc ≥ 0
then
Ac > 0 =⇒ εn+1 >0 c
Pn As ρnc > 0 and εnc > 0 then Ac > εnc − nc Cv ρc 1 for ideal gas γ−1 ρnc εnc Thus Cv < = 1 Pcn for JWL gas n) c γ − 1 + fρ(ρ n εn
⇒
Ac > 0
c c
July 21st, 2014
Franc¸ois Vilar
Positivity-preserving cell-centered scheme
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CCLS
Descriptions
1st order
Schemes
Ac >
P? ρcn+1
CCDG numerical results
Conclusion
Positivity
Stability
for stiffened EOS
If Bc ≥ 0 then Ac > Ac = (εnc − Since εnc − Thus
High-order
Time step constraint
P? ρnc ) (1 P? ρnc
Cv
c
− (γ − 1)
>0
1 γ−1
P? ρn+1 c
vcn+1 − vcn P? ) + vcn ρn+1 c Ac > (εnc −
then =⇒
Ac >
P? ρn+1 c
P? ρnc ) (1
− (γ − 1)Cv ) +
P? ρcn+1
Discrete entropy inequality λc Bc = εn+1 − Ac = εn+1 − εnc + Pcn ( ρ1 )n+1 − ( ρ1 )nc c c c
Entropy T dS = dε + Pd( ρ1 ) ≥ 0 July 21st, 2014
Franc¸ois Vilar
Gibbs identity + second law of thermodynamics Positivity-preserving cell-centered scheme
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CCLS
Descriptions
1st order
Schemes
Bc ≥ 0 X
Bc =
High-order
Time step constraint
p∈Q(∂ωc )
Mpc V p V p −
λc ( 2
CCDG numerical results
Conclusion
Positivity
X
Stability
Mpc V p )2
p∈Q(∂ωc )
Np
Mpc =
X n=1
X
Zpn lpn npn ⊗ npn
Mpc V p V p =
Np X
X
Zpn lpn (V p npn )2 =
p∈Q(∂ωc ) n=1
p∈Q(∂ωc )
Np
X
Re-numbering:
X
p∈Q(∂ωc ) n=1 Nc X
ψpn =
X
Np X
Zpn lpn Xp2n
p∈Q(∂ωc ) n=1 Nc X
ψi
i=1
Nc λc X Zi Zj li lj Xi Xj (ni nj ) = HX X , 2 i=1 i,j=1 Zi li (1 − λc Zi li ), if i = j, t 2 where X = (X1 , . . . , XNc ) and Hij = λ − c Z Z l l (n n ), if i 6= j. i j i j i j 2
Bc =
July 21st, 2014
Zi li Xi2 −
Franc¸ois Vilar
Positivity-preserving cell-centered scheme
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CCLS
Descriptions
1st order
Schemes
High-order
Time step constraint
CCDG numerical results Positivity
Conclusion Stability
Theorem If H is symmetric diagonally dominant with non-negative diagonal entries then H is positive semi-definite (thanks to Gerschgorin theorem)
Bc ≥ 0 If λc ≤
2 Zi li
If λc ≤ X j
Thus if
Hii ≥ 0
then 2
Zj lj |ni nj |
|Hii | −
then
j
∆t ≤
July 21st, 2014
ac Lc
|Hij | ≥ 0
then
Bc ≥ 0
j
Acoustic impedance If
j6=i
mc X 1 Zj lj 2
2 λc ≤ X ⇐⇒ ∆t ≤ Zj lj
vcn
X
Zc = ρc ac
where Lc = Franc¸ois Vilar
1 2
P
j lj
then
Bc ≥ 0
Positivity-preserving cell-centered scheme
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CCLS
Descriptions
Schemes
1st order
High-order
CCDG numerical results
Time step constraint
Conclusion
Positivity
Stability
Positivity-preserving property Finally, for the first-order finite volume cell-centered Lagrangian schemes, if 1
2
Wnc ∈ G ∆t ≤ Cv
|
X
p∈Q(∂ωc )
3
vn ∆t ≤ c , ac Lc
Then
Wn+1 ∈G c
July 21st, 2014
vcn n n U np lpc npc |
with Lc =
,
with
Cv < min 1,
1 n
c) γ − 1+ fρ(ρ n εn
c c
1 2
X
lpc ,
GLACE
p∈P(ωc ) 1 2
X
lpp+ ,
EUCCLHYD
p∈P(ωc ) 1 2
X
X
lq|pp+ . CCDG
p∈P(ωc ) q∈Q(pp+ )
1 n 1 n+1 n n ) − ( ) and εn+1 − ε + P ( c c c ρ c ρ c ≥0 Franc¸ois Vilar
Positivity-preserving cell-centered scheme
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CCLS
Descriptions
1st order
Schemes
High-order
Time step constraint
CCDG numerical results
Conclusion
Positivity
Stability
Norm definitions kψkL1 =
Z
ρ0 |ψ| dV =
Ω
Z kψkL2 =
0
2
Z ω
ρ |ψ| dv
12
ρ ψ dV
Z =
Ω
2
12
ρ ψ dv ω
Stability analysis For sake of simplicity periodic boundary conditions (PBC) are considered n ψhn is the piecewise constant numerical solution such as ψh| = ψcn ω c
We assume the initial solution vector W0c = (( ρ1 )0c , U 0c , ec0 )t on cell ωc is computed through Z 1 ρ0 (X ) W0 (X ) dV , W0c = mc Ωc where W0 = ( ρ10 , U 0 , e0 )t and ρ10 , U 0 , e0 respectively are the initial specific volume, velocity and total energy July 21st, 2014
Franc¸ois Vilar
Positivity-preserving cell-centered scheme
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CCLS
Descriptions
Schemes
1st order
High-order
Time step constraint
CCDG numerical results
Conclusion
Positivity
Stability
Specific volume 1 1 |( )nc | = ( )nc ρ ρ X X X 1 1 mc ( )n−1 (since PBC + lpc npc = 0) Conservation mc ( )nc = c ρ ρ c c Positivity
c∈C(p)
X X 1 1 1 1 k( )nh kL1 = mc |( )nc | = mc |( )n−1 | = k( )n−1 kL1 c ρ ρ ρ ρ h c c
Total energy |ecn | = ecn (since εnc > 0 ⇐⇒ ecn > 12 (U nc )2 ≥ 0) X X X Conservation mc ecn = mc ecn−1 (since PBC + F pc = 0) Positivity
c
c
kehn kL1 = July 21st, 2014
X c
Franc¸ois Vilar
mc |ecn | =
c∈C(p)
X c
mc |ecn−1 | = kehn−1 kL1 Positivity-preserving cell-centered scheme
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CCLS
Descriptions
1st order
Schemes
High-order
CCDG numerical results
Time step constraint
Positivity
Conclusion Stability
Kinetic energy and velocity K = 21 U 2 n 2 1 2 (U c )
specific kinetic energy
< ecn
=⇒
1 2
X
mc (U nc )2
