Add-ons to the compatible staggered Lagrangian scheme and other unspoken details
` 1 R. Loubere 1 Institut
´ de Mathematique de Toulouse (IMT) and CNRS, Toulouse, France ECCOMAS, September 2012
` R. Loubere (IMT and CNRS)
Add-ons to the compat. stag. Lagr. scheme
ECCOMAS, September 2012
1 / 24
Plan
Introduction and motivation 2D Lagrangian Staggered Hydrodynamics scheme Subcell formalism Specifics : Artificial viscosity, subpressure forces Properties
Some “facts” and deaper studies Lagrangian subcell ? Internal (and volume) consistency ? Stability ? Accuracy ?
Conclusions and perspectives
` R. Loubere (IMT and CNRS)
Add-ons to the compat. stag. Lagr. scheme
ECCOMAS, September 2012
2 / 24
Introduction and motivation Why do we still analyse a staggered Lagrangian scheme from the 50’s ? 2D Staggered Lagrangian scheme for hydrodynamics dates back to von Neumann, RichtmyerJ. Appl. Phy. 1950)], Schultz, Wilkins [Green book (1964)] era. later improved by many authors in national labs or academy important subcell based compatible discretization of div/grad [Burton, Caramana, Shashkov (1998)] improved artificial viscosity, hourglass filters, accuracy time/space, axisymetric geo. coupling with slide line, materials, diffusion, elastoplasticity, etc. “engine” of many ALE codes most of all this scheme has been and still is routinely used ! =⇒ Need to deeply understand its behaviors ! to explain already known features to chose between different “versions” to measure the relative importance of “improvements” to fight back, justify or simply understand urban legends ` R. Loubere (IMT and CNRS)
Add-ons to the compat. stag. Lagr. scheme
ECCOMAS, September 2012
3 / 24
2D Lagrangian Staggered Hydro scheme Governing equations 2D gas dynamics equations ρ
d dt
1 −∇·U =0 ρ
ρ
d U + ∇P = 0 dt
ρ
d ε + P∇ · U = 0 dt
2
Equation of state EOS P = P(ρ, ε), where ε = E − U2 . Internal energy equation can be viewed as an entropy evolution equation (Gibbs relation TdS = dε + Pd ρ1 ≥ 0) d d 1 d ε+P ≥0 ρ ε + P∇ · U = ρ dt dt dt ρ Trajectory equations dX = U(X (t), t), dt
X (0) = x,
Lagrangian motion of any point initially located at position x.
` R. Loubere (IMT and CNRS)
Add-ons to the compat. stag. Lagr. scheme
ECCOMAS, September 2012
4 / 24
2D Lagrangian Staggered Hydro scheme Preliminaries Staggered placement of variables Point velocity U p , cell-centered density ρc and internal energy εc Subcells are Lagrangian volumes Subcell mass mcp is constant in time so are cell/point masses
p+
Ωc c
mc =
X
mcp ,
p∈P(c)
mp =
X
mcp ,
Ω cp p
c∈C(p)
L cp Ncp p−
Compatible discretization Given total energy definition and momentum discretization (Newton’s 2nd law) imply energy discretization as sufficient condition Cornerstone : subcell force F cp that acts from subcell Ωcp on p. compile pressure gradient F cp = −Pc Lcp N cp , artificial visco, anti-hourglass, elasto forces. X Galilean invariance and/or momentum conservation implies F cp = 0 p∈P(c) ` R. Loubere (IMT and CNRS)
Add-ons to the compat. stag. Lagr. scheme
ECCOMAS, September 2012
5 / 24
2D Lagrangian Staggered Hydro scheme Discretization Time discretization : t n −→ t n+1 Originaly staggered placement of variable in time U n+1/2 and ρn , εn . Improvement gained by same time location U n , ρn , εn . Side effect : This helped total energy conservation. Predictor-Corrector P/C type of scheme is very often considered. Predictor step is often used as to time center the pressure for correction step. Very seldom : GRP, ADER to reduce the cost of a two-step P/C process Space discretization : Ωp , Ωc X d Vc − Lcp N cp · U p = 0 dt
or
d X p = Up, dt
X d F cp Up + dt
=
0
X d εc − F cp · U p dt
=
0
p∈P(c)
mp
X p (0) = x p
c∈C(p)
mc
p∈P(c)
` R. Loubere (IMT and CNRS)
Add-ons to the compat. stag. Lagr. scheme
ECCOMAS, September 2012
6 / 24
2D Lagrangian Staggered Hydro scheme Properties General grid formulation GCL First order accurate scheme in space on non-regular grid, Conservation of mass, momentum, total energy Expected properties Expected (internal) consistency Expected second-order accuracy in time Expected stability under classical CFL condition Biblio ` [1] Volume consistency in Staggered Grid Lagrangian Hydrodynamics Schemes, JCP, Volume 227, Pages 3731-3737 R. Loubere, M. Shashkov, B. Wendroff, ` [2] On stabiliy analysis of staggered schemes, A.L. Bauer, R. Loubere, B. Wendroff, SINUM. Vol 46 Issue 2 (2008) [3] The Internal Consistency, Stability, and Accuracy of the Discrete, Compatible Formulation of Lagrangian Hydrodynamics, JCP, ` Volume 218, Pages 572-593 A.L. Bauer, D.E. Burton, E.J. Caramana, R. Loubere, M.J. Shashkov, P.P. Whalen
` R. Loubere (IMT and CNRS)
Add-ons to the compat. stag. Lagr. scheme
ECCOMAS, September 2012
7 / 24
Internal consistency
General remark The equations are essentially created in discrete form, as opposed to being the discretization of a system of PDE’s. As such, one may or may not be able to rigorously take the continuum limit to obtain the latter ; this depends on the kinds of forces that are employed to resolve shocks and to counteract spurious grid motions. Ambiguity of cell volume definition Results from requiring both total energy conservation and the modeling of the internal energy d d advance from the differential equation dt ε + p dt (1/ρ) = 0 under assumptions Vc can be computed from X p for all p ∈ P(c) U p is constant for all t ∈ [t n ; t n+1 ], so that X p (t) = X np + U p (t − t n ) There exist a coordinate and a compatible cell volume which may be different !
` R. Loubere (IMT and CNRS)
Add-ons to the compat. stag. Lagr. scheme
ECCOMAS, September 2012
8 / 24
Internal consistency Ambiguity of cell volume definition
Implied coordinate cell volume Z t n+1 dVc dt Vcn+1 − Vcn = n dt t
=
X p∈P(c)
=
X
Z up
t n+1 tn
∂Vc dt + vp ∂xp
Z
t n+1 tn
∂Vc dt ∂yp
up Acp + vp Bcp
p∈P(c)
with A, B are rectangular sparce matrices. Remark Not simple average of integrands unless for Cartesian geometry.
` R. Loubere (IMT and CNRS)
Add-ons to the compat. stag. Lagr. scheme
ECCOMAS, September 2012
9 / 24
Internal consistency Ambiguity of cell volume definition Implied coordinate cell volume X
Vcn+1 − Vcn =
up Acp + vp Bcp
p∈P(c)
Implied compatible cell volume Discrete momentum + total energy conservation implicitely defines mp (upn+1 − upn ) −
X
mp (vpn+1 − vpn ) −
Pc acp = 0,
c∈C(p)
mc (εn+1 − εnc ) + Pc c
X
Pc bcp = 0
c∈C(p)
X
up acp + vp bcp = 0
p∈P(c)
= with (acp , bcp ) = ∆t Lcp N cp . For adiabatic flows the entropy S satisfies T dS dt
dε dt
+ P dV = 0. dt
mc εn+1 − εnc + Pc Vcn+1 − Vcn = 0 c
` R. Loubere (IMT and CNRS)
Add-ons to the compat. stag. Lagr. scheme
ECCOMAS, September 2012
10 / 24
Internal consistency Ambiguity of cell volume definition Implied coordinate cell volume X
Vcn+1 − Vcn =
up Acp + vp Bcp
p∈P(c)
Implied compatible cell volume Discrete momentum + total energy conservation implicitely defines mp (upn+1 − upn ) −
X
mp (vpn+1 − vpn ) −
Pc acp = 0,
c∈C(p)
mc (εn+1 − εnc ) + Pc c
X
Pc bcp = 0
c∈C(p)
X
up acp + vp bcp = 0
p∈P(c)
= with (acp , bcp ) = ∆t Lcp N cp , for adiabatic flows the entropy S satisfies T dS dt
dε dt
+ P dV = 0. dt
X mc εn+1 − εnc + Pc up Acp + vp Bcp = 0 c p∈P(c)
` R. Loubere (IMT and CNRS)
Add-ons to the compat. stag. Lagr. scheme
ECCOMAS, September 2012
10 / 24
Internal consistency Ambiguity of cell volume definition Condition for uniqueness of cell volume definition Same volume definition if Acp = acp ,
and
Bcp = bcp
∀c, p
along with total energy conservation and PdV work. But a, b correspond to your prefered discrete gradient and A, B are given by the geometry ! Do the matrices match for different geometry and classical discrete gradient ? 1D Cartesian - Yes 1D cylindrical - No unless (time centering grid vectors + force=0) 1D spherical - No unless (time centering + 1D vector manipulation) 2D Cartesian - No unless (time centering + force=0). 2D cylindrical r − z - No Remark : 2D Cartesian analysis shows that the difference is small (O(∆t 3 ) for one time step)
` R. Loubere (IMT and CNRS)
Add-ons to the compat. stag. Lagr. scheme
ECCOMAS, September 2012
11 / 24
Internal consistency Wendroff’s idea [(JCP, 227, 2010)] Derive A, B for different geometries and deduce appropriate discrete gradient. 1D spherical : cell half-index i + 21 , vertices ri , ri+1 , cell volume Vi+ 1 = A
V n+1 i+ 12
−
z Z
n Vi+ 1 2
=
ui+1
t n+1 tn
2
i+ 1 ,i+1 2
}| n ri+1
{
2 + ui+1 (t − t ) dt −ui n
2 rin 2 + rin+1 + rin rin+1 − ∆t 3 2 = A 1 n+1 2 n+1 n n ∆t i+ ,k ri+1 + ri+1 + ri+1 ri+1 3 2 0
Matrix A is given by
z Z
t n+1 tn
1 3 A
3 − r3 . ri+1 i i+ 1 ,i 2
}|
rin
{ 2 + ui (t − t ) dt
if
k =i
if
k =i +1
if
k 6= i, k 6= i + 1
n
Imposing ai± 1 ,i ≡ Ai± 1 ,i leads to 2
mi (uin+1
−
uin )
2
= Ai+ 1 ,i pi+ 1 + Ai− 1 ,i pi− 1 = −∆t 2
2
2
2
rin
2
2 + rin+1 + rin rin+1 Pi+ 1 − Pi− 1 2 2 3
−→ This is the good discrete gradient. ` R. Loubere (IMT and CNRS)
Add-ons to the compat. stag. Lagr. scheme
ECCOMAS, September 2012
12 / 24
Internal consistency 2D cylindrical r − z : quad. cell Vj , nodes (ri , zi ), i = 1, . . . , 4. Define
+
Ri→j
=
Zi→j
=
n n n+1 n+1 n+1 n+1 n n 2ri + rj zj − zi + 2ri + rj zj − zi n o n n n n n+1 n+1 n+1 n+1 +2 2ri + rj zj − zi + 2ri + rj zj − zi , n n n+1 n+1 n+1 n+1 n n 2ri + rj rj − ri + 2ri + rj rj − ri n o n n n n n+1 n+1 n+1 n+1 +2 2ri + rj rj − ri + 2ri + rj rj − ri ,
∆t n Vjn+1 − Vjn = 36 u1 [R1→4 − R1→2 ] + u2 [R2→3 − R2→1 ] + u3 [R3→4 − R3→2 ] + u4 [R4→3 − R4→1 ] o v1 [Z1→4 − Z1→2 ] + v2 [Z2→3 − Z2→1 ] + v3 [Z3→4 − Z3→2 ] + v4 [Z4→3 − Z4→1 ] ,
[R1→4 − R1→2 ] defines Ajp for p global index of vertex 1, [Z1→4 − Z1→2 ] defines Bjp A, B being defined, it uniquely implies the discretizations of discrete gradient with a = A, b = B. −→ This is the good discrete gradient.
` R. Loubere (IMT and CNRS)
Add-ons to the compat. stag. Lagr. scheme
ECCOMAS, September 2012
13 / 24
Internal consistency Numerical scheme n+1 n n Initialization : Pc = Pcn , an+1 cp = acp , bcp = bcp
0- Outer iterations : 0- Inner consistency iterations : Pressure Pc fixed solve the implicit system 1-2 1- Velocity X X mp (upn+1 − upn ) − Pc an+1 mp (vpn+1 − vpn ) − Pc bn+1 cp = 0, cp = 0 c∈C(p)
2- Position and acp , bcp xpn+1 = xpn + ∆t
c∈C(p)
upn + upn+1 2
= 0,
ypn+1 = ypn + ∆t
vpn + vpn+1 2
=0
3- Exit when convergence is reached for xp , yp , up , vp 1- Compute new cell volume Vcn+1 and deduce internal energy mc (εn+1 − εnc ) + Pc (Vcn+1 − Vcn ) = 0 c 2- Deduce new pressure Pcn+1 and Pc = 3- Exit when convergence is reached for ` R. Loubere (IMT and CNRS)
1 (Pcn+1 2 n+1 εc
+ Pcn )
Add-ons to the compat. stag. Lagr. scheme
ECCOMAS, September 2012
14 / 24
Internal consistency Numerical scheme
Remarks These schemes are indexed by (#outer, #inner) Classical P/C staggered compatible scheme is a (2, 1) scheme. For 2D axisymetric problem the Cartesian geometrical vectors are modified but this can not fulfill volume consistency and total energy conservation. Conversely our proposed scheme is a (2, ∞) scheme which enjoys these properties.
` R. Loubere (IMT and CNRS)
Add-ons to the compat. stag. Lagr. scheme
ECCOMAS, September 2012
15 / 24
Internal consistency Numerical scheme
Numerical results : Coggeshall adabatic compression in 2D r − z geometry L1 Entropy error
L1 Density err
0.0014
L1 Energy err
0.014 11x51 21x101 41x201 81x401
0.0012
0.003 11x51 21x101 41x201 81x401
0.012
0.001
11x51 21x101 41x201 81x401 0.0025
0.01
0.0006
L1 energy error
L1 density error
L1 entropy error
0.002 0.0008
0.008
0.006
0.0015
0.001
0.0004 0.004 0.0002
0.0005 0.002
0 0
0 0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0
0.1
0.2
0.3
Time
0.4
0.5
0.6
0.7
0
0.1
0.2
0.3
Time
2.5e-05
0.4
0.5
0.6
0.7
Time
0.014 11x51 21x101 41x201 81x401
0.003 11x51 21x101 41x201 81x401
0.012
11x51 21x101 41x201 81x401 0.0025
2e-05 0.01
1e-05
L1 energy error
L1 density error
L1 entropy error
0.002 1.5e-05
0.008
0.006
0.0015
0.001 0.004 5e-06
0.0005 0.002
0
0
0 0
0.1
0.2
0.3
0.4
0.5
Time
` R. Loubere (IMT and CNRS)
0.6
0.7
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
Time
Add-ons to the compat. stag. Lagr. scheme
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
Time
ECCOMAS, September 2012
16 / 24
Stability Which stability ? If the continuum system has no growing solutions, the discretized form should also contains no growing solutions. Predictor-corrector scheme In general the prediction step only serves as to predict a time advanced pressure P ∗ = αP predicted + (1 − α)P n with t ∗ ∈ [t n ; t n+1 ]. Scheme #1 Correction
Prediction 1- Predict U n+∗ = U np + ∆t f (Pcn ), and p U n+1/2 =
1 (U n 2
U n+1/2 =
+ U n+∗ ) n+1/2
2- Predict X pn+∗ = Xpn + ∆tU p 3- Compute 4- Predict 5- Predict
Vcn+∗ , ρcn+∗
` R. Loubere (IMT and CNRS)
1 (U n 2
+ U n+1 ) n+1/2
2- compute X n+1 = Xpn + ∆tU p p 3- Compute
n+1/2 = εnc + ∆t f (U p , Pcn ) Pc∗ ≡ αPcn+∗ + (1 − α)Pcn
εcn+∗
1- Compute U n+1 = U np + ∆t f (Pc∗ ), and p
Vcn+1 , ρn+1 c n+1/2
4- Compute εn+1 = εnc + ∆t f (U p c
, Pc∗ )
5- Compute Pcn+1
Add-ons to the compat. stag. Lagr. scheme
ECCOMAS, September 2012
17 / 24
Stability Scheme #1 1- Predict U n+∗ = U np + ∆t f (Pcn ), and p U n+1/2 =
1 (U n 2
2- Predict X n+∗ = Xpn + p
n+1/2
5- Predict
≡
+ U n+1 )
2- compute X n+1 = Xpn + ∆tU p p 3- Compute Vcn+1 , ρn+1 c
4- Predict εcn+∗ = εnc + ∆t f (U p αPcn+∗
1 (U n 2
n+1/2
n+1/2 ∆tU p
3- Compute Vcn+∗ , ρcn+∗ Pc∗
1- Compute U n+1 = U np + ∆t f (Pc∗ ), and p U n+1/2 =
+ U n+∗ )
+ (1 −
, Pcn )
α)Pcn
n+1/2
4- Compute εn+1 = εnc + ∆t f (U p c
, Pc∗ )
5- Compute Pcn+1
Scheme #2 1- Compute U n+1 = U np + ∆t f (Pc∗ ), and p
12- Predict X pn+∗ = Xpn + ∆tU np 3- Compute Vcn+∗ , ρcn+∗ 4- Predict εcn+∗ = εnc + ∆t f (U np , Pcn ) 5- Predict Pc∗ ≡ αPcn+∗ + (1 − α)Pcn
U n+1/2 =
+ U n+1 ) n+1/2
2- compute X n+1 = Xpn + ∆tU p p 3- Compute Vcn+1 , ρn+1 c
n+1/2
4- Compute εn+1 = εnc + ∆t f (U p c 5- Compute
` R. Loubere (IMT and CNRS)
1 (U n 2
, Pc∗ )
Pcn+1
Add-ons to the compat. stag. Lagr. scheme
ECCOMAS, September 2012
18 / 24
Stability von Neumann stability study on 2D wave model
2D wave equation (as a model) du ∂p = , dt ∂x
dv ∂p = , dt ∂y
dp ∂u ∂v = + . dt ∂x ∂ ye
Prelims Rectangular scheme, periodic BCs, staggered placement of variables ; cell centered pi+1/2,j+1/2 and nodal ui,j , vi,j . Mid-edge values are interpolated values pi+ 1 ,j+1 = 12 pi+ 1 ,j+ 3 + pi+ 1 ,j− 1 , and ui+ 1 ,j+1 = 12 ui,j+1 + ui+1,j+1 , λx = ∆t/∆x and 2
2
2
2
2
2
Any variable w defined at two time levels tn+1 > tn on a point or in a cell, we define at an intermediate time n + κ w n+κ = κw n+1 + (1 − κ) w n ,
` R. Loubere (IMT and CNRS)
0 ≤ κ ≤ 1.
Add-ons to the compat. stag. Lagr. scheme
ECCOMAS, September 2012
19 / 24
Stability von Neumann stability study Fully implicit staggered scheme n+α n+1 n+1 n+α n+α n n + λx pn+α − p + λ ui,j = ui,j , v = v p − p , y i,j i,j i− 12 ,j i+ 12 ,j i,j+ 21 i,j− 21 n − v n+β pn+11 1 = pi+ u n+β 1 − u n+β1 + λy v n+β . 1 ,j+ 1 + λx 1 1 i+ 2 ,j+ 2
2
M=
∗
Qx u
0 0 −Qx∗
Qx Qy , 0
0 0 −Qy∗
(Qx p)i,j
=
i+ 1 ,j+ 1 2 2
=
i,j+ 2
i+1,j+ 2
2
1
λx Λ= 0 0
i+ 2 ,j
0 0 . 1
0 λy 0
1
p
i+ 1 ,j+ 1 2 2
2 2
i+ 2 ,j+1
+p
i+ 1 ,j− 1 2 2
−p
i− 1 ,j+ 1 2 2
−p
i− 1 ,j− 1 2 2
ui,j + ui,j+1 − ui+1,j − ui+1,j+1 .
Hence the implicit scheme also writes w n+1 = w n + ΛMΛw α,β . Theorem The fully implicit scheme is stable for any λx,y is α ≥ 2 anb β ≥ 2. ` R. Loubere (IMT and CNRS)
Add-ons to the compat. stag. Lagr. scheme
ECCOMAS, September 2012
20 / 24
Stability von Neumann stability study P/C staggered scheme #1 n+1 ue i,j n+1 ve i,j n+1 pe 1 i+ ,j+ 1 2 2
Predictor step : n n Qx p
=
ui,j + λx
=
n vi,j
=
n p 1 i+ ,j+ 1 2 2
i,j
n + λy Qy p
i,j
n+1
,
ui,j
,
n+1 vi,j
∗ n+β − λx Qx u
i+ 1 ,j+ 1 2 2
∗ n+β −λy Qy v
i+ 1 ,j+ 1 2 2
von Neumann analysis : pn
i+ 1 ,j+ 1 2 2
=
n p 1 i+ ,j+ 1 2 2
i,j
i,j
∗ n+β − λx Qx u
i+ 1 ,j+ 1 2 2
i+ 1 ,j+ 1 2 2
7−→ p0 e
θ(n∆t)+i 2δ (i+ 1 )∆x +2γ (j+ 1 )∆y 2 2 ,
1 − αΦ2x
−αΦx Φy
−αΦx Φy
1 − αΦ2y iΦy
,
=
n vi,j
.
S=
1 − αβ(Φ2x + Φ2y )
,
n+α + λy Qy p
ui,j + λx
∗ n+β −λy Qy v
iΦx
n+1 p 1 i+ ,j+ 1 2 2
Corrector step : n n+α Qx p
=
1 − αβ(Φ2x + Φ2y )
θ complex, δ, γ reals
iΦx 1 − αβ(Φ2x + Φ2y ) iΦy
.
1 − αβ Φ2x + Φ2y
1 + αβ 2 (Φ2x + Φ2y )2 − β Φ2x + Φ2y
.
Setting Φx = 2λx sin ξ cos η and Φy = 2λy sin η cos ξ, we further study the boundness of numerical radius R(S) = supw | hSw, wi |, with hw, wi = 1. ` R. Loubere (IMT and CNRS)
Add-ons to the compat. stag. Lagr. scheme
ECCOMAS, September 2012
21 / 24
Stability von Neumann stability study Theorem The 2D staggered rectangular scheme #1 and #2 are stable if α ≥ 4αβ max λ2x , λ2y ≤ 1 and unstable if α < 21 and β < 12 . Numerical tests
1
1 , 2
β≥
1 2
and
Exact 2D Lagrang code
1 0.8 0.8
CFL
CFL
0.6 0.6
0.4 0.4 Exact "=0.5 0.6 0.75 0.9 1.0 max CFL = 1
0.2
0 0
0.2
0.2
0 0.4
0.6
0.8
1
0
0.2
0.4
0.6
0.8
1
!
!
2D wave equations
2D Euler equations, β = 1/2,
On a 100 × 100 mesh one runs 105 cycles and compute the total kinetic energy K λ (t n ) = 21
P n 2 n 2 ui,j + vi,j for a
given CFL number λ at a given time t n . It must remain at the square of machine precision, about 10−28 ∼ 10−30
` R. Loubere (IMT and CNRS)
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ECCOMAS, September 2012
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Conclusion and Perspectives
Conclusions Compatible staggered Lagrangian scheme is old and venerable but presents some features that need to be pointed out Inconsistency of cell volume definition Particular stability diagram Moot points : subcells are Lagrangian object ? P/C scheme is 2nd order ? Perspectives Moot points : subcells are Lagrangian object ? P/C scheme is 2nd order ? impact of artificial viscosity ?
` R. Loubere (IMT and CNRS)
Add-ons to the compat. stag. Lagr. scheme
ECCOMAS, September 2012
23 / 24
Acknowledgments
THANK YOU ! This research was supported in parts by ANR JCJC “ALE INC(ubator) 3D”.
` R. Loubere (IMT and CNRS)
Add-ons to the compat. stag. Lagr. scheme
ECCOMAS, September 2012
24 / 24
