Hydrodynamic applications using ReALE method T.Harribey1 ,
J.Breil 1 , 1
M. Shashkov3
CELIA, Universite´ Bordeaux 1, FRANCE 2
3
P.H. Maire2 ,
C.E.A. CESTA, FRANCE
Los Alamos National Laboratory, United States
[email protected] [email protected]
10th September 2012
Harribey-Breil (CELIA)
10th September 2012
1 / 19
Introduction and motivations Motivations We are interested in the simulation of shock waves interaction in the context of Lagrangian hydrodynamics
Difficulty Pure Lagrangian computation have strong limitation for large deformations ALE methods are a solution but still have limitations especially when strong vorticity occurs and when complex geometries are considered
Solution The use of a Reconnection based ALE (ReALE) method allow to keep Lagrangian feature in an ALE framework We propose here a new remapping method (flux approach) dedicated to ReALE
Harribey-Breil (CELIA)
10th September 2012
2 / 19
ReALE Flowchart
I NITIALIZATION Initial distribution of physical variables over the initial grid
R EMAPPING t n+1 = t n + ∆t
Remap from the Lagrangian mesh onto the reconnected mesh
L AGRANGIAN P HASE Solving Lagrangian equations Node-centered approximate Riemann solver Update Lagrangian grid and variables
R ECONNECTION BASED R EZONING Definition of new generator location Creation of the reconnected polygonal mesh
Harribey-Breil (CELIA)
10th September 2012
3 / 19
Lagrangian phase Lagrangian Euler equations For a volume V moving at the velocity U, conservative equations write: Z d ρdV = 0, dt ZV Z d dV − ∇ · UdV = 0, dt ZV VZ d ρUdV + ∇PdV = 0, dt ZV ZV d ρEdV + ∇ · (PU)dV = 0, dt V V
Thermodynamical closures P = P(ρ, ε) and ε = E − |U|2 /2
Harribey-Breil (CELIA)
Local kinematic equation d X=U dt
10th September 2012
4 / 19
Reconnection-based rezone strategy Voronoi Tesselation Set of generators: Gc = (xc , yc ) Voronoi cell: Ωc = {X = (x, y) : |X − Gc | < |X − Gj |, for all j 6= c} Y
Gcn+1 is calcuted using a convex combination between the new Lagrangian-like position of the Gcn+1,lag and the centroid Xn+1 c Lagrangian cell at time t n+1 : Gcn+1 = Gn+1,lag + ωc Xn+1 − Gn+1,lag c c c
• • ◦
◦
•
• ◦
eY
•
Ωc
◦
◦
Gc ◦
◦
•
•
ωc ∈ [0; 1] is constructed using invariants of the right Cauchy Green strain tensor.
•
eX
X
` , P.H.M AIRE , M.S HASHKOV, J.B REIL , S. G ALERA, ReALE: A reconnection-based R. L OUB ERE arbitrary-LagrangianEulerian method, JCP, 229, 47244761 (2010). Harribey-Breil (CELIA)
10th September 2012
5 / 19
Remapping phase Main goal of remapping Remap the value of ψ = (ρ, ρU, ρE) on the new rezoned mesh to get ψe Usually a cell-intersection method is used, here we have delopped a method based on a flux approach
Swept-intersection-based method p1 c+ f p2
•
+ cf
• •
Qs e f →e f f
•
• e p1
•
• ep2
•
c+ p2
• •
p1
•
f
Qs f →e p
c
• ep
•
•
•
face to face
e c
•
•
•
•
• ep2
•
•
• c
e c
c
• •
•
p
•
face to point
•
e f Qs p1 p→e f e
e c
•
• •
+ cf
•
• •
point to face
M. K UCHARIK , M. S HASHKOV, Extension of efficient, swept-integration-based conservative remapping method for meshes with changing connectivity, Int. J. Numer. Meth. Fluids, 56, 1359-1365 (2008). Harribey-Breil (CELIA)
10th September 2012
6 / 19
Remapping phase Face to face projection Given Ψ a physical variable defined on the domain {c}, its expression in cell e c is: Z X s 1 Ψec = ΨdX where Vec = Vc + Qf →ef Vec ec
• c+ f
p2
p1
•
+ cf
•
• e p1
f ∈c
Qs f →e f
•
e f
•
• ep2
•
If e c + denotes the cell neighbor with e c:
e c
c
•
•
Z XZ 1 ΨdX + Ψec = Vec c Qs
•
f ∈c
•
Ψec + Qs
f →e f
= {p1, e p1 , e p2 , p2 }
(old) Lagrangian mesh (new) Reconnected mesh
ΨdX
f →e f
Z XZ 1 = ΨdX − Vec + c+ Qs f ∈c
ΨdX
f →e f
Upwind approximation: Z Qs e f →f
Harribey-Breil (CELIA)
R Q s Ψc + (X)dX if Q s e ≥ 0 f →f ΨdX = R f →ef Q s Ψc (X)dX otherwise f →e f
10th September 2012
7 / 19
Voronoi-like reconnection Two projection steps
Illustration •• • •
c4
• • e c4
p2
•• •
•
•
• ef f
e p1 e c1 c1
• •
•
••
• •
• •
e c3 c3 e p2
e c c44 •
p2
• •
p1 e c2
•
c2
•
c1 e c1
•
•
•
e p1
e c4 c4 •
•
e p2
f s Qf →ep
Qfs→ep 1 e c2
p2 • Qs
e c3
c1
2
•
c3
p2 →e f Qs e p1 →f
e c1
p1
c2
•
e p1
e c2
•
e p2 e c3
e f •
c3 p1
c2
Flux estimation If Qfs→ep1 > Qfs→ep2 then cref = c1 , else cref = c3 . Then we have: Ψce1
Ψce2
Z 1 = Ψc1 + Ψcref (X)dX Vce1 Qs e f →p1 Z 1 = Ψc2 − Ψcref (X)dX Vce2 Qs p1 →e f
Harribey-Breil (CELIA)
Ψce3
Ψce4
Z 1 = Ψ c3 − Ψcref (X)dX Vce3 Qs e f →p2 Z 1 = Ψ c4 + Ψcref (X)dX Vce4 Qs p2 →e f
10th September 2012
8 / 19
Numerical test: periodic vortex Description We consider a [0, 1]2 domain in which we initialize the density field using Ψ(x, y ) = 2 + cos(2πx) sin(2πy ). We impose that the generators move under the nonlinear unsteady velocity field defined by the following stream function (T = 3): 1 πt F= sin2 (πx) sin2 (πy ) cos 4π T Grid 82 162 322 642
Lexact 1 1.07 × 10−1 2.47 × 10−2 5.15 × 10−3 1.10 × 10−3
Lswept 1 1.07 × 10−1 2.48 × 10−2 5.18 × 10−3 1.11 × 10−3
Order 2.11 2.26 2.22
t exact (s) 7.7 × 100 5.3 × 101 4.8 × 102 4.6 × 103
t swept (s) 3.5 × 100 1.3 × 101 9.1 × 101 8.3 × 102
Comparison of relative L1 error and CPU time t between exact intersection and our flux method Harribey-Breil (CELIA)
10th September 2012
9 / 19
Schardin test case: Presentation Description Schardin test case deals with shock wave diffraction on a triangular body. This shock generates curved reflected and diffracted waves.
Configuration xp,0
xp,f
xs,i
0.05
Ms = 1.3 θ
P0 = 0.05 MPa
0.02
ρ0 = 1.2 kg.m−3
0 0.01921
−0.05 −0.059
γ = 1.4 Results area
0.01
0.05
θ = 55◦
0.15
Cell number: 23892 xp,0 : xp,f : xs,i :
the initial lagrangian position of the piston the final position of the piston the position of the incident shock at t = 0
tpiston = 343µs tinteraction = 305µs
H; S CHARDIN, High frequency cinematography in the shock tube, J. Phot. Sci., 5, 17-19 (1957). Harribey-Breil (CELIA)
10th September 2012
10 / 19
Schardin test case: Numerical simulation
Harribey-Breil (CELIA)
10th September 2012
11 / 19
Schardin test case: Experimental validation t = 40 µs
t = 150 µs
t = 270 µs
Comparison between experimental shadowgraphs and computed isopycnics at different times
Harribey-Breil (CELIA)
10th September 2012
12 / 19
Falcovitz test case: Presentation Description Falcovitz test case deals with the interaction between a planar shock wave and a square cavity which create many reflected shock waves xp,f
xp,0
xs,i
Ms = 1.3
0.05
P0 = 98285 Pa Results area 0.
ρ0 = 1.2 kg.m−3 γ = 1.4
−0.05 −0.2443
xp,0 : xp,f : xs,i :
−0.1
−0.05
0
0.05
the initial lagrangian position of the piston the final position of the piston the position of the incident shock at t = 0 with xs,i = −0.0055 m
0.16
Cell number: 15748 tpiston = 542µs tinteraction = 420µs
O. I GRA , J. FALCOVITZ , H. R EICHENBACH , W. H EILIG . Experimental and numerical study of the interaction between a planar shock wave and square cavity, J. Fluid. Mech., 313,105-130 (1996). Harribey-Breil (CELIA)
10th September 2012
13 / 19
Falcovitz test case: Numerical simulation
Harribey-Breil (CELIA)
10th September 2012
14 / 19
t = 280 µs
t = 140 µs
t = 40 µs
Falcovitz test case: Experimental validation
Comparison between experimental shadowgraphs and computed isopycnics at different times Harribey-Breil (CELIA)
10th September 2012
15 / 19
Double wedge test case: Presentation Description The shock is regularly reflected from the first wedge and the incited-reflected shock pattern is reflected from the second wedge. xp,f Results area
xp,0
Wedge angle: θ = 55◦
xs,i
0.141
Ms = 1.488 P0 = 98500 Pa ρ0 = 1.2 kg.m−3 Material: γ = 1.4
θ 0. −0.475
xp,0 : xp,f : xs,i :
−0.01 0.
0.0536
the initial lagrangian position of the piston the final position of the piston the position of the incident shock at t = 0
Harribey-Breil (CELIA)
0.105
Mesh: 29355 cells tpiston = 1.08ms tinteraction = 216µs
10th September 2012
16 / 19
Double wedge test case: Simulation & Validation
Harribey-Breil (CELIA)
10th September 2012
17 / 19
Conclusions and perspectives
Summary An accurate reconnection-based ALE technique Validation of the new swept-intersection-method dedicated to the remapping on several hydrodynamic simulations 5 times faster than the classical cell-intersection method with the same accuracy
Future works Adapt this method to multi-material simulations (hybrid remapping) Add/remove generators dynamically (AMR-like) Validate this approach on laser applications Using ReALE in Combination with Lagrangian and ALE Methods
Harribey-Breil (CELIA)
10th September 2012
18 / 19
Using ReALE in Combination with Lagrangian and ALE Methods ReALE combine with Lagrangian and ALE Methods
Predetermined regions with Lagrangian boundaries with ReALE inside Dynamically changing regions with Lagrangian boundaries with ReALE inside ALE buffer region Harribey-Breil (CELIA)
10th September 2012
19 / 19
