A nominally second-order cell-centered Lagrangian scheme for simulating elastic-plastic flows on twodimensional unstructured grids
ECCOMAS 2012 Vienna, Austria | Pierre-Henri Maire CEA/CESTA ` \ , B. RebourcetM R. Abgrall] , J. Breil[ , R. Loubere ] INRIA Bordeaux Sud Ouest, France [ UMR CELIA, Universite´ Bordeaux I, France \ CNRS IMT Toulouse, France M CEA-DIF, Bruyeres ` Le Chatel, France SEPTEMBER 10th, 2012
Outline
1
Introduction
2
Governing equations
3
Spatial discretization
4
Numerical results
5
Conclusion and Perspectives
P.-H. Maire | ECCOMAS | SEPTEMBER 10th, 2012 | PAGE 1/37
Motivations and context Cell-centered Lagrangian scheme for elastic-plastic flows Elastic-Perfectly Plastic materials model defined in the seminal paper [Wilkins (MCP, 1964)] and discretized by means of a staggered scheme Extension of EUCCLHYD scheme [Maire (CAF, 2011)] to EPP flows
Finite volume discretization on 2D moving unstructured grids Geometric conservation law compatibility requirement Sub-cell force-based discretization thermodynamically consistent High-order extension by means of the Generalized Riemann Problem (GRP) methodology [Ben-Artzi and Falcovitz (JCP, 1984)]
Review of existing works in the Lagrangian framework ´ (JCP, 2010)] Cell-centered FV for hyperelastic flows [Kluth and Despres, Cell-centered FV [Barlow, (AWE, 2012)], [Sambasivan, (LANL, 2012)] High-order curvilinear FEM [Dobrev et al. (LLNL, 2012)] P.-H. Maire | ECCOMAS | SEPTEMBER 10th, 2012 | PAGE 2/37
Elastic-plastic equations in Lagrangian form Conservation equations d 1 ( ) − ∇ · V = 0, dt ρ dV ρ − ∇ · T = 0, dt dE ρ − ∇ · (TV ) = 0. dt ρ
Notation T = −P Id + S, Cauchy stress tensor 1 D = (∇V + ∇V t ), strain rate tensor 2 1 D0 = D − tr(D)Id , deviatoric part 3 1 W = (∇V − ∇V t ). spin tensor 2
Thermodynamic closure by means of the equation of state P = P(ρ, ε), where, ε = E −
1 | V |2 . 2
Assumptions for plasticity modeling [Gurtin et al., (Mechanics and Thermodynamics of continua,2010)] D = De + Dp , additive decomposition into elastic and plastic strain rates tr(Dp ) = 0, plastic strain rate is deviatoric S : Dp ≥ 0, where, R : Q = tr(Rt Q); dissipation inequality P.-H. Maire | ECCOMAS | SEPTEMBER 10th, 2012 | PAGE 3/37
Elastic-plastic equations in Lagrangian form Constitutive law for the deviatoric stress D S = 2µ(D0 − Dp ), rate form of the Hooke’s law Dt D d S = S + SW − WS, Jaumann stress rate Dt dt The Jaumann stress rate ensures that the constitutive law obeys the principle of material frame independence (objectivity).
Plastic flow rule of von Mises type for perfectly plastic materials r f =| S | −
2 0 Y ≤ 0, 3
plastic boundedness inequality for the yield function
S Dp = χ(Np : D)Np , plastic strain rate; Np = plastic flow direction |S| ( 0 if f < 0 or if f = 0 and (Np : D) ≤ 0, χ= switching parameter 1 if f = 0 and (Np : D) > 0. P.-H. Maire | ECCOMAS | SEPTEMBER 10th, 2012 | PAGE 4/37
Interpretation of the Jaumann stress rate Rotation associated to the spin tensor W The unique tensor solution, Ω(t), to the initial valued problem d Ω = WΩ, Ω(0) = Id , dt satisfies Ωt Ω(t) = Id for all t > 0, thus it is a rotation.
Expression of the Jaumann rate in terms of the rotation Ω d t D d S = S + SW − WS = Ω (Ω SΩ) Ωt . Dt dt dt The Jaumann rate of S is the image by the rotation, Ω, of the material derivative of the pre-image of S by the same rotation.
Comments about stress rate formulation Objective stress rate definition is somewhat arbitrary Objectivity preserving time discretization Occurence of non-conservative terms for multi-dimensional flows P.-H. Maire | ECCOMAS | SEPTEMBER 10th, 2012 | PAGE 5/37
About thermodynamic consistency Time rate of change of specific entropy η Recalling Gibbs relation θdη = dε + Pd( ρ1 ) and employing specific internal energy definition ε = E − 12 | V |2 leads to dη = S : D. dt The Wilkins model does not preserve entropy for smooth elastic flows. ρθ
New definition of internal energy [Gavriluk (JCP), 2007] Introducing elastic energy εe =
1 4µρ (S
: S) and re-defining ε = E − εe − 21 V 2
dη (S : S) d = (µρ). dt 4µ2 ρ dt Furthermore, in the elastic regime, the yield criterion ensures that 2 (S : S) 1 Y0 0≤ ≤ 1 for most solids. 4µ2 ρ 6ρ µ For aluminium, µ = 27.6 109 Pa and Y 0 = 300 106 Pa. ρθ
P.-H. Maire | ECCOMAS | SEPTEMBER 10th, 2012 | PAGE 6/37
About thermodynamic consistency Comments In the present work, we keep the original form of the Wilkins model Namely, internal energy is defined as ε = E − 12 V 2 In spite of its thermodynamic inconsistency, Wilkins model is largely used in engineering sciences it allows to incorporate easily plasticity effects it is accurate in the domain of solid dynamics for metals
Our main motivation is to provide an alternative discretization for this model based on cell-centered FV methods
Dissipation inequality satisfied by the Wilkins model Rate of internal energy reads ρ dtd ε − T : D = 0 Recalling the additive decomposition D = De + Dp and the dissipation inequality S : Dp ≥ 0 yields the following inequality ρ
d ε − T : De ≥ 0. dt P.-H. Maire | ECCOMAS | SEPTEMBER 10th, 2012 | PAGE 7/37
Control volume formulation Conservation equations over the moving domain ω(t) d dt
Z
d dt
Z
d dt
Z
dv −
ω(t)
Z
V · n ds = 0,
∂ω(t)
ρV dv −
Z
ρE dv −
Z
Tn ds = 0,
ω(t)
∂ω(t)
ω(t)
TV · n ds = 0.
∂ω(t)
Constitutive law d dt
Z
Z ρS dv =
ω(t)
ρ[2µ(D0 − Dp ) − (SW − WS)] dv ,
ω(t)
thanks to the Reynolds formula, i.e.,
R ω(t)
ρ dS dt dv =
d dt ( ω(t)
R
ρS dv ).
Trajectory equation d x = V (x(t), t), dt
x(0) = X . P.-H. Maire | ECCOMAS | SEPTEMBER 10th, 2012 | PAGE 8/37
Finite Volume discretization Moving cell ωc (t)
Semi-discrete equations
n
p ( ρ1c , V c, Ec, Sc)
∂ωc
Vp
Z d 1 mc ( ) − V · n ds = 0, dt ρc ∂ωc (t) Z d Tn ds = 0, mc V c − dt ∂ωc (t) Z d mc Ec − TV · n ds = 0, dt ∂ωc (t) dSc = 2µc (D0,c − Dpc ) − (Sc Wc − Wc Sc ). dt
ωc
Trajectory equation d x p = V p (x p (t), t), dt
x p (0) = X p
Geometric conservation law compatibility requirement Knowing that mc = ρc (t)vc (t), spatial discretization of the first equation must be compatible with the trajectory equation. P.-H. Maire | ECCOMAS | SEPTEMBER 10th, 2012 | PAGE 9/37
GCL compatibility Cell triangulation p+
npp+
Volume computation vc (t) =
lpcnpc
X
(x p × x p+ ) · e z
p∈P(c)
lpp+
Vp
Volume rate
p ωc
np−p
lp−p
p−
O
X ∂vc d d vc = · x p, dt ∂x p dt p∈P(c) X = lpc npc · V p p∈P(c)
Mimetic discretization of the divergence operator DIVc (V ) =
1 d 1 X vc = lpc npc · V p . vc dt vc p∈P(c)
Geometric identity:
P
p∈P(c) lpc npc
=0 P.-H. Maire | ECCOMAS | SEPTEMBER 10th, 2012 | PAGE 10/37
Mimetic discretization of D and W Discrete velocity tensor gradient over a polygonal cell ωc 1 GRADc (V ) = vc
Z
1 ∇V dv = v c ωc
Z
V ⊗ n ds =
ωc
1 X lpc V p ⊗ npc . vc p∈P(c)
Discrete strain rate and spin tensors Dc =
1 [GRADc (V ) + GRADtc (V )], 2
Wc =
1 [GRADc (V ) − GRADtc (V )]. 2
Comments Mimetic discretization exact for linear velocity field It captures exactly rigid body motion tr(Dc ) = DIVc (V ) and tr(Wc ) = 0.
Discrete tensor identity vc [DIVc (QA)−Ac ·DIVc (Qt )−Qtc :GRADc (A)]=
P
p∈P(c) lpc [(Qpc −Qc )(Ap −Ac )]·npc ,
where A is a vector and Q a tensor. P.-H. Maire | ECCOMAS | SEPTEMBER 10th, 2012 | PAGE 11/37
Sub-cell force-based discretization Sub-cell definition
Cell partition into sub-cells
p+
[
ωc = 11111111111 00000000000 00000000000 11111111111 00000000000 11111111111 p 00000000000 11111111111 00000000000 11111111111 00000000000 11111111111 00000000000 11111111111 ωpc 00000000000 11111111111 00000000000 11111111111 00000000000 11111111111 00000000000 11111111111 00000000000 11111111111 00000000000 11111111111
ωpc ,
p∈P(c)
∂ωc =
[
∂ωpc ∩ ∂ωc .
p∈P(c)
Sub-cell force definition
ωc
F pc = −
p−
Z Tn ds ∂ωpc ∩∂ωc
Sub-cell contribution to total energy flux −
Z ∂ωpc ∩∂ωc
TV · n ds ≈ −
!
Z Tn ds
· V p = F pc · V p .
∂ωpc ∩∂ωc
P.-H. Maire | ECCOMAS | SEPTEMBER 10th, 2012 | PAGE 12/37
Sub-cell force-based discretization Semi-discrete equations
Notation lpcnpc F
( ρ1c , V c, ec, Sc)
pc 11111111111 00000000000 Vp 00000000000 11111111111 00000000000 11111111111 00000000000 11111111111 p 00000000000 11111111111 00000000000 11111111111 ωpc 00000000000 11111111111 00000000000 11111111111 00000000000 11111111111 00000000000 11111111111 00000000000 11111111111 00000000000 11111111111
mc
X d 1 ( )− lpc npc · V p = 0, dt ρc p∈P(c)
X d F pc = 0, mc V c − dt p∈P(c)
X d mc Ec − F pc · V p = 0, dt p∈P(c)
ωc
dSc = 2µc (D0,c − Dpc ) − (Sc Wc − Wc Sc ). dt
Trajectory equation d x p = V p, dt
x p (0) = X p .
It remains to determine V p and F pc invoking thermodynamic consistency and total energy conservation P.-H. Maire | ECCOMAS | SEPTEMBER 10th, 2012 | PAGE 13/37
Thermodynamic consistency Dissipation inequality reads d dt
Z ωc
ρε dv −
Z
T : De ≥ 0.
ωc
Semi-discrete expression of the rate of internal energy mc
X d εc − vc (Tc : Dec ) = vc (Sc : Dpc ) + (F pc − lpc Tc npc ) · (V p − V c ). dt p∈P(c)
Sufficient condition to satisfy the dissipation inequality Recalling that the plastic work rate, (Sc : Dpc ) ≥ 0, the dissipation inequality holds provided that the sub-cell force expresses as F pc = lpc Tc npc + Mpc (V p − V c ), where Mpc is a 2 × 2 matrix characterized by Mpc is positive semidefinite, i.e. Mpc V · V ≥ 0,
∀ V ∈ R2 .
[Mpc ] = length × mass density × velocity P.-H. Maire | ECCOMAS | SEPTEMBER 10th, 2012 | PAGE 14/37
Thermodynamic consistency Semi-discrete expression of the rate of entropy mc θ c
X d ηc = vc (Sc : Dc ) + Mpc (V p − V c ) · (V p − V c ). dt p∈P(c)
Semi-discrete entropy inequality Recalling that, Mpc is positive semidefinite, leads to the inequality d mc θc ηc ≥ vc (Sc : Dc ). dt
Hydrodynamic limit: S = 0 and T = −PId Sub-cell force boils down to F pc = −lpc Pc npc + Mpc (V p − V c ). Dissipation inequality collapses to the classical entropy inequality d d mc θc ηc = mc εc + Pc vc tr(Dc ) ≥ 0. dt dt P.-H. Maire | ECCOMAS | SEPTEMBER 10th, 2012 | PAGE 15/37
Conservation principles Conservation of total energy over the domain D X c
where
F ?p
d mc Ec = dt
Z
X
TV · n ds =
∂D
F ?p · V p ,
p∈∂D
is a prescribed corner force acting onto boundary point p.
Substituting specific total energy equation yields X X c
p∈P(c)
F pc · V p =
X
F ?p · V p .
p∈∂D
Interchanging the order of the double sum X ? X X ( F pc ) · V p = F p · V p, p
c∈C(p)
p∈∂D
where C(p) is the set of cells surrounding point p. P.-H. Maire | ECCOMAS | SEPTEMBER 10th, 2012 | PAGE 16/37
Conservation principles Total energy balance amounts to the variational formulation ∀V p ∈ R2 ,
X
(
X
F pc ) · V p +
p∈D o c∈C(p)
X
(
X
F pc ) · V p =
p∈∂D c∈C(p)
X
F ?p · V p ,
p∈∂D
where D0 is the interior of the domain D.
Total energy conservation holds if and only if ∀ p ∈ Do ,
X
F pc = 0,
inner points
F pc = F ?p .
outer points
c∈C(p)
∀ p ∈ ∂D,
X c∈C(p)
These conditions allow us to show that 1
Momentum is conserved
2
Nodal velocity satisfies a vectorial equation since F pc = lpc Tc npc + Mpc (V p − V c ) P.-H. Maire | ECCOMAS | SEPTEMBER 10th, 2012 | PAGE 17/37
Node-centered solver Nodal velocity V p satisfies the 2 × 2 system ∀ p ∈ Do ,
Mp V p =
X
(−lpc Tc npc + Mpc V c ) ,
inner points
(−lpc Tc npc + Mpc V c ) − F ?p .
outer points
c∈C(p)
∀ p ∈ ∂D,
Mp V p =
X c∈C(p)
It admits always a unique solution provided that Mp =
P
c∈C(p)
Mpc > 0.
Main features of the proposed FV cell-centered scheme General sub-cell force formalism based on mimitic discretization GCL compatibility Thermodynamic consistency Total energy and momentum conservation Dissipation characterized by the corner matrix Mpc P.-H. Maire | ECCOMAS | SEPTEMBER 10th, 2012 | PAGE 18/37
Node-centered solver Plane-wave analysis of the Wilkins model Hyperbolicity is ensured granted that In the limit
0
Y0 µ
1
1, the acoustic tensor of the Wilkins model collapses to
Y µ
H(n) = aL2 (n ⊗ n) + aT2 [Id − (n ⊗ n)] ,
where | n |= 1,
where aL and aT are the longitudinal and transverse elastic wave speeds 4µ µ aL2 = a2 + , aT2 = . 3ρ ρ
Notation p+
Corner matrix and force
n+ pc + lpc
1
p ωc
T+ pc T− pc
(ρc, V c, Tc)
lpcnpc Vp − lpc
n− pc
± ± 2 M± pc = ρc lpc [H(npc )] , + Mpc = M− pc + Mpc , − − − + + + F pc = lpc Tpc npc + lpc Tpc npc 1
± ± 2 (T± pc − Tc )npc = ρc [H(npc )] (V p − V c ) p−
P.-H. Maire | ECCOMAS | SEPTEMBER 10th, 2012 | PAGE 19/37
Summary of the semi-discrete scheme Equations for ( ρ1c , V c , Ec ) X d 1 mc ( ) − lpc npc · V p = 0, dt ρc p∈P(c)
mc
X d Vc − F pc = 0, dt p∈P(c)
X d mc Ec − F pc · V p = 0. dt
Force and nodal velocity F pc = lpc Tc npc + Mpc (V p − V c ), X Mp V p = (−lpc Tc npc + Mpc V c ) , c∈C(p)
Tc = −Pc Id + Sc , Pc = P(ρc , εc ), εc = Ec −
p∈P(c)
1 2 V . 2 c
Constitutive law d Sc = 2µc (D0,c − Dpc ) − (Sc Wc − Wc Sc ), dt r 2 0 Y ≤ 0, fc =| Sc | − 3 Sc Dpc = χc (Npc : Dc )Npc , where Npc = . | Sc | P.-H. Maire | ECCOMAS | SEPTEMBER 10th, 2012 | PAGE 20/37
High-order one-step time discretization Formal second-order time discretization mc
1 ρn+1 c
1 − n ρc
n+ 21
1
X
− ∆t
(lpc npc )n+ 2 · V p
= 0,
p∈P(c)
X n+ 1 mc V n+1 − V nc − ∆t F pc 2 = 0, c p∈P(c) n+ 12
X
mc Ecn+1 − Ec − ∆t n
F pc
n+ 12
· Vp
= 0.
p∈P(c)
Trajectory equation n+ 12
x n+1 = x np + ∆tV p p
,
x 0p = X p
Time-centered fluxes by Taylor expansion n+ 1 Vp 2
=
V np
∆t + 2
d Vp dt
n ,
n+ 1 F pc 2
=
F npc
∆t + 2
d F pc dt
n .
P.-H. Maire | ECCOMAS | SEPTEMBER 10th, 2012 | PAGE 21/37
Total energy conservation for HO scheme Balance of total energy over the domain D X
mc (Ecn+1 − Ecn ) =
Z
c ?,n+ 21
where F p
t n+1
Z
tn
∂D
?,n+ 12
X
TV · n ds dt =
Fp
n+ 12
· Vp
,
p∈∂D
is the time-centered evaluation of boundary force at p.
Total energy balance amounts to the variational formulation X
(
X
n+ 1
n+ 12
F pc 2 ) · V p
+
p∈D o c∈C(p)
X
(
X
n+ 1
n+ 12
F pc 2 ) · V p
p∈∂D c∈C(p)
=
X
?,n+ 12
Fp
n+ 21
· Vp
.
p∈∂D
Total energy conservation holds if and only if ∀ p ∈ Do ,
X
n+ 12
F pc
= 0,
inner points
c∈C(p)
∀ p ∈ ∂D,
X
n+ 12
F pc
?,n+ 12
= Fp
.
outer points
c∈C(p) P.-H. Maire | ECCOMAS | SEPTEMBER 10th, 2012 | PAGE 22/37
Total energy conservation for HO scheme Recalling that time-centered sub-cell force writes n+ 1 F pc 2
F npc
=
∆t + 2
dF pc dt
n .
Total energy conservation at discrete level provided that ∀ p ∈ Do ,
X c∈C(p)
∀ p ∈ ∂D,
X c∈C(p)
X dF pc n = 0, dt c∈C(p) X dF pc n dF p ?,n ?,n = Fp , = . dt dt
F npc = 0,
F npc
c∈C(p)
Knowing that F pc = lpc Tc npc + Mpc (V p − V c ), this leads to 1 2
One node-centered solver for computing V np n dV One node-centered solver for computing dt p P.-H. Maire | ECCOMAS | SEPTEMBER 10th, 2012 | PAGE 23/37
Constitutive law time discretization Elastic prediction to satisfy incremental objectivity n+ 12
Sn+1,tr − Q[Wc c
n+ 1
](∆t)Snc Qt [Wc 2 ](∆t) = n+ 1 ∆t n+ 1 n+ 1 ∆t 2µc ∆tQ[Wc 2 ]( )D0,c 2 Qt [Wc 2 ]( ). 2 2 Here, the rotation, Q, is given in terms of the Pade´ approximant of exp(tA) −1 t t Q[A](t) = Id − A Id + A , 2 2 where, A is a skew symmetric tensor.
Plastic correction by means of the Wilkins radial return algorithm Sn+1 c
Sn+1,tr c = q2 0 3Y
if f (Sn+1,tr )≤0 c Sn+1,tr c |Sn+1,tr | c
if f (Sn+1,tr ) > 0, c
q where the yield function, f , reads f (S) =| S | − 23 Y 0 . P.-H. Maire | ECCOMAS | SEPTEMBER 10th, 2012 | PAGE 24/37
Numerical results Mie-Gruneisen equation of state P(ρ, ε) = ρ0 a02 f (η) + ρ0 Γ0 ε, f (η) = Ushock
(η − 1)[η − 12 Γ0 (η − 1)] ρ , where η = [η − s(η − 1)]2 ρ0 = a0 + sUpart .
Sound speed a2 =
∂P ∂ρ
+ ε
P ρ2
∂P ∂ε
, ρ
Γ0 P , ρ0 η 2 η + (s − Γ0 )(η − 1) f 0 (η) = . [η − s(η − 1)]3 2 aMG = a02 f 0 (η) +
P.-H. Maire | ECCOMAS | SEPTEMBER 10th, 2012 | PAGE 25/37
EPP piston problem [Udaykumar (JCP, 2003)] Data for copper ρ0 = 8930 kg/m3 , a0 = 3940 m/s, Γ0 = 2, s = 1.49, µ = 45 109 Pa, Y0 = 90 106 Pa.
Set up Computational domain: (x, y ) ∈ [0, 1] × [0, 0.1] Initial data: (P0 , V 0 ) = (105 Pa, 0) Piston velocity at left boundary: Vpiston = 20 m/s Stopping time: tend = 150 µs
Comparison with analytical solution : two-wave structure Leading shock: elastic precursor, Second shock: plastic shock. P.-H. Maire | ECCOMAS | SEPTEMBER 10th, 2012 | PAGE 26/37
EPP piston problem [Udaykumar (JCP, 2003)] Convergence analysis at stopping time for density 8975 analytical 100 200 400
8970
8965
8960
8955
8950
8945
8940
8935
8930 0
0.1
0.2
0.3
0.4
0.5 x
0.6
0.7
0.8
0.9
1
P.-H. Maire | ECCOMAS | SEPTEMBER 10th, 2012 | PAGE 27/37
Flyer Al plates [Wilkins (MCP, 1964)] Data for aluminium ρ0 = 2785 kg/m3 , a0 = 5328 m/s, Γ0 = 2, s = 1.338, µ = 27.6 109 Pa, Y0 = 300 106 Pa.
Set up Computational domain: (x, y ) ∈ [0, 50 10−3 ] × [0, 5 10−3 ] The two plates: x ∈ [0, 5 10−3 ] and x ∈ [5 10−3 , 50 10−3 ] Initial data: (P0 , V 0 ) = (10−6 Pa, 800e x ), for x ∈ [0, 5 10−3 ] (P0 , V 0 ) = (10−6 Pa, 0), for x ∈ [5 10−3 , 50 10−3 ] Pressure boundary condition at x = 0: P ? = P0 Stopping time: tend = 0.5 µs P.-H. Maire | ECCOMAS | SEPTEMBER 10th, 2012 | PAGE 28/37
Flyer Al plates [Wilkins (MCP, 1964)] Density at tend
Deviatoric stress at tend 2.5e+08
3000
deviator
density 2e+08 1.5e+08
2950
1e+08
2900
5e+07
2850
-5e+07
2800
-1.5e+08
0
-1e+08
-2e+08 -2.5e+08
2750 0
0.005
0.01
0.015
0.02
0.025 x
0.03
0.035
0.04
0.045
0
0.05
0.005
0.01
0.015
0.02
0.025 x
0.03
0.035
0.04
0.045
0.05
Comments nx = 500 uniformly spaced cells Elastic precursor and plastic shock Plastic rarefaction and elastic rarefaction P.-H. Maire | ECCOMAS | SEPTEMBER 10th, 2012 | PAGE 29/37
Be shell collapse [Howell and Ball, JCP(2002)] Data for beryllium ρ0 = 1845 kg/m3 , a0 = 12870 m/s, Γ0 = 2, s = 1.124, µ = 151 109 Pa, Y0 = 330 106 Pa.
Set up Computational domain: (r , φ) ∈ [20 10−3 , 80 10−3 ] × [0, 2π] Initial data: (P0 , V 0 ) = (10−6 Pa, 417.1 Rr i e r ) Pressure boundary condition: P ? = P0 Stopping time: tend = 130 µs
Comparison with analytical solution 1D analytical solution under the assumption of incompressiblity Initial kinetic energy is dissipated through plastic work Shell comes to rest at a final stopping radius P.-H. Maire | ECCOMAS | SEPTEMBER 10th, 2012 | PAGE 30/37
Be shell collapse [Howell and Ball, JCP(2002)] Energy balance w.r.t time 4e+05
energy (J)
3e+05 internal energy 10x8 kinetic energy 10x8 total energy 10x8 internal energy 20x16 kinetic energy 20x16 total energy 20x16 internal energy 40x32 kinetic energy 40x32 total energy 40x32
2e+05
1e+05
0
0
0.0001
5e-05 time (s)
P.-H. Maire | ECCOMAS | SEPTEMBER 10th, 2012 | PAGE 31/37
Be shell collapse [Howell and Ball, JCP(2002)] Inner and outer stopping radius 0.1
inner radius 10x8 outer radius 10x8 inner radius 20x16 outer radius 20x16 inner radius 40x32 outer radius 40x32 analytical inner radius analytical outer radius
0.09
radius (m)
0.08
0.07
0.06
0.05
0.04
0
2e-05
4e-05
6e-05 time (s)
8e-05
0.0001
0.00012
P.-H. Maire | ECCOMAS | SEPTEMBER 10th, 2012 | PAGE 32/37
Be shell collapse [Howell and Ball, JCP(2002)] Final grid
Initial grid 0.1 0.09
0.07
0.08
0.06
0.07
0.05 0.06
0.04
0.05 0.04
0.03
0.03
0.02 0.02
0.01 0.01 0 0
0.02
0.04
0.06
0.08
0.1
0 0
0.01
0.02
0.03
0.04
0.05
0.06
0.07
Comments Symmetry preservation Thickening of the shell during its collapse due to radial convergence P.-H. Maire | ECCOMAS | SEPTEMBER 10th, 2012 | PAGE 33/37
2D projectile impact [Kluth (JCP, 2010)] Set up Computational domain: (x, y ) ∈ [0, 5] × [−0.5, 0.5] Initial data: (P0 , V 0 ) = (10−6 Pa, −150 m/s e x ) Uniform Cartesian mesh: 100 × 20 Wall boudary condition at x = 0 Pressure boundary condition: P ? = P0 Stopping time: tend = 5 ms
Comments Counterpart of the Taylor problem in planar geometry This problem has no analytical solution neither experimental results Good test to assess the robustness of the scheme on various unstructured grids P.-H. Maire | ECCOMAS | SEPTEMBER 10th, 2012 | PAGE 34/37
2D projectile impact [Kluth (JCP, 2010)] Structured grid at final time: 200 × 40 uniformly spaced cells 0.5 0 −0.5 0
0.5
1
1.5
2
2.5
3
3.5
4
4.5
Unstructured grid at final time: 2000 polygonal cells 0.5 0 −0.5 0
0.5
1
1.5
2
2.5
3
3.5
4
4.5
P.-H. Maire | ECCOMAS | SEPTEMBER 10th, 2012 | PAGE 35/37
2D projectile impact [Kluth (JCP, 2010)] Length versus time
Energy balance
5 Al target length 50x10 Al target length 100x20 Al target length 200x40 Al target length unstructured
1.5e+08
1e+08
5e+07
length (m)
energy (J)
4.9 internal energy 50x10 kinetic energy 50x10 internal energy 100x20 kinetic energy 100x20 internal energy 200x40 kinetic energy 200x40 internal energy unstructured kinetic energy unstructured total energy
4.8
4.7
0
0
0.001
0.002
0.003 time (s)
0.004
0.005
4.6
0
0.001
0.002
0.003
0.004
0.005
time (s)
Comments Convergence analysis Kinetic energy entirely converted into internal energy through plastic dissipation The length of the rod tends toward the limit Lfinal ∼ 4.62 m P.-H. Maire | ECCOMAS | SEPTEMBER 10th, 2012 | PAGE 36/37
Conclusion and Perspectives Cell-centered Lagrangian scheme for EPP flows EUCCLHYD scheme (Explicit Unstructured Cell-Centered LAgrangian HYDrodynamics) GCL compatibility Sub-cell force-based discretization Thermodynamic consistency Total energy and momentum conservation Node-centered solver
GRP high-order extension ` and B. Rebourcet, A nominally P.-H. Maire, R. Abgrall, J. Breil, R. Loubere second-order cell-centered Lagrangian scheme for simulating elastic-plastic flows on two-dimensional unstructured grids; INRIA Research Report 7975, May 2012.
Perspectives Extension of plasticity flow rule (hardening) Extension to axisymmetric geometry Extension to a new model which ensures consistency with the second law of thermodynamics [P. Le Tallec (2011)] P.-H. Maire | ECCOMAS | SEPTEMBER 10th, 2012 | PAGE 37/37
