A Cell-centered Finite Volume Lagrangian method for ...

A Cell-centered Finite Volume Lagrangian method for solving elastic-plastic flows in two-dimensional axisymmetric geometry

MULTIMAT Conference, Wurzburg, Germany | Pierre-Henri Maire ¨ Isabelle Bertron] , Bernardo RebourcetM ] CEA-CESTA, Le Barp, France M CEA-DIF, Bruyeres ` Le Chatel, France SEPTEMBER 7th to 11th , 2015

Outline 1

Introduction

2

Governing equations

3

Control volume discretization

4

Area weighted discretization Sub-cell force-based discretization Node-centered solver

5

Numerical results Blake problem Verney problem Taylor problem

6

Conclusion and Perspectives

I. Bertron, P.-H. Maire and B. Rebourcet CEA | MULTIMAT 2015 | SEPTEMBER 7th to 11th , 2015 | PAGE 1/32

Motivations Cell-centered Lagrangian method for solid dynamics Hypoelastic model introduced in the seminal work [Wilkins (MCP 1964)] Extension of the EUCCLHYD scheme to cylindrical geometry Area Weighted discretization to preserve rotational symmetry

Main features of EUCCLHYD scheme [Maire (JCP 2009)] Finite Volume (FV) discretization on moving grid Geometric conservation law (GCL) compatibility Total energy conservation and thermodynamic consistency Godunov-type node-centered solver for numerical fluxes computation

Works dedicated to hypoelasticity in cylindrical geometry Cell-centered FV [Sambasivan et al. (LA-UR-12-23177) and (JCP 2013)] Cell-centered FV [Burton et al. (LA-UR-13-23155)] Cell-centered FV [Barlow (Multimat 2013)] High-order curvilinear FEM [Dobrev et al. (JCP 2014)] I. Bertron, P.-H. Maire and B. Rebourcet CEA | MULTIMAT 2015 | SEPTEMBER 7th to 11th , 2015 | PAGE 2/32

Wilkins hypoelastic-plastic model Conservation laws d 1 ρ ( ) − ∇ · V = 0, dt ρ dV ρ − ∇ · T = 0, dt dE ρ − ∇ · (TV ) = 0, dt where E = ε + 21 V 2 .

Constitutive law T = −p Id + S, p = p(ρ, ε) dS = 2µ(D0 − Dp ) − (SW − WS), dt D = sym(∇V ), D0 = dev(D), W = skew(∇V ).

von Mises yield condition r f =| S | −

2 0 Y ≤ 0. 3

Plastic flow rule for elastic perfectly plastic material S Dp = χ(Np : D)Np , Np = , plastic flow direction, |S| ( 0 if f < 0 or if f = 0 and (Np : D) ≤ 0, χ= 1 if f = 0 and (Np : D) > 0. I. Bertron, P.-H. Maire and B. Rebourcet CEA | MULTIMAT 2015 | SEPTEMBER 7th to 11th , 2015 | PAGE 3/32

Notation related to axisymmetric geometry Assumptions

Pseudo-Cartesian frame y(r)

Rotational symmetry about x axis Pseudo-radius R(y ) = 1 − α + αy

l(t) 11111111111111111111111 00000000000000000000000

11111111111111111111111 00000000000000000000000 00000000000000000000000 11111111111111111111111 00000000000000000000000 11111111111111111111111 00000000000000000000000 11111111111111111111111 ω 2D 00000000000000000000000 11111111111111111111111 00000000000000000000000 11111111111111111111111 00000000000000000000000 11111111111111111111111 00000000000000000000000 11111111111111111111111 00000000000000000000000 11111111111111111111111 a(t) 00000000000000000000000 11111111111111111111111 00000000000000000000000 11111111111111111111111 00000000000000000000000 11111111111111111111111 00000000000000000000000 11111111111111111111111 00000000000000000000000 11111111111111111111111 00000000000000000000000 11111111111111111111111 00000000000000000000000 11111111111111111111111

α = 0 planar geometry α = 1 pylindrical geometry

R(y) ey O

θ x(z)

ex

The control volume ω is the torus obtained rotating planar zone ω 2D about x-axis

Notation Orthonormal basis (ex , ey , eθ ) Line element dl Area element da = dxdy;

Area of control volume Aω =

R ω

da

Surface element ds = R(y) dl Volume element dv = R(y ) da;

Volume of control volume Vω =

R ω

Rda

I. Bertron, P.-H. Maire and B. Rebourcet CEA | MULTIMAT 2015 | SEPTEMBER 7th to 11th , 2015 | PAGE 4/32

Grad and Div in cylindrical coordinates Vector divergence

Vector gradient ∇V =

∂u ∂x ∂v ∂y

∂u ∂y ∂v ∂y

0

0

0

αv R

0

! =

1 ∂(Ru) R ∂x 1 ∂(Rv ) R ∂y

1 ∂(Ru) αu R ∂y − R 1 ∂(Rv ) αv R ∂y − R

0

0

0

αv R

0

! ,

where V = uex + v ey .

∂u ∂v αv + + ∂x ∂y R   1 ∂(Ru) ∂(Rv ) = + R ∂x ∂y

∇·V=

Tensor divergence ∂Txy Txy ∂Tyx ∂Tyy Tyy Tθθ ∂Txx + +α )ex + ( + +α )ey − α ey , ∂x ∂y R ∂x ∂y R R 1 ∂ ∂ 1 ∂ ∂ Tθθ = [ (RTxx ) + (RTxy )]ex + [ (RTyx ) + (RTyy )]ey − α ey , R ∂x ∂y R ∂x ∂y R

∇ · T =(

where the second-order tensor, T is defined by T = Txx (ex ⊗ ex ) + Txy (ex ⊗ ey ) + Tyx (ey ⊗ ex ) + Tyy (ey ⊗ ey ) + Tθθ (eθ ⊗ eθ ) I. Bertron, P.-H. Maire and B. Rebourcet CEA | MULTIMAT 2015 | SEPTEMBER 7th to 11th , 2015 | PAGE 5/32

Volume integrals of Div and Grad Control volume expressions Z Z Z ω

Z Zω

∇ · T dv = Z ∇V dv =

TnR dl − α

∂ω 2D

V ⊗ nR dl − α

Zω

ω 2D

∇ · V dv =



2D

Teθ · eθ da ey  Z V ⊗ ey da + α

ω 2D



V · ey da eθ ⊗ eθ

ω 2D

Z

V · nR dl since ∇ · V = tr(∇V) Z Taking T = Id leads to the geometrical identity ω 2D

ω

Z nR dl = α

ω 2D

Area weighted quadratures Z Z Z ω

Z ω

∇ · T dv ≈ Rω Z ∇V dv ≈ Rω



Tn dl + α ∂ω 2D

ω 2D

V ⊗ n dl + α

Zω

2D

 da ey

ω 2D

Z

Tey da − α Teθ · eθ da ω 2D  V · ey da eθ ⊗ eθ

 ey

ω 2D

Here, Rω is the mean radius of ω defined by Rω =

Vω . Aω

I. Bertron, P.-H. Maire and B. Rebourcet CEA | MULTIMAT 2015 | SEPTEMBER 7th to 11th , 2015 | PAGE 6/32

Control volume discretization Moving cell ωc (t)

Semi-discrete conservation laws

n

Z p

( ρ1c , V c, Ec)

∂ωc ωc

Vp

mc =

ρ dv and ωc (t)

dmc = 0, dt

Z d 1 ( )− V · nR dl = 0, GCL dt ρc ∂ωc Z Z d TnR dl + α Tθθ da ey = 0, mc Vc − dt ∂ωc ωc Z d mc Ec − Tn · VR dl = 0. dt ∂ωc mc

Remarks 1

Geometric Conservation Law (GCL) since mc = ρc Vc (t)

2

Compatibility between GCL and trajectory equation of vertex xp d xp = Vp (xp (t), t), xp (0) = Xp dt I. Bertron, P.-H. Maire and B. Rebourcet CEA | MULTIMAT 2015 | SEPTEMBER 7th to 11th , 2015 | PAGE 7/32

GCL compatibility Volume computation

Cell triangulation p+ dxp dt = V p

Triangulation of the polygonal cell [ ωc = TOpp+

p

p∈P(c)

ωc

p−

O

Cell volume reads P (R +Rp +R + ) Vc (t) = p∈P(c) O 6 p [xp (t) × xp+ (t)] · eθ

Time rate of change of volume Vc is expressed in terms of the vertex position, xp , then chain rule yields X ∂Vc d d Vc = · xp . dt ∂xp dt p∈P(c)

Here,

∂Vc ∂xp

is the corner vector related to point p and cell c I. Bertron, P.-H. Maire and B. Rebourcet CEA | MULTIMAT 2015 | SEPTEMBER 7th to 11th , 2015 | PAGE 8/32

GCL compatibility Corner vector expressions

Cell triangulation p+

n+ pc

+ Rpc

p

+ lpc

ωc

− lpc

Vp − Rpc

n− pc

O

p−

∂Vc − − − + + + = Rpc lpc npc + Rpc lpc npc ∂xp Rp− + 2Rp Rp+ + 2Rp + − ; Rpc = Rpc = 3 3 1 − − lpc npc = (xp − xp− ) × eθ 2 1 + + lpc npc = (xp+ − x) × eθ 2

Semi-discrete volume equation [Whalen (JCP 1996)] X d − − − + + + Vc − (Rpc lpc npc + Rpc lpc npc ) · Vp = 0 dt p∈P(c)

± NB: in planar geometry Rpc = 1 and we recover the classical GCL. I. Bertron, P.-H. Maire and B. Rebourcet CEA | MULTIMAT 2015 | SEPTEMBER 7th to 11th , 2015 | PAGE 9/32

GCL compatibility Compatible discretization of the divergence operator over ωc DIV c (V) =

1 X 1 dVc − − − + + + = (Rpc lpc npc + Rpc lpc npc ) · Vp Vc dt Vc p∈P(c)

The discrete divergence operator preserves linear velocity fields Axc + b · ey , Rc where xc is the planar cell centroid and Rc = Vc /Ac the cell radius.

If V(x) = Ax + b then DIV c (V) = tr(A) + α

The demonstration relies on the following geometric identities X

− − − + + + Rpc lpc npc + Rpc lpc npc = αAc ey ,

p∈P(c)

X

(2)

(2)

− − − + + + (Rpc lpc npc + Rpc lpc npc ) ⊗ xp = (1 − α)Ac Id + αVc Id + αAc ey ⊗ xc ,

p∈P(c) (2)

where Id denotes the bidimensional unit tensor. I. Bertron, P.-H. Maire and B. Rebourcet CEA | MULTIMAT 2015 | SEPTEMBER 7th to 11th , 2015 | PAGE 10/32

CV discretization of the constitutive law Semi-discrete constitutive law d Sc = 2µc (D0,c − Dpc ) − (Sc Wc − Wc Sc ) dt D0,c = dev(Dc ); Dc = sym[GRADc (V)] and Wc = skew[GRADc (V)]

CV discretization of the velocity gradient X

Vc [GRADc (V)] =

− − − + + + Vp ⊗ (Rpc lpc npc + Rpc lpc npc ) − α

p∈P(c)

+α

X

X

Apc (Vp ⊗ ey )

p∈P(c)

Apc (Vp · ey )(eθ ⊗ eθ ),

p∈P(c) − − + + where Apc = 12 (lpc npc + lpc npc ) · (xp − xc ) is the subcell area.

Comments tr[GRADc (V)] = DIV c (V) and preservation of linear fields The CV discretization does not preserve rotational symmetry [Burton et al. (LA-UR-13-23155)] I. Bertron, P.-H. Maire and B. Rebourcet CEA | MULTIMAT 2015 | SEPTEMBER 7th to 11th , 2015 | PAGE 11/32

Area weighted discretization Moving cell ωc (t)

Semi-discrete conservation laws

n

p

Vp

( ρ1c , V c, Ec)

∂ωc ωc

Z d 1 mc ( ) − V · nR dl = 0, GCL dt ρc ∂ω Z c Z d mc Vc − Rc Tn dl + α Tey da dt ∂ωc ωc Z −α Tθθ da ey = 0, ωc Z d mc Ec − Tn · VR dl = 0. dt ∂ωc

Velocity gradient for the constitutive law Vc [GRADAW c (V)] = Rc

X

X

− − + + Vp ⊗(lpc npc +lpc npc )+α

p∈P(c)

Apc (Vp ·ey )(eθ ⊗eθ )

p∈P(c)

Rotational symmetry preservation Linear velocity fields preservation for Rc = Vc /Ac Lack of consistency with the GCL since tr[GRADAW c (V)] 6= DIV c (V) I. Bertron, P.-H. Maire and B. Rebourcet CEA | MULTIMAT 2015 | SEPTEMBER 7th to 11th , 2015 | PAGE 12/32

Sub-cell force-based discretization Cell partition into sub-cells

Sub-cell definition p+

m± midpoint of edge [p, p± ] m+

Sub-cell ωpc : quad. {c m− p m+ }

n+ pc + Tpc

Vp − Tpc

c

Sub-cell force definition Numerical trace T± pc defined by Z ± ± ± Tn dl = lpc Tpc npc

n− pc m−

[p,m± ]

Sub-cell force defined by ωc

− − − + + + Fpc = lpc Tpc npc + lpc Tpc npc

p−

Total energy flux

Momentum flux Z Rc

Tn dl = Rc ∂ωc

X p∈P(c)

Fpc

Z ∂ωc

Tn · VR dl ≈

X

Rp Fpc · Vp

p∈P(c)

I. Bertron, P.-H. Maire and B. Rebourcet CEA | MULTIMAT 2015 | SEPTEMBER 7th to 11th , 2015 | PAGE 13/32

Sub-cell force-based discretization Sub-cell definition

Semi-discrete equations

p+

mc n+ pc + Tpc

p∈P(c)

Vp − Tpc

n− pc

c

ωc

X d 1 − − − + + + ( )− (Rpc lpc npc + Rpc lpc npc ) · Vp = 0, dt ρc

X d F pc − αAc (Tc ey − Tθθ ey ) = 0, mc Vc − Rc dt p∈P(c)

mc p−

d Ec − dt

X

Rp F pc · Vp = 0,

p∈P(c)

− − − + + + where sub-cell force Fpc = lpc Tpc npc + lpc Tpc npc .

Numerical fluxes, Fpc and Vp , shall be defined invoking 1

Analogy with planar geometry for Fpc definition Fpc = lpc Tc npc + Mpc (Vp − Vc ), − − + + where Mpc is the viscous corner matrix and lpc npc = lpc npc + lpc npc .

2

Conservation of total energy I. Bertron, P.-H. Maire and B. Rebourcet CEA | MULTIMAT 2015 | SEPTEMBER 7th to 11th , 2015 | PAGE 14/32

Total energy conservation Motivation regarding total energy conservation Sub-cell force F pc is a priori discontinuous at node p and total energy is not conserved We shall determine V p enforcing the conservation of total energy

Balance of total energy over the computational domain D X c

mc

d Ec = dt

Z

X

Tn · RV dl =

∂D

Rp F?p · Vp .

p∈∂D

Here, F?p is a prescribed corner force acting onto boundary point p ∈ ∂D.

Substitution of total energy equation in energy balance X X c

X p

Rp Fpc · Vp =

Rp

c∈C(p)

Rp F?p · Vp ,

p∈∂D

p∈P(c)

X

X

Fpc · Vp =

X

Rp F?p · Vp .

p∈∂D

Here, C(p) is the set of cells surrounding point p. I. Bertron, P.-H. Maire and B. Rebourcet CEA | MULTIMAT 2015 | SEPTEMBER 7th to 11th , 2015 | PAGE 15/32

Total energy conservation Balance of total energy for the semi-discrete scheme Total energy conservation holds provided that X X X X X ∀V p ∈ R2 , Rp ( Fpc ) · Vp + Rp ( Fpc ) · Vp = Rp F?p · Vp . p∈D o

c∈C(p)

p∈∂D

c∈C(p)

p∈∂D

Balance of total energy holds if and only if ∀ p ∈ Do ,

X

F pc = 0,

inner nodes

c∈C(p)

∀ p ∈ ∂D,

X

F pc = F ?p .

boundary nodes

c∈C(p)

Comments 1

Since sub-cell force F pc = lpc Tc npc + Mpc (V p − V c ), nodal velocity V p is defined as being the unique solution of one of the above systems

2

Conservation of total momentum is not ensured I. Bertron, P.-H. Maire and B. Rebourcet CEA | MULTIMAT 2015 | SEPTEMBER 7th to 11th , 2015 | PAGE 16/32

Node-centered solver Nodal velocity V p satisfies one of the systems either p ∈ Do ,

X

Mp V p =

(−lpc Tc npc + Mpc V c ) ,

inner points

c∈C(p)

or p ∈ ∂D,

Mp V p =

X

(−lpc Tc npc + Mpc V c ) − F ?p . boundary points

c∈C(p)

Granted that Mp =

P

c∈C(p)

Mpc is SPD, V p is always uniquely defined.

Comments Construction of a family of CCFV Lagrangian schemes characterized by GCL compatibility Total energy conservation Rotational symmetry preservation

Numerical fluxes are evaluated by means of a node-centered solver that is exactly the same as the one derived in planar geometry Numerical dissipation governed by the viscous corner matrix Mpc I. Bertron, P.-H. Maire and B. Rebourcet CEA | MULTIMAT 2015 | SEPTEMBER 7th to 11th , 2015 | PAGE 17/32

Node-centered solver Viscous corner matrix − − + + ± ± ± ± Mpc = ρc (lpc Apc + lpc Apc ) and A± pc = aL,c (npc ⊗ npc ) + aT ,c [Id − (npc ⊗ npc )],

aL and aT are the longitudinal and transverse elastic wave speeds ± A± pc is the square root of the acoustic tensor related to npc

Notation

Sub-cell force and nodal stresses

p+

− − − + + + Fpc = lpc Tpc npc + lpc Tpc npc − T− pc − Tc = ρc Apc (Vp − Vc ),

n+ pc + Tpc

Vp − Tpc

n− pc

c

+ T+ pc − Tc = ρc Apc (Vp − Vc ).

Main properties Mpc is SPD

ωc

p−

For µ = 0, we recover the hydrodynamic nodal solver

I. Bertron, P.-H. Maire and B. Rebourcet CEA | MULTIMAT 2015 | SEPTEMBER 7th to 11th , 2015 | PAGE 18/32

Hybrid velocity gradient discretization The AW velocity gradient is not consistent with the GCL Vc [GRADAW c (V)] =

X

− − + + Vp ⊗ Rc (lpc npc + lpc npc ) + αApc (Vp · ey )(eθ ⊗ eθ )

p∈P(c)

Vc tr[GRADAW c (V)]

=

X

− − + + Vp · Rc (lpc npc + lpc npc ) + αApc (Vp · ey )

p∈P(c)

Vc DIV c (V) =

X

− − − + + + Vp · (Rpc lpc npc + Rpc lpc npc ) 6= Vc tr[GRADAW c (V)]

p∈P(c)

Consistency is enforced by modifying the hoop term as follows X

Vc [GRADAWH (V)] = c

− − + + Vp ⊗ Rc (lpc npc + lpc npc )

p∈P(c)

+[ This modification leads to

X dVc − − + + − Vp · Rc (lpc npc + lpc npc )](eθ ⊗ eθ ) dt p∈P(c)

tr[GRADAWH (V)] = c

1 dVc = DIV c (V) Vc dt

I. Bertron, P.-H. Maire and B. Rebourcet CEA | MULTIMAT 2015 | SEPTEMBER 7th to 11th , 2015 | PAGE 19/32

Blake problem [Kamm, LA-UR-09-01255] Description of the problem Elastic response of a medium containing a spherical cavity of radius R0 Boundary of the cavity subject to a driving pressure p0 Small strain regime Material parameters: ρ (density), λ and µ (Lame´ coefficients)

Analytical solution Displacement, velocity and stresses derived from the potential φ(R, t) ( 0 0 if ξ = t − R−R ≤ 0, aL φ(R, t) = H(ξ) if ξ > 0,    p0 R0 α H(ξ) = − 1 − exp(−αξ) cos(βξ) + sin(βξ) . ρ(α2 + β 2 ) R β r a2 a2 Here, α = 2 aL RT 0 , β = α a2L − 1, aL2 = λ+2µ and aT2 = µρ . ρ T

I. Bertron, P.-H. Maire and B. Rebourcet CEA | MULTIMAT 2015 | SEPTEMBER 7th to 11th , 2015 | PAGE 20/32

Blake problem 45 × 18 polar grid

Set up Radius: R0 = 0.1 m Density: ρ = 3 103 kg/m3 Lame´ coefficient: λ = 25 109 Pa Lame´ coefficient: µ = 25 109 Pa Computational domain: (R, θ) ∈ [0.1, 1] × [0, π2 ] Polar grids: 45 × 18, 90 × 36, 180 × 72 Stopping time: tend = 1.6 10−4 s

Boundary conditions Pressure boundary condition at R = R0 (driving pressure p0 ) Wall boundary condition at R = 1.1 since the front of the stress wave at tend is located at Rend = 0.9 m I. Bertron, P.-H. Maire and B. Rebourcet CEA | MULTIMAT 2015 | SEPTEMBER 7th to 11th , 2015 | PAGE 21/32

Blake problem: exact and numerical results Radial stress deviator

Pressure

0 Analytical solution 45x18 grid 90x36 grid 180x72 grid

60000

Analytical solution 45x18 grid 90x36 grid 180x72 grid

-2e+05

-4e+05 Srr (Pa)

Pressure (Pa)

40000

20000

-6e+05 0 -8e+05 -20000 0

0,2

0,4

0,6

0,8

1

Radius (m)

-1e+06

0

0,2

0,4

0,6

0,8

Radius (m)

Comments The current value of the driving pressure implies a displacement on the order of 10−6 m at the cavity boundary at the final time Small strain assumption is observed Reasonable convergence towards the exact solution for both pressure and radial stress deviator I. Bertron, P.-H. Maire and B. Rebourcet CEA | MULTIMAT 2015 | SEPTEMBER 7th to 11th , 2015 | PAGE 22/32

1

Blake problem: exact and numerical results Pressure snapshot at t = tend for the finer grid 720 × 288 Analytical solution 720x288 grid

60000

Pressure (Pa)

40000

20000

0

-20000 0

0,2

0,4

0,6

0,8

1

Radius (m) I. Bertron, P.-H. Maire and B. Rebourcet CEA | MULTIMAT 2015 | SEPTEMBER 7th to 11th , 2015 | PAGE 23/32

Verney problem Description of the problem Spherical shell collapse given an initial inwardly directed radial velocity Kinetic energy of the shell converted to internal energy via plastic work Analytic values for inner and outer stopping shell radii [Verney, (1968)]

Set up [Kamm et al., LA-14379 (2008)] Material: beryllium (Mie Gruneisen EOS and EPP model) ¨ Initial radii: Ri = 8 10−2 m (inner) and Ro = 10 10−2 m (outer)  2 Initial data: (p0 , V0 ) = [10−6 Pa, −V0 Rr i er ] Velocity magnitude chosen as V0 = 675.036 m/s to give final radii ri = 3.0 10−2 m (inner) and ro = 8.015 10−2 m (outer) Stopping time: tend = 110 µs

Features tested Ability to accurately predict the inner and outer stopping radii Abality to preserve the spherical symmetry of the computational grid I. Bertron, P.-H. Maire and B. Rebourcet CEA | MULTIMAT 2015 | SEPTEMBER 7th to 11th , 2015 | PAGE 24/32

Verney problem Initial grid

Final grid

Comments Symmetry preservation is ensured Thickening of the shell during its collapse due to radial convergence I. Bertron, P.-H. Maire and B. Rebourcet CEA | MULTIMAT 2015 | SEPTEMBER 7th to 11th , 2015 | PAGE 25/32

Verney problem Total, kinetic, internal energies vs. time for the 100 × 90 grid 3e+05

2,5e+05

Energy

2e+05 Total energy Internal energy Kinetic energy

1,5e+05

1e+05

50000

0

0

5e-05 Time (s)

0,0001

I. Bertron, P.-H. Maire and B. Rebourcet CEA | MULTIMAT 2015 | SEPTEMBER 7th to 11th , 2015 | PAGE 26/32

Verney problem Grid convergence analysis of the inner radius 0,08 25x90 grid 50x90 grid 100x90 grid 200x90 grid

Radius (m)

0,07

0,06

0,05

0,04

0,03

0

5e-05 Times (s)

0,0001

I. Bertron, P.-H. Maire and B. Rebourcet CEA | MULTIMAT 2015 | SEPTEMBER 7th to 11th , 2015 | PAGE 27/32

Impact of a copper rod against a rigid wall Material data for copper Mie-Gruneisen EOS ρ0 = 8930 kg/m3 , ¨ Linear hardening

Y = A + B ˙p

Set up [Sambasivan et al., JCP (2013)] Cylindrical rod of length lrod = 32.4 10−3 m and radius rrod = 3.2 10−3 m Computational domain: (x, y ) ∈ [lrod , rrod ] Initial velocity: V0 = −227ex m/s Boundary conditions: rigid wall at x = 0, symmetry at y = 0 and free surface boundary condition elsewhere Stopping time: tend = 80 µs

Comments Dissipation of kinetic energy through plastic work Comparison between cell-centered and staggered numerical results I. Bertron, P.-H. Maire and B. Rebourcet CEA | MULTIMAT 2015 | SEPTEMBER 7th to 11th , 2015 | PAGE 28/32

Impact of a copper rod against a rigid wall Map of p at t = 20 µs

Map of p at t = 40 µs

Map of p at t = 60 µs

Map of p at t = 80 µs

I. Bertron, P.-H. Maire and B. Rebourcet CEA | MULTIMAT 2015 | SEPTEMBER 7th to 11th , 2015 | PAGE 29/32

Impact of a copper rod against a rigid wall Plastic strain map: cell-centered (top) vs staggered (bottom)

I. Bertron, P.-H. Maire and B. Rebourcet CEA | MULTIMAT 2015 | SEPTEMBER 7th to 11th , 2015 | PAGE 30/32

Impact of a copper rod against a rigid wall Position of head and tail

Velocity of head and tail

0,04 Head cell-centered scheme Head staggered scheme Tail cell-centered scheme Tail staggered scheme

Head cell-centered scheme Head staggered scheme Tail cell-centered scheme Tail staggered scheme

400

Velocity (m/s)

Position (m)

0,03

0,02

200

0

0,01 -200

0

0

2e-05

4e-05 Time (s)

6e-05

8e-05

0

2e-05

4e-05 Time (s)

6e-05

Quantitative comparaisons Case Current Staggered Sambasivan

Final length (mm) 21.30 21.09 21.58

Final base radius (mm) 7.19 7.6 6.99

Max. p 3.6 3.9 3.16

I. Bertron, P.-H. Maire and B. Rebourcet CEA | MULTIMAT 2015 | SEPTEMBER 7th to 11th , 2015 | PAGE 31/32

8e-05

Conclusion and Perspectives EUCCLHYD scheme extension Cell-centered FV formulation for hypoelasticity in cylindrical geometry Area Weighted (hybrid) discretization Rotational symmetry preservation Total energy conservation

Perspectives More validation (strong need of analytical test cases) Development of the Control Volume formulation Comparison of the AW and CV numerical methods Hyperelasticity extension in 3D geometry (cf. G. George talk) Hyperelasticity and plasticity at large strains [ANR SNIHYPER (S. Gavrilyuk, IUSTI Marseille)] I. Bertron, P.-H. Maire and B. Rebourcet CEA | MULTIMAT 2015 | SEPTEMBER 7th to 11th , 2015 | PAGE 32/32