A first order hyperbolic framework for large strain computational computational solid dynamics in OpenFOAM

1. Introduction 2. Reversible elastodynamics 3. Numerical technique 4. Numerical results 5. Conclusions
A ﬁrst order hyperbolic framework for large strain
computational solid dynamics: An upwind cell centred
Total Lagrangian scheme
Jibran Haider a, b
, Chun Hean Lee a
, Antonio J. Gil a
, Javier Bonet c
& Antonio Huerta b
a
Zienkiewicz Centre for Computational Engineering (ZCCE)
College of Engineering, Swansea University, UK
b
Laboratori de Càlcul Numèric (LaCàN),
Universitat Politèchnica de Catalunya (UPC BarcelonaTech), Spain
c
University of Greenwich, London, UK
20th
May 2016
Website: http://www.jibranhaider.weebly.com
Jibran Haider Fast solid dynamics in OpenFOAM CoMe Seminar, UPC BarcelonaTech 1/61

Scheme of presentation
1 Introduction
Motivation
Mixed solid dynamics
OpenFOAM
2 Reversible elastodynamics
Governing equations
3 Numerical technique
Finite Volume Methodology
Angular momentum preserving scheme
4 Numerical results
5 Conclusions

1 Introduction
Motivation
OpenFOAM
Governing equations
4 Numerical results
5 Conclusions

1.1 Motivation
Solid dynamics
Objectives:
• Simulate fast-transient solid dynamic problems.
• Develop an efﬁcient library of low order numerical schemes.
Displacement based FEM/FVM formulations:
• Linear tetrahedral elements suffer from:
× Volumetric locking in nearly incompressible materials.
× Reduced order of convergence for stresses and strains.
× Poor performance in bending and shock dominated
scenarios.
Proposed mixed formulation [Haider et al., 2016]:
• An upwind cell-centred FVM Total Lagrangian scheme.
• Entitled TOtal Lagrangian Upwind Cell-centred FVM for
Hyperbolic conservation laws (TOUCH).
Utilises an explicit time integration scheme.
Programmed in the open-source CFD software OpenFOAM.
0 0.5 1
0
0.5
1
1.5
X-Coordinate
Y-Coordinate
t=0.03s
-1
-0.5
0
0.5
1x 10
7
-0.5 0 0.5 1 1.5
0
0.5
1
1.5
X-Coordinate
Y-Coordinate
t=0.0006s
Q1-P0 FEM
0 0.5 1
0
0.5
1
1.5
X-CoordinateY-Coordinate
t=0.03s
-1
-0.5
0
0.5
1x 10
7
-0.5 0 0.5 1 1.5
0
0.5
1
1.5
X-Coordinate
Y-Coordinate
t=0.0006s
Upwind FVM
Haider J, Lee CH, Gil AJ, Bonet J. A ﬁrst order hyperbolic framework for large strain computational solid dynamics: An upwind cell centred Total
Lagrangian scheme. International Journal for Numerical Methods in Engineering 2016; (In Press), doi: 10.1002/nme.5293

1.2 Mixed solid dynamics
Adapted CFD technologies
(LLNL - 1991) Upwind cell centred FVM (1D) - [Trangenstein & Colella]
(DU - 1994) Upwind cell centred FVM (2D) - [Trangenstein]
(UPMC - 2010) Cell centred FVM (2D Hyperelastic GLACE) - [Kluth & Després]
(SU - 2012) Two-step Taylor-Galerkin FEM (2D) - [Karim, Lee, Gil & Bonet]
(MIT - 2012) Hybridizable Discontinuous Galerkin FEM (2D) - [Nguyen & Peraire]
(SU - 2013) Upwind cell centred FVM (2D) - [Lee, Gil & Bonet]
(SU - 2013) JST vertex centred FVM (3D) - [Aguirre, Gil, Bonet & Carreño]
(SU - 2013) Petrov-Galerkin FEM (3D) - [Lee, Gil & Bonet]
(CEA - 2013) Cell centred FVM (2D EUCCLHYD) - [Maire, Abgrall, Breil, Loubére & Rebourcet]
(SU - 2014) Fractional step Petrov-Galerkin FEM (3D) - [Gil, Lee, Bonet & Aguirre]
(DU - 2015) D-VMS FEM (3D) - [Scovazzi, Carnes & Zeng]
(SU - 2015) Polyconvex Petrov-Galerkin FEM (3D) - [Bonet, Gil, Lee, Aguirre & Ortigosa]
(SU - 2015) Upwind vertex centred FVM (3D) - [Aguirre, Gil, Bonet & Lee]
(SU - 2016) Polyconvex (Fractional) Hydrocode (3D) - [Gil, Lee, Bonet & Ortigosa]
(SU - 2016) Upwind cell centred FVM (3D TOUCH) - [Haider, Lee, Gil & Bonet, In press]
(SU - 2016) JST SPH (3D) - [Lee, Gil, Greto, Kulasegaram & Bonet, Under review]

1.3 OpenFOAM
Solid dynamics in OpenFOAM
What is OpenFOAM?
• An open-source CFD software package in C++.
• Based on the cell centred Finite Volume Method.
• Limited functionality for solid dynamics.
v = 100 m/s
(0.5, 0.5, 0.5)
(−0.5, −0.5, −0.5)
[Punch cube]
OpenFOAM solvers [Jasak & Weller, 2000]
× Linear elastic material
× Small strain deformation
TOUCH scheme [Haider, 2016; Lee, 2013]
Complex constitutive models
Large strain deformation
Jasak H, Weller H. Application of the ﬁnite volume method and unstructured meshes to linear elasticity. International Journal for Numerical
Methods in Engineering 2000; 48(2):267–287
Lee CH, Gil AJ, Bonet J. Development of a cell centred upwind ﬁnite volume algorithm for a new conservation law formulation in structural
dynamics. Computers and Structures 2013; 118:13–38

1 Introduction
Motivation
OpenFOAM
Governing equations
4 Numerical results
5 Conclusions

2.1 Governing equations
Total Lagrangian formulation
Conservation Laws
1. Linear momentum:
∂p
∂t
= 0 · P(F) + ρ0b; p = ρ0v
2. Deformation gradient:
∂F
∂t
= 0 ·
1
ρ0
p ⊗ I
Additional equations
3. Total energy:
∂E
∂t
= 0 ·
1
ρ0
P
T
p − Q + s
4. Geometry update:
∂x
∂t
=
1
ρ0
p; x = X + u
An appropriate constitutive model is required to close the system.

2.1 Governing equations
Total Lagrangian formulation
First order hyperbolic system:
∂U
∂t
= 0 · F(U) + S
where
U =







p
F
E
x







; F =







P(F)
1
ρ0
p ⊗ I
1
ρ0
PT p − Q
0







; S =







ρ0b
0
s
1
ρ0
p







.
First order hyperbolic system widely used in CFD.
Develop an efﬁcient second order numerical scheme for mixed transient solid dynamics.

1 Introduction
Motivation
OpenFOAM
Governing equations
4 Numerical results
5 Conclusions

3.1 Methodology
Finite volume methodology
Steps:
1. Reconstruction
• Spatial discretisation
• Linear reconstruction procedure
2. Flux computation
• Lagrangian contact dynamics
• Acoustic Riemann solver
3. Involutions
• Constrained transport schemes
4. Evolution
• Time discretisation
Constrained transport schemes are widely used in Magnetohydrodynamics (MHD).

3.1 Methodology: Reconstruction
Spatial discretisation
Conservation equations for an arbitrary element:
dUe
dt
=
1
Ωe
0 Ωe
0
∂FI
∂XI
dΩ0 −→ ∀ I = 1, 2, 3;
=
1
Ωe
0 ∂Ωe
0
FI NI
FN
dA (Gauss Divergence theorem)
≈
1
Ωe
0
f∈Λ
f
e
g∈Λ
g
f
F
C
Neg
(U
+
, U
−
) Ceg
Only 1 Gauss quadrature point per face!
dUe
dt
=
1
Ωe
0
f∈Λ
f
e
F
C
Nef
(U
+
, U
−
) Cef
Ceg
e
FC
Neg
Ce f
e
FC
Ne f

Standard face-based CC-FVM
dUe
dt
=
1
Ωe
0



f∈Λ
f
e
F
C
Nef
(U
−
f , U
+
f ) Cef



e FC
Ne f
Ce f Ωe
0
Node-based CC-FVM [Kluth & Després, 2010]
dUe
dt
=
1
Ωe
0



a∈Λa
e
F
C
Nea
(U
−
a , U
+
a ) Cea



FC
Nea
Cea
Ωe
0
e
Kluth G, Després B. Discretization of hyperelasticity on unstructured mesh with a cell-centered Lagrangian scheme. Journal of Computational
Physics 2010; 229(24):9092–9118

Standard face-based CC-FVM
dUe
dt
=
1
Ωe
0



f∈Λ
f
e
F
C
Nef
(U
−
f , U
+
f ) Cef



dpe
dt
=
1
Ωe
0
f∈Λ
f
e
t
C
f Cef
dFe
dt
=
1
Ωe
0
f∈Λ
f
e
1
ρ0
p
C
f ⊗ Cef
dEe
dt
=
1
Ωe
0
f∈Λ
f
e
1
ρ0
p
C
f · t
C
f Cef
dxe
dt
=
1
ρ0
pe
Node-based CC-FVM [Kluth & Després, 2010]
dUe
dt
=
1
Ωe
0



a∈Λa
e
F
C
Nea
(U
−
a , U
+
a ) Cea



dpe
dt
=
1
Ωe
0 a∈Λa
e
t
C
ea Cea
dFe
dt
=
1
Ωe
0 a∈Λa
e
1
ρ0
p
C
a ⊗ Cea
dEe
dt
=
1
Ωe
0 a∈Λa
e
1
ρ0
p
C
a · t
C
ea Cea
dxa
dt
=
1
ρ0
pa
Kluth G, Després B. Discretization of hyperelasticity on unstructured mesh with a cell-centered Lagrangian scheme. Journal of Computational
Physics 2010; 229(24):9092–9118

Godunov’s method
• Piecewise constant representation in every cell.
• Methodology is ﬁrst order accurate in space.
x
y
U
Ue
Uα1
Uα4
Uα2
Uα3
(a) Piecewise constant values ×
x
y
U
Uα4
Uα3
Uα2
Uα1
Ue
Uα3
Uα4
(b) Linear reconstruction
× First order simulations suffer from excessive numerical dissipation.
A linear reconstruction procedure is essential to increase spatial accuracy.

Linear reconstruction procedure
Gradient operator:
• Classical least squares minimisation procedure.
Π(Ge) =
1
2 α∈Λα
e
1
deα
2
[Uα − (Ue + Ge · deα)]
2
;
deα = Xα − Xe; ˆdeα =
deα
deα
.
Ge =


α∈Λα
e
ˆdeα ⊗ ˆdeα


−1
α∈Λα
e
Uα − Ue
deα
ˆdeα
Linear extrapolation to ﬂux integration point:
U{f,a} = Ue + Ge · X{f,a} − Xe
de1α2
e1
α1
α2
α3
α4
αf1
αf2
αf3αf4
αf5
e2
de2α4
Gradient correction procedure:
• Necessary for the satisfaction of monotonicity through Barth and Jespersen limiter (φe).
U{f,a} = Ue + φe Ge(Ue, Uα) · X{f,a} − Xe
Ensures that the spatial discretisation is second order accurate.

3.1 Methodology: Flux computation
Lagrangian contact dynamics
Rankine-Hugoniot jump conditions:
c U = F N
where = +
− −
c p = t
c F =
1
ρ0
p ⊗ N
c E =
1
ρ0
P
T
p · N
X, x
Y, y
Z, z
Ω+
0
Ω−
0
N+
N−
n−
n+
Ω+(t)
Ω−(t)
φ+
φ−
n−
n+
c−
s
c+
s
c+
pc−
p
Time t = 0
Time t

Acoustic Riemann solver
Contact ﬂux:
F
C
N =




tC
1
ρ0
pC
⊗ N
1
ρ0
tC
· pC




where p
C
= p
C
n + p
C
t
t
C
= t
C
n + t
C
t
Jump condition for linear momentum equation:
c p = t
Normal jump → cp pn = tn
Tangential jump → cs pt = tt
p+
n , t+
np−
n , t−
n
c+
pc−
p
pC
n , tC
n
x
t
Normal jump
p+
t , t+
tp−
t , t−
t
c+
sc−
s
pC
t , tC
t
x
t
Tangential jump

Acoustic Riemann solver
Riemann ﬂux:
p
C
= (n ⊗ n)
c−
p p−
+ c+
p p+
c−
p + c+
p
+ (I − n ⊗ n)
c−
s p−
+ c+
s p+
c−
s + c+
s
pC
Ave
+ (n ⊗ n)
t+
− t−
c−
p + c+
p
+ (I − n ⊗ n)
t+
− t−
c−
s + c+
s
pC
Stab
.
t
C
= (n ⊗ n)
c+
p t−
+ c−
p t+
c−
p + c+
p
+ (I − n ⊗ n)
c+
s t−
+ c−
s t+
c−
s + c+
s
tC
Ave
(n ⊗ n)
c−
p c+
p (p+
− p−
)
c−
p + c+
p
+ (I − n ⊗ n)
c−
s c+
s (p+
− p−
)
c−
s + c+
s
tC
Stab
.

Boundary ﬂux
1. Fixed boundary:
p
C
= pB
t
C
= t
−
− c
−
p (n ⊗ n) + c
−
s (I − n ⊗ n) (pB − p
−
)
2. Free boundary:
p
C
= p
−
+
1
c−
p
(n ⊗ n) +
1
c−
s
(I − n ⊗ n) (tB − t
−
)
t
C
= tB
3. Symmetric (tangentially sliding) boundary:
p
C
= (I − n ⊗ n) p
−
+
1
c−
s
(tB − t
−
)
t
C
= (n ⊗ n) t
−
− c
−
p p
−
+ (I − n ⊗ n)tB
4. Skew-symmetric (normally sliding) boundary:
p
C
= (n ⊗ n) p
−
+
1
c−
s
(tB − t
−
)
t
C
= (I − n ⊗ n) t
−
− c
−
p p
−
+ (n ⊗ n)tB
c+
p = c+
s = ∞
p−
pB
c+
p = c+
s = 0
p−
tB
p+
n = 0, c+
p = ∞, c+
s = 0
p−
tB
p+
t = 0, c+
p = 0, c+
s = ∞
p−
tB

3.1 Methodology: Involutions
Constrained transport schemes
Compatibility condition [Lee, Gil & Bonet, 2013]: CURL F = 0
Face based F update
1. Standard Godunov’s scheme:
dFe
dt
=
1
Ωe
0
f∈Λ
f
e
pC
f
ρ0
⊗ Cef X
2. Constrained-TOUCH scheme:
dFe
dt
=
1
Ωe
0
f∈Λ
f
e
˜pC
f (pC
f )
ρ0
⊗ Cef
3. Penalised-TOUCH scheme:
Fe = Fe
Update from Godunov’s scheme
+ ξF ( 0 x − Fe)
Lee CH, Gil AJ, Bonet J. Development of a cell centred upwind ﬁnite volume algorithm for a new conservation law formulation in structural
dynamics. Computers and Structures 2013; 118:13–38

3.1 Methodology: Involutions
Godunov-type FVM: constrained-TOUCH
Constrained-TOUCH update:
dFe
dt
=
1
Ωe
0
f∈Λ
f
e
˜pC
f (pC
f )
ρ0
⊗ Cef
1. Cell averaged linear momentum:
˜pe =
1
Λ
f
e
f∈Λ
f
e
p
C
f
2. Local gradient operator:
Ge =




f∈Λ
f
e
ˆdef ⊗ ˆdef




−1
f∈Λ
f
e


pC
f − ˜pe
def

ˆdef
3. Nodal linear momentum
pea = ˜pe + φeGe(Xa − Xe); pa =
1
Λe
a e∈Λe
a
pea
4. Modiﬁed contact linear momentum
˜p
C
f =
1
Λa
f a∈Λa
f
pa
Ge
˜pepe
= Linear reconstruction + Riemann solver
˜pC
f
pa
pC
f
= Gradient calculation through averaged
= Extrapolation / interpolation to nodes
= Interpolation to face centers
linear momentum

3.1 Methodology: Evolution
Temporal discretisation
dUe
dt
= ˙Ue
Runge-Kutta time integration:
1
st
RK stage −→ U
∗
e = U
n
e + ∆t ˙U
n
e (U
n
e , t
n
)
2
nd
RK stage −→ U
∗∗
e = U
∗
e + ∆t ˙U
∗
e (U
∗
e , t
n+1
)
U
n+1
e =
1
2
U
n
e + U
∗∗
e
with stability constraint:
∆t = αCFL
hmin
cp,max
An explicit Total Variation Diminishing Runge-Kutta scheme widely used in CFD context.
Ensures that the formulation is second order accurate in time.
Monolithic time update for geometry.

3.2 Conservation of angular momentum
Global Lagrangian projection procedure
Functional:
Π ˜˙pe, λAng, λLin =
1
2 e
Ω
e
0
˜˙pe − ˙pe · ˜˙pe − ˙pe + λAng ·
e
Ω
e
0 T
e
Torque − Xe × ˜˙pe
+ λLin ·
e
T
e
Force − Ω
e
0
˜˙pe
where Xe =
xn
e → 1st
RK stage
xn+1/2
e + ∆t
2ρ0
pe → 2nd
RK stage
Te
Force =
f∈Λ
f
e
tC
f Cef
Te
Torque =
f∈Λ
f
e
xf × tC
f Cef
Modiﬁed rate of linear momentum:
˜˙pe = ˙pe + λAng × Xe + λLin
Lagrange multipliers:



e
Ωe
0 [Xe ⊗ Xe − (Xe · Xe)I] −
e
Ωe
0
ˆXe
e
Ωe
0
ˆXe −
e
Ωe
0






λAng
λLin


 =



e
Ωe
0 Xe × ˙pe − Te
Torque
e
Ωe
0 ˙pe − Te
Force


 ; ˆXe
ik
= Eijk [Xe]j

1 Introduction
Motivation
OpenFOAM
Governing equations
4 Numerical results
5 Conclusions

4.1 Cable
Shock dominated scenario
L = 10m
x
P(t)
Problem description: Linear elastic material, ρ0 = 8000 kg/m3
, E = 200 GPa, ν = 0, αCFL = 0.5,
Discretisation = 100 cells and P(L, t) = −5 × 107
Pa.

4.1 Cable
Shock dominated scenario
Stress evolution at mid cable (x = 5 m)
0 0.002 0.004 0.006 0.008 0.01 0.012
−12.5
−10
−7.5
−5
−2.5
0
2.5
5
x 10
7
Time (sec)
Stress(Pa)
Analytical
TOUCH (1st order)
TOUCH (2nd order w/o limiter)
TOUCH (2nd order with limiter)
JST VCFVM
6 7 8 9 10
x 10
−3
−7.5
−5
−2.5
0
2.5
x 10
7
Time (sec)
Stress(Pa)
Demonstrates the ability of the formulation to handle shock problems.
, E = 200 GPa, ν = 0, αCFL = 0.5,
Discretisation = 100 cells and P(L, t) = −5 × 107
Pa.

4.1 Cable
Mesh convergence
Velocity at t = 34.47 s
10
−2
10
−1
10
0
10
−8
10
−7
10
−6
10
−5
10
−4
10
−3
Grid Size (m)
VelocityError
slope = 1
L1
norm (1st order)
L2
norm (1st order)
slope = 2
L1
norm (2nd order)
L2
norm (2nd order)
Stress at t = 34.47 s
10
−2
10
−1
10
0
10
−8
10
−7
10
−6
10
−5
10
−4
10
−3
Grid Size (m)
StressError
slope = 1
L1
norm (1st order)
L2
norm (1st order)
slope = 2
L1
norm (2nd order)
L2
norm (2nd order)
Demonstrates the expected convergence of the available schemes for all variables.
, E = 1 GPa, ν = 0.3 αCFL = 0.5 and
P(L, t) = 1 × 10−3
[sin ((πt)/20 − π/2) + 1] Pa.

4.2 Low dispersion cube
Mesh convergence
X, x
Y, y
Z, z
(0, 0, 0)
(1, 1, 1)
Displacements scaled 300 times
t = 0 s t = 2 ms t = 4 ms t = 6 ms
Pressure (Pa)
Boundary conditions:
1. Symmetric conditions at:
X = 0, Y = 0, Z = 0
2. Skew-symmetric conditions at:
X = 1, Y = 1, Z = 1
Analytical solution:
u(X, t) = U0 cos
√
3
2
cdπt





A sin
πX1
2 cos
πX2
2 cos
πX3
2
B cos
πX1
2 sin
πX2
2 cos
πX3
2
C cos
πX1
2 cos
πX2
2 sin
πX3
2





where U0 = 5 × 10
−4
m; cd =
λ + 2µ
ρ0
; A = B = C = 1.
Problem description: Unit side cube, linear elastic material, ρ0 = 1100 kg/m3
, E = 17 MPa, ν = 0.3
and αCFL = 0.3.

Mesh convergence (Constrained-TOUCH)
10
−2
10
−1
10
0
10
−7
10
−6
10
−5
10
−4
Grid Size (m)
L2NormError
vx
vy
vZ
Slope = 2
10
−2
10
−1
10
0
10
−7
10
−6
10
−5
10
−4
Grid Size (m)
L2NormError
Pxx
Pyy
Pzz
Slope = 2
Demonstrates second order convergence using constrained-TOUCH scheme.
, E = 17 MPa, ν = 0.3
and αCFL = 0.3.

Mesh convergence (Penalised-TOUCH)
10
−2
10
−1
10
0
10
−7
10
−6
10
−5
10
−4
Grid Size (m)
L2NormError
vx
vy
vZ
Slope = 2
10
−2
10
−1
10
0
10
−7
10
−6
10
−5
10
−4
Grid Size (m)
L2NormError
Pxx
Pyy
Pzz
Slope = 2
Demonstrates second order convergence using penalised-TOUCH scheme.
, E = 17 MPa, ν = 0.3
and αCFL = 0.3.

4.3 Spinning Plate
Structured vs unstructured elements
X, x
Y, y
(0.5, 0.5, 0)
ω0 = [0, 0, Ω]T
(−0.5, −0.5, 0)
Time = 0.15 s
(a) Structured 20 × 20 cells (b) Unstructured 484 cells
Pressure (Pa)
Demonstrates the ability of the framework to handle unstructured grids.
Problem description: Unit side square, nearly incompressible hyperelastic neo-Hookean material,
ρ0 = 1000 kg/m3
, E = 17 MPa, ν = 0.45 and αCFL = 0.3 and Ω = 105 rad/s.

4.3 Spinning Plate
Structured vs unstructured elements
X, x
Y, y
(0.5, 0.5, 0)
ω0 = [0, 0, Ω]T
(−0.5, −0.5, 0)
Displacement of point X = [0.5, 0.5, 0]T
0 0.025 0.05 0.075 0.1 0.125 0.15 0.175 0.2
−1.5
−1.25
−1
−0.75
−0.5
−0.25
0
0.25
0.5
0.75
1
Time (sec)
Displacement(m)
ux
structured
uy
structured
ux
unstructured
uy
unstructured
Demonstrates the ability of the framework to handle unstructured grids.
ρ0 = 1000 kg/m3
, E = 17 MPa, ν = 0.45, αCFL = 0.3 and Ω = 105 rad/s.

4.3 Spinning Plate
Momentum conservation
Global angular momentum
0 0.05 0.1 0.15 0.2
0
0.5
1
1.5
2
2.5
3
3.5
x 10
4
Time (sec)
Angularmomentum(N.m.s
−1
)
A
x
(structured)
A
y
(structured)
A
z
(structured)
Ax
(unstructured)
A
y
(unstructured)
A
z
(unstructured)
Global linear momentum
0 0.05 0.1 0.15 0.2
−5
−4
−3
−2
−1
0
1
2
3
4
5
x 10
−11
Time (sec)
Linearmomentum(kg.m.s−1
)
p
x
(structured)
p
y
(structured)
p
z
(structured)
p
x
(unstructured)
py
(unstructured)
p
z
(unstructured)
Conservation the angular and linear momenta of the system.
ρ0 = 1000 kg/m3
, E = 17 MPa, ν = 0.45 and αCFL = 0.3 and Ω = 105 rad/s.

4.4 L-shaped block
Y, y
(0, 10, 0)
X, x
Z, z
F1(t) F2(t)
(6, 0, 0)
(3, 3, 3)
[L-shaped block]
F1(t) = −F2(t) = (150, 300, 450)
T
p(t)
p(t) =



t 0 ≤ t < 2.5
5 − t 2.5 ≤ t < 5
0 t ≥ 5
[Simo, 1992]
t = 2.5 s t = 5 s
t = 7.5 s t = 10 s
Pressure (Pa)
Problem description: Nearly incompressible hyperelastic neo-Hookean material, ρ0 = 1000 kg/m3
,
E = 50.05 kPa, ν = 0.3 and αCFL = 0.3.

4.4 L-shaped block
0 5 10 15 20 25 30
−3
−2
−1
0
1
2
3
x 10
5
Time (sec)
AngularMomentum(N.m.s−1
)
Ax
A
y
Az
A
x
w/o AMPA
Ay
w/o AMPA
Az
w/o AMPA
0 5 10 15 20 25 30
−6
−4
−2
0
2
4
6
x 10
−11
Time (sec)
LinearMomentum(kg.m.s
−1
)
px
py
pz
Conservation the angular and linear momenta of the system.
,
E = 50.05 kPa, ν = 0.3 and αCFL = 0.3.

4.4 L-shaped block
Energy conservation
Global total energy
0 5 10 15 20 25 30 35 40
0
1
2
3
4
5
x 10
4
Time (sec)
Energy(J)
Total energy
6x10x3
12x20x6
24x40x12
Lower numerical dissipation with mesh reﬁnement.
,
E = 50.05 kPa, ν = 0.3 and αCFL = 0.3.

4.4 L-shaped block
Cuboid elements
Mesh reﬁnement along z axis at t = 7.5 s
(a) 6 × 10 × 4 cells (b) 6 × 10 × 8 cells (c) 6 × 10 × 16 cells
Proposed methodology performs well for high aspect ratio elements.
,
E = 50.05 kPa, ν = 0.3 and αCFL = 0.3.

4.5 Bending Column
Bending dominated scenario
X, x
Y, y
(−0.5, 0, 0.5)
(0.5, 6, −0.5)
Z, z
L = 6m
v0 = [V Y/L, 0, 0]T
[Bending column]
Mesh convergence at t = 1.5 s
Pressure (Pa)
Eliminates bending difﬁculty.
,
E = 17 MPa, ν = 0.45, αCFL = 0.3 and V = 10 m/s.

4.5 Bending Column
Enhanced reconstruction
Time = 0.5 s
(b) Standard reconstruction (b) Enhanced reconstruction
Pressure (Pa)
,

4.5 Bending Column
Comparison with hyperelastic-GLACE scheme
Evolution of tip displacement (X = [0.5, 6, 0.5]T
)
0 1 2 3 4 5 6 7
−4
−2
0
2
4
6
8
Time (sec)
xdisplacement(m)
Constrained TOUCH (4x24x4)
GLACE (4x24x4)
GLACE (8x48x8)
GLACE (16x96x16)
,

4.6 Twisting Column
Highly non-linear problem
X, x
Y, y
(−0.5, 0, 0.5)
(0.5, 6, −0.5)
Z, z
ω0 = [0, Ω sin(πY/2L), 0]T
L
[Twisting column - Reﬁnement]
Mesh reﬁnement at t = 0.1 s
(a) 4 × 24 × 4 (b) 8 × 48 × 8 (c) 40 × 240 × 40
(a) 4 × 24 × 4
(b) 8 × 48 × 8
(c) 40 × 240 × 40
Pressure (Pa)
Demonstrates the robustness of the numerical scheme
,
E = 17 MPa, ν = 0.45, αCFL = 0.3 and Ω = 105 rad/s.

4.6 Twisting Column
Mesh convergence
X, x
Y, y
(−0.5, 0, 0.5)
(0.5, 6, −0.5)
Z, z
ω0 = [0, Ω sin(πY/2L), 0]T
L
Column height at t = 0.1 s
0 20 40 60 80
6
6.02
6.04
6.06
6.08
Cells along x and z axes
Height(m)
constrained−TOUCH
Demonstrates mesh convergence of the numerical scheme
,

4.6 Twisting Column
Constrained-TOUCH vs hyperelastic GLACE
X, x
Y, y
(−0.5, 0, 0.5)
(0.5, 6, −0.5)
Z, z
ω0 = [0, Ω sin(πY/2L), 0]T
L
[Twisting column - Comparison]
t = 0.1 s
(a) Constrained-TOUCH (b) GLACE (c) GLACE (Fine)
Pressure (Pa)
,

4.6 Twisting Column
Comparison of various alternative numerical schemes
t = 0.1 s
(a) C-TOUCH (b) P-TOUCH (c) GLACE (d) B-bar (e) upwind-VC (f) JST-VC (g) PG-FEM (h) Hu-Washizu
Pressure (Pa)
,

4.6 Twisting Column
Comparison of various alternative numerical schemes
t = 0.1 s
(a) C-TOUCH (b) P-TOUCH (c) B-bar (d) PG-FEM (e) Hu-Washizu (f) Q2-Q1 FEM
Pressure (Pa)
,

4.6 Twisting Column
Standard FVM update (penalised-TOUCH with ξF = 0)
t = 0.01 s t = 0.04 s t = 0.07 s t = 0.1 s t = 0.13 s t = 0.16 s
Pressure (Pa)
,

4.6 Twisting Column
Increased initial angular velocity
t = 0.02 s t = 0.04 s t = 0.06 s t = 0.08 s t = 0.1 s
Pressure (Pa)
,

4.7 Taylor Impact
von-Mises plasticity
X, x
Y, y
v0
(−0.0032, 0, 0)
(0.0032, 0.0324, 0)
Z, z
r0
[Taylor impact]
[Taylor impact - Radius]
Evolution of pressure wave
t = 0.1 µs t = 0.2 µs t = 0.3 µs t = 0.4 µs t = 0.5 µs t = 0.6 µs
Pressure (Pa)
Demonstrates the ability of the algorithm to simulate plastic behaviour.
Problem description: Hyperelastic-plastic material, ρ0 = 8930 kg/m3
, E = 117 GPa, ν = 0.35,
αCFL = 0.3, ¯τ0
y = 0.4 GPa, H = 0.1 GPa and v0 = −227 m/s.

4.7 Taylor Impact
Comparison of cell centred FVM schemes
X, x
Y, y
v0
(−0.003, 0, 0.003)
(0.003, 0.03, −0.003)
Z, z
t = 60 µs
(a) Constrained-TOUCH (b) Penalised-TOUCH (c) HyperelasticGLACE
Plastic strain p
, E = 117 GPa, ν = 0.35,
αCFL = 0.3, ¯τ0
y = 0.4 GPa, H = 0.1 GPa and v0 = −227 m/s.

4.7 Taylor Impact
X, x
Y, y
v0
(−0.003, 0, 0.003)
(0.003, 0.03, −0.003)
Z, z
t = 60 µs
(a) Constrained-TOUCH (b) Penalised-TOUCH (c) HyperelasticGLACE
Pressure (Pa)
, E = 117 GPa, ν = 0.35,
αCFL = 0.3, ¯τ0
y = 0.4 GPa, H = 0.1 GPa and v0 = −227 m/s.

4.7 Taylor Impact
X, x
Y, y
v0
(−0.003, 0, 0.003)
(0.003, 0.03, −0.003)
Z, z
x coordinate of the point X = [0.003, 0, 0]T
0 2 4 6 8
x 10
−5
3
4
5
6
7
8
x 10
−3
Time (sec)
x−coordinate(m)
Constrained−TOUCH (6x30x6)
Constrained−TOUCH (12x60x12)
Penalised−TOUCH (6x30x6)
Penalised−TOUCH (12x60x12)
Hyperelastic−GLACE (6x30x6)
, E = 117 GPa, ν = 0.35,
αCFL = 0.3, ¯τ0
y = 0.4 GPa, H = 0.1 GPa and v0 = −227 m/s.

4.8 Bar rebound
Contact problem
X, x
Y, y
v0
(−0.0032, 0, 0)
(0.0032, 0.0324, 0)
Z, z
r0
0.004
[Bar rebound]
t = 6 ms t = 9 ms t = 12 ms t = 18 ms t = 24 ms
Pressure (Pa)
Demonstrates the ability of the algorithm to simulate contact problems.
Problem description: Nearly incompressible neo-Hookean material, ρ0 = 8930 kg/m3
, E = 585 MPa,
ν = 0.45, αCFL = 0.3 and v0 = −150 m/s.

4.8 Bar rebound
Height evolution
X, x
Y, y
v0
(−0.0032, 0, 0)
(0.0032, 0.0324, 0)
Z, z
r0
0.004
y Displacement of the points X = [0, 0.0324, 0]T
and X = [0, 0, 0]T
0 0.5 1 1.5 2 2.5 3
x 10
−4
−20
−16
−12
−8
−4
0
4
8
x 10
−3
Time (sec)
yDispacement(m)
Top (2880 cells)
Top (23040 cells)
Bottom (2880 cells)
Bottom (23040 cells)
Demonstrates the ability of the algorithm to simulate contact problems.
, E = 585 MPa,
ν = 0.45, αCFL = 0.3 and v0 = −150 m/s.

4.9 Ring impact
Contact problem
X, x
Y, y
(−0.03, 0)
0.004
v0
(0, −0.04)
[Ring impact]
t = 10 µs t = 15 µs t = 20 µs
t = 25 µs t = 30 µs t = 35 µs
Pressure (Pa)
Demonstrates the ability of the algorithm to simulate contact behaviour.
, E = 1 MPa,
ν = 0.4, αCFL = 0.3 and v0 = −0.59 m/s.

4.9 Ring impact
Energy conservation
X, x
Y, y
(−0.03, 0)
0.004
v0
(0, −0.04)
Global total energy
0 0.01 0.02 0.03 0.04
0.25
0.3
0.35
0.4
0.45
0.5
Time (sec)
Energy(J)
Total energy
K.E + P.E (420 cells)
K.E + P.E (25760 cells)
Lower numerical dissipation with mesh reﬁnement.
, E = 1 MPa,
ν = 0.4, αCFL = 0.3 and v0 = −0.59 m/s.

4.9 Ring impact
0 0.01 0.02 0.03 0.04
−1.5
−1
−0.5
0
0.5
1
1.5
Time (sec)
Linearmomentum(kg.m.s
−1
)
px
(420 cells)
py
(420 cells)
pz
(420 cells)
p
x
(3660 cells)
py
(3660 cells)
pz
(3660 cells)
px
(25760 cells)
py
(25760 cells)
pz
(25760 cells)
0 0.01 0.02 0.03 0.04
−0.6
−0.4
−0.2
0
0.2
0.4
0.6
0.8
Time (sec)
Angularmomentum(N.m.s−1
)
Ax
(420 cells)
Ay
(420 cells)
Az
(420 cells)
Ax
(3660 cells)
A
y
(3660 cells)
Az
(3660 cells)
Ax
(25760 cells)
Ay
(25760 cells)
Az
(25760 cells)
Conservation of linear and angular momenta of the system.
, E = 1 MPa,
ν = 0.4, αCFL = 0.3 and v0 = −0.59 m/s.

4.10 Torus impact
Contact problem
[Torus impact]
Problem description: Neo-Hookean material, ρ0 = 1000 kg/m3
, E = 1 MPa, ν = 0.4, αCFL = 0.3
and v0 = −3 m/s.

1 Introduction
Motivation
OpenFOAM
Governing equations
4 Numerical results
5 Conclusions

Conclusions and further research
Conclusions
• Upwind FVM is presented for fast solid dynamic simulations within the OpenFOAM environment.
• Linear elements can be used without usual volumetric and bending locking.
• Velocities and stresses display the same rate of convergence.
On-going work
• Investigation into an advanced Roe’s Riemann solver with robust shock capturing algorithm.
• Extension to multiple body and self contact.
• Ability to handle tetrahedral elements.
Future work
• Implementation of an Arbitrary Lagrangian-Eulerian (ALE) formulation.
• Extension to Fluid-Struction Interaction (FSI) problems.

References
1. Haider J, Lee CH, Gil AJ, Bonet J. A first order hyperbolic framework for large strain computational solid dynamics: An
upwind cell centred Total Lagrangian scheme. International Journal for Numerical Methods in Engineering 2016; (In Press),
doi: 10.1002/nme.5293.
2. Bonet J, Gil AJ, Lee CH, Aguirre M, Ortigosa R. A first order hyperbolic framework for large strain computational solid
dynamics. Part I: Total Lagrangian isothermal elasticity. Computer Methods in Applied Mechanics and Engineering 2015;
283:689–732.
3. Aguirre M, Gil AJ, Bonet J, Lee CH. An upwind vertex centred Finite Volume solver for Lagrangian solid dynamics. Journal of
Computational Physics 2015; 300:387–422.
4. Aguirre M, Gil AJ, Bonet J, Carreño AA. A vertex centred finite volume Jameson–Schmidt–Turkel (JST) algorithm for a mixed
conservation formulation in solid dynamics. Journal of Computational Physics 2014; 259:672–699.
5. Gil AJ, Lee CH, Bonet J, Aguirre M. A stabilised Petrov–Galerkin formulation for linear tetrahedral elements in compressible,
nearly incompressible and truly incompressible fast dynamics. Computer Methods in Applied Mechanics and Engineering
2014; 276:659–690.
6. Lee CH, Gil AJ, Bonet J. Development of a stabilised Petrov–Galerkin formulation for conservation laws in Lagrangian fast
solid dynamics. Computer Methods in Applied Mechanics and Engineering 2014; 268:40–64.
7. Lee CH, Gil AJ, Bonet J. Development of a cell centred upwind finite volume algorithm for a new conservation law
formulation in structural dynamics. Computers and Structures 2013; 118:13–38.

References
APPENDIX

References
Eigenvalue structure
Vector of conserved variables:
U = p F E x
T
Flux vector:
FN = PN 1
ρ0
p ⊗ N 1
ρ0
t · p 0
T
Flux Jacobian matrix:
AN =
∂FN
∂U
=









03×3 CN 03×1 03×3
1
ρ0
IN 09×9 09×1 09×3
1
ρ0
PN 1
ρ0
pT
CN 0 01×3
03×3 03×9 03×1 03×3









;
where [CN ]ijJ = CiIjJNI
CiIjJ =
∂PiI
∂FjJ
[IN ]iIj = δijNI
Right eigenvector:
ANRα = cαRα where Rα = p
R
α F
R
α E
R
α x
R
α
T
Eigen decomposition:
cαp
R
α = CN : F
R
α
cαF
R
α =
1
ρ0
p
R
α ⊗ N
ρ0c
2
αp
R
α = CNN p
R
α
where [CNN ]ij = [C]iIjJNI NJ

References
Acoustic wave speeds
Neo-Hookean constitutive model:
CNN = −
2
3
µJ
−5/3
F [(HFN) ⊗ (FN) + (FN) ⊗ (HFN)]
+ µJ
−2/3
F I +
5
9
µJ
−8/3
F (F : F) + κ (HFN) ⊗ (HFN)
= γ1 m ⊗ m
∗
+ m
∗
⊗ m + γ2I + γ3m ⊗ m
Eigen-system:
ρ0c
2
αp
R
α = CNN p
R
α
= γ1 m ⊗ m
∗
+ m
∗
⊗ m + γ2I + γ3m ⊗ m p
R
α
Longitudinal wave speed:
c1,2 = ±cp
cp =
2γ1
JF
λ + γ3
JF
λ
2
+ γ2
ρ0
Shear wave speed:
c3,4 = c5,6 = ±cs
cs =
γ2
ρ0

References
Riemann solver
Upwind Riemann solver:
F
C
N = F
C
NAve
+ F
C
NStab
=
1
2
FN(U
−
f ) + FN(U
+
f ) −
1
2
U
+
f
U
−
f
|AN| dU
Upwinding stabilisation
Decomposition of ﬂux Jacobian matrix:
|AN| =
1
2
6
α=1
|cα|RαL
T
α
=
1
2
(R1, . . . , R6)








cp 0 0 0 0 0
0 cp 0 0 0 0
0 0 cs 0 0 0
0 0 0 cs 0 0
0 0 0 0 cs 0
0 0 0 0 0 cs












LT
1
.
.
.
LT
6




=
1
2
cp(R1L
T
1 + R2L
T
2 ) + cs(R3L
T
3 + R4L
T
4 + R5L
T
5 + R6L
T
6 )

References
Pressure instabilities?
t = 0.125 s
(a) Cell values (4) (b) Nodal values (4) (c) Cell values (6) (d) Cell values (8) (e) Cell values (16)
Pressure (Pa)
,

A first order hyperbolic framework for large strain computational computational solid dynamics in OpenFOAM

Recommended

Recommended

More Related Content

What's hot

What's hot (18)

Viewers also liked

Viewers also liked (20)

Similar to A first order hyperbolic framework for large strain computational computational solid dynamics in OpenFOAM

Similar to A first order hyperbolic framework for large strain computational computational solid dynamics in OpenFOAM (20)

Recently uploaded

Recently uploaded (20)

A first order hyperbolic framework for large strain computational computational solid dynamics in OpenFOAM