Large strain computational solid dynamics: An upwind cell centred Finite Volume Method

Introduction Governing equations Numerical methodology Results Conclusions
Large strain computational solid dynamics:
An upwind cell centred Finite Volume Method
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
Laboratory of Computational Methods and Numerical Analysis (LaCàN),
Universitat Politèchnica de Catalunya (UPC BarcelonaTech), Spain
c
University of Greenwich, London, UK
World Congress in Computational Mechanics (24th
- 29th
July 2016)
MS 703: Advances in Finite Element Methods for Tetrahedral Mesh Computations
http://www.jibranhaider.weebly.com
Funded by the Erasmus Mundus Programme and International Association for Computational Mechanics
Jibran Haider (Swansea University, UK & UPC, Spain) WCCM XII & APCOM VI (Seoul, Korea) 1

Scheme of presentation
1. Introduction
2. Governing equations
3. Numerical methodology
4. Results
5. Conclusions

Fast transient solid dynamics
Displacement based FEM/FVM formulations
• Linear tetrahedral elements suffer from:
× Locking in nearly incompressible materials.
× First order for stresses and strains.
× Poor performance in shock scenarios.
Proposed mixed formulation [Haider et al., 2016]
• First order conservation laws similar to the one
used in CFD community.
• Entitled TOtal Lagrangian Upwind Cell-centred
FVM for Hyperbolic conservation laws (TOUCH).
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-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
Upwind FVM
Aim is to bridge the gap between CFD and computational solid dynamics.

1. Introduction
4. Results
5. Conclusions

Total Lagrangian formulation
Conservation laws
• Linear momentum
∂p
∂t
= 0 · P(F) + ρ0b; p = ρ0v
• Deformation gradient
∂F
∂t
= 0 ·
1
ρ0
p ⊗ I ; CURL F = 0
Additional equations
• Total energy
∂E
∂t
= 0 ·
1
ρ0
PT
p − Q + s
An appropriate constitutive model is required to close the system.

Hyperbolic system
First order conservation laws
∂U
∂t
= 0 · F(U) + S
U =




p
F
E



 ; F =




P(F)
1
ρ0
p ⊗ I
1
ρ0
PT p − Q



 ; S =




ρ0b
0
s




• Geometry update
∂x
∂t
=
1
ρ0
p; x = X + u
Adapt CFD technology to the proposed formulation.
Develop an efﬁcient low order numerical scheme for transient solid dynamics.

1. Introduction
Spatial discretisation
Flux computation
Involutions
Evolution
4. Results
5. Conclusions

1. Introduction
Flux computation
Involutions
Evolution
4. Results
5. Conclusions

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
FINI
FN
dA (Gauss Divergence theorem)
≈
1
Ωe
0 f∈Λf
e
FC
Nef
Cef
e FC
Ne f
Ce f Ωe
0
Traditional cell centred Finite Volume Method
dUe
dt
=
1
Ωe
0



f∈Λf
e
FC
Nef
Cef


 ; FC
Nef
=





tC
1
ρ0
pC ⊗ N
1
ρ0
tC · pC






1. Introduction
Flux computation
Involutions
Evolution
4. Results
5. Conclusions

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
PT
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
Jump condition for linear momentum
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
Upwinding numerical stabilisation
p
C
=
c−
p p−
n + c+
p p+
n
c−
p + c+
p
+
c−
s p−
t + c+
s p+
t
c−
s + c+
s
pC
Ave
+
t+
n − t−
n
c−
p + c+
p
+
t+
t − t−
t
c−
s + c+
s
pC
Stab
t
C
=
c+
p t−
n + c−
p t+
n
c−
p + c+
p
+
c+
s t−
t + c−
s t+
t
c−
s + c+
s
tC
Ave
+
c−
p c+
p (p+
n − p−
n )
c−
p + c+
p
+
c−
s c+
s (p+
t − p−
t )
c−
s + c+
s
tC
Stab
Linear reconstruction procedure + limiter (monotonicity) for U−,+
.

1. Introduction
Flux computation
Involutions
Evolution
4. Results
5. Conclusions

Godunov-type FVM
Standard FV update (CURL F = 0)
dFe
dt
=
1
Ωe
0
f∈Λ
f
e
pC
f
ρ0
⊗ Cef X
Constrained FV update (CURL F = 0)
[Dedner et al., 2002; Lee et al., 2013]
dFe
dt
=
1
Ωe
0
f∈Λ
f
e
˜pC
f
ρ0
⊗ Cef
• Algorithm is entitled ’C-TOUCH’.
pe
pC
f −→
˜pe
Ge

˜pC
f
←−
pa
Constrained transport schemes are widely used in Magnetohydrodynamics (MHD).

1. Introduction
Flux computation
Involutions
Evolution
4. Results
5. Conclusions

Time integration
Two stage Runge-Kutta time integration
1st
RK stage −→ U∗
e = Un
e + ∆t ˙U
n
e(Un
e, tn
)
2nd
RK stage −→ U∗∗
e = U∗
e + ∆t ˙U
∗
e (U∗
e , tn+1
)
Un+1
e =
1
2
(Un
e + U∗∗
e )
with stability constraint:
∆t = αCFL
hmin
cp,max
; cp,max = max
a
(ca
p)
An explicit Total Variation Diminishing Runge-Kutta time integration scheme.
Monolithic time update for geometry.

1. Introduction
4. Results
Mesh convergence
Highly non-linear problem
Von-Mises plasticity
Contact problems
5. Conclusions

1. Introduction
4. Results
Mesh convergence
Contact problems
5. Conclusions

Low dispersion cube
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 at:
X = 0, Y = 0, Z = 0
2. Skew-symmetric 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





Problem description: Unit side cube, linear elastic material, ρ0 = 1100 kg/m3
, E = 17 MPa, ν = 0.3
and αCFL = 0.3.

Low dispersion cube: Mesh convergence
Velocity at t = 0.004 s
10
−2
10
−1
10
0
10
−7
10
−6
10
−5
10
−4
Grid Size (m)
L2NormError
vx
vy
vZ
Slope = 2
Stress at t = 0.004 s
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 for velocities and stresses.
Problem description: Unit side cube, linear elastic material, ρ0 = 1100 kg/m3
, E = 17 MPa, ν = 0.3
and αCFL = 0.3.

1. Introduction
4. Results
Mesh convergence
Contact problems
5. Conclusions

Twisting column
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]
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
Problem description: Nearly incompressible neo-Hookean material, ρ0 = 1100 kg/m3
, E = 17 MPa,
ν = 0.45, αCFL = 0.3 and Ω = 105 rad/s.

Comparison of various alternative numerical schemes
t = 0.1 s
C-TOUCH P-TOUCH B-bar Taylor Hood PG-FEM Hu-Washizu JST-SPH SUPG-SPH
Pressure (Pa)
Problem description: Nearly incompressible hyperelastic neo-Hookean material, ρ0 = 1100 kg/m3
,
E = 17 MPa, ν = 0.495, αCFL = 0.3 and Ω = 105 rad/s.

1. Introduction
4. Results
Mesh convergence
Contact problems
5. Conclusions

Taylor impact
X, x
Y, y
v0
(−0.0032, 0, 0)
(0.0032, 0.0324, 0)
Z, z
r0
[Taylor impact]
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.

1. Introduction
4. Results
Mesh convergence
Contact problems
5. Conclusions

Bar rebound
X, x
Y, y
v0
(−0.0032, 0, 0)
(0.0032, 0.0324, 0)
Z, z
r0
0.004
[Bar rebound]
t = 3 ms t = 6 ms t = 12 ms t = 18 ms t = 27 ms
Pressure (Pa)
Demonstrates the ability of the algorithm to simulate contact problems.
, E = 585 MPa,
[Lahiri et al., 2010] ν = 0.45, αCFL = 0.3 and v0 = −150 m/s.

Bar rebound
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,
[Lahiri et al., 2010] ν = 0.45, αCFL = 0.3 and v0 = −150 m/s.

Torus impact
[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
4. Results
5. Conclusions

Conclusions and further research
Conclusions
• Upwind CC-FVM is presented for fast solid dynamic simulations within the OpenFOAM
environment.
• Linear elements can be used without usual 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.

References
Published / accepted
• J. Haider, C. H. Lee, A. J. Gil and J. Bonet. "A first order hyperbolic framework for large strain computational solid
dynamics: An upwind cell centred Total Lagrangian scheme", IJNME (2016), DOI: 10.1002/nme.5293.
• A. J. Gil, C. H. Lee, J. Bonet and R. Ortigosa. "A first order hyperbolic framework for large strain computational solid
dynamics. Part II: Total Lagrangian compressible, nearly incompressible and truly incompressible elasticity",
CMAME (2016); 300: 146-181.
• J. Bonet, A. J. Gil, C. H. Lee, M. Aguirre and R. Ortigosa. "A first order hyperbolic framework for large strain
computational solid dynamics. Part I: Total Lagrangian isothermal elasticity", CMAME (2015); 283: 689-732.
• M. Aguirre, A. J. Gil, J. Bonet and C. H. Lee. "An upwind vertex centred Finite Volume solver for Lagrangian solid
dynamics", JCP (2015); 300: 387-422.
• C. H. Lee, A. J. Gil and J. Bonet. "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.
Under review
• C. H. Lee, A. J. Gil, G. Greto, S. Kulasegaram and J. Bonet. "A new Jameson-Schmidt-Turkel Smooth Particle
Hydrodynamics algorithm for large strain explicit fast dynamics, CMAME.
• C. H. Lee, A. J. Gil, J. Bonet and S. Kulasegaram. "An efficient Streamline Upwind Petrov-Galerkin Smooth Particle
Hydrodynamics algorithm for large strain explicit fast dynamics, CMAME.
In preparation
• J. Haider, C. H. Lee, A. J. Gil, A. Huerta and J. Bonet. "Contact dynamics in OpenFOAM, JCP.
• J. Bonet, A. J. Gil, C. H. Lee, A. Huerta and J. Haider. "Adapted Roe’s Riemann solver in explicit fast solid
dynamics, JCP.
http://www.jibranhaider.weebly.com/research

Large strain computational solid dynamics: An upwind cell centred Finite Volume Method

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