Large strain computational solid dynamics in open-source CFD software package in OpenFOAM.

1. Large strain solid dynamics in OpenFOAM
Jibran Haidera, b
, Dr. Chun Hean Leea
, Dr. Antonio J. Gila
,
Prof. Javier Bonetc
& Prof. Antonio Huertab
a
Zienkiewicz Centre for Computational Engineering, Swansea University, UK
b
Laboratori de C`alcul Num`eric (LaC`aN), UPC BarcelonaTech, Spain
c
University of Greenwich, London, UK
Research outline
Objectives:
• Simulate fast-transient solid dynamic problems.
• Develop a fast and efficient low order numerical
scheme.
Key features:
An upwind cell-centred FVM Total Lagrangian scheme (TOUCH).
Utilises an explicit Runge-Kutta time integrator.
Programmed in the open-source CFD software OpenFOAM.
Overcomes the shortcomings of linear tetrahedral
elements in standard displacement based
FEM/FVM formulations:
• Equal order of convergence for velocities and stresses.
• No volumetric locking for nearly incompressible materials.
• Excellent performance in bending and shock dominated
scenarios.
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 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
Q1-P0 FEM Proposed FVM
First order conservation laws
1. Linear momentum:
2. Deformation gradient:
3. Total energy:
d
dt Ω0
p dΩ0 =
∂Ω0
t dA +
Ω0
ρ0b dΩ0
d
dt Ω0
F dΩ0 =
∂Ω0
p
ρ0
⊗ N dA
d
dt Ω0
E dΩ0 =
∂Ω0
p
ρ0
· t dA −
∂Ω0
Q · N dA +
Ω0
s dΩ0
• Hyperbolic laws in differential form:
∂U
∂t
=
∂FI
∂XI
+ S, ∀ I = 1, 2, 3
Cell centred FVM discretisation
Standard face-based CC-FVM
e FC
Ne f
Ce f Ωe
0
dUe
dt
=
1
Ωe
0
f∈Λf
e
FC
Nef
(U−
f , U+
f ) Cef
Node-based CC-FVM
FC
Nea
Cea
Ωe
0
e
dUe
dt
=
1
Ωe
0 a∈Λa
e
FC
Nea
(U−
a , U+
a ) Cea
• Gradient calculation through least squares minimisation −→ Ge
• Satisfaction of monotonicity through Barth and Jespersen limiter −→ φe
• Linear reconstruction procedure for second order spatial accuracy −→ U+,−
(φe, Ge)
Lagrangian contact dynamics
Contact flux:
FC
N = FINI =




tC
1
ρ0
pC
⊗ N
1
ρ0
pC
· tC
− Q · N




Acoustic Riemann solver:
FC
N = FC
NAve
+ FC
NStab
=
1
2
FN (U−
f ) + FN (U+
f ) −
1
2
U+
f
U−
f
|AN | dU
Upwinding stabilisation
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
Explicit time integration
Total Variation Diminishing Runge-Kutta scheme:
1st
RK stage −→ Ue = Un
e + ∆t ˙U
n
e (Un
e , tn
)
2nd
RK stage −→ Ue = Ue + ∆t ˙Ue(Ue, tn+1
)
Un+1
e =
1
2
(Un
e + Ue )
with stability criterion:
∆t = αCFL
hmin
cmax
p
Numerical results
Shock scenario
6 7 8 9 10
x 10
−3
−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
Mesh convergence
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)
Structured vs Unstructured
Pressure (Pa)
Complex twisting
Pressure (Pa)
Flapping structure
Pressure (Pa)
Von Mises plasticity
Constrained-TOUCH Penalised-TOUCH Hyperelastic-GLACE
Plastic strain
Bar rebound
Pressure (Pa)
Torus impact
Pressure (Pa)
On-going work
1. An advanced Roe’s Riemann solver.
2. Robust shock capturing algorithm.
3. Ability to handle tetrahedral elements.
Future work
1. Extension to Fluid-Structure Interaction
(FSI) problems.
2. Implementation of Arbitrary
Lagrangian-Eulerian (ALE) formulation.
References
[1] 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, International Journal for Numerical Methods
in Engineering, 109(3) : 407–456, 2017.
[2] 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, 118 : 13–38, 2013.
Website: http://www.jibranhaider.weebly.com Email:{m.j.haider,c.h.lee,a.j.gil}@swansea.ac.uk