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Large strain solid dynamics in OpenFOAM
Jibran Haider a, b, Dr. Chun Hean Lee a, Dr. Antonio J. Gil a,Prof. Javier Bonet c & Prof. Antonio Huerta b
a Zienkiewicz Centre for Computational Engineering, Swansea University, UKb Laboratori de Calcul Numeric (LaCaN), 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:
X An upwind cell-centred FVM Total Lagrangian scheme (TOUCH).
X Utilises an explicit Runge-Kutta time integrator.
X Programmed in the open-source CFD software OpenFOAM.
X 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 dominatedscenarios.
0 0.5 1
0
0.5
1
1.5
X-Coordinate
Y-C
oord
inate
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-C
oord
inate
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
=∂F I
∂XI+ S, ∀ I = 1, 2, 3
Cell centred FVM discretisation
Standard face-based CC-FVM
e FCNe f
‖Ce f‖ Ωe0
dU e
dt=
1
Ωe0
∑f∈Λf
e
FCN ef
(U−f ,U+f ) ‖Cef‖
Node-based CC-FVM
FCNea
‖Cea‖
Ωe0
e
dU e
dt=
1
Ωe0
∑a∈Λa
e
FCN ea
(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:
FCN = F INI =
tC
1ρ0pC ⊗N
1ρ0pC · tC −Q ·N
Acoustic Riemann solver:
FCN = 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−sc+s
c+pc−p
Time t = 0
Time t
Explicit time integrationTotal Variation Diminishing Runge-Kutta scheme:
1st RK stage −→ U?e = Un
e + ∆t Un
e (Une , t
n)
2nd RK stage −→ U??e = U?
e + ∆t U?
e(U?e, t
n+1)
Un+1e =
1
2(Un
e + U??e )
with stability criterion:
∆t = αCFLhmin
cmaxp
Numerical results
Shock scenario
6 7 8 9 10
x 10−3
−7.5
−5
−2.5
0
2.5
5x 10
7
Time (sec)
Str
ess
(Pa)
AnalyticalTOUCH (1st order)TOUCH (2nd order w/o limiter)TOUCH (2nd order with limiter)JST VCFVM
Mesh convergence
10−2 10−1 100
10−8
10−7
10−6
10−5
10−4
10−3
Grid Size (m)
Str
ess
Err
or
slope = 1L1 norm (1st order)
L2 norm (1st order)
slope = 2L1 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 ArbitraryLagrangian-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
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