The document describes a numerical methodology for simulating large strain solid dynamics using OpenFOAM. It proposes using a total Lagrangian formulation and first-order conservation laws similar to computational fluid dynamics to model solid mechanics problems involving large deformations. A cell-centered finite volume method is used for spatial discretization along with Riemann solvers and linear reconstruction to capture fluxes. A two-stage Runge-Kutta scheme is employed for time integration. Results are presented demonstrating the method's ability to handle problems involving mesh convergence, enhanced reconstruction, highly nonlinear behavior, plasticity, contact, unstructured meshes, and complex geometries.

1. Introduction Governing equations Numerical methodology Results Conclusions
Large strain solid dynamics in OpenFOAM
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
The 4th Annual OpenFOAM User Conference (11th
- 13th
October 2016)
12 th
October 2016
http://www.jibranhaider.weebly.com
Funded by the Erasmus Mundus SEED PhD Programme and ESI Group
Jibran Haider (Swansea University, UK & UPC, Spain) 4th OpenFOAM User Conference (Cologne, Germany) 1

2. Introduction Governing equations Numerical methodology Results Conclusions
Research group at Swansea University
Dr. Antonio J. Gil
Associate Professor
Dr. Chun Hean Lee
Research Fellow
Prof. Javier Bonet
University of Greenwich
Prof. Antonio Huerta
UPC BarcelonaTech
Dr. Rogelio
Ortigosa
Postdoc
Jibran Haider
Research Assistant
Osama I.
Hassan
Research Assistant
Roman Poya
Research Assistant
Emilio G. Blanco
Research Assistant
Ataollah
Ghavamian
Research Assistant
Jibran Haider (Swansea University, UK & UPC, Spain) 4th OpenFOAM User Conference (Cologne, Germany) 2

3. Introduction Governing equations Numerical methodology Results Conclusions
Scheme of presentation
1. Introduction
2. Governing equations
3. Numerical methodology
4. Results
5. Conclusions
Jibran Haider (Swansea University, UK & UPC, Spain) 4th OpenFOAM User Conference (Cologne, Germany) 3

4. Introduction Governing equations Numerical methodology Results Conclusions
Scheme of presentation
1. Introduction
2. Governing equations
3. Numerical methodology
4. Results
5. Conclusions
Jibran Haider (Swansea University, UK & UPC, Spain) 4th OpenFOAM User Conference (Cologne, Germany) 4

5. Introduction Governing equations Numerical methodology Results Conclusions
Fast transient dynamics
Objectives
• Simulate fast-transient solid dynamic problems.
• Develop an industry-driven library of low order numerical
schemes.
Solid dynamics in OpenFOAM [Jasak & Weller, 2000]
× Standard displacement based implicit dynamics
× Linear elastic material with small strain deformation
× Locking in nearly incompressible scenarios
× First order convergence for stresses and strains
× Poor performance in shock dominated scenarios
OpenFOAM solid mechanics community [Ivankovic et al.]
• [Cardiff et al., 2012; 2014; 2016] −→ displacement based + pressure instabilities +
moderate strains + ....
Jibran Haider (Swansea University, UK & UPC, Spain) 4th OpenFOAM User Conference (Cologne, Germany) 5

6. Introduction Governing equations Numerical methodology Results Conclusions
Proposed solid formulation
• 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.
TOUCH scheme
[Haider et al., 2016; Lee et al., 2013]
Mixed explicit dynamics
Complex constitutive models
Large strain deformation
No bending and volumtric locking
Second order convergence for stresses and
strains
v = 100 m/s
(0.5, 0.5, 0.5)
(−0.5, −0.5, −0.5)
[Punch cube]
Aim is to bridge the gap between CFD and computational solid dynamics.
Jibran Haider (Swansea University, UK & UPC, Spain) 4th OpenFOAM User Conference (Cologne, Germany) 6

7. Introduction Governing equations Numerical methodology Results Conclusions
Scheme of presentation
1. Introduction
2. Governing equations
3. Numerical methodology
4. Results
5. Conclusions
Jibran Haider (Swansea University, UK & UPC, Spain) 4th OpenFOAM User Conference (Cologne, Germany) 7

8. Introduction Governing equations Numerical methodology Results 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.
Jibran Haider (Swansea University, UK & UPC, Spain) 4th OpenFOAM User Conference (Cologne, Germany) 8

9. Introduction Governing equations Numerical methodology Results Conclusions
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.
Jibran Haider (Swansea University, UK & UPC, Spain) 4th OpenFOAM User Conference (Cologne, Germany) 9

10. Introduction Governing equations Numerical methodology Results Conclusions
Scheme of presentation
1. Introduction
2. Governing equations
3. Numerical methodology
Spatial discretisation
Flux computation
Involutions
Evolution
4. Results
5. Conclusions
Jibran Haider (Swansea University, UK & UPC, Spain) 4th OpenFOAM User Conference (Cologne, Germany) 10

11. Introduction Governing equations Numerical methodology Results Conclusions
Scheme of presentation
1. Introduction
2. Governing equations
3. Numerical methodology
Spatial discretisation
Flux computation
Involutions
Evolution
4. Results
5. Conclusions
Jibran Haider (Swansea University, UK & UPC, Spain) 4th OpenFOAM User Conference (Cologne, Germany) 11

12. Introduction Governing equations Numerical methodology Results Conclusions
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
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





Jibran Haider (Swansea University, UK & UPC, Spain) 4th OpenFOAM User Conference (Cologne, Germany) 12

13. Introduction Governing equations Numerical methodology Results Conclusions
Scheme of presentation
1. Introduction
2. Governing equations
3. Numerical methodology
Spatial discretisation
Flux computation
Involutions
Evolution
4. Results
5. Conclusions
Jibran Haider (Swansea University, UK & UPC, Spain) 4th OpenFOAM User Conference (Cologne, Germany) 13

14. Introduction Governing equations Numerical methodology Results 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
Jibran Haider (Swansea University, UK & UPC, Spain) 4th OpenFOAM User Conference (Cologne, Germany) 14

15. Introduction Governing equations Numerical methodology Results Conclusions
Acoustic Riemann solver
Jump condition for linear momentum
c p = t
Normal jump → cp pn = tn
Tangential jump → cs pt = tt
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
How do we obtain U−,+
?
Jibran Haider (Swansea University, UK & UPC, Spain) 4th OpenFOAM User Conference (Cologne, Germany) 15

16. Introduction Governing equations Numerical methodology Results Conclusions
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.
Jibran Haider (Swansea University, UK & UPC, Spain) 4th OpenFOAM User Conference (Cologne, Germany) 16

17. Introduction Governing equations Numerical methodology Results Conclusions
Linear reconstruction procedure
Gradient operator:
• Classical least squares minimisation procedure.
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.
Jibran Haider (Swansea University, UK & UPC, Spain) 4th OpenFOAM User Conference (Cologne, Germany) 17

18. Introduction Governing equations Numerical methodology Results Conclusions
Scheme of presentation
1. Introduction
2. Governing equations
3. Numerical methodology
Spatial discretisation
Flux computation
Involutions
Evolution
4. Results
5. Conclusions
Jibran Haider (Swansea University, UK & UPC, Spain) 4th OpenFOAM User Conference (Cologne, Germany) 18

19. Introduction Governing equations Numerical methodology Results 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).
Jibran Haider (Swansea University, UK & UPC, Spain) 4th OpenFOAM User Conference (Cologne, Germany) 19

20. Introduction Governing equations Numerical methodology Results Conclusions
Scheme of presentation
1. Introduction
2. Governing equations
3. Numerical methodology
Spatial discretisation
Flux computation
Involutions
Evolution
4. Results
5. Conclusions
Jibran Haider (Swansea University, UK & UPC, Spain) 4th OpenFOAM User Conference (Cologne, Germany) 20

21. Introduction Governing equations Numerical methodology Results 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.
Jibran Haider (Swansea University, UK & UPC, Spain) 4th OpenFOAM User Conference (Cologne, Germany) 21

22. Introduction Governing equations Numerical methodology Results Conclusions
Scheme of presentation
1. Introduction
2. Governing equations
3. Numerical methodology
4. Results
Mesh convergence
Enhanced reconstruction
Highly non-linear problem
Von-Mises plasticity
Contact problems
Unstructured meshes
Complex geometries
5. Conclusions
Jibran Haider (Swansea University, UK & UPC, Spain) 4th OpenFOAM User Conference (Cologne, Germany) 22

23. Introduction Governing equations Numerical methodology Results Conclusions
Scheme of presentation
1. Introduction
2. Governing equations
3. Numerical methodology
4. Results
Mesh convergence
Enhanced reconstruction
Highly non-linear problem
Von-Mises plasticity
Contact problems
Unstructured meshes
Complex geometries
5. Conclusions
Jibran Haider (Swansea University, UK & UPC, Spain) 4th OpenFOAM User Conference (Cologne, Germany) 23

24. Introduction Governing equations Numerical methodology Results 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





Jibran Haider (Swansea University, UK & UPC, Spain) 4th OpenFOAM User Conference (Cologne, Germany) 24
Problem description: Unit side cube, linear elastic material, ρ0 = 1100 kg/m3
, E = 17 MPa, ν = 0.3
[Haider et al., 2016] and αCFL = 0.3.
[Aguirre et al., 2014]

25. Introduction Governing equations Numerical methodology Results Conclusions
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.
Jibran Haider (Swansea University, UK & UPC, Spain) 4th OpenFOAM User Conference (Cologne, Germany) 25
Problem description: Unit side cube, linear elastic material, ρ0 = 1100 kg/m3
, E = 17 MPa, ν = 0.3
[Haider et al., 2016] and αCFL = 0.3.
[Aguirre et al., 2014]

26. Introduction Governing equations Numerical methodology Results Conclusions
Scheme of presentation
1. Introduction
2. Governing equations
3. Numerical methodology
4. Results
Mesh convergence
Enhanced reconstruction
Highly non-linear problem
Von-Mises plasticity
Contact problems
Unstructured meshes
Complex geometries
5. Conclusions
Jibran Haider (Swansea University, UK & UPC, Spain) 4th OpenFOAM User Conference (Cologne, Germany) 26

27. Introduction Governing equations Numerical methodology Results Conclusions
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.
Jibran Haider (Swansea University, UK & UPC, Spain) 4th OpenFOAM User Conference (Cologne, Germany) 27
Problem description: Nearly incompressible hyperelastic neo-Hookean material, ρ0 = 1100 kg/m3
,
[Haider et al., 2016] E = 17 MPa, ν = 0.45, αCFL = 0.3 and V = 10 m/s.

28. Introduction Governing equations Numerical methodology Results Conclusions
Bending dominated scenario
Time = 0.5 s
(a) OpenFOAM least square gradient (b) Enhanced least square gradient
Pressure (Pa)
Jibran Haider (Swansea University, UK & UPC, Spain) 4th OpenFOAM User Conference (Cologne, Germany) 28
Problem description: Nearly incompressible hyperelastic neo-Hookean material, ρ0 = 1100 kg/m3
,
[Haider et al., 2016] E = 17 MPa, ν = 0.45, αCFL = 0.3 and V = 10 m/s.

29. Introduction Governing equations Numerical methodology Results Conclusions
Scheme of presentation
1. Introduction
2. Governing equations
3. Numerical methodology
4. Results
Mesh convergence
Enhanced reconstruction
Highly non-linear problem
Von-Mises plasticity
Contact problems
Unstructured meshes
Complex geometries
5. Conclusions
Jibran Haider (Swansea University, UK & UPC, Spain) 4th OpenFOAM User Conference (Cologne, Germany) 29

30. Introduction Governing equations Numerical methodology Results 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 - Refinement]
[Twisting column - Comparison]
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
Jibran Haider (Swansea University, UK & UPC, Spain) 4th OpenFOAM User Conference (Cologne, Germany) 30
Problem description: Nearly incompressible neo-Hookean material, ρ0 = 1100 kg/m3
, E = 17 MPa,
[Haider et al., 2016] ν = 0.45, αCFL = 0.3 and Ω = 105 rad/s.
[Gil et al., 2014]

31. Introduction Governing equations Numerical methodology Results Conclusions
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)
Jibran Haider (Swansea University, UK & UPC, Spain) 4th OpenFOAM User Conference (Cologne, Germany) 31
Problem description: Nearly incompressible hyperelastic neo-Hookean material, ρ0 = 1100 kg/m3
,
[Haider et al., 2016] E = 17 MPa, ν = 0.495, αCFL = 0.3 and Ω = 105 rad/s.
[Lee et al., 2016]

32. Introduction Governing equations Numerical methodology Results Conclusions
Scheme of presentation
1. Introduction
2. Governing equations
3. Numerical methodology
4. Results
Mesh convergence
Enhanced reconstruction
Highly non-linear problem
Von-Mises plasticity
Contact problems
Unstructured meshes
Complex geometries
5. Conclusions
Jibran Haider (Swansea University, UK & UPC, Spain) 4th OpenFOAM User Conference (Cologne, Germany) 32

33. Introduction Governing equations Numerical methodology Results Conclusions
Taylor impact
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.
Jibran Haider (Swansea University, UK & UPC, Spain) 4th OpenFOAM User Conference (Cologne, Germany) 33
Problem description: Hyperelastic-plastic material, ρ0 = 8930 kg/m3
, E = 117 GPa, ν = 0.35,
[Aguirre et al., 2014] αCFL = 0.3, ¯τ0
y = 0.4 GPa, H = 0.1 GPa and v0 = −227 m/s.
[Lee et al., 2014]

34. Introduction Governing equations Numerical methodology Results Conclusions
Scheme of presentation
1. Introduction
2. Governing equations
3. Numerical methodology
4. Results
Mesh convergence
Enhanced reconstruction
Highly non-linear problem
Von-Mises plasticity
Contact problems
Unstructured meshes
Complex geometries
5. Conclusions
Jibran Haider (Swansea University, UK & UPC, Spain) 4th OpenFOAM User Conference (Cologne, Germany) 34

35. Introduction Governing equations Numerical methodology Results 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.
Jibran Haider (Swansea University, UK & UPC, Spain) 4th OpenFOAM User Conference (Cologne, Germany) 35
Problem description: Nearly incompressible neo-Hookean material, ρ0 = 8930 kg/m3
, E = 585 MPa,
[Lahiri et al., 2010] ν = 0.45, αCFL = 0.3 and v0 = −150 m/s.

36. Introduction Governing equations Numerical methodology Results Conclusions
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.
Jibran Haider (Swansea University, UK & UPC, Spain) 4th OpenFOAM User Conference (Cologne, Germany) 36
Problem description: Nearly incompressible neo-Hookean material, ρ0 = 8930 kg/m3
, E = 585 MPa,
[Lahiri et al., 2010] ν = 0.45, αCFL = 0.3 and v0 = −150 m/s.

37. Introduction Governing equations Numerical methodology Results Conclusions
Torus impact
[Torus impact]
t = 2 ms t = 4 ms t = 8 ms
t = 17 ms t = 28 ms t = 28 ms
Pressure (Pa)
Jibran Haider (Swansea University, UK & UPC, Spain) 4th OpenFOAM User Conference (Cologne, Germany) 37
Problem description: Neo-Hookean material, ρ0 = 1000 kg/m3
, E = 1 MPa, ν = 0.4, αCFL = 0.3 and
v0 = −3 m/s.

38. Introduction Governing equations Numerical methodology Results Conclusions
Scheme of presentation
1. Introduction
2. Governing equations
3. Numerical methodology
4. Results
Mesh convergence
Enhanced reconstruction
Highly non-linear problem
Von-Mises plasticity
Contact problems
Unstructured meshes
Complex geometries
5. Conclusions
Jibran Haider (Swansea University, UK & UPC, Spain) 4th OpenFOAM User Conference (Cologne, Germany) 38

39. Introduction Governing equations Numerical methodology Results Conclusions
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.
Jibran Haider (Swansea University, UK & UPC, Spain) 4th OpenFOAM User Conference (Cologne, Germany) 39
Problem description: Unit side square, nearly incompressible hyperelastic neo-Hookean material,
[Haider et al., 2016] ρ0 = 1000 kg/m3
, E = 17 MPa, ν = 0.45 and αCFL = 0.3 and Ω = 105 rad/s.

40. Introduction Governing equations Numerical methodology Results Conclusions
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
u
y
structured
u
x
unstructured
u
y
unstructured
Demonstrates the ability of the framework to handle unstructured grids.
Jibran Haider (Swansea University, UK & UPC, Spain) 4th OpenFOAM User Conference (Cologne, Germany) 40
Problem description: Unit side square, nearly incompressible hyperelastic neo-Hookean material,
[Haider et al., 2016] ρ0 = 1000 kg/m3
, E = 17 MPa, ν = 0.45, αCFL = 0.3 and Ω = 105 rad/s.

41. Introduction Governing equations Numerical methodology Results Conclusions
Flapping device
1. Introduction
2. Governing equations
3. Numerical methodology
4. Results
Mesh convergence
Enhanced reconstruction
Highly non-linear problem
Von-Mises plasticity
Contact problems
Unstructured meshes
Complex geometries
5. Conclusions
Jibran Haider (Swansea University, UK & UPC, Spain) 4th OpenFOAM User Conference (Cologne, Germany) 41

42. Introduction Governing equations Numerical methodology Results Conclusions
Flapping device
t = 0 ms t = 25 ms t = 50 ms t = 75 ms
t = 100 ms t = 125 ms t = 175 ms t = 200 ms
Pressure (Pa)
[Flapping device]
Jibran Haider (Swansea University, UK & UPC, Spain) 4th OpenFOAM User Conference (Cologne, Germany) 42
Problem description: Nearly incompressible hyperelastic neo-Hookean material, ρ0 = 1000 kg/m3
,
E = 17 MPa, ν = 0.45, αCFL = 0.3.

43. Introduction Governing equations Numerical methodology Results Conclusions
Complex twisting
[Complex twisting]
t = 5 ms t = 10 ms t = 15 ms
t = 20 ms t = 25 ms t = 30 ms
Pressure (Pa)
Jibran Haider (Swansea University, UK & UPC, Spain) 4th OpenFOAM User Conference (Cologne, Germany) 43
Problem description: Nearly incompressible hyperelastic neo-Hookean material, ρ0 = 1000 kg/m3
,
E = 17 MPa, ν = 0.45, αCFL = 0.3.

44. Introduction Governing equations Numerical methodology Results Conclusions
Scheme of presentation
1. Introduction
2. Governing equations
3. Numerical methodology
4. Results
5. Conclusions
Jibran Haider (Swansea University, UK & UPC, Spain) 4th OpenFOAM User Conference (Cologne, Germany) 44

45. Introduction Governing equations Numerical methodology Results Conclusions
Conclusions and on-going work
Conclusions
• Upwind cell centred 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.
• Extension to fluid-structure interaction problems.
Jibran Haider (Swansea University, UK & UPC, Spain) 4th OpenFOAM User Conference (Cologne, Germany) 45

46. Introduction Governing equations Numerical methodology Results Conclusions
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.
• 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 (2016); 311: 71-111.
• 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, O. I. Hassan, J. Bonet and S. Kulasegaram. "An efficient Streamline Upwind Petrov-Galerkin
Smooth Particle Hydrodynamics algorithm for large strain explicit fast dynamics, CMAME (2016).
In preparation
• J. Haider, C. H. Lee, A. J. Gil, A. Huerta and J. Bonet. "Contact dynamics in OpenFOAM, JCP.
• A. J. Gil, J. Bonet, C. H. Lee, J. Haider and A. Huerta. "Adapted Roe’s Riemann solver in explicit fast solid
dynamics, JCP.
More information at: http://www.jibranhaider.weebly.com/research
Jibran Haider (Swansea University, UK & UPC, Spain) 4th OpenFOAM User Conference (Cologne, Germany) 46