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Outline Introduction Reversible elastodynamics Numerical technique Numerical results Conclusions
A first order conservation law formulation for fastsolid dynamics in OpenFOAM
Jibran Haider a, Antonio J. Gil a, Javier Bonet a, Chun Hean Lee a, Antonio Huerta b
a Zienkiewicz Centre for Computational Engineering (ZC2E)College of Engineering, Swansea University, UK
b Laboratory of Computational Methods and Numerical Analysis (LaCáN)Polytechnic University of Catalonia, Spain
ACME 201523rd conference on Computational Mechanics, UK
Solids & Structures 5
Jibran Haider (ACME 2015, Swansea University, UK) 10th April 2015
Outline Introduction Reversible elastodynamics Numerical technique Numerical results Conclusions
Outline
1 IntroductionMotivationOpenFOAM
2 Reversible elastodynamicsGoverning equations
3 Numerical techniqueSpace-time discretisationContact fluxInvolution
4 Numerical resultsMesh convergence2D results3D results
5 Conclusions
Jibran Haider (ACME 2015, Swansea University, UK) 10th April 2015
Outline Introduction Reversible elastodynamics Numerical technique Numerical results Conclusions
Outline
1 IntroductionMotivationOpenFOAM
2 Reversible elastodynamicsGoverning equations
3 Numerical techniqueSpace-time discretisationContact fluxInvolution
4 Numerical resultsMesh convergence2D results3D results
5 Conclusions
Jibran Haider (ACME 2015, Swansea University, UK) 10th April 2015
Outline Introduction Reversible elastodynamics Numerical technique Numerical results Conclusions
Introduction
Motivation
• Fast solid dynamics
• Large strain deformation
• Shock propagation
• Impact mechanics
• Solid dynamics formulations
Displacementbased
formulations
Mixedformulation
• Incompressibility × X• Performance in
bending problems × X• Shock capturing
ability × X• Convergence of
stresses/strains × X• Conservation of
angular momentum X -
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
-0.5 0 0.5 1 1.5
0
0.5
1
1.5
X-Coordinate
Y-C
oord
inate
t=0.0006s
-5
0
5x 10
9
FEM
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
-0.5 0 0.5 1 1.5
0
0.5
1
1.5
X-Coordinate
Y-C
oord
inate
t=0.0006s
-5
0
5x 10
9
FVM
Jibran Haider (ACME 2015, Swansea University, UK) 10th April 2015
Outline Introduction Reversible elastodynamics Numerical technique Numerical results Conclusions
Introduction
Motivation
• Fast solid dynamics
• Large strain deformation
• Shock propagation
• Impact mechanics
• Solid dynamics formulations
Displacementbased
formulations
Mixedformulation
• Incompressibility × X• Performance in
bending problems × X• Shock capturing
ability × X• Convergence of
stresses/strains × X• Conservation of
angular momentum X -
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
-0.5 0 0.5 1 1.5
0
0.5
1
1.5
X-Coordinate
Y-C
oord
inate
t=0.0006s
-5
0
5x 10
9
FEM
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
-0.5 0 0.5 1 1.5
0
0.5
1
1.5
X-Coordinate
Y-C
oord
inate
t=0.0006s
-5
0
5x 10
9
FVM
Jibran Haider (ACME 2015, Swansea University, UK) 10th April 2015
Outline Introduction Reversible elastodynamics Numerical technique Numerical results Conclusions
Introduction
OpenFOAM
• An open source CFD software package in C++
• Based on Cell Centered Finite Volume Method
• Limited functionality for solid dynamics
v = 500 m/s
(0.5, 0.5, 0.5)
(−0.5,−0.5,−0.5)
Tensile rubber cube
Existing solid solver in OpenFOAM
× Linear elastic material
× Small strain deformation
Mixed Formulation in OpenFOAM
X Hyper elastic material
X Large strain deformation
[Animation - 3D]
Jibran Haider (ACME 2015, Swansea University, UK) 10th April 2015
Outline Introduction Reversible elastodynamics Numerical technique Numerical results Conclusions
Introduction
OpenFOAM
• An open source CFD software package in C++
• Based on Cell Centered Finite Volume Method
• Limited functionality for solid dynamics
v = 500 m/s
(0.5, 0.5, 0.5)
(−0.5,−0.5,−0.5)
Tensile rubber cube
Existing solid solver in OpenFOAM
× Linear elastic material
× Small strain deformation
Mixed Formulation in OpenFOAM
X Hyper elastic material
X Large strain deformation
[Animation - 3D]
Jibran Haider (ACME 2015, Swansea University, UK) 10th April 2015
Outline Introduction Reversible elastodynamics Numerical technique Numerical results Conclusions
Introduction
OpenFOAM
• An open source CFD software package in C++
• Based on Cell Centered Finite Volume Method
• Limited functionality for solid dynamics
v = 500 m/s
(0.5, 0.5, 0.5)
(−0.5,−0.5,−0.5)
Tensile rubber cube
Existing solid solver in OpenFOAM
× Linear elastic material
× Small strain deformation
Mixed Formulation in OpenFOAM
X Hyper elastic material
X Large strain deformation
[Animation - 3D]
Jibran Haider (ACME 2015, Swansea University, UK) 10th April 2015
Outline Introduction Reversible elastodynamics Numerical technique Numerical results Conclusions
Introduction
OpenFOAM
• An open source CFD software package in C++
• Based on Cell Centered Finite Volume Method
• Limited functionality for solid dynamics
v = 500 m/s
(0.5, 0.5, 0.5)
(−0.5,−0.5,−0.5)
Tensile rubber cube
Existing solid solver in OpenFOAM
× Linear elastic material
× Small strain deformation
Mixed Formulation in OpenFOAM
X Hyper elastic material
X Large strain deformation
[Animation - 3D]
Jibran Haider (ACME 2015, Swansea University, UK) 10th April 2015
Outline Introduction Reversible elastodynamics Numerical technique Numerical results Conclusions
Outline
1 IntroductionMotivationOpenFOAM
2 Reversible elastodynamicsGoverning equations
3 Numerical techniqueSpace-time discretisationContact fluxInvolution
4 Numerical resultsMesh convergence2D results3D results
5 Conclusions
Jibran Haider (ACME 2015, Swansea University, UK) 10th April 2015
Outline Introduction Reversible elastodynamics Numerical technique Numerical results Conclusions
Governing equations
First order conservation formulation
• Conservation of linear momentum:∂ρ0v∂t−∇0 · P(F) = ρ0b
• Conservation of deformation gradient:∂F∂t−∇0 · (v⊗ I) = 0
• Conservation of energy:∂E∂t−∇0 ·
(PT v− Q
)= s
• Or in standard form:
∂U∂t
+∇0 ·F(U) = S
A constitutive model is needed to complete the coupled system A first order hyperbolic system similar to the one in CFD Our aim is to develop low order numerical schemes for Total Lagrangian fast solid dynamics
Jibran Haider (ACME 2015, Swansea University, UK) 10th April 2015
Outline Introduction Reversible elastodynamics Numerical technique Numerical results Conclusions
Governing equations
First order conservation formulation
• Conservation of linear momentum:∂ρ0v∂t−∇0 · P(F) = ρ0b
• Conservation of deformation gradient:∂F∂t−∇0 · (v⊗ I) = 0
• Conservation of energy:∂E∂t−∇0 ·
(PT v− Q
)= s
• Or in standard form:
∂U∂t
+∇0 ·F(U) = S
A constitutive model is needed to complete the coupled system A first order hyperbolic system similar to the one in CFD Our aim is to develop low order numerical schemes for Total Lagrangian fast solid dynamics
Jibran Haider (ACME 2015, Swansea University, UK) 10th April 2015
Outline Introduction Reversible elastodynamics Numerical technique Numerical results Conclusions
Governing equations
First order conservation formulation
• Conservation of linear momentum:∂ρ0v∂t−∇0 · P(F) = ρ0b
• Conservation of deformation gradient:∂F∂t−∇0 · (v⊗ I) = 0
• Conservation of energy:∂E∂t−∇0 ·
(PT v− Q
)= s
• Or in standard form:
∂U∂t
+∇0 ·F(U) = S
A constitutive model is needed to complete the coupled system A first order hyperbolic system similar to the one in CFD Our aim is to develop low order numerical schemes for Total Lagrangian fast solid dynamics
Jibran Haider (ACME 2015, Swansea University, UK) 10th April 2015
Outline Introduction Reversible elastodynamics Numerical technique Numerical results Conclusions
Governing equations
First order conservation formulation
• Conservation of linear momentum:∂ρ0v∂t−∇0 · P(F) = ρ0b
• Conservation of deformation gradient:∂F∂t−∇0 · (v⊗ I) = 0
• Conservation of energy:∂E∂t−∇0 ·
(PT v− Q
)= s
• Or in standard form:
∂U∂t
+∇0 ·F(U) = S
A constitutive model is needed to complete the coupled system A first order hyperbolic system similar to the one in CFD Our aim is to develop low order numerical schemes for Total Lagrangian fast solid dynamics
Jibran Haider (ACME 2015, Swansea University, UK) 10th April 2015
Outline Introduction Reversible elastodynamics Numerical technique Numerical results Conclusions
Governing equations
First order conservation formulation
• Conservation of linear momentum:∂ρ0v∂t−∇0 · P(F) = ρ0b
• Conservation of deformation gradient:∂F∂t−∇0 · (v⊗ I) = 0
• Conservation of energy:∂E∂t−∇0 ·
(PT v− Q
)= s
• Or in standard form:
∂U∂t
+∇0 ·F(U) = S
A constitutive model is needed to complete the coupled system A first order hyperbolic system similar to the one in CFD Our aim is to develop low order numerical schemes for Total Lagrangian fast solid dynamics
Jibran Haider (ACME 2015, Swansea University, UK) 10th April 2015
Outline Introduction Reversible elastodynamics Numerical technique Numerical results Conclusions
Outline
1 IntroductionMotivationOpenFOAM
2 Reversible elastodynamicsGoverning equations
3 Numerical techniqueSpace-time discretisationContact fluxInvolution
4 Numerical resultsMesh convergence2D results3D results
5 Conclusions
Jibran Haider (ACME 2015, Swansea University, UK) 10th April 2015
Outline Introduction Reversible elastodynamics Numerical technique Numerical results Conclusions
Finite Volume Methodology
Space-time discretisation
• The conservation laws are spatially semi-discretised by astandard Cell Centered Finite Volume Method(CCFVM):
dU e
dt= −
1Ve
∑f∈e
[FCN ]f Af
• An explicit 2 step Total Variation DiminishingRunge-Kutta (TVD-RK) time integrator is utilised:
U∗n+1 = Un + ∆t Un
U∗n+2 = U∗n+1 + ∆t U∗n+1
Un+1 =12
(Un + U∗n+2
)with stability constraint:
∆t = αCFLhmin
Up,max
e
Ve
Af
[FCN ]f
Just one Gauss point per face!
Jibran Haider (ACME 2015, Swansea University, UK) 10th April 2015
Outline Introduction Reversible elastodynamics Numerical technique Numerical results Conclusions
Finite Volume Methodology
Space-time discretisation
• The conservation laws are spatially semi-discretised by astandard Cell Centered Finite Volume Method(CCFVM):
dU e
dt= −
1Ve
∑f∈e
[FCN ]f Af
• An explicit 2 step Total Variation DiminishingRunge-Kutta (TVD-RK) time integrator is utilised:
U∗n+1 = Un + ∆t Un
U∗n+2 = U∗n+1 + ∆t U∗n+1
Un+1 =12
(Un + U∗n+2
)with stability constraint:
∆t = αCFLhmin
Up,max
e
Ve
Af
[FCN ]f
Just one Gauss point per face!
Jibran Haider (ACME 2015, Swansea University, UK) 10th April 2015
Outline Introduction Reversible elastodynamics Numerical technique Numerical results Conclusions
Riemann Solver
Generalised Riemann Solver
• Rankine-Hugoniot relation:
U J ρ0v K = −J P KN
U J F K = −J v K⊗ N
U J E K = −J PT v K · N
where JU K = U+c − U
−c
• Linear reconstruction procedure to enhance thespatial discretisation:
U+,−c = Ue + Ge · (Xc − Xe)
where the local gradient operator is defined as:
Ge =
[∑α∈e
νeα ⊗ νeα
]−1 ∑α∈e
(Uα − Ue
deα
)νeα
() Time
, ,
()
, ,
,
Time = 0
= = −
= = −
,
Jibran Haider (ACME 2015, Swansea University, UK) 10th April 2015
Outline Introduction Reversible elastodynamics Numerical technique Numerical results Conclusions
Riemann Solver
Generalised Riemann Solver
• Rankine-Hugoniot relation:
U J ρ0v K = −J P KN
U J F K = −J v K⊗ N
U J E K = −J PT v K · N
where JU K = U+c − U
−c
• Linear reconstruction procedure to enhance thespatial discretisation:
U+,−c = Ue + Ge · (Xc − Xe)
where the local gradient operator is defined as:
Ge =
[∑α∈e
νeα ⊗ νeα
]−1 ∑α∈e
(Uα − Ue
deα
)νeα
() Time
, ,
()
, ,
,
Time = 0
= = −
= = −
,
Jibran Haider (ACME 2015, Swansea University, UK) 10th April 2015
Outline Introduction Reversible elastodynamics Numerical technique Numerical results Conclusions
Riemann Solver
Generalised Riemann Solver
• Rankine-Hugoniot relation:
U J ρ0v K = −J P KN
U J F K = −J v K⊗ N
U J E K = −J PT v K · N
where JU K = U+c − U
−c
• Linear reconstruction procedure to enhance thespatial discretisation:
U+,−c = Ue + Ge · (Xc − Xe)
where the local gradient operator is defined as:
Ge =
[∑α∈e
νeα ⊗ νeα
]−1 ∑α∈e
(Uα − Ue
deα
)νeα
() Time
, ,
()
, ,
,
Time = 0
= = −
= = −
,
Jibran Haider (ACME 2015, Swansea University, UK) 10th April 2015
Outline Introduction Reversible elastodynamics Numerical technique Numerical results Conclusions
Riemann Solver
Contact flux
• Contact flux at an interface:
FCN(U+,U−) =
−tC−vC ⊗ N−tC · vC
• Contact values for a homogeneous body(Up = U−p = U+
p , Us = U−s = U+s ):
vC =12
[v− + v+
]︸ ︷︷ ︸
unstable flux
+1
2ρ0
[1
Up(n ⊗ n) +
1Us
(I − n ⊗ n)] [
P+ − P−]
N︸ ︷︷ ︸stabilising term
tC =12
[P− + P+
]N︸ ︷︷ ︸
unstable flux
+ρ0
2[Up(n ⊗ n) + Us(I − n ⊗ n)] [v+ − v−]︸ ︷︷ ︸
stabilising term
v+ = 0, U+p = U+
s = ∞
v−(t)
v+n = 0, U+p = ∞, U+
s = 0
v−(t)
v+t = 0, U+p = 0, U+
s = ∞
v−(t)
U+p = U+
s = 0
v−(t)
t
Jibran Haider (ACME 2015, Swansea University, UK) 10th April 2015
Outline Introduction Reversible elastodynamics Numerical technique Numerical results Conclusions
Riemann Solver
Contact flux
• Contact flux at an interface:
FCN(U+,U−) =
−tC−vC ⊗ N−tC · vC
• Contact values for a homogeneous body(Up = U−p = U+
p , Us = U−s = U+s ):
vC =12
[v− + v+
]︸ ︷︷ ︸
unstable flux
+1
2ρ0
[1
Up(n ⊗ n) +
1Us
(I − n ⊗ n)] [
P+ − P−]
N︸ ︷︷ ︸stabilising term
tC =12
[P− + P+
]N︸ ︷︷ ︸
unstable flux
+ρ0
2[Up(n ⊗ n) + Us(I − n ⊗ n)] [v+ − v−]︸ ︷︷ ︸
stabilising term
v+ = 0, U+p = U+
s = ∞
v−(t)
v+n = 0, U+p = ∞, U+
s = 0
v−(t)
v+t = 0, U+p = 0, U+
s = ∞
v−(t)
U+p = U+
s = 0
v−(t)
t
Jibran Haider (ACME 2015, Swansea University, UK) 10th April 2015
Outline Introduction Reversible elastodynamics Numerical technique Numerical results Conclusions
Involution
Local constraint preserving scheme• Compatibility condition
[ Torrilhon(2005), Lee-Gil-Bonet(2013) ]:
∇0 × F = 0
• An adapted curl-free updated scheme:
dFe
dt=∑a∈e
[va ⊗∇0Na]
• Bilinear shape functions:
Na =18
(1 + ξξa)(1 + ηηa)(1 + µµa)
• Nodal velocity:
va = vRe + G(vC
) · (Xa − Xe)
where:
vRe =
1∑f∈e
dfe
∑f∈e
[dfevC
f
]; dfe = |XC − Xe|
G(vC) =
∑f∈e
νfe ⊗ νfe
−1 ∑f∈e
vCf − vR
e
dfe
νfe
∆X
∆Y
ve
1
2
3
vC
vaG (vC )
1 = Linear reconstruction + Riemann solver
2 = Velocity gradient (Least square minimisation)
3 = Interpolation/Extrapolation to nodes
Jibran Haider (ACME 2015, Swansea University, UK) 10th April 2015
Outline Introduction Reversible elastodynamics Numerical technique Numerical results Conclusions
Involution
Local constraint preserving scheme• Compatibility condition
[ Torrilhon(2005), Lee-Gil-Bonet(2013) ]:
∇0 × F = 0
• An adapted curl-free updated scheme:
dFe
dt=∑a∈e
[va ⊗∇0Na]
• Bilinear shape functions:
Na =18
(1 + ξξa)(1 + ηηa)(1 + µµa)
• Nodal velocity:
va = vRe + G(vC
) · (Xa − Xe)
where:
vRe =
1∑f∈e
dfe
∑f∈e
[dfevC
f
]; dfe = |XC − Xe|
G(vC) =
∑f∈e
νfe ⊗ νfe
−1 ∑f∈e
vCf − vR
e
dfe
νfe
∆X
∆Y
ve
1
2
3
vC
vaG (vC )
1 = Linear reconstruction + Riemann solver
2 = Velocity gradient (Least square minimisation)
3 = Interpolation/Extrapolation to nodes
Jibran Haider (ACME 2015, Swansea University, UK) 10th April 2015
Outline Introduction Reversible elastodynamics Numerical technique Numerical results Conclusions
Involution
Local constraint preserving scheme• Compatibility condition
[ Torrilhon(2005), Lee-Gil-Bonet(2013) ]:
∇0 × F = 0
• An adapted curl-free updated scheme:
dFe
dt=∑a∈e
[va ⊗∇0Na]
• Bilinear shape functions:
Na =18
(1 + ξξa)(1 + ηηa)(1 + µµa)
• Nodal velocity:
va = vRe + G(vC
) · (Xa − Xe)
where:
vRe =
1∑f∈e
dfe
∑f∈e
[dfevC
f
]; dfe = |XC − Xe|
G(vC) =
∑f∈e
νfe ⊗ νfe
−1 ∑f∈e
vCf − vR
e
dfe
νfe
∆X
∆Y
ve
1
2
3
vC
vaG (vC )
1 = Linear reconstruction + Riemann solver
2 = Velocity gradient (Least square minimisation)
3 = Interpolation/Extrapolation to nodes
Jibran Haider (ACME 2015, Swansea University, UK) 10th April 2015
Outline Introduction Reversible elastodynamics Numerical technique Numerical results Conclusions
Involution
Local constraint preserving scheme• Compatibility condition
[ Torrilhon(2005), Lee-Gil-Bonet(2013) ]:
∇0 × F = 0
• An adapted curl-free updated scheme:
dFe
dt=∑a∈e
[va ⊗∇0Na]
• Bilinear shape functions:
Na =18
(1 + ξξa)(1 + ηηa)(1 + µµa)
• Nodal velocity:
va = vRe + G(vC
) · (Xa − Xe)
where:
vRe =
1∑f∈e
dfe
∑f∈e
[dfevC
f
]; dfe = |XC − Xe|
G(vC) =
∑f∈e
νfe ⊗ νfe
−1 ∑f∈e
vCf − vR
e
dfe
νfe
∆X
∆Y
ve
1
2
3
vC
vaG (vC )
1 = Linear reconstruction + Riemann solver
2 = Velocity gradient (Least square minimisation)
3 = Interpolation/Extrapolation to nodes
Jibran Haider (ACME 2015, Swansea University, UK) 10th April 2015
Outline Introduction Reversible elastodynamics Numerical technique Numerical results Conclusions
Involution
Local constraint preserving scheme• Compatibility condition
[ Torrilhon(2005), Lee-Gil-Bonet(2013) ]:
∇0 × F = 0
• An adapted curl-free updated scheme:
dFe
dt=∑a∈e
[va ⊗∇0Na]
• Bilinear shape functions:
Na =18
(1 + ξξa)(1 + ηηa)(1 + µµa)
• Nodal velocity:
va = vRe + G(vC
) · (Xa − Xe)
where:
vRe =
1∑f∈e
dfe
∑f∈e
[dfevC
f
]; dfe = |XC − Xe|
G(vC) =
∑f∈e
νfe ⊗ νfe
−1 ∑f∈e
vCf − vR
e
dfe
νfe
∆X
∆Y
ve
1
2
3
vC
vaG (vC )
1 = Linear reconstruction + Riemann solver
2 = Velocity gradient (Least square minimisation)
3 = Interpolation/Extrapolation to nodes
Jibran Haider (ACME 2015, Swansea University, UK) 10th April 2015
Outline Introduction Reversible elastodynamics Numerical technique Numerical results Conclusions
Outline
1 IntroductionMotivationOpenFOAM
2 Reversible elastodynamicsGoverning equations
3 Numerical techniqueSpace-time discretisationContact fluxInvolution
4 Numerical resultsMesh convergence2D results3D results
5 Conclusions
Jibran Haider (ACME 2015, Swansea University, UK) 10th April 2015
Outline Introduction Reversible elastodynamics Numerical technique Numerical results Conclusions
1D Cable
Smooth Sinusoidal Loading: 1D mesh convergenceProblem description: Linear elastic material, ρ0 = 1 kg/m3, E = 1 GPa, ν = 0.3 αCFL = 0.5,
P(t) = 1× 10−3 [sin ((πt)/20− π/2) + 1] N
L = 10m
x
P (t)
10−2 10−1 10010−8
10−7
10−6
10−5
10−4
10−3
10−2
10−1
Grid Size (m)
Err
or
Contact Velocity at t = 34.4757 sec
L1 norm error (Forward Euler, 1st Order)
L2 norm error (Forward Euler, 1st Order)
L1 norm error (2−Step RK, 2nd Order)
L2 norm error (2−step RK, 2nd Order)
Slope = 1Slope = 2
10−2 10−1 10010−8
10−7
10−6
10−5
10−4
10−3
10−2
10−1
Grid Size (m)
Err
or
Contact Stress at t = 34.4757 sec
L1 norm error (Forward Euler, 1st Order)
L2 norm error (Forward Euler, 1st Order)
L1 norm error (2−Step RK, 2nd Order)
L2 norm error (2−step RK, 2nd Order)
Slope = 1Slope = 2
X 1D convergence analysis by means of the L1 & L2 norm has been carried out at t ≈ 34.5 sX Demonstrates the expected accuracy of the available schemes for all variables
Jibran Haider (ACME 2015, Swansea University, UK) 10th April 2015
Outline Introduction Reversible elastodynamics Numerical technique Numerical results Conclusions
1D Cable
Smooth Sinusoidal Loading: 1D mesh convergenceProblem description: Linear elastic material, ρ0 = 1 kg/m3, E = 1 GPa, ν = 0.3 αCFL = 0.5,
P(t) = 1× 10−3 [sin ((πt)/20− π/2) + 1] N
L = 10m
x
P (t)
10−2 10−1 10010−8
10−7
10−6
10−5
10−4
10−3
10−2
10−1
Grid Size (m)
Err
or
Contact Velocity at t = 34.4757 sec
L1 norm error (Forward Euler, 1st Order)
L2 norm error (Forward Euler, 1st Order)
L1 norm error (2−Step RK, 2nd Order)
L2 norm error (2−step RK, 2nd Order)
Slope = 1Slope = 2
10−2 10−1 10010−8
10−7
10−6
10−5
10−4
10−3
10−2
10−1
Grid Size (m)
Err
or
Contact Stress at t = 34.4757 sec
L1 norm error (Forward Euler, 1st Order)
L2 norm error (Forward Euler, 1st Order)
L1 norm error (2−Step RK, 2nd Order)
L2 norm error (2−step RK, 2nd Order)
Slope = 1Slope = 2
X 1D convergence analysis by means of the L1 & L2 norm has been carried out at t ≈ 34.5 sX Demonstrates the expected accuracy of the available schemes for all variables
Jibran Haider (ACME 2015, Swansea University, UK) 10th April 2015
Outline Introduction Reversible elastodynamics Numerical technique Numerical results Conclusions
2D Spinning Plate
Spinning Plate
Problem description: Unit side plate, nearly incompressible hyperelastic Neo-Hookean material,ρ0 = 1100 kg/m3, E = 17 MPa, ν = 0.45,Discretisation = 20×20 cells, ∆t = 1× 10−4s, αCFL = 0.5, ω0 = 105 rad/s
x
y
ω0
1 m
0 0.05 0.1 0.15 0.2−500
0
500
1000
1500
2000
Time (sec)
Mom
entu
m (
N.m
.s;k
g.m
.s−1
)
Linear momentum xLinear momentum yLinear momentum zAngular momentum
X Demonstrates the conservation of angular momentum
Jibran Haider (ACME 2015, Swansea University, UK) 10th April 2015
Outline Introduction Reversible elastodynamics Numerical technique Numerical results Conclusions
2D Spinning Plate
Spinning Plate
Problem description: Unit side plate, nearly incompressible hyperelastic Neo-Hookean material,ρ0 = 1100 kg/m3, E = 17 MPa, ν = 0.45,Discretisation = 20×20 cells, ∆t = 1× 10−4s, αCFL = 0.5, ω0 = 105 rad/s
x
y
ω0
1 m
0 0.05 0.1 0.15 0.2−500
0
500
1000
1500
2000
Time (sec)
Mom
entu
m (
N.m
.s;k
g.m
.s−1
)
Linear momentum xLinear momentum yLinear momentum zAngular momentum
X Demonstrates the conservation of angular momentum
Jibran Haider (ACME 2015, Swansea University, UK) 10th April 2015
Outline Introduction Reversible elastodynamics Numerical technique Numerical results Conclusions
3D Bending Column
Bending dominated scenarioProblem description: Nearly incompressible hyperelastic Neo-Hookean material, ρ0 = 1100 kg/m3,
E = 17 MPa, ν = 0.45, αCFL = 0.3, V = 10 m/s
x
y
z
1m1m
L = 6m
v0 = [V y/L, 0, 0]T
[Animation - 3D]
X The formulation eliminates the bending difficulty
Jibran Haider (ACME 2015, Swansea University, UK) 10th April 2015
Outline Introduction Reversible elastodynamics Numerical technique Numerical results Conclusions
3D Bending Column
Bending dominated scenarioProblem description: Nearly incompressible hyperelastic Neo-Hookean material, ρ0 = 1100 kg/m3,
E = 17 MPa, ν = 0.45, αCFL = 0.3, V = 10 m/s
x
y
z
1m1m
L = 6m
v0 = [V y/L, 0, 0]T
[Animation - 3D]
X The formulation eliminates the bending difficulty
Jibran Haider (ACME 2015, Swansea University, UK) 10th April 2015
Outline Introduction Reversible elastodynamics Numerical technique Numerical results Conclusions
3D Twisting Column
Highly non-linear problem
Problem description: Nearly incompressible hyperelastic Neo-Hookean material, ρ0 = 1100 kg/m3,E = 17 MPa, ν = 0.45, αCFL = 0.3, Ω = 105 rad/s
x
y
z
L
ω0 = [0,Ωsin(πy/2L), 0]T
(−0.5, 0,−0.5)
(0.5, 6, 0.5)
[Animation - 3D]
X Demonstrates the robustness of the numerical scheme
Jibran Haider (ACME 2015, Swansea University, UK) 10th April 2015
Outline Introduction Reversible elastodynamics Numerical technique Numerical results Conclusions
3D Twisting Column
Highly non-linear problem
Problem description: Nearly incompressible hyperelastic Neo-Hookean material, ρ0 = 1100 kg/m3,E = 17 MPa, ν = 0.45, αCFL = 0.3, Ω = 105 rad/s
x
y
z
L
ω0 = [0,Ωsin(πy/2L), 0]T
(−0.5, 0,−0.5)
(0.5, 6, 0.5)
[Animation - 3D]
X Demonstrates the robustness of the numerical scheme
Jibran Haider (ACME 2015, Swansea University, UK) 10th April 2015
Outline Introduction Reversible elastodynamics Numerical technique Numerical results Conclusions
Outline
1 IntroductionMotivationOpenFOAM
2 Reversible elastodynamicsGoverning equations
3 Numerical techniqueSpace-time discretisationContact fluxInvolution
4 Numerical resultsMesh convergence2D results3D results
5 Conclusions
Jibran Haider (ACME 2015, Swansea University, UK) 10th April 2015
Outline Introduction Reversible elastodynamics Numerical technique Numerical results Conclusions
Conclusions and further research
Conclusions• An upwind first order conservation law formulation based on the finite volume method, has been
presented for fast solid dynamic simulations within the OpenFOAM environment
• Linear elements can be used without usual volumetric and bending difficulties
• Velocities (or displacements) and stresses (or strains) display the same rate of convergence
On-going work• Introduction of an additional conservation law for the Jacobian of the deformation gradient
• Introduction of a plasticity model
References· 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 (2013) 13-38.
· M. Aguirre, A. J. Gil, J. Bonet and A. Arranz Carreño. A vertex centred Finite Volume Jameson-Schmidt-Turkel (JST)algorithm for a mixed conservation formulation in solid dynamics, Journal of Computational Physics, 259 (2014)672-699.
· A. J. Gil, C. H. Lee, J. Bonet and M. Aguirre. A stabilised Petrov-Galerkin formulation for linear tetrahedral elementsin compressible, nearly incompressible and truly incompressible fast dynamics, Computer Methods in AppliedMechanics and Engineering, 279 (2014) 659-690
· J. Bonet, A. J. Gil, C. H. Lee, M. Aguirre and R. Ortigosa. A first order hyperbolic framework for large straincomputational solid dynamics: Part 1 Total Lagrangian Isothermal Elasticity, 283 (2015) 689-732.
· J. Haider, A. J. Gil, J. Bonet and C. H. Lee. A first order conservation law formulation for fast solid dynamics inOpenFOAM, Journal of Computational Physics, In preparation
Jibran Haider (ACME 2015, Swansea University, UK) 10th April 2015
Outline Introduction Reversible elastodynamics Numerical technique Numerical results Conclusions
Conclusions and further research
Conclusions• An upwind first order conservation law formulation based on the finite volume method, has been
presented for fast solid dynamic simulations within the OpenFOAM environment
• Linear elements can be used without usual volumetric and bending difficulties
• Velocities (or displacements) and stresses (or strains) display the same rate of convergence
On-going work• Introduction of an additional conservation law for the Jacobian of the deformation gradient
• Introduction of a plasticity model
References· 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 (2013) 13-38.
· M. Aguirre, A. J. Gil, J. Bonet and A. Arranz Carreño. A vertex centred Finite Volume Jameson-Schmidt-Turkel (JST)algorithm for a mixed conservation formulation in solid dynamics, Journal of Computational Physics, 259 (2014)672-699.
· A. J. Gil, C. H. Lee, J. Bonet and M. Aguirre. A stabilised Petrov-Galerkin formulation for linear tetrahedral elementsin compressible, nearly incompressible and truly incompressible fast dynamics, Computer Methods in AppliedMechanics and Engineering, 279 (2014) 659-690
· J. Bonet, A. J. Gil, C. H. Lee, M. Aguirre and R. Ortigosa. A first order hyperbolic framework for large straincomputational solid dynamics: Part 1 Total Lagrangian Isothermal Elasticity, 283 (2015) 689-732.
· J. Haider, A. J. Gil, J. Bonet and C. H. Lee. A first order conservation law formulation for fast solid dynamics inOpenFOAM, Journal of Computational Physics, In preparation
Jibran Haider (ACME 2015, Swansea University, UK) 10th April 2015
Outline Introduction Reversible elastodynamics Numerical technique Numerical results Conclusions
THANK YOU
Jibran Haider (ACME 2015, Swansea University, UK) 10th April 2015