large strain computational solid dynamics: an upwind cell centred finite volume method
TRANSCRIPT
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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
August 2, 2016
Jibran Haider (Swansea University, UK & UPC, Spain) WCCM XII & APCOM VI (Seoul, Korea) 1
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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) WCCM XII & APCOM VI (Seoul, Korea) 2
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Introduction Governing equations Numerical methodology Results 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 oneused in CFD community.
• Entitled TOtal Lagrangian Upwind Cell-centredFVM for Hyperbolic conservation laws (TOUCH).
X Programmed in the open-source CFD softwareOpenFOAM.
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
Q1-P0 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
Upwind FVM
Aim is to bridge the gap between CFD and computational solid dynamics.
Jibran Haider (Swansea University, UK & UPC, Spain) WCCM XII & APCOM VI (Seoul, Korea) 3
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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) WCCM XII & APCOM VI (Seoul, Korea) 4
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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 ·(
1ρ0
p⊗ I)
; CURL F = 0
Additional equations
• Total energy
∂E∂t
= ∇0 ·(
1ρ0
PT p− Q)
+ s
An appropriate constitutive model is required to close the system.
Jibran Haider (Swansea University, UK & UPC, Spain) WCCM XII & APCOM VI (Seoul, Korea) 5
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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)
1ρ0
p⊗ I1ρ0
(PT p)− Q
; S =
ρ0b
0
s
• Geometry update
∂x∂t
=1ρ0
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) WCCM XII & APCOM VI (Seoul, Korea) 6
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Introduction Governing equations Numerical methodology Results Conclusions
Scheme of presentation
1. Introduction
2. Governing equations
3. Numerical methodologySpatial discretisationFlux computationInvolutionsEvolution
4. Results
5. Conclusions
Jibran Haider (Swansea University, UK & UPC, Spain) WCCM XII & APCOM VI (Seoul, Korea) 7
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Introduction Governing equations Numerical methodology Results Conclusions
Scheme of presentation
1. Introduction
2. Governing equations
3. Numerical methodologySpatial discretisationFlux computationInvolutionsEvolution
4. Results
5. Conclusions
Jibran Haider (Swansea University, UK & UPC, Spain) WCCM XII & APCOM VI (Seoul, Korea) 8
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Introduction Governing equations Numerical methodology Results Conclusions
Spatial discretisation
Conservation equations for an arbitrary element
dU e
dt=
1Ωe
0
∫Ωe
0
∂F I
∂XIdΩ0 −→ ∀ I = 1, 2, 3;
=1
Ωe0
∫∂Ωe
0
F INI︸ ︷︷ ︸FN
dA (Gauss Divergence theorem)
≈1
Ωe0
∑f∈Λf
e
FCNef‖Cef ‖
e FCNe f
‖Ce f‖ Ωe0
Traditional cell centred Finite Volume Method
dU e
dt=
1Ωe
0
∑f∈Λf
e
FCNef‖Cef ‖
; FCNef
=
tC
1ρ0
pC ⊗ N1ρ0
tC · pC
Jibran Haider (Swansea University, UK & UPC, Spain) WCCM XII & APCOM VI (Seoul, Korea) 9
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Introduction Governing equations Numerical methodology Results Conclusions
Scheme of presentation
1. Introduction
2. Governing equations
3. Numerical methodologySpatial discretisationFlux computationInvolutionsEvolution
4. Results
5. Conclusions
Jibran Haider (Swansea University, UK & UPC, Spain) WCCM XII & APCOM VI (Seoul, Korea) 10
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Introduction Governing equations Numerical methodology Results Conclusions
Lagrangian contact dynamics
Rankine-Hugoniot jump conditions
c JU K = JF K N
where JK = + −−wc J p K = J t K
c J F K =1ρ0
J p K⊗ N
c J E K =1ρ0
J PT p K · N
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
Jibran Haider (Swansea University, UK & UPC, Spain) WCCM XII & APCOM VI (Seoul, Korea) 11
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Introduction Governing equations Numerical methodology Results Conclusions
Acoustic Riemann solver
Jump condition for linear momentum
cJpK = JtK
Normal jump→ cpJpnK = JtnKTangential jump→ csJptK = JttK
p+n , t+np−
n , t−n
c+pc−ppC
n , tCn
x
t
Normal jump
p+t , t+tp−
t , t−t
c+sc−s pCt , tC
t
x
t
Tangential jump
Upwinding numerical stabilisation
pC=
[c−p p−n + c+p p+n
c−p + c+p
]+
[c−s p−t + c+s p+t
c−s + c+s
]︸ ︷︷ ︸
pCAve
+
[t+n − t−nc−p + c+p
]+
[t+t − t−tc−s + c+s
]︸ ︷︷ ︸
pCStab
tC =
[c+p t−n + c−p t+n
c−p + c+p
]+
[c+s t−t + c−s t+t
c−s + c+s
]︸ ︷︷ ︸
tCAve
+
[c−p c+p (p+n − p−n )
c−p + c+p
]+
[c−s c+s (p+t − p−t )
c−s + c+s
]︸ ︷︷ ︸
tCStab
Linear reconstruction procedure + limiter (monotonicity) for U−,+.
Jibran Haider (Swansea University, UK & UPC, Spain) WCCM XII & APCOM VI (Seoul, Korea) 12
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Introduction Governing equations Numerical methodology Results Conclusions
Scheme of presentation
1. Introduction
2. Governing equations
3. Numerical methodologySpatial discretisationFlux computationInvolutionsEvolution
4. Results
5. Conclusions
Jibran Haider (Swansea University, UK & UPC, Spain) WCCM XII & APCOM VI (Seoul, Korea) 13
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Introduction Governing equations Numerical methodology Results Conclusions
Godunov-type FVM
Standard FV update (CURL F 6= 0)
dFe
dt=
1Ωe
0
∑f∈Λ
fe
pCf
ρ0⊗ Cef X
Constrained FV update (CURL F = 0)[Dedner et al., 2002; Lee et al., 2013]
dFe
dt=
1Ωe
0
∑f∈Λ
fe
pCf
ρ0⊗ Cef X
• Algorithm is entitled ’C-TOUCH’.
pe
pCf −→
pe
Ge
ypC
f
←−
pa
Constrained transport schemes are widely used in Magnetohydrodynamics (MHD).
Jibran Haider (Swansea University, UK & UPC, Spain) WCCM XII & APCOM VI (Seoul, Korea) 14
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Introduction Governing equations Numerical methodology Results Conclusions
Scheme of presentation
1. Introduction
2. Governing equations
3. Numerical methodologySpatial discretisationFlux computationInvolutionsEvolution
4. Results
5. Conclusions
Jibran Haider (Swansea University, UK & UPC, Spain) WCCM XII & APCOM VI (Seoul, Korea) 15
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Introduction Governing equations Numerical methodology Results Conclusions
Time integration
Two stage Runge-Kutta time integration
1st RK stage −→ U∗e = Une + ∆t Un
e(Une , t
n)
2nd RK stage −→ U∗∗e = U∗e + ∆t U∗e (U∗e , tn+1)
Un+1e =
12
(Une + U∗∗e )
with stability constraint:
∆t = αCFLhmin
cp,max; cp,max = max
a(ca
p)
X An explicit Total Variation Diminishing Runge-Kutta time integration scheme.
X Monolithic time update for geometry.
Jibran Haider (Swansea University, UK & UPC, Spain) WCCM XII & APCOM VI (Seoul, Korea) 16
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Introduction Governing equations Numerical methodology Results Conclusions
Scheme of presentation
1. Introduction
2. Governing equations
3. Numerical methodology
4. ResultsMesh convergenceHighly non-linear problemVon-Mises plasticityContact problems
5. Conclusions
Jibran Haider (Swansea University, UK & UPC, Spain) WCCM XII & APCOM VI (Seoul, Korea) 17
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Introduction Governing equations Numerical methodology Results Conclusions
Scheme of presentation
1. Introduction
2. Governing equations
3. Numerical methodology
4. ResultsMesh convergenceHighly non-linear problemVon-Mises plasticityContact problems
5. Conclusions
Jibran Haider (Swansea University, UK & UPC, Spain) WCCM XII & APCOM VI (Seoul, Korea) 18
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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
2cdπt
)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) WCCM XII & APCOM VI (Seoul, Korea) 19
Problem description: Unit side cube, linear elastic material, ρ0 = 1100 kg/m3, E = 17 MPa, ν = 0.3and αCFL = 0.3.
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Introduction Governing equations Numerical methodology Results Conclusions
Low dispersion cube: Mesh convergence
Velocity at t = 0.004 s
10−2
10−1
100
10−7
10−6
10−5
10−4
Grid Size (m)
L2
No
rm E
rro
r
vx
vy
vZ
Slope = 2
Stress at t = 0.004 s
10−2
10−1
100
10−7
10−6
10−5
10−4
Grid Size (m)
L2
No
rm E
rro
r
Pxx
Pyy
Pzz
Slope = 2
X Demonstrates second order convergence for velocities and stresses.
Jibran Haider (Swansea University, UK & UPC, Spain) WCCM XII & APCOM VI (Seoul, Korea) 20
Problem description: Unit side cube, linear elastic material, ρ0 = 1100 kg/m3, E = 17 MPa, ν = 0.3and αCFL = 0.3.
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Introduction Governing equations Numerical methodology Results Conclusions
Scheme of presentation
1. Introduction
2. Governing equations
3. Numerical methodology
4. ResultsMesh convergenceHighly non-linear problemVon-Mises plasticityContact problems
5. Conclusions
Jibran Haider (Swansea University, UK & UPC, Spain) WCCM XII & APCOM VI (Seoul, Korea) 21
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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]
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)
X Demonstrates the robustness of the numerical scheme
Jibran Haider (Swansea University, UK & UPC, Spain) WCCM XII & APCOM VI (Seoul, Korea) 22
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.
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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) WCCM XII & APCOM VI (Seoul, Korea) 23
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.
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Introduction Governing equations Numerical methodology Results Conclusions
Scheme of presentation
1. Introduction
2. Governing equations
3. Numerical methodology
4. ResultsMesh convergenceHighly non-linear problemVon-Mises plasticityContact problems
5. Conclusions
Jibran Haider (Swansea University, UK & UPC, Spain) WCCM XII & APCOM VI (Seoul, Korea) 24
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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]
Evolution of pressure wave
t = 0.1µs t = 0.2µs t = 0.3µs t = 0.4µs t = 0.5µs t = 0.6µs
Pressure (Pa)
X Demonstrates the ability of the algorithm to simulate plastic behaviour.
Jibran Haider (Swansea University, UK & UPC, Spain) WCCM XII & APCOM VI (Seoul, Korea) 25
Problem description: Hyperelastic-plastic material, ρ0 = 8930 kg/m3, E = 117 GPa, ν = 0.35,αCFL = 0.3, τ 0
y = 0.4 GPa, H = 0.1 GPa and v0 = −227 m/s.
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Introduction Governing equations Numerical methodology Results Conclusions
Scheme of presentation
1. Introduction
2. Governing equations
3. Numerical methodology
4. ResultsMesh convergenceHighly non-linear problemVon-Mises plasticityContact problems
5. Conclusions
Jibran Haider (Swansea University, UK & UPC, Spain) WCCM XII & APCOM VI (Seoul, Korea) 26
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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)
X Demonstrates the ability of the algorithm to simulate contact problems.
Jibran Haider (Swansea University, UK & UPC, Spain) WCCM XII & APCOM VI (Seoul, Korea) 27
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.
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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
8x 10
−3
Time (sec)
y D
isp
acem
ent
(m)
Top (2880 cells)Top (23040 cells)Bottom (2880 cells)Bottom (23040 cells)
X Demonstrates the ability of the algorithm to simulate contact problems.
Jibran Haider (Swansea University, UK & UPC, Spain) WCCM XII & APCOM VI (Seoul, Korea) 28
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.
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Introduction Governing equations Numerical methodology Results Conclusions
Torus impact
[Torus impact]
Jibran Haider (Swansea University, UK & UPC, Spain) WCCM XII & APCOM VI (Seoul, Korea) 29
Problem description: Neo-Hookean material, ρ0 = 1000 kg/m3, E = 1 MPa, ν = 0.4, αCFL = 0.3 andv0 = −3 m/s.
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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) WCCM XII & APCOM VI (Seoul, Korea) 30
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Introduction Governing equations Numerical methodology Results Conclusions
Conclusions and further research
Conclusions
• Upwind CC-FVM is presented for fast solid dynamic simulations within the OpenFOAMenvironment.
• 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.
Jibran Haider (Swansea University, UK & UPC, Spain) WCCM XII & APCOM VI (Seoul, Korea) 31
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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.
• A. J. Gil, C. H. Lee, J. Bonet and R. Ortigosa. "A first order hyperbolic framework for large strain computational soliddynamics. 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 straincomputational 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 soliddynamics", 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 newconservation 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 ParticleHydrodynamics 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 soliddynamics, JCP.
http://www.jibranhaider.weebly.com/research
Jibran Haider (Swansea University, UK & UPC, Spain) WCCM XII & APCOM VI (Seoul, Korea) 32