A stabilised Petrov-Galerkin formulation for linear tetrahedral elements in compressible, nearly incompressible and truly incompressible fast dynamics

Outline Motivation Reversible elastodynamics Petrov-Galerkin formulation Numerical results Conclusions
A Stabilised Petrov-Galerkin Formulation For Linear
Tetrahedral Elements In Compressible, Nearly Incompressible
and Truly Incompressible Fast Dynamics
Chun Hean Lee1, Antonio J. Gil2, Javier Bonet3, Miquel Aguirre4
Zienkiewicz Centre for Computational Engineering (ZC2E)
College of Engineering, Swansea University, UK
Advances in Finite Element Methods for Tetrahedral Mesh Computations I (MS209A)
11th World Congress on Computational Mechanics (WCCM XI)
1 https://www.researchgate.net/profile/Chun_Hean_Lee2/
2 http://www.swansea.ac.uk/staff/academic/engineering/gilantonio/
3 http://www.swansea.ac.uk/staff/academic/engineering/bonetjavier/
4 https://www.researchgate.net/profile/Miquel_Aguirre
CHL-AJG-JB-MA (MS209A: Advances in Finite Element Methods for Tetrahedral Mesh Computations I) 20th - 25th July 2014

Outline
1 Motivation
2 Reversible elastodynamics
Balance principles
Conservation laws
3 Petrov-Galerkin formulation
Petrov-Galerkin spatial discretisation
Perturbed test function space
Temporal discretisation
Incompressible and nearly incompressible formulation
Fractional-step formulation
4 Numerical results
5 Conclusions and further research

Motivation
• Standard solid dynamic formulations:
× Linear tetrahedral elements behave poorly in nearly incompressible
and bending dominated scenarios
× Uniform and selective reduced integrated linear hexahedral elements
suffer from respected hourglassing and pressure instabilities
× Convergence of stresses and strains is only first order
× Shock capturing technologies are poorly developed
Time integrators are robust and preserve angular momentum
Extensive availability of commercial packages (ANSYS, Altair
HyperWorks, LS-DYNA, ABAQUS, . . .)
• Mixed conservation law formulation:
Express as first order conservation laws enabling the use of
standard CFD discretisation process
Permits the use of linear tetrahedra, as well as enhanced linear
hexahedra, for solid dynamics without locking difficulties
Achieves optimal convergence with equal orders in velocities and
stresses
Take advantage of the conservative formulation to introduce
state-of-the-art discontinuity-capturing operator
× Enhance existing time integrators to preserve angular momentum

Balance principles
First order conservation formulation
• Consider the standard dynamic equilibrium equation:
∂ρ0v
∂t
− DIVP(F, J) = ρ0b
• To alleviate bending difﬁculty, the conservation law for the deformation
gradient can be incorporated:
∂F
∂t
− DIV (v ⊗ I) = 0
• To avoid volumetric locking, the conservation law for the Jacobian can be
added:
∂J
∂t
− DIV HT
F v = 0; HF = (detF)F−T
Constitutive model is needed to complete the coupled system

Governing equation
Conservation laws
• The mixed equations can be written as a system of ﬁrst order conservation laws:
∂
∂t


ρ0v
F
J

 + DIV


−P(F, J)
−v ⊗ I
−HT
F v

 =


ρ0b
0
0


• More generally, if the energy equation is added:
∂
∂t




ρ0v
F
J
ET



 + DIV




−P(F, J)
−v ⊗ I
−HT
F v
Q − PT
v



 =




ρ0b
0
0
s




• Or in standard form:
∂U
∂t
+DIVF(U) = S; U =




ρ0v
F
J
ET



 ; F =




−P(F, J)
−v ⊗ I
−HT
F v
Q − PT
v



 ; S =




ρ0b
0
0
s




Our aim is to develop a library of second order numerical schemes for a mixed
conservation law formulation of fast solid dynamics using existing CFD technologies

Governing equation
CFD formulations for fast solid dynamics
• Following are the stabilised numerical methodologies recently developed for fast
solid dynamics using mixed formulation:
Swansea University Research Group (Led by Prof. Javier Bonet and Dr. Antonio J.
Gil)
· Two-Step Taylor-Galerkin (2TG) Formulation [Karim, Lee, Gil and Bonet, 2011]
· Total Variation Diminishing (TVD) Upwind Cell Centred Finite Volume Method
(CCFVM) [Lee, Gil and Bonet, 2012]
· Jameson-Schmidt-Turkel (JST) Vertex Centred Finite Volume Method (VCFVM)
[Aguirre, Gil, Bonet and Carreño, 2013]
· Stabilised Petrov-Galerkin (PG) Finite Element Method [Lee, Gil and Bonet, 2013]
· Fractional-Step Petrov-Galerkin (PG) Framework [Gil, Lee, Bonet and Aguirre, 2014]
M.I.T Research Group (Led by Prof. Jaime Peraire)
· Hybridizable Discontinuous Galerkin (HDG) Method [Nguyen and Peraire, 2012]

Petrov-Galerkin spatial discretisation
Stabilised Petrov-Galerkin formulation
• Variational statement of Bubnov-Galerkin formulation (unstable):
V0
δV · R dV = 0; R =
∂U
∂t
+ DIVF − S; δV =


δv
δP
δq


• Integration by parts gives:
V0
δV ·
∂U
∂t
dV =
V0
F : 0δV dV −
∂V0
δV · FN dA +
V0
δV · S dV
• Deﬁne stabilised Petrov-Galerkin (PG) formulation satisfying Second Law of
Thermodynamics:
V0
δVst
· R dV = 0; δVst
=


δvst
δPst
δqst



Perturbation
• Stabilised test function space is generally defined by
δVst
= δV + τT ∂FI
∂U
T
∂δV
∂XI
Perturbation
• Define flux Jacobian matrix:
∂FI
∂U
=



03×3 −CI −κ [HF ]I
− 1
ρ0
II 09×9 09×1
− 1
ρ0
HI −
∂(v·[HF ]I )
∂F
0



• Assuming τ (intrinsic time scale) a diagonal matrix for simplicity:
δVst
:=


δvst
δPst
δqst

 =



δv −
τpF
ρ0
DIVδP −
τpJ
ρ0
HF 0δq
δP − τFpC : 0δv − τFJ (v ⊗ 0δq) :
∂HF
∂F
δq − τJpκHF : 0δv


 ; δP = C : δF
• Bubnov-Galerkin is recovered by setting stabilisation τ = 0

Petrov-Galerkin stabilisation
• Weak statement of stabilised Petrov-Galerkin (PG) formulation:
0 =
V0
δV +
∂F
∂U
τ
T
0δV · R dV
=
V0
δV · R dV
Bubnov-Galerkin
+
V0
∂F
∂U
τR : 0δV dV
Petrov-Galerkin stabilisation
• Integration by parts gives:
V0
δV·
∂U
∂t
dV =
V0
F −
∂F
∂U
τR
Fst
: 0δV dV−
∂V0
δV·FN dA+
V0
δV·S dV
• The stabilised ﬂux Fst
can be more generally deﬁned as (equivalent to
Variational Multi-Scale (VMS) stabilisation):
Fst
= F(Ust
); Ust
= U + U ; U = −τR

Finite element discretisation
• Using standard linear ﬁnite element interpolation for velocity, deformation gradient
and Jacobian renders:
v =
a
va
Na
; F =
a
Fa
Na
; J =
a
Ja
Na
b
Mab ˙pb
=
∂V0
Na
tB dA +
V0
Na
ρ0b dV −
V0
P(Fst
, Jst
) 0Na
dV
b
Mab ˙F
b
=
∂V0
Na
(vB ⊗ N) dA −
V0
vst
F ⊗ 0Na
dV
b
Mab ˙Jb =
∂V0
Na
(vB · HF N) dA −
V0
vst
J · HF 0Na
dV
• By construction the stabilised deformation gradient, Jacobian and velocities are:
Fst
= F + τFp 0v − ˙F ; Jst
= J + τJp DIV HT
F v − ˙J
vst
F = v +
τpF
ρ0
(DIVP + ρ0b − ˙p) ; vst
J = v +
τpJ
ρ0
(DIVP + ρ0b − ˙p)

Finite element discretisation
• Using standard linear ﬁnite element interpolation for velocity, deformation gradient
and Jacobian renders:
v =
a
va
Na
; F =
a
Fa
Na
; J =
a
Ja
Na
b
Mab ˙pb
=
∂V0
Na
tB dA +
V0
Na
ρ0b dV −
V0
P(Fst
, Jst
) 0Na
dV
• By construction the stabilised deformation gradient, Jacobian and velocities are:
Fst
= F + τFp 0v − ˙F + ξF ( 0x − F)
Jst
= J + τJp DIV HT
F v − ˙J + ξJ (det 0x − J)
• To reduce implicitness of the resulting formulation additional time-integrated
residual-based artiﬁcial diffusions can be added

Explicit time marching scheme
Time Integration
• Integration in time is achieved by means of an explicit two-stage Total Variation
Diminishing Runge-Kutta (TVD-RK) time integrator:
U
(1)
n+1 = Un + ∆t ˙Un
U
(2)
n+2 = U
(1)
n+1 + ∆t ˙U
(1)
n+1
Un+1 =
1
2
Un + U
(2)
n+2
together with a stability constraint
∆t = αCFL
hmin
Un
max
; Un
max = max
a
Un
p,a
• Introduce Lagrange multiplier correction to preserve the angular momentum [Lee,
Gil, Bonet, 2013]

Incompressible and nearly incompressible formulation
• Time steps are very small given the presence of a very large value of Poisson’s
ratio in near incompressible solids
• Fully incompressible limit cannot be modelled
• Using standard fractional-step formulation renders:
ρ0 vint − vn
∆t
− DIVPn
dev − DIV pn
HFn − ρ0bn
= 0
Fn+1
− Fn
∆t
− 0vn
= 0
ρ0 vn+1 − vint
∆t
− DIV pn+1
− pn
HFn = 0
• Incompressiblity constraint gives
pn+1 − pn
κ∆t
− HFn : 0vint
−
∆t
ρ0
HFn : 0 DIV pn+1
− pn
HFn = 0

Finite element discretisation: fractional-step formulation
• Using Petrov Galerkin stabilisation the predictor step becomes:
b
Mab
Fn+1
b − Fn
b
∆t
=
∂V0
Na(vB ⊗ N) dA −
V0
vn
⊗ 0Na dV
b
Mab
ρ0 vint
b − vn
b
∆t
=
∂V0
NatBdA +
V0
Naρ0bn
dV −
V0
Pn
Fst
, pst
0NadV
• Project the velocity onto a space of divergence-free:
b
1
κ
Mab +
∆t2
ρ0
Kab
pn+1
b − pn
b
∆t
dV =
∂V0
HT
Fn vB ·NNa dV−
V0
HT
Fn vst
· 0NadV
• Update velocity:
b
Mab
ρ0 vn+1
b − vint
b
∆t
=
V0
Na DIV pn+1
− pn
HT
Fn dV

1D Cable
1D mesh convergence
· Problem description: L = 10m, ρ0 = 1Kg/m3, E = 1Pa, ν = 0, αCFL = 0.5, P = 1 × 10−3EXP(−0.1(t − 13)2)N,
τFp = 0.5∆t, τpF = ξF = 0
2
1
2
1
1D convergence analysis by means of the L2
norm has been carried out at t = 40s
Demonstrates the expected accuracy of the available schemes for all variables
The use of both slope limiter and lumped mass matrix maintains the expected order of
convergence

3D L-shaped block
Angular momentum preserving example
· Problem description: L-shaped block, ρ0 = 1000Kg/m3, E = 5.005 × 104Pa, ν = 0.3, αCFL = 0.3, τFp = 0.5∆t,
τpJ = 0.2∆t, ξJ = 0.5
µ
κ
, τpF = τJp = τFJ = ξF = 0, lumped mass contribution
1X
2X
3X
T(3,3,3)
T(0,10,3)
T(6,0,0)
)t(1F
)t(2F
J. C. Simo, N. Tarnow, K. K. Wong. Exact energy-momentum
conserving algorithms and symplectic schemes for nonlinear
dynamics, CMAME 100, 63-116 (1992)
• Imposed external forces at faces {X1 = 6,
X2 = 10} described as
· F1(t) = − F2(t) = η(t) (150, 300, 450)T
η(t) =



t, 0 ≤ t < 2.5
5 − t, 2.5 ≤ t < 5
0, t ≥ 5
• Free BC at all sides
• Suitable for long term dynamic response
· Angular Momentum
· Total energy (summation of kinetic and
potential energies)
[MOVIE]
Study the conservation properties of the proposed formulation

3D Slender beam
Detrimental locking effects
· Problem description: Column 1 × 1 × 20, ρ0 = 1.1Mg/m3, E = 0.017GPa, ν = 0.49999, αCFL = 0.3, linear variation of
velocity ﬁeld V0 = 2m/s, τFp = 0.5∆t, τpJ = 0.2∆t, ξJ = 0.5
µ
κ
, τpF = τJp = τFJ = ξF = 0, lumped mass
contribution
J. Bonet, H. Marriott, O. Hassan. An averaged nodal deformation
gradient linear tetrahedral for large strain explicit dynamic
applications, COMMUN NUMER METH EN 17, 551-561 (2001)
• Imposed linear variation in velocity ﬁeld
described as
· v(X) = (V0X3/L, 0, 0)T
; V0 = 2m/s
• Thin structures in bending-dominated
scenario
• Nearly incompressible material behaviour
with Poisson ratio ν = 0.49999
• Eliminate shear and volumetric locking
effects and the appearance of pressure
instabilities
[MOVIE]
Assess the performance of the PG formulation in the case of near incompressibility

3D Twisting column
Fractional step formulation
· Problem description: Column 1 × 1 × 6, ρ0 = 1.1Mg/m3, E = 0.017GPa, ν = 0.499, αCFL = 0.3, sinusoidal rotational
velocity ﬁeld Ω = 100m/s, lumped mass contribution
p-F PG p-F-J PG Fractional step
Assess the performance of the fractional step method in the case of near incompressibility

3D Taylor impact bar
Classical benchmark impact problem
· Problem description: Copper bar, L0 = 3.24 cm, r0 = 0.32 cm, v0 = (0, 0, −227) m/s. von Mises hyperelastic-plastic
material with ρ0 = 8930Kg/m3, E = 117GPa, ν = 0.35, ¯τ0
y = 0.4GPa, H = 0.1GPa, αCFL = 0.3, 1361 nodes, lumped
mass contribution
0V
= 03X
0L
0r
Radius and length (in cm) at t = 80µs
Methods Radius Length
Standard 4-Node Tet. 0.555 -
8-Node Hex. (P1/P0) 0.695 2.148
4-Node ANP Tet. (P1/P1-projection) 0.699 -
4-Node Mixed Tet. (P1/P1-stabilised) 0.700 2.156
J. Bonet, A. Burton. A simple average nodal pressure tetrahedral element for
incompressible and nearly incompressible dynamic explicit applications, COMMUN
NUMER METH EN 14, 437-449 (1998)
O. C. Zienkiewicz, J. Rojek, R. L. Taylor, M Pastor. Triangles and tetrahedra in explicit
dynamic codes for solids, INT J NUMER METH ENG 43, 565-583 (1998)
[MOVIE]
Assess the performance within the context of contact/impact mechanics

Conclusions and further research
Conclusions
• A stabilised Petrov-Galerkin formulation is presented for the numerical simulations
of fast dynamics in large deformations
• Linear tetrahedral elements can be used without usual volumetric and bending
difﬁculties
• Velocities (or displacements) and stresses display the same rate of convergence
On-going works
• Standard CFD techniques for discontinuity capturing operator can be
incorporated [Scovazzi et al., 2007]
• Sophisticated constitutive models (Mie-Gruneisen) can be employed [Aguirre et al.,
Under review]
• Industrial applications including crash, impact analysis and explosion modelling

Publications
Journal publications
· 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.
· I. A. Karim, C. H. Lee, A. J. Gil and J. Bonet. A Two-Step Taylor Galerkin formulation for fast
dynamics, Engineering Computations 31 (2014) 366-387.
· C. H. Lee, A. J. Gil and J. Bonet. Development of a stabilised Petrov-Galerkin formulation for a
mixed conservation law formulation in fast solid dynamics, CMAME 268 (2013) 40-64.
· 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, JCP 259 (2014) 672-699.
· A. J. Gil, C. H. Lee, J. Bonet and M. Aguirre. A stabilised Petrov-Galerkin formulation for linear
tetrahedral elements in compressible, nearly incompressible and truly incompressible fast
dynamics, CMAME 276 (2014) 659-690.
Under review
· M. Aguirre, A. J. Gil, J. Bonet and C. H. Lee. An edge based vertex centred upwind finite
volume method for Lagrangian solid dynamics. JCP. Under review.
· 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 1 Total Lagrangian Isothermal Elasticity.
CMAME. Under review.

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