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1. A two-step Taylor-Galerkin
formulation for fast dynamics
Izian Abd. Karim
Department of Civil Engineering, Universiti Putra Malaysia,
Serdang, Malaysia, and
Chun Hean Lee, Antonio J. Gil and Javier Bonet
College of Engineering, Swansea University, Swansea, UK
Abstract
Purpose – The purpose of this paper is to present a new stabilised low-order finite element
methodology for large strain fast dynamics.
Design/methodology/approach – The numerical technique describing the motion is formulated
upon the mixed set of first-order hyperbolic conservation laws already presented by Lee et al. (2013)
where the main variables are the linear momentum, the deformation gradient tensor and the total
energy. The mixed formulation is discretised using the standard explicit two-step Taylor-Galerkin
(2TG) approach, which has been successfully employed in computational fluid dynamics (CFD).
Unfortunately, the results display non-physical spurious (or hourglassing) modes, leading to the
breakdown of the numerical scheme. For this reason, the 2TG methodology is further improved
by means of two ingredients, namely a curl-free projection of the deformation gradient tensor and the
inclusion of an additional stiffness stabilisation term.
Findings – A series of numerical examples are carried out drawing key comparisons between the
proposed formulation and some other recently published numerical techniques.
Originality/value – Both velocities (or displacements) and stresses display the same rate of
convergence, which proves ideal in the case of industrial applications where low-order discretisations tend
to be preferred. The enhancements introduced in this paper enable the use of linear triangular (or bilinear
quadrilateral) elements in two dimensional nearly incompressible dynamics applications without locking
difficulties. In addition, an artificial viscosity term has been added into the formulation to eliminate the
appearance of spurious oscillations in the vicinity of sharp spatial gradients induced by shocks.
Keywords Finite element method, Conservation laws, Fast dynamics, Low order, Riemann solver,
Taylor-Galerkin
Paper type Research paper
1 Introduction
Fast solid dynamics is currently an extremely active field of research with notable
engineering problems of high practical relevance: nearly incompressible large strains
dynamics, impact and contact problems, propagation of shock waves and moving
boundaries (Bonet and Wood, 2008; Bower, 2010; Toro, 2006; Leveque, 2002; Belytschko
et al., 2000). Industry still presents many open problems with regard to the selection
of the most effective numerical methodology for specific applications. Modern
commercial finite element codes, such as LS-DYNA3D (Hallquist, 1998, 2003),
ABAQUS/explicit (Hibbitt et al., 2000) and PAM-CRASH (Pam, 2002), have been
developed taking advantage of the classical displacement-based formulation in
The current issue and full text archive of this journal is available at
www.emeraldinsight.com/0264-4401.htm
Received 21 December 2012
Revised 30 May 2013
Accepted 5 June 2013
Engineering Computations:
International Journal for
Computer-Aided Engineering and
Software
Vol. 31 No. 3, 2014
pp. 366-387
r Emerald Group Publishing Limited
0264-4401
DOI 10.1108/EC-12-2012-0319
The first author acknowledges the financial support received from the Ministry of Higher
Education (Malaysia) and the University of Putra Malaysia (Serdang). The third author
acknowledges the financial support received through “The Leverhulme Prize” awarded by The
Leverhulme Trust, United Kingdom.
366
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31,3

2. conjunction with explicit time stepping schemes. The formulation is discretised first in
space using the Bubnov-Galerkin Finite Element Method (FEM) (Zienkiewicz et al.,
2007), usually, with a low-order approximation. The resulting semi-discrete ordinary
differential equations are then integrated in time by employing a family of Newmark
time integrators (Hughes, 2000; Bathe, 1996). However, this classical numerical
methodology presents a series of drawbacks.
From the time discretisation standpoint, the use of Newmark type integrators is
known to be inefficient in shock wave dominated problems, showing a tendency for
high-frequency numerical noise to persist in the solution (Wood et al., 1980; Chung and
Hulbert, 1993; Adams and Wood, 1983). From the spatial discretisation point of view,
it is well known that the use of linear interpolation in the FEM leads to second-order
convergence for the primary variable (displacements) but one order less for derived
variables (i.e. strains and stresses) (Aguirre et al., 2014; Mukherjee et al., 2012),
requiring a posteriori stress improvement recovery procedure (Payen and Bathe, 2011,
2012) if the latter variables are of interest.
Another shortcoming of employing low-order linear triangles in two dimensions
(or tetrahedras in three dimensions) is the appearance of locking effects in nearly
incompressible deformations and bending dominated scenarios (Bonet and Wood,
2008; Neto et al., 2008). Unfortunately, many practical engineering applications, such as
complex human body impact modelling and metal forming processes, can often only be
meshed using this type of discretisation (Benson, 1992). To overcome these difficulties,
a variety of numerical strategies have been proposed, namely hp-refinement (Szabo and
Babuska, 1991; Babuska et al., 1981; Duster et al., 2003; Heisserer et al., 2008) and the
use of multi-field Fraeijs de Veubeke-Hu-Washizu (FdVHW) type variational principles
(Washizu, 1975). This latter approach is favoured in commercial codes where low-order
interpolation is preferred. In particular, the mean dilatational formulation (Nagtegaal
et al., 1974), a special case of the FdVHW approach, uses constant interpolation to
describe the volumetric deformation (Hallquist, 1998, 2003; Hibbitt et al., 2000;
Pam, 2002).
The mean dilatational formulation cannot be employed when linear interpolation is
used to describe the kinematics, as linear finite elements already make use of a single
Gauss point for numerical quadrature. In this case, some form of projection to reduce
the number of volumetric constraints is used (Nagtegaal et al., 1974; Argyris et al.,
1974; Malkus and Hughes, 1978; Simo et al., 1985; de Souza Neto et al., 1996). Bonet and
Burton (1998) suggested the approximation of the volumetric strain energy functional
by means of averaged nodal pressures in terms of tributary nodal volumes, whilst
the deviatoric component remains treated in the standard manner. However, the
resulting solution was reported to behave poorly in bending dominated scenarios.
Dohrmann et al. (2000) proposed a new linear tetrahedra by applying a nodal averaging
process to the whole small strain tensor and Bonet et al. (2001) extended this
application to the large strain regime. The resulting formulation still suffered
from artificial mechanisms similar to hourglassing (Lahiri et al., 2010; Puso and
Solberg, 2006).
The objective of this paper is to introduce an effective low-order stabilised two-step
Taylor-Galerkin (2TG) framework for the numerical simulation of fast solid dynamics,
aiming to resolve the shortcomings mentioned above. Following reference Lee et al.,
2013, a new mixed formulation is summarised in the form of a system of first-order
hyperbolic conservation laws, where the linear momentum, the deformation gradient
tensor and the total energy of the system are regarded as the three main conservation
367
A 2TG
formulation

3. variables (Lee et al., 2013). Crucially, the formulation enables the convergence of the
stresses at the same rate as the velocities (or displacements), which proves ideal in
the case of low-order discretisations of interest in industrial applications. Perhaps
more importantly, the new formulation permits the use of linear elements in
nearly incompressible regimes without volumetric or bending difficulties (Lee et al.,
2012, 2013).
This paper is organised as follows. Section 2 introduces a new mixed conservation
law formulation for reversible elastodynamics within the context of large
deformations. Section 3 describes the development of a curl-free 2TG methodology
accomplished by introducing appropriate artificial stabilisation paremeters, which
consequently avoids the appearance of non-physical spurious modes in the solution.
Section 4 shows various possible boundary conditions derived through the use of the
Rankine-Hugoniot jump conditions in the presence of a contact or discontinuity.
In Section 5, a series of numerical examples are compared with some alternative
numerical techniques to demonstrate the robustness of the proposed methodology.
Finally, some concluding remarks are drawn in Section 6.
2. Governing equations for reversible elastodynamics
Let us consider a mapping function f to describe the deformation of a continuum body
from a referential (material) domain X 2 V R3
to a current (spatial) domain x 2
vðtÞ R3
at time t, namely x ¼ f(X,t). Following references (Lee et al., 2012, 2013), the
deformation process under isothermal considerations can be described by a set of
Lagrangian conservation laws as:
qp
qt
=0 P ¼ r0b ð1aÞ
qF
qt
=0
1
r0
p I
¼ 0 ð1bÞ
qET
qt
=0
1
r0
PT
p
¼ 0 ð1cÞ
where p ¼ r0v is the linear momentum, F is the deformation gradient tensor
and ET is the total energy. Here, r0 is the material density, v is the velocity vector,
b is the body force per unit mass, P is the first Piola-Kirchhoff stress tensor,
I is the identity tensor and =0 denotes the material gradient operator in the
undeformed space. Notice that the heat flux and heat source terms have been
neglected in expression (1c) for simplicity. The above balance laws (Equations (1a-1c))
can be combined into a single system of first-order hyperbolic conservation
equations defined by:
qU
qt
þ
qFI
qXI
¼ S; 8I ¼ 1; 2; 3 ð2Þ
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4. with:
U ¼
p1
p2
p3
F11
F12
F13
F21
F22
F23
F31
F32
F33
ET
0
B
B
B
B
B
B
B
B
B
B
B
B
B
B
B
B
B
B
B
B
B
B
@
1
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
A
; FI ¼
P1I
P2I
P3I
dI1p1=r0
dI2p1=r0
dI3p1=r0
dI1p2=r0
dI2p2=r0
dI3p2=r0
dI1p3=r0
dI2p3=r0
dI3p3=r0
PiI pi=r0
0
B
B
B
B
B
B
B
B
B
B
B
B
B
B
B
B
B
B
B
B
B
B
@
1
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
A
; S ¼
r0b1
r0b2
r0b3
0
0
0
0
0
0
0
0
0
0
0
B
B
B
B
B
B
B
B
B
B
B
B
B
B
B
B
B
B
B
B
B
B
@
1
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
A
ð3Þ
where dIJ (I, J ¼ 1,2,3) is the Kronecker delta tensor. To complete the coupled system
(Equation (1a-1c)), a closure equation is introduced by means of an appropriate frame
indifferent constitutive model (based upon the underlying material response) relating
stress field to deformation (i.e. P ¼ P (F )). A detailed description of this mathematical
formulation, including the eigenstructure of the problem for various constitutive
hyperelastic laws (see references Lee et al., 2013; Lee, 2012; Izian, 2011).
3. Stabilised 2TG methodology
It is well known that the use of the classical Bubnov-Galerkin projection (Zienkiewicz
et al., 2007) for the discretisation of the spatial derivatives in system 2 leads to a
non-coercive approximation (Morgan and Peraire, 1998), allowing for the appearance of
spurious modes in the solution. To overcome this problem, additional numerical
diffusion can be added to the relevant formulation to provide stability, whilst, more
importantly, still maintaining the accuracy of the original approximation (Donea and
Huerta, 2004). In this section, we propose a 2TG (Lohner et al., 1984) methodology in
conjunction with a curl-free projection of the deformation gradient tensor and an
additional stiffness stabilisation term, suitable for the underlying formulation
(Equation (1a-1c)), which performs effectively in the case of nearly incompressible
large deformations. The 2TG methodology is a computationally efficient adaptation of
its counterpart One-Step Taylor-Galerkin (1TG) scheme (Donea, 1984). The latter
requires the evaluation of the flux Jacobian matrix AI ¼ qFI =qU.
Following the works of Lohner et al. (1984), Peraire et al. (1986), Mabssout and
Pastor (2003), Mabssout et al. (2006), Karim et al. (2011) and Bonet and Gil (2008), for
the evolution of the conservation variables from time step tn
to time step tn þ 1
, it is first
necessary to predict U at half time step t(n þ 1)/2
(predictor step) by employing the
conventional forward Euler time integrator as:
Uðnþ1Þ=2
Un
þ
Dt
2
qUn
qt
¼ Un
Dt
2
ðqFI ðUn
ÞÞ
qXI
Sn
ð4Þ
369
A 2TG
formulation

5. The above expression can be particularised for the set of conservation variables of
interest in this paper as:
vðnþ1Þ=2
¼ vn
þ
Dt
2r0
=0 PðFn
Þ þ r0bn
ð Þ ð5aÞ
Fðnþ1Þ=2
¼ Fn
þ
Dt
2
=0vn
ð5bÞ
E
ðnþ1Þ=2
T ¼ En
T þ
Dt
2
=0 PT
ðFn
Þvn
ð5cÞ
In this work, we restrict our attention to hyperelastic isothermal processes, which
allows for the decoupling of the energy Equation (5c) from the rest of the system
(Equations (5a and 5b)). However, as pointed out in Mukherjee et al. (2012) and Izian
(2011), the predicted deformation gradient F (n þ 1)/2
(Equation (5b)) exhibits non-physical
low-energy modes (which can be detected via an eigenvalue analysis of the discrete
tangent matrix; Bonet et al., 2001). For this reason, an augmented stabilised
deformation gradient F (n þ 1)/2
st is proposed by introducing an additional stiffness
stabilisation term into expression (5b), which aims to eliminate these spurious modes:
F
ðnþ1Þ=2
st ¼ Fn
þ
Dt
2
=0vn
þ a =0xn
Fn
ð Þ ð6Þ
where a is a non-dimensional parameter, typically in the range [0,0.5]. Notice that
a ¼ 0 recovers the 2TG predictor step (Equation (5b)), whereas a ¼ 1 leads to the
standard finite element formulation for F (n þ 1)/2
(i.e. F ðnþ1Þ=2
¼ =0xðnþ1Þ=2
with
xðnþ1Þ=2
¼ xn
þ ðDt=2Þ vn
Þ, which exhibits overly stiff behaviour (Bonet and Wood,
2008). It is now possible to evolve the unknown variables to tn þ 1
(also known as
corrector step) by utilising expression (4a) for U(n þ 1)/2
as (i.e. classical mid-point rule):
Unþ1
Un
þ Dt
qUðnþ1Þ=2
qt
¼ Un
Dt
qFI ðUðnþ1Þ=2
Þ
qXI
Sðnþ1Þ=2
!
ð7Þ
From the spatial discretisation point of view, the finite element formulation is
established starting from the variational or weak form of the underlying partial
differential equations. The inner product of above expression (7) with an appropriate
work conjugate virtual field (test function), is computed to give:
dWdv ¼
Z
V
dv
Dp
Dt
=0 PðF
ðnþ1Þ=2
st Þ r0bðnþ1Þ=2
dV ¼ 0; Dp ¼ pnþ1
pn
ð8aÞ
dWdP ¼
Z
V
dP :
DF
Dt
=0vðnþ1Þ=2
dV ¼ 0; DF ¼ Fnþ1
Fn
ð8bÞ
where dv is the virtual velocity field and dP is the virtual first Piola-Kirchhoff stress
field. Following a standard isoparametric FEM methodology (Bonet and Wood, 2008),
the unknown variables U, as well as the virtual fields dv and dP, can be discretised in
370
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31,3

6. terms of discrete nodal values (i.e. Ua, dva, dPa) and suitable shape functions Na, where
a ¼ {1, y, Nn}, Nn being the number of nodes in a non-overlapping tessellation of finite
elements (e) of volume V(e)
, resulting in:
dW
ðeÞ
dv ðU; NadvaÞ ¼ dva
Z
VðeÞ
Na
Dp
Dt
=0 PðF
ðnþ1Þ=2
st Þ r0bðnþ1Þ=2
dV
2
6
4
3
7
5 ð9Þ
Furthermore, the contribution to dWdvðU; NadvaÞ from all neighbouring elements (e)
converging on node aðe3aÞ is:
dWdvðU; NadvaÞ ¼
X
e
e3a
dW
ðeÞ
dv ðU; NavaÞ ¼ 0 ð10Þ
Finally, the contribution to dWdvðU; dvÞ from all nodes a in the finite element mesh is:
dWdvðU; dvÞ ¼
X
a
dWdvðU; NadvaÞ ¼ 0 ð11Þ
Since the local virtual work dWdv must be satisfied for any arbitrary virtual nodal
velocities dva, expression (10) emerges as:
Z
V
Na
Dp
Dt
=0 P
ðnþ1Þ=2
st r0bðnþ1Þ=2
dV ¼ 0; P
ðnþ1Þ=2
st :¼ PðF
ðnþ1Þ=2
st Þ ð12Þ
Rearranging and applying the Gauss divergence theorem to (Equation (12)) gives:
Z
V
Na
Dp
Dt
dV ¼
Z
qV
Nat
ðnþ1Þ=2
B dA
Z
V
P
ðnþ1Þ=2
st =0NadV þ
Z
V
Nar0bðnþ1Þ=2
dV ð13Þ
where t
ðnþ1Þ=2
B ¼ P
ðnþ1Þ=2
st N is the traction vector and N is the outward unit normal on
the boundary qV. In problems which are highly non-linear, such as shock-dominated
scenarios, the stabilised first Piola-Kirchhoff stress P
ðnþ1Þ=2
st in (Equation (13)) is
replaced with a total stress P
ðnþ1Þ=2
T which incorporates a viscous linear stress term
Pðnþ1Þ=2
v (see Section 3.2). For completeness, we then further utilise the interpolation for
p given by SbNb pb to result in:
X
b
Mab
Dpb
Dt
¼
Z
qV
Nat
ðnþ1Þ=2
B dA
Z
V
P
ðnþ1Þ=2
T =0NadV þ
Z
V
Nar0bðnþ1Þ=2
dV ð14Þ
where Mab ¼
R
V NaNbdV
I. An analogous derivation can also be followed for
dWdP , to yield:
X
b
Mab
DFb
Dt
¼
Z
qV
Nav
ðnþ1Þ=2
B N dA
Z
V
vðnþ1Þ=2
=0NadV ð15Þ
Here, vB is the corresponding velocity vector at the boundary (see Section 4).
Expressions (14) and (15) introduce suitable stabilisation through the terms v(n þ 1)/2
371
A 2TG
formulation

7. and F (n þ 1)/2
(see equations (5a,b) and (6)). It is worthwhile to mention that as was
reported in Lee et al. (2012), the 2TG formulation can be viewed (under certain
circumstances) as a particular case of a stabilised Petrov-Galerkin (PG) FEM (Brooks
and Hughes, 1982; Hughes et al., 2010; Hughes and Tezduyar, 1984; Hughes and Mallet,
1986; Shakib et al., 1991; Hughes et al., 1987; Tezduyar and Hughes, 1982, 1983). The
proposed 2TG formulation is schematically summarised in a flowchart in Figure 1.
3.1 Two-dimensional orthogonal curl-free projection
Unlike the standard displacement-based formulation (Hughes, 2000) where the
deformation gradient F is computed from the primary displacement field, in this new
mixed formulation (1), F is one of the primary variables. Hence, satisfaction of the
Saint-Venant compatibility conditions for large strains (=0 F ¼ 0) must be fulfilled.
Nodal values
Elemental nodal values
Gauss points for elemental volume integrals
Gauss points for elemental boundary integrals
1
n
2
n
3
n
I,1
n
I,2
n
I,3
n
1
(e),(n+1)/2
2
(e),(n+1)/2
3
(e),(n+1)/2
gp
(e),(n+1)/2
, I,gp
(e),(n+1)/2
2
n+1
1
n+1
3
n+1
(e)
(a)
(b)
(c) (d)
(e)
(f)
Notes: (a) Given nodal unknown variables at time n, Un
a; (b) compute
fluxes Fn
I,a at each node; (c) predict the elemental unknown variables at
half time step Ua
(e),(n+1)/2 where a ∈ (e); (d) interpolate the elemental
unknown variables and fluxes at the Gauss points of volume and boundary
integrals (i.e. Ugp
(e),(n+1)/2 and FI,gp
(e),(n+1)/2); (e) compute the right-hand-side
of Equations (14) and (15); and (f) update the unknown variables at each
node Ua
n+1
Figure 1.
Flowchart of the
two-step Taylor-Galerkin
methodology
372
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31,3

8. Otherwise, the appearance of non-physical spurious modes will accumulate in time and
lead to the failure of the numerical scheme (Lee et al., 2013). In an attempt to solve this
problem, an efficient method is here proposed in order to correct the deformation
gradient by projecting it onto a curl-free space without the need to introduce any
additional extra variables within the system. First, note that the curl operator of F in
two dimensions can be written as:
=0 F ¼ F=?
0 ; F ¼
F11 F12
F21 F22
; =?
0 ¼
q
qX2
q
qX1
!
ð16Þ
Introducing the interpolation for F at element (e) given by
P
a
a2e
NaFa into the above
curl-free constraint gives:
=0 F ¼ F=?
0 ¼
X
a
a2e
Fa=?
0 Na; =?
0 Na ¼
qNa
qX2
qNa
qX1
!
ð17Þ
Notice that =?
0 Na denotes the orthogonal vector to the gradient of the shape function
at node a, as =?
0 Na =0Na ¼ 0. It is now possible to introduce a general functional P
defined by:
Pð ^
Fa; leÞ ¼
1
2
X
a
a2e
ð ^
Fa FaÞ : ð ^
Fa FaÞ
0
@
1
A þ le
X
a
a2e
^
Fa=?
0 Na
0
@
1
A ð18Þ
where F̂a is the corrected deformation gradient at node a and ke is an elemental
Lagrange multiplier vector introduced to satisfy the elemental curl-free constraint.
The stationary condition of the above functional (18) with respect to ke and F̂a
will be considered separately. Note firstly that the derivative of P with respect to ke
gives:
qP
qle
¼
X
a
a2e
^
Fa=?
0 Na
0
@
1
A ¼ 0 ð19Þ
implying that the spurious elemental curl-error modes are completely removed by
providing the corrected deformation gradient F̂a, yet to be defined. To obtain F̂a, the
derivative of (18) with respect to F̂a is computed as:
qP
q ^
Fa
¼
X
a
a2e
ð ^
Fa FaÞ þ
X
a
a2e
le =?
0 Na ¼ 0 ð20Þ
Rearranging the above expression yields:
^
Fa ¼ Fa le =?
0 Na
|fflfflfflfflfflfflffl{zfflfflfflfflfflfflffl}
correction term
ð21Þ
373
A 2TG
formulation

9. The remaining unknown parameter to be evaluated in the above equation is the
Lagrange multiplier ke. Inserting equation (21) into (19) for F̂a, results in:
le ¼
P
a
a2e
Fa=?
0 Na
P
a
a2e
=?
0 Na =?
0 Na
ð22Þ
3.2 Viscous formulation
To alleviate high-frequency modes in the solution, an artificial viscosity term can be
added (Izian, 2011; Donea, 1984). This term is applied to the entire computational
domain without the use of a discontinuity sensor (Zienkiewicz et al., 2005). The amount
of viscosity required varies proportionally to the order of the approximating
shape functions (Bonet and Gil, 2008). A simple dissipative formulation is derived
based upon:
sv ¼ Cv : d ð23Þ
where rv is the viscous Cauchy stress tensor, d is the rate of deformation tensor and Cv
is the simplest linearised fourth order constitutive tensor:
Cv ¼ lvI I þ mvðI þ ^
IÞ; lv ¼ Cll; mv ¼ Cmm; kv ¼ lv þ
2
3
mv ð24Þ
where the above tensors I I; I and ^
I can be written in index notation as:
½I IiIjJ ¼ diI djJ ; I
½ iIjJ ¼ dijdIJ ; ½^
IiIjJ ¼ diJ dIj ð25Þ
Here, kv is the volumetric viscosity, mv is the shear viscosity and Cl and Cm are viscosity
coefficients (with intrinsic time scale) for the Lamé constants. Additive decomposition
of the strain rate tensor into its volumetric and deviatoric contributions and
subsequent pull back (Bonet and Wood, 2008) renders the first Piola-Kirchhoff stress
tensor as:
Pv ¼ Jmv FT
ð=0vÞT
FT
þ ð=0vÞF1
FT
2
3
FT
: =0v
FT
|fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl}
shear
þ Jkv FT
: =0v
FT
|fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl}
volumetric
ð26Þ
3.3 A complete set of conservation laws including viscous fluxes
The full hyperbolic system of conservation equations, incorporating possible viscous
stresses Pv (see formula (26)) into expression (1a) can be summarised as:
qp
qt
=0 PT ¼ r0b ð27aÞ
374
EC
31,3

10. qF
qt
=0
1
r0
p I
¼ 0 ð27bÞ
qET
qt
=0 ð
1
r0
PT
TpÞ ¼ 0 ð27cÞ
where PT ¼ P þ Pv. Consequently, the above set of conservation laws can be cast in a
standard compact form as:
qU
qt
þ
qFI
qXI
þ
qFv
I
qXI
¼ S; 8I ¼ 1; 2; 3 ð28Þ
where the corresponding components of viscous fluxes Fv
I are defined by:
Fv
I ¼
Pv
1I U; =0U
ð Þ
Pv
2I U; =0U
ð Þ
Pv
3I U; =0U
ð Þ
0
0
0
0
0
0
0
0
0
Pv
iI pi=r0
0
B
B
B
B
B
B
B
B
B
B
B
B
B
B
B
B
B
B
B
B
B
B
@
1
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
A
ð29Þ
and U, FI and S are as previously defined in expression (3).
4. Non-linear Riemann solver: contact flux
This section describes the computation of the numerical boundary fluxes {tB, vB}
appearing in expressions (14) and (15), which are derived through the Rankine-Hugoniot
jump relationship (Toro, 2006; Leveque, 2002; Donea and Huerta, 2004). In Lagrangian
contact mechanics, it is often the case that two surfaces initially (material configuration)
resting apart can eventually come into contact in the current (spatial) configuration
(see Figure 2). Numerically, contacts may also arise from the use of discontinuous
interpolations for problem variables at a given interface, referred to as the Riemann
problem (Lee et al., 2013). The impact (or interpolation discontinuity) will generate
shock waves travelling from the contact interface into each of the contacting domains. In
case of frictionless contact, the shock waves travel with volumetric speed Up and the
normal components of the momentum and traction vectors after contact must be
identical for both domains (Belytschko et al., 2000). Recall that the Rankine-Hugoniot
jump condition for the linear momentum variable is given by Lee (2012) and Izian (2011).
U ½p
½
¼ ½½PTN ð30Þ
375
A 2TG
formulation

11. The expression above can be further expanded to become:
UL
p ðpL
n pC
n Þ ¼ tL
n tC
n ð31aÞ
UR
p ðpR
n pC
n Þ ¼ ðtR
n tC
n Þ ð31bÞ
where pL
n and pR
n denote the left and right normal components of the momentum
vector before contact, that is pL;R
n ¼ pL;R
n. Analogously, tL,R
n describe the normal
components of the traction vector before contact tL;R
n ¼ n PL;R
T N
. Note that the
surface normal is defined outwards for the left body and inwards for the right body
so as to define a unique traction vector after contact t C
. For instance, N ¼ N L
¼ N R
and n ¼ nL
¼ nR
. Combining expressions (31a) and (31b) for pC
n and tC
n and assuming
that Up is independent of the jump magnitude, renders:
pC
n ¼
UL
p pL
n þ UR
p pR
n
UL
p þ UR
p
þ
tR
n tL
n
UL
p þ UR
p
ð32aÞ
tC
n ¼
UL
p UR
p
UL
p þ UR
p
tL
n
UL
p
þ
tR
n
UR
p
!
þ
UL
p UR
p
UL
p þ UR
p
ðpR
n pL
nÞ ð32bÞ
An analogous derivation can now be followed for infinite friction contact, where the
shock waves will be travelling with shear speed Us and the tangential components of
Time t
R
, pR
, FR
L
,pL
,FL
VR
nL
v R
(t)
Up
R
Us
R
NR
NL
VL
Time t = 0
Up
L
Us
L
V L (t)
X2, x2
X1, x1
Figure 2.
Contact generated
shock waves
376
EC
31,3

12. the corresponding momentum and traction vectors after contact are:
pC
t ¼
UL
s pL
t þ UR
s pR
t
UL
s þ UR
s
þ
tR
t tL
t
UL
s þ UR
s
ð33aÞ
tC
t ¼
UL
s UR
s
UL
s þ UR
s
tL
t
UL
s
þ
tR
t
UR
s
þ
UL
s UR
s
UL
s þ UR
s
ðpR
t pL
t Þ ð33bÞ
With the aid of the above expressions (see equations (32) and (33)), the complete
contact momentum and traction vectors are defined as pC
¼ pC
t þ pC
n n and
tC
¼ tC
t þ tC
n n, respectively. The upwind contact conditions {tC
, pC
} can be
particularised for any set of boundary conditions (Lee et al., 2013) under
consideration (see Figure 3).
4.1 Two-dimensional strong boundary conditions
Problem variables such as linear momentum p and deformation gradient F (or first
Piola-Kirchhoff stress tensor PT) are corrected at every step of the time integration
process, ensuring that these variables satisfy the exact boundary conditions. In
general, three different types of boundary conditions are often encountered, which are
described below.
4.1.1 Free surface case. In this case, no corrections need to be applied to the linear
momentum p or the deformation gradient F (see Figure 3(a)). However, a weak
boundary condition can be applied to F through the use of a corrected first Piola-
Kirchhoff stress tensor P
#T so that the traction vector at the boundary is in equilibrium
with an applied traction vector tApp
, as follows:
^
PT ¼ PT þ ðtApp
t s PTNÞs N þ ðtApp
n n PTNÞn N ð34Þ
where tApp
n ¼ tApp
n, tApp
t ¼ tApp
s and s and n are the spatial tangential and normal
vectors, respectively.
4.1.2 Rigid wall case. In this case, the linear momentum (see Figure 3(b)) vanishes
due to the non-slip condition (Leveque, 2002).
p ¼ 0 ð35Þ
Free case
tt
App
t App
VL
(t) VL
(t) VL
(t)
Sticking case Sliding case
(a) (b) (c)
Figure 3.
Boundary conditions
377
A 2TG
formulation

13. In addition, the material tangential vector T is not allowed to rotate or stretch. This
enables the deformation gradient F̂ to be corrected as:
^
F ¼ F ðN FTÞN T
|fflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflffl}
rotation
þ ð1 T FTÞT T
|fflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl}
stretching
ð36Þ
4.1.3 Sliding surface case. For this case (see Figure 3(c)), the kinematic constraints are
defined by:
^
p ¼ ðI N NÞp ð37Þ
and:
^
F ¼ F ðN FTÞN T ð38Þ
The correction for the corresponding first Piola-Kirchhoff stress tensor can then be
obtained as:
^
PT ¼ PT ðT PTNÞT N ð39Þ
5. Numerical examples
In this section, a series of computational examples will be presented in order to
demonstrate the performance of the proposed methodology. The use of a stabilised
2TG method is compared against the standard Bubnov-Galerkin FEM (in conjunction
with a classical Newmark trapezoidal type integrator) and some recently proposed
numerical alternatives, namely the TVD upwind Cell Centred Finite Volume Method
(CCFVM) (Lee et al., 2013) and the stabilised PG methodology (Lee et al., 2012).
5.1 One dimensional pile case
A one dimensional linear elastic steel pile of length L¼ 10 with unit cross-sectional area, as
reported in Lee et al. (2012, 2013), is considered in order to verify the numerical accuracy of
the proposed 2TG formulation. The bottom end of the pile is fixed and a forcing function is
applied at its free top end. A convergence analysis by means of the L1-norm and the L2-
norm has been carried out on a sequence of grids (Leveque, 2002). To guarantee a smooth
solution for the problem, we employ a CN
exponential forcing function defined by:
PðL; tÞ ¼ Cexp x t 13
ð Þ2
; tX0 ð40Þ
where the constant parameters C ¼ 1 103
and x ¼ 0.1, respectively. In this
particular case, Young’s modulus E and density r0 are taken to be unity and the Poisson’s
ratio is chosen as n ¼ 0. A comparison between the stabilised 2TG methodology and
alternative numerical schemes, namely TVD upwind CCFVM (Lee et al., 2013) and PG
(Lee et al., 2012), at a particular time, is performed (see Figure 4) for various variables (i.e.
velocity and axial stress). The introduction of a slope limiter in the CCFVM (Piecewise
Linear Reconstruction (PLR) þ limiter) and the lumped mass contribution in the
stabilised FEM (i.e. PG and 2TG) produce relatively less accurate results (in comparison
to the CCFVM with PLR and the stabilised FEM using consistent mass), whilst
maintaining the expected second order of convergence for linear shape functions. Notice
that the contribution of the consistent mass in the stabilised FEM provides an accuracy
superior to its finite volume counterpart (imposing PLR) on the same structured grid.
378
EC
31,3

14. 5.2 Spinning plate
A unit thickness square plate without any constraints is made of a nearly incompressible
rubber material with Young’s modulus E ¼ 1.7 107
Pa, density r0 ¼ 1.1 103
kg/m3
and Poisson’s ratio n ¼ ð1 m=kÞ=2 ¼ 0:45. The plate is released without imposing any
initial deformation (by means of F ¼ I ) but with an initial angular velocity of
O ¼ 105 rad/s (see Figure 5). The initial velocity field relative to the origin is:
v0
ðXÞ ¼ x X; x ¼ ð0; 0; OÞT
; X ¼ ðX1; X2; 0ÞT
ð41Þ
Note that the initial conditions used here are such that there is no steady-state solution,
even in a rotating reference frame. The main objective is to examine the conservation
properties of the proposed 2TG methodology imposing lumped mass contribution.
Figure 6 illustrates the results using a discretisation of 10 10 2 equal triangular
elements for the 2TG curl-free formulation (lumped mass, Cm ¼ Dt, Cl ¼ 0, a ¼ 0.05).
The figure depicts the time histories of linear momentum L, angular momentum A,
kinetic energy K, elastic energy c and total energy K þ c. The linear momentum,
L ¼
R
V pdV, remains zero at all times with no movement of the centre of mass being
developed (Gonzalez and Stuart, 2008), whereas the total angular momentum within
10–2
Velocity at t=40 s Velocity at t=40 s
2TG (Consistent)
PG (Consistent)
2TG (Lumped)
PG (Lumped)
CCFVM (PLR)
CCFVM (PLR + Limiter)
2TG (Consistent)
PG (Consistent)
2TG (Lumped)
PG (Lumped)
CCFVM (PLR)
CCFVM (PLR + Limiter)
2TG (Consistent)
PG (Consistent)
2TG (Lumped)
PG (Lumped)
CCFVM (PLR)
CCFVM (PLR + Limiter)
2TG (Consistent)
PG (Consistent)
2TG (Lumped)
PG (Lumped)
CCFVM (PLR)
CCFVM (PLR + Limiter)
10–4
L
1
-Norm
Error
10–6
10–8
10–3
10–2
Grid Size
Stress at t=40 s Stress at t=40 s
10–1
100
10–2
10–4
10–6
10–8
10–3
10–2
Grid Size
Notes: Results obtained with P(L, t) = C exp ( (t – 13)2), where C = 1×10–3 and = –0.1.
First column shows the L1-norm convergence and second column shows the L2-norm
convergence. First row shows the velocity error and second row illustrates the stress error.
The linear elastic constitutive model is used and material properties are Poisson's ratio = 0,
Young's modulus E = 1, density 0 = 1 and CFL = 0.4. Stabilising parameters of PG
(consistent and lumped mass matrices): F = 0.5Δt, p = = 0. Stabilising parameters of 2TG
(consistent and lumped mass matrices): C = 0.1Δt, C = 0 = = 0
10–1
100
10–2
10–4
10–6
10–8
10–3
10–2
Grid Size
10–1
100
10–2
10–4
10–6
10–8
10–3
10–2
Grid Size
10–1
100
L
2
-Norm
Error
L
2
-Norm
Error
L
1
-Norm
Error
1
2
1
2
1
2
1
2
Figure 4.
One dimensional pile case
(exponential loading)
379
A 2TG
formulation

15. the system A ¼
R
V k x p k dV, is expected to be conserved at its initial value during
time integration. A series of deformed shapes obtained from the 2TG formulation is
found to be in perfect agreement with the recently proposed PG methodology (Lee
et al., 2012) imposing consistent mass contribution (see Figure 7). Notice that, in this
particular example, there is no qualitative difference in the solutions regardless of the
type of mass matrix being used.
5.3 Punching test
A flat square rubber plate of unit side length is constrained with rollers at the bottom
and on the left and right-hand-sides (see reference Lee et al., 2013). The right half of
the domain experiences a prescribed punch velocity vpunch ¼ 100 m/s (see Figure 8).
A nearly incompressible rubber material is chosen with properties Young’s modulus
E ¼ 1.7 107
Pa, Poisson’s ratio n ¼ 0.45 and material density r0 ¼ 1.1 103
kg/m3
.
The aim of this example is to show that the introduction of the proposed formulation
X2
X1
Ω
Figure 5.
Spinning plate
3 × 10
4
Linear X-Momentum
Linear Y-Momentum
Angular Momentum
2.5
× 105
15 Potential
Kinetic
Total
10
Energy
5
0
0 0.05 0.1 0.15
Time
0.2 0.25 0.3
2
1.5
1
0.5
Momentum
0.05 0.1 0.15
Time
Notes: (a) shows the evolution of linear momentum and angular momentum; and
(b) demonstrates the time histories of kinetic, potential and total energy. The nearly
incompressible Neo-Hookean (NH) constitutive model is used and the material properties are
Poisson's ratio = 0.45, Young's modulus E = 1.7 × 107 Pa, density 0 = 1.1 × 103 kg/m3 and
CFL = 0.4. Discretisation of 10 × 10 × 2 triangular elements. Time step Δt ≈ 1 × 10–4 s.
Two-step Taylor-Galerkin methodology (lumped mass matrix, C = Δt, C = 0, = 0.05)
imposing curl-free projection technique
0.2 0.25 0.3
0
0
–0.5
(a) (b)
Figure 6.
Spinning plate: results
obtained with angular
velocity O ¼ 105rad=s
380
EC
31,3

16. 0.5
2
×106
×106
1.5
1
0.5
0
t=0.1 s t=0.1 s
0.5
0
0
t=0.15 s t=0.15 s
t=0.2 s
t=0.2 s
t=0.25 s t=0.25 s
–0.5
–0.5
0.5
2
×106
1.5
1
0.5
0
0.5
0
0
–0.5
–0.5
0.5
2
×106
1.5
1
0.5
0
0.5
0
0
–0.5
–0.5
0.5
2
×106
1.5
1
0.5
0
0.5
0
0
–0.5
–0.5
0.5
2
×106
1.5
1
0.5
0
0.5
0
0
–0.5
–0.5
0.5
2
×106
1.5
1
0.5
0
0.5
0
0
–0.5
–0.5
0.5
2
×106
1.5
1
0.5
0
0.5
Notes: (a) PG formulation (consistent mass, F = Δt, p = 0, = 0.05); and
(b) two-step Taylor-Galerkin methodology imposing curl-free projection
technique (lumped mass matrix, C = Δt, C = 0, = 0.05). Results obtained
with angular velocity = 105 rad/s. This example is run with the nearly
incompressible Neo-Hookean (NH) constitutive model and material properties
are such that Poisson's ratio = 0.45, Young's modulus E = 1.7×107Pa, density
0 = 1.1×103 kg/m3 and CFL ≈ 0.4. Discretisation of 10 × 10 × 2 triangular
elements. Time step Δt = 1 × 10–4 s
0
0
–0.5
–0.5
0.5
2
1.5
1
0.5
0
0.5
0
0
–0.5
–0.5
(a) (b)
Figure 7.
Spinning plate: sequence
of pressure distribution of
deformed shapes
381
A 2TG
formulation

17. eliminates locking effects and the appearance of spurious pressure checkerboard
modes in the case of near incompressibility. Various numerical techniques including
standard Bubnov-Galerkin FEM, CCFVM and PG formulations are employed for
comparison purposes. Figure 9 depicts the comparison of the deformed shapes at
a particular time using the dicretisation of 10 10 equal quadrilateral elements. It is
clear that the standard FEM exhibits volumetric locking (see Figure 9(a)). To rectify
X1
X2
punch
Figure 8.
A punch case
1
0.5
(a) (b)
(c) (d)
0
0 0.5
t=0.04 s t=0.04 s
1 0 0.5 1
1
0.5
0
1
0.5
0
–0.5
–1
1
× 107 × 107
0.5
0
–0.5
–1
1
0.5
0
0 0.5
Notes: (a) Standard FEM procedure; (b) mean dilatation technique;
(c) CCFVM imposing piecewise linear reconstruction; and (d) two-step
Taylor-Galerkin methodology imposing curl-free projection technique
(consistent mass matrix, C = Δt, C = 0, = 0.1). Initial compressive
velocity punch = 100 m/s is applied. A rubber plate is used and its material
properties are Poisson's ratio = 0.45, Young's modulus E = 1.7 × 107 Pa
and density 0 = 1.1 × 103 kg/m3. Discretisation of 10 × 10 equal quadrilateral
elements. Time step Δt = 5 × 10–5 s
t=0.04 s t=0.04 s
1 0 0.5 1
1
0.5
0
1
0.5
0
–0.5
–1
1
× 107 × 107
0.5
0
–0.5
–1
Figure 9.
Punch test case: deformed
shapes at a particular
time t
382
EC
31,3

18. this, the mean dilatational formulation is usually employed but unfortunately, the
results obtained contain non-physical pressure modes as depicted in Figure 9(b).
These shortcomings (i.e. pressure instability and locking effects) can be completely
removed by utilising effective numerical strategies (i.e. CCFVM and 2TG formulations)
based upon the new conservation law formulation (see Figure 9(c) and (d)). References
Lee et al. (2012) and Mukherjee et al. (2012) report that the use of the consistent mass
matrix is of critical importance as it gives more accurate results than its counterpart
diagonal (or lumped) mass. Figure 10(a), (b) and (c) show the qualitative difference
between the contributions of both consistent and lumped mass matrices, where the latter
exhibits inaccurate behaviour due to the presence of dispersive error. It is useful to notice
that the inclusion of the a stabilising parameter given in Equation (6) eliminates
non-physical low-energy modes that appeared in the solution (see Figure 10(d)).
1.5
(a) (b)
(c) (d)
t=0.03 s
1
0.5
0
0 0.5 1
–1
–0.5
0
0.5
1
× 107
1.5
t=0.03 s
1
0.5
0
0 0.5 1
–1
–0.5
0
0.5
1
× 107
1.5
t=0.03 s
1
0.5
0
0 0.5 1
–1
–0.5
0
0.5
1
× 107
1.5
t=0.03 s
1
0.5
0
0 0.5
Notes: (a) PG formulation (consistent mass, F = Δt, P = 0, = 0.1);
(b) two-step Taylor Galerkin methodology imposing curl-free projection
technique (consistent mass, C = Δt, C = 0, = 0.1); (c) two-step
Taylor-Galerkin methodology imposing curl-free projection technique (lumped
mass, C = Δt, C = 0, = 0.1); and (d) two-step Taylor-Galerkin methodology
imposing curl-free projection technique (consistent mass, C = Δt, C = 0, = 0).
Initial compressive velocity punch = 100 m/s is applied. A rubber plate is used
and its material properties are Poisson's ratio = 0.45, Young's modulus
E = 1.7 × 107 Pa and density 0 = 1.1 × 103 kg/m3. Discretisation of 10 × 10 × 2
triangular elements. Time step Δt = 5 × 10–5s
1
–1
–0.5
0
0.5
1
× 107
Figure 10.
Punch test case: deformed
shapes at a particular time t
383
A 2TG
formulation

19. 6. Conclusions
This paper introduces a curl-free 2TG methodology for the simulation of transient
solid dynamics problems. A system of first-order conservation laws is used to describe
the equations of motion in the context of large deformations. As both linear momentum
and deformation gradient are used as primary variables of the system, equal order of
approximation is achieved in both fields, leading to second order convergence for
stresses when using linear shape functions. The proposed methodology for linear
quadrilateral and triangular elements compares well against other strategies already
published by the authors, namely CCFVM (Lee et al., 2013) and Stablised Petrov
Galerkin (Lee et al., 2012) and improves existing displacement-based methodologies in
terms of locking and pressure field distribution.
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Corresponding author
Dr Antonio J. Gil can be contacted at: a.j.gil@swansea.ac.uk
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