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An acoustic Riemann solver for large strain computational contact
dynamics
Callum J. Runcie a, Chun Hean Lee a,1, Jibran Haider b, Antonio J. Gil c, Javier Bonet d
aGlasgow Computational Engineering Centre, James Watt School of Engineering, University of Glasgow,
United Kingdom
bCambridge Flow Solutions Ltd., Cambridge Science Park, Cambridge CB4 0GZ, UK
cZienkiewicz Centre for Computational Engineering, Faculty of Science and Engineering, Swansea University,
Bay Campus, SA1 8EN, United Kingdom
dCentre Internacional de M`etodes Num`erics en Enginyeria (CIMNE), Universitat Polit`ecnica de Catalunya,
08034 Barcelona, Campus Norte UPC C/Gran Capit´an S/N, Spain
Abstract
This paper presents a vertex-centred finite volume algorithm for the explicit dynamic analy-
sis of large strain contact problems. The methodology exploits the use of a system of first
order conservation equations written in terms of the linear momentum and a triplet of geo-
metric deformation measures (comprising the deformation gradient tensor, its co-factor and its
Jacobian) together with their associated jump conditions. The latter can be used to derive sev-
eral dynamic contact models ensuring the preservation of hyperbolic characteristic structure
across solution discontinuities at the contact interface, a clear advantage over the standard
quasi-static contact models where the influence of inertial effects at the contact interface is
completely neglected. Taking advantage of the conservative nature of the formalism, both ki-
netic (traction) and kinematic (velocity) contact interface conditions are explicitly enforced at
the fluxes through the use of appropriate jump conditions. Specifically, the kinetic condition
is enforced in the usual linear momentum equation, whereas the kinematic condition can now
be easily enforced in the geometric conservation equations without requiring a computationally
demanding iterative algorithm. Additionally, a Total Variation Diminishing shock capturing
technique can be suitably incorporated in order to improve dramatically the performance of
the algorithm at the vicinity of shocks. Moreover, and to guarantee stability from the spatial
discretisation standpoint, global entropy production is demonstrated through the satisfaction
of semi-discrete version of the classical Coleman–Noll procedure expressed in terms of the time
rate of the so-called Hamiltonian energy of the system. Finally, a series of numerical examples is
examined in order to assess the performance and applicability of the algorithm suitably imple-
mented in OpenFOAM. The knowledge of the potential contact loci between contact interfaces
is assumed to be known a priori.
Keywords: Explicit contact dynamics, Conservation laws, large strain, Riemann solver,
OpenFOAM, Shocks
1Corresponding author: chunhean.lee@glasgow.ac.uk
Preprint submitted to International Journal for Numerical Methods in Engineering August 1, 2022
1. Introduction
In computational mechanics, the numerical modelling of contact and impact phenomena
has been a major field of interest in industry, including numerous applications such as vehicle
crash testing [1], prototype testing or manufacture processes [2, 3]. These very complex prob-
lems are typically characterised by highly nonlinear deformation behaviour with accompanying
non-smooth response (or shocks) caused by transitions between various contact modes such
as separation-to-contact, stick-to-slip, slip/stick-to-separation. Such problems must be solved
by ensuring the satisfaction of linear momentum conservation equation (complemented by ap-
propriate initial and boundary conditions) for each body individually, whilst at the same time
enforcing the additional set of (kinematic and kinetic) contact interface conditions, that govern
the interaction of these bodies.
When considering a model with frictionless contact, these interface conditions act to pre-
vent interpenetration of the bodies (kinematic condition) and to insure compressive interaction
normal to the interface (kinetic condition). One challenging aspect, is that impenetrability
cannot be expressed as an evolution (or algebraic) equation and so requires special numeri-
cal treatment. The most common techniques addressing this issue include penalty method,
Lagrange multiplier method, or a combination of both. In the penalty method [4], the impen-
etrability constraint is enforced as a penalty normal traction along the contact surface. The
disadvantage of the penalty approach is that the enforcement of the impenetrability condition
is only approximate and its effectiveness depends on the selection of the user-defined penalty
parameters. If the value of the penalty parameter is too small, unpredictable amount of inter-
penetration would be observed. However, the penalty parameter cannot be arbitrarily large,
as this can generate ill-conditioned systems that may require extremely small time steps for
stability [5, 6]. The correct choice for this parameter is key to success of the algorithm. For
the Lagrange multiplier method [7, 8], the multipliers must be approximated and solved at the
contact interface with the constraint such that the normal component of the traction must be
compressive. The disadvantage of the method is that it requires the construction of a separate
independent mesh and also requires the introduction of additional regularisation techniques
necessary in obtaining robust solutions. Such regularisation procedures are usually ad-hoc and
are not motivated by physical arguments. A popular example of regularisation is the addition
of von Neumann’s artificial viscosity [9, 10] to the Euler fluid equations to smear shocks over
several computational cells. In absence of this artificial viscosity, central difference solutions to
the Euler equation in the vicinity of shocks are oscillatory, eventually leading to the breakdown
of a numerical scheme.
One of the earliest attempts at enforcing contact interface conditions via the physically-
motivated jump conditions that derive from the linear momentum conservation equation and
kinematic compatibility can be traced back to the work of Abedi and Haber [11]. In particular,
a space-time Discontinuous Galerkin finite element method for elasto-dynamic contact was pre-
sented. The presented examples were restricted to the case of small strain linear elasticity in
two dimensions. Moreover, it is not yet clear if the overall finite element algorithm would sat-
isfy the classical Coleman–Noll procedure in order to guarantee the production of non-negative
entropy. On another front, some interesting works have also been explored using the Computa-
tional Fluid Dynamics (CFD) platform “OpenFOAM” via the use of displacement-based finite
volume discretisation [12, 13], with special attention paid to the quasi-static simulation based
on the use of penalty method [14, 15] and lubricated contact models [16].
Aiming to resolve the shortcomings described above, the main goal of this paper is to explore
2
the solution of contact dynamics utilising a set of first order conservation equations [17–20],
combined with the associated jump conditions across moving shocks2[21–33]. Building upon
previous work developed by the authors [34, 35], a mixed methodology is presented in the form
of a system of hyperbolic conservation laws, where the linear momentum and the minors of
deformation (the deformation gradient tensor, its co-factor and its Jacobian) are regarded as
the main conservation variables of this mixed approach. Taking advantage of this formalism,
appropriate kinetic and kinematic contact interface conditions can be suitably enforced at the
boundary fluxes of the underlying hyperbolic system by means of the Rankine-Hugoniot jump
conditions. For instance, in the case of frictionless contact, the normal traction is enforced in the
standard manner, that is, in the boundary fluxes of linear momentum conservation equation.
One crucial advantage of solving the geometric conservation equations in this context is such
that we now have the luxury of explicitly enforcing the normal component of the contact velocity
in the associated boundary fluxes. On the other hand, upon the use of Rankine-Hugoniot jump
conditions [36], we can then naturally derive a series of dynamic contact models typically
required in the simulation of contact problems. The objective of this paper is to present a
complete set of continuum Riemann-based type solutions for contact and separation (derived
based on a system of first order conservation laws) assuming a priori knowledge of the potential
contact loci. Physically, Riemann solutions describe correct fluxes in the form of traction and
velocity at the contact interface. In linear elasticity, we can show that by enforcing appropriate
boundary fluxes at the point of contact through the use of jump conditions would lead to exact
energy transfer, provided the shock wave travels at the speed of sound.
From a spatial discretisation point of view, a vertex based finite volume algorithm [34] in
conjunction with (piecewise) linear reconstruction is employed. Additionally, a shock capturing
technique [37] can also be incorporated in order to dramatically improve the resolution of the
field variables at the vicinity of shocks. No ad-hoc algorithmic regularisation procedures are
needed. Insofar as contact-impact introduces discontinuities in the solution, the use of explicit
time integrators is preferred (see Chapter 10 in [5]) as neither linearisation nor a Newton’s
method is required. With this in mind, from a temporal discretisation standpoint, we use
the explicit type of two-stage Runge-Kutta time integrator. A crucial aspect that requires
special attention is that of the stability of the overall algorithm [38]. This can be demonstrated
by monitoring over time the Hamiltonian energy of the system, ensuring the production of
entropy throughout the entire simulation. The overall methodology is shown to be capable of
handling contact-impact problems without excessive spurious modes, even in the case of nearly
incompressible elasticity and elasto-plasticity. Examples presented in the paper are specifically
chosen in order to illustrate the capability of the proposed framework addressing spurious
oscillations in problems with shocks (or spatial jumps) without resorting to ad-hoc dissipative
correction. Another contribution of the current work is to carry out its implementation into
the OpenFOAM platform, widely accepted these days by industry.
The paper is broken-down into the following sections. Section 2 starts by summarising
the Total Lagrangian formulation of the conservation laws to be solved, comprising the linear
momentum and the three geometric conservation laws. Section 3 provides the exact solution
of the simple one dimensional two-bar impact derived from the associated jump conditions in
linear elasticity. This leads to exact energy transfer from one bar to the other after contact
2First order conservation equations with moving shocks are typically used in the solution of fluid dynamics
problems where solution patterns incorporating shock waves are frequently encountered.
3
without energy loss. Motivated by this, the section continues deriving a set of interface contact
conditions (velocity and stress) applicable to multi-dimensional contact problems. Section 4
presents the second law of thermodynamics written in terms of the so-called Hamiltonian free
energy. Section 5 describes the computational methodology of the vertex centred finite volume
method. A proof of entropy production is included as a necessary condition for stability at
the semi-discrete level. Section 6 includes the algorithmic flowchart of the resulting numerical
scheme where special attention is paid to the procedure in addressing non-matching mesh
interface. Section 7 presents a set of numerical examples to assess the accuracy and stability of
the computational framework, with detailed comparison with other verified finite element code
such as Abaqus. Section 8 presents some concluding remarks.
2. First order hyperbolic system for solid dynamics
Consider the three dimensional deformation of an isothermal body of material density ρR
moving from its initial undeformed configuration ΩV, with boundary ∂ΩVdefined by an outward
unit normal N, to a current deformed configuration Ωv(t) at time t, with boundary ∂Ωv(t)
defined by an outward unit normal n. The time dependent motion ϕ(X, t) of the body can be
described by the following system of Total Lagrangian global conservation laws [35, 39–52]
d
dt ZΩV
UdΩV+Z∂ΩV
FNdA =ZΩV
SdΩVin ΩV,(1)
with the surface flux vector being defined as FN=P3
I=1 FINI. Here, Uis the vector of
conservation variables, FIis the flux vector at I-th material direction and Sis the source
term. Their components are
U=
p
F
H
J
,FI=−
P EI
v⊗EI
F(v⊗EI)
H: (v⊗EI)
,S=
fR
0
0
0
,(2)
with the Cartesian material coordinate basis being defined
E1=
1
0
0
;E2=
0
1
0
;E3=
0
0
1
.(3)
In terms of notations, p=ρRvis the linear momentum per unit material volume, vrepresents
the velocity field, fRis the body force per unit material volume, {F,H, J}are the triplet
deformation measures representing deformation gradient tensor, its co-factor and its Jacobian,
Prepresents the first Piola-Kirchhoff stress tensor. The symbol represents the tensor cross
product between vectors and/or second order tensors, see [41, 53, 54].
Given the fact that the above system (1) has more equations than needed, suitable com-
patibility conditions (also known as involutions [45, 46, 55]) are necessary, namely
CURLF=0; DIVH=0.(4)
CURL and DIV represent the material curl and divergence operators carried out with respect
to the material configuration.
4
For smooth functions, expression (1) is equivalent to the set of first order local differential
equations described as
∂U
∂t +
3
X
I=1
∂FI
∂XI
=Sin ΩV.(5)
The above local form implies that the variables describing the state of the solid in motion (such
as, velocity vand stresses P) are continuous functions throughout the solid. In other words, it
is always possible to find their spatial derivatives as required by the divergence operators that
appear in equation (5). This is indeed usually the case but situations may arise when these
variables experience sudden jumps in value, that is, they become discontinous across surfaces
which move across the body. These jumps are known as shocks and are the result of sudden
physical phenomena such as contact-impact problems.
To acccount for shock phenomena, the integral equation (1) also leads to the following jump
conditions across a discontinuity surface with normal Npropagating with speed U, that is
U[[U]] = [[FN]].(6)
These jump conditions are sometimes referred to as Rankine-Hugoniot equations [56, 57] de-
scribing the behaviour of a material across a shock. These conditions can then be particularised
for the set of conservation variables considered in this paper, namely the linear momentum and
the triplet of deformation measures
U[[p]] = −[[P]]N; (7a)
U[[F]] = −[[v]] ⊗N; (7b)
U[[H]] = −F([[v]] ⊗N) ; (7c)
U[[J]] = −H: ([[v]] ⊗N).(7d)
Here, [[•]] = [•]+−[•]−denotes the jump operator between the right and left states of a
discontinuous surface.
For the particular case of a reversible process, the closure of system (1) requires the intro-
duction of a suitable constitutive law relating the stress tensor Pwith the triplet of geometric
strain measures {F,H,J}, obeying the principle of objectivity [5, 58, 59] and thermodynamic
consistency (via the Coleman-Noll procedure [60]). In this work, a Mooney-Rivin model is
employed and is summarised in Remark 2 for completeness. Finally, for a complete definition
of the initial boundary value problem, initial and boundary (essential and natural) conditions
must be specified as appropriate.
Remark 1: For the conservation of mass, the material density at the reference configuration
cannot be a function of time, that is ∂ρR
∂t = 0 [61]. This implies that ρRis given by the initial
conditions of the solids and it remains constant throughout the motion, and does not need to
be considered as part of the unknowns in system (1) to be solved in time. Since there is no
possible flow of mass across the physical interface (meaning, the associated normal flux vector
vanishes), the jump condition associated with the mass conservation becomes U[[ρR]] = 0. This
intrinsically implies that the material density at the reference configuration must be continuous
across a shock. Equation (7a) thus reduces to
UρR[[v]] = −[[P]]N.(8)
5
Remark 2: In this work, and without loss of generality, we consider a Mooney-Rivlin model
such that the strain energy density is defined as a convex multi-variable function Wof the
deformation measures {F,H, J}[42, 62] as
W(F,H, J) = ζJ−2/3(F:F) + ξJ−2(H:H)3/2−3ζ+√3ξ+κ
2(J−1)2,(9)
where ζ,ξand κ(bulk modulus) are positive material parameters. Appropriate values for
the material parameters ζand ξcan be defined in terms of the shear modulus µ, that is,
2ζ+ 3√3ξ=µ. It is now possible to express the first Piola Kirchhoff stress tensor as [41]
P=ΣF+ΣHF+ ΣJH,(10)
where the conjugate stresses {ΣF,ΣH,ΣJ}are defined by
ΣF=∂W
∂F= 2ζJ−2/3F;ΣH=∂W
∂H= 3ξJ −2(H:H)1/2H(11)
and
ΣJ=∂W
∂J =−2
3ζJ−5/3(F:F)−2ξJ−3(H:H)3/2+κ(J−1).(12)
For ξ= 0, the Mooney–Rivlin model described above (9) degenerates into the so-called nearly
incompressible neo-Hookean model. In order to model irrecoverable plastic behaviour, the
standard rate-independent von-Mises plasticity model [58] with isotropic hardening is used and
the basic structure was already summarised in Algorithm 1 in [45].
Remark 3: It is often necessary to obtain expressions for the symmetric Kirchhoff (or Cauchy)
stress tensor since it is needed either to express plasticity models or to display the solution re-
sults. Such expressions can be easily obtained from the following standard relationship between
these tensors [58]
Jσ=τ=P F T.(13)
To achieve this, substitution of (10) into (13) for P, and following the procedure described in
Reference [41], gives the resulting expression for the Kirchhoff (or Cauchy) stress
Jσ=τ=τF+τHI+τJI,(14)
where
τF=ΣFFT;τH=ΣHHT;τJ=JΣJ.(15)
3. Contact-Impact conditions
3.1. Motivation: the local one-dimensional contact solution
In order to motivate the more complex contact-impact solutions (e.g. stick-slip-separation
transition) developed in this section, consider first a simple one-dimensional case comprising
two bars, as illustrated in Figure 1a, where the left bar is travelling with a given velocity
6
Figure 1: Wave solution for one-dimensional two-bar impact at different time: (a) t0= 0, (b) t1=δ/v0, (c)
t2=t1+L/(2cp), (d) t3=t1+L/cp, (e) t4=t1+ 3L/(2cp) and (f) t5=t1+ 2L/cp.δis the gap separation
between two bar at t0. Left column represents the velocity profile vxand right column represents the stress
profile PxX (not the traction).
7
v0impacts the right bar which is at rest. When the contact between two bars takes place,
the resulting contact-impact motion is governed by a reduced set of (one-dimensional) jump
conditions described in system (7) as
cpρR[[vx]] = −[[PxX ]]NX; (16a)
cp[[FxX ]] = −[[vx]]NX.(16b)
For ease of understanding, both bars are assumed to be made of the same linear elastic
material defined as PxX = (λ+ 2µ) (FxX −1) [58]. With this linear model, and noting that
[[PxX ]] = (λ+ 2µ) [[FxX ]], we can then obtain the shock wave speed cpby substituting expression
(16b) into (16a). Expression (16a) after some algebra becomes
cpρR[[vx]] = −(λ+ 2µ) [[FxX ]]NX=(λ+ 2µ)
cp
[[vx]],(17)
and which after rearranging gives3
cp=sλ+ 2µ
ρR
.(18)
This corresponds to the speed of the sound wave in the bar which can be obtained considering
the classical wave propagation theory [61].
Once the shock wave speed is determined, attention is now focussed on the evaluation
of the velocity (kinematic) and traction (kinetic) at the contact-impact scenario. The same
evaluation procedure would also be repeated when considering the separation process. When
contact is made between two points of the bar, shock waves are generated and travel in opposite
directions along each bar (from the contact point to the free end of the bar) as shown in Figures
1a-1c. When such a compressive stress wave reaches the end of a bar, wave reflection occurs
(see Figure 1d). The reflective wave varies depending on the actual physical boundary of the
problem under consideration. In the case of a free end (i.e. traction is zero and particle velocity
is doubled), the reflected wave becomes a tensile stress wave which is an inverted image of the
compressive stress wave. The frequency and amplitude of the velocity wave however in this
case remains unchanged in reflection. Finally, as soon as the tensile stress wave arrives at the
contact point, both bars would undergo the separation process. The above procedure describing
the wave evolution for a two-bar impact in one dimension is graphically represented in Figure
1 for clarity.
Let us now focus on the mathematical solutions of contact-impact state between two bars.
Upon contact, both instantaneous velocity vC
xand traction tC
xare obtained by applying the
jump condition (16a) between the values of variables before and after the impact through
appropriate initial conditions, to give
cpρRv0−vC
x= 0 −tC
x; (19a)
cpρR0−vC
x=−(0 −tC
x).(19b)
The first equation (19a) corresponds to the jump relation between the left bar (travelling with
a given speed v−
x=v0) and the contact point, whereas the second equation (19b) refers to the
3When transverse deformation is neglected in the model, the speed of sound reduces to cp=qE
ρR. This can
be achieved by simply setting the value of Poisson’s ratio equal to zero.
8
jump between the right bar (at rest v+
x= 0) and the contact point. Additionally, both ends
of the bars are traction-free right before contact, which implies that t−
x=t+
x= 0. Solving the
above system (19) analytically gives the common (or continuous) velocity and traction at the
contact point for both bars
vC
x=1
2v0;tC
x=−cpρR
2v0.(20)
This is generally known as contact-stick mode in one dimension.
Following the same procedure described above, it is also possible to determine the release
velocity for each of the bars right after separation. In this separation mode, the release traction
must be zero ensuring the traction-free compatibility condition. Focussing first on the left bar,
the release velocity can be achieved by introducing the jump condition (16a) between the values
of variables before and after separation to give
cpρRv−
x−vC,−
x=t−
x−0.(21)
Suitable conditions for velocity and traction right before separation must be enforced. In
this specific demonstration, and referring to Figure 1e, we use the values of v−
x=t−
x= 0 at
the contact point prior to separation. With these at hand, the associated release velocity in
expression (21) becomes null, shown as below
vC,−
x=v−
x−t−
x
cpρR
= 0.(22)
Analogously, we can now repeat the same demonstration for the right bar. The corresponding
jump condition in relation to the right bar follows
cpρRv+
x−vC,+
x=−t+
x−0.(23)
Using the exact same value of the contact traction as for the left bar (that is, t+
x= 0) and the
velocity to be v+
x=v0, the above expression yields
vC,+
x=v+
x+t+
x
cpρR
=v0+ 0 = v0.(24)
Observing the fact that the release velocity for the left bar is null (22) and for the right bar
is v0(24), this represents a complete transfer of kinetic energy (and also total energy) from
the left bar to the right bar at the point of contact without energy loss. This is illustrated in
Figures 1a and 1f.
From the point of view of hyperbolic differential equations, the sudden impact between
two bars results in a Riemann problem in each bar with a simple analytical solution. The
boundary fluxes at the point of impact, namely the velocity and traction, must be compatible
with the jump conditions. In linear elasticity, this compatibility leads to exact energy balance4
since the shocks wave travels at the speed of sound. From a mathematical standpoint, the
elastodynamic contact problem is well posed without the need to introduce artificial ad-hoc
dissipative effects, provided that the first order conservation equations (1) together with the
associated jump conditions (7) are used. The sections that follow will conceptually extend the
above simple local contact solution to three dimensional local solution with a possibility to
consider bi-material between contact.
4Exact energy transfer at the contact interface holds when an exact Riemann solver is used.
9
3.2. Extension to general contact procedure
It is worthwhile recalling the general solution process for the contact algorithm (for instance,
stick-slip-separation transition) that would ensure the satisfaction of Karush-Kuhn-Tucker con-
dition [5, 11]. To begin with, it is instructive to first determine the trial solution assuming
contact-stick mode. Such trial solution is used as a criteria in the subsequent development, to
check if two bodies are in contact or separate.
The contact between two bodies will only take place if the following two conditions hold
tC,trial
n<0; δ= 0.(25)
The first (kinetic) condition ensures that the normal component of the trial contact traction
tC,trial
nmust be in compression, whereas the second (kinematic) condition ensures two bodies
are in contact, that is the normal separation δbetween points of contact is zero. We next need
to examine the nature of the contact motion, whether they are in stick mode or slip mode,
depending on the tangential friction introduced in the model. To achieve this, it is possible to
introduce a slip criterion Φ accounting for the difference between the value of a trial tangential
traction vector tC,trial
tand a tangential friction arising from isotropic Coulomb friction [5], that
is
Φ = ∥tC,trial
t∥ − k⟨−tC
n⟩,(26)
with the magnitude (or norm) of a vector being defined as ∥[•]∥=p([•]·[•]). Here, kis
the Coulomb friction coefficient and the symbol ⟨•⟩ =1
2([•] + |[•]|) in the above equation
represents the positive part of the scalar value. The value of the slip criterion determines the
transition between contact-stick and contact-slip modes. If Φ ≤0, contact-stick conditions
hold given the fact that the computed tangential force does not exceed the Coulomb limit.
Otherwise, when the value of Φ >0, we then accept the solution to be in contact-slip mode.
Finally, the transition from either contact-slip or contact-stick to separation would happen
when tC,trial
n≥0 (that is, the violation of the kinetic condition), even with the value of δ= 0.
The overall procedure described above is summarised in Algorithm 1. This however would
require the evaluation of contact traction and velocity associated with various dynamic contact
models involved, and which will be presented in the following sections.
Algorithm 1: General procedure for stick-to-slip-to-separation transition
if δ= 0 then
Obtain trial contact-stick traction: tC,trial =tC(34a) (see Section 3.2.1)
Determine the normal contact traction: tC,trial
n=n·tC,trial
if tC,trial
n<0then
Check slip criterion: Φ = ∥tC,trial
t∥ − k⟨−tC
n⟩
if Φ≤0then
Contact-stick mode: tC(34a) and vC(34b) (see Section 3.2.1)
else
Contact-slip mode: tC(41) and vC(40) (see Section 3.2.2)
end
else
Separation mode: vC(49) and tC=0(see Section 3.2.3)
end
else
Not in contact: vC(49) and tC=0(see Section 3.2.3)
end
10
Figure 2: Contact-impact generated shock waves in multi-dimensions.
3.2.1. Contact-stick condition
Motivated by the one-dimensional problem illustrated in Section 3.1, we now extend the
concept to multi-dimensions by postulating that impact between two bodies travelling at differ-
ent speeds leads to a common velocity and traction at the point of contact as shown in Figure
2. The normal components of the velocity and traction at the point of contact are defined as
vn=v·n;tn=t·n= (P N )·n.(27)
Both of these values are likely to be different for the left and right bodies right before contact
and are therefore denoted as v−
nand t−
nfor the left body and v+
nand t+
nfor the right body. The
common values after contact are denoted by vC
nand tC
n. Note first that the impact will generate
two types of shock waves travelling from the contact point into each of the two bodies.
In the case of frictionless contact, the generated shock waves will travel with volumetric
speed Up. The evaluation of the common contact velocity and traction vectors is governed by
the jump conditions across the two shocks, obtained by applying equation (8) on each body as
follows
U−ρ−
R[[v]]−=−[[P]]−(−N−); U+ρ+
R[[v]]+=−[[P]]+(−N+).(28)
Note that the negative sign in front of N−and N+are necessary as the shocks propagate into
the body in directions opposite to N−and N+. Multiplying the above expressions by a unique
normal vector gives
U−
pρ−
Rv−
n−vC
n=t−
n−tC
n; (29a)
U+
pρ+
Rv+
n−vC
n=−t+
n−tC
n.(29b)
The difference in sign between expressions (29a) and (29b) is because nis normal to the surface
of the left body and hence t+
n=−n·(P+N+) and tC
n=−n·PCN+, whereas t−
n=n·(P−N−)
11
and tC
n=n·(PCN−). Expressions (29) represent a system of two equations for four unknowns,
namely tC
nand vC
n(expressed in terms of the left and right normal tractions and velocity before
the impact) and also {U−
p, U+
p}the speeds of the shocks after impact. Unfortunately, the shock
speeds in equation (29) are in general also a function of the unknowns tC
nand vC
nrendering
the system of equations highly nonlinear. A much simpler case emerges when the speed of the
shock after contact is assumed to be equal to the speed of sound cp(18) (derived on the basis of
the linear elastic model), which only depends on the material properties under consideration.
This is usually referred to as an acoustic Riemann solver [56, 57] widely known in the field of
Computational Fluid Dynamics. Doing this will give closed-form expressions for the normal
components of velocity vC
nand traction tC
nas
vC
n=c−
pρ−
Rv−
n+c+
pρ+
Rv+
n
c−
pρ−
R+c+
pρ+
R
+t+
n−t−
n
c−
pρ−
R+c+
pρ+
R
;
tC
n=c−
pρ−
Rc+
pρ+
R
c−
pρ−
R+c+
pρ+
R t−
n
c−
pρ−
R
+t+
n
c+
pρ+
R!+c−
pρ−
Rc+
pρ+
R
c−
pρ−
R+c+
pρ+
Rv+
n−v−
n.
(30)
In the situation where friction is present to a sufficient degree to prevent relative sliding,
similar common tangential components of the velocity and traction can be derived. Conse-
quently shear shocks are also generated and are, again, assumed to be identical to the simple
shear wave speed obtained via linear elasticity [45]
cs=rµ
ρR
.(31)
If vtand ttare defined as
vt=v−vnn;tt=t−tnn,(32)
a similar derivation for the common tangential traction and velocity vectors gives
vC
t=c−
sρ−
Rv−
t+c+
sρ+
Rv+
t
c−
sρ−
R+c+
sρ+
R
+t+
t−t−
t
c−
sρ−
R+c+
sρ+
R
;
tC
t=c−
sc+
sρ−
Rρ+
R
c−
sρ−
R+c+
sρ+
Rt−
t
c−
sρ−
R
+t+
t
c+
sρ+
R+c−
sc+
sρ−
Rρ+
R
c−
sρ−
R+c+
sρ+
Rv+
t−v−
t,
(33)
where c−
sand c+
sare the left and right body shear shock speeds.
Finally, the complete common velocity and traction vectors at the contact point can be
combined
tC=tC
nn+tC
t;vC=vC
nn+vC
t.(34)
This is typically known as contact-stick mode. Numerically, expressions (34) can be viewed
as the summation of the average states (unstable) and the associated upwinding stabilisation
terms depending on the jumps. This has been extensively exploited by the authors in devel-
oping stabilised methods with the objective to improve the numerical solutions by alleviating
unwanted spurious hour-glassing and pressure instabilities [34, 43, 45, 47–49].
Provided the interface conditions (25) and also the slip function Φ ≤0 (26) hold, we
accept the contact-stick solution (34) as the actual local solution for contact. Otherwise, we
must investigate other possible contact models such as contact-slip or separation. This will be
presented in the next section.
12
Remark 4: It is also useful to consider the case where the jump in traction (or the first Piola
Kirchhoff stress P(10)) is dominated by the jump in the pressure component of the stress
(which in this case is related to ΣJ), whilst the rest of the components of the stress {ΣF,ΣH}
can be neglected. This is indeed the case when attempting to model problems with predominant
nearly incompressible behaviour. Use of (10) in conjunction with the Nanson’s rule HAveN=
ΛHn(where ΛHis the ratio between the current area and the undeformed area), enables the
jump in the traction vector to be
t+−t−= [[t]] = [[P]]N= [[ΣF+ΣHF+ ΣJH]]N(35a)
≈[[ΣJ]] HAveN(35b)
= [[ΣJ]]ΛHn.(35c)
Neglecting the stress components ΣFand ΣHimplies that [[ΣF+ΣHF]]N=0. The jump
in the traction vector in the normal direction can now be derived by multiplying (35c) with a
normal vector nto yield
t+
n−t−
n=n·[[t]] = [[ΣJ]]ΛH.(36)
It is also interesting to notice that the jump in the tangential component of the traction vanishes.
This is easily shown as below
t+
t−t−
t= [[tt]] = [[t]] −[[tn]]n(37a)
= [[ΣJ]]ΛHn−[[ΣJ]]ΛHn=0,(37b)
by making use of expressions (32b), (36) and (35c).
Remark 5:
When considering the exact same material properties on the left and right sides at a point
of contact, the density and the shock wave speeds are identical and constant for both sides,
namely ρ−
R=ρ+
R=ρRand c−
p=c+
p=cpand c−
s=c+
s=cs. Enforcing these conditions in (30)
and (33) especially in the case of nearly incompressible materials (see Remark 4) yields
vC
n=1
2v−
n+v+
n+1
2ρRcp
[[ΣJ]]ΛH;tC
n=1
2t−
n+t+
n+ρRcp
2v+
n−v−
n,(38a)
vC
t=1
2v−
t+v+
t;tC
t=1
2t−
t+t+
t+ρRcs
2v+
t−v−
t.(38b)
Remark 6: Consider a contact system where the material on the right side is much stiffer than
the material on the left side. Under this circumstance, both pressure and shear shock wave
speeds on the stiffer material are approximated to be c+
p≈ ∞ and c+
s≈ ∞, which, upon
substitution into equations (30) and (33), gives
vC
n=v+
n;vC
t=v+
t;tC
n=t−
n+c−
pρ−
R(v+
n−v−
n); tC
t=t−
t+c−
sρ−
R(v+
t−v−
t).(39)
Observe the fact that only the velocity of the stiffer side, that is v+, enters the solutions.
These solutions indeed coincide with the Dirichlet boundary conditions already discussed in
[45], where the velocity v+is prescribed on the boundary of the domain. For instance, when
considering no-slip wall boundary condition, the values of v+
tand v+
nare set to zero.
13
3.2.2. Contact-slip conditions
When the magnitude of the tangential traction described in (33) exceeds the Coulomb fric-
tion limit, that is Φ >0 (26), relative sliding between two surfaces is then allowed. This
phenomenon is known as contact-slip mode. In this mode, only normal component of the ve-
locity is continuous across the contact surface between the left and right bodies. The tangential
components of the velocity however may suffer jumps. Mathematically, the complete velocity
field associated with slip mode for both the left and right surfaces are postulated as
vC,−=vC
nn+vC,−
t;vC,+=vC
nn+vC,+
t,(40)
with vC
nbeing defined in (30a). The remaining variables to be determined are the respective
tangential velocity components {vC,−
t,vC,+
t}.
In order to achieve this, it is instructive to consider the slip traction vector to be
tC=tC
nn+tB
t;tB
t=k⟨−tC
n⟩n⊥;n⊥=−tC
t
∥tC
t∥.(41)
With regard to the first term on the right hand side of the above equation, the normal compo-
nent of the contact traction tC
nmust be in compression (i.e. its value must be strictly negative)
and its expression remains exactly the same as the one described in contact-stick mode (30b).
The second term tB
trepresents the tangential frictional traction arising from the Coulomb
model of friction and is in the direction n⊥opposing the motion predicted using the tangen-
tial traction (33) in stick mode. Use of (41) in conjunction with (28), enables the tangential
components of slip velocity vectors to become
c−
sρ−
Rv−
t−vC,−
t=t−
t−tB
t;c+
sρ+
Rv+
t−vC,+
t=−t+
t−tB
t.(42a)
Re-arranging the above expressions render
vC,−
t=v−
t+tB
t−t−
t
c−
sρ−
R
;vC,+
t=v+
t−tB
t−t+
t
c+
sρ+
R
.(43)
It is useful to notice that frictionless contact condition (also known as symmetric condition)
can be easily recovered by enforcing the value of frictional coefficient k= 0 in (41), which in
turn implies that tB
t=0.
3.2.3. Separation conditions
In the current work, we assume homogeneous prescribed traction (that is, traction-free
conditions) in separation mode, but non-vanishing prescribed tractions due to viscous fluid
loading are also possible. This reveals the fact that the traction after separation must ensure
traction-free compatibility conditions, namely
tC,−=tC,+=0.(44)
With this, we are now in a position to proceed with the evaluation of release (or post-separation)
velocity on respective surfaces by re-applying equation (8) between the values of variables before
and after separation, repeated below again for convenience
U−ρ−
R[[v]]−=−[[P]]−−N−;U+ρ+
R[[v]]+=−[[P]]+−N+.(45)
14
The normal components of the release velocities are obtained by multiplying expressions above
with the normal vector n
c−
pρ−
Rv−
n−vC,−
n=t−
n−0; c+
pρ+
Rv+
n−vC,+
n=−t+
n−0,(46)
and which, after rearranging, becomes
vC,−
n=v−
n−t−
n
c−
pρ−
R
;vC,+
n=v+
n+t+
n
c+
pρ+
R
.(47)
In line with linear elasticity theory, above expressions coincide with the expressions shown in
(22) and (24). A similar derivation for the tangential velocity vectors can now follow
vC,−
t=v−
t−t−
t
c−
sρ−
R
;vC,+
t=v+
t+t+
t
c+
sρ+
R
.(48)
Combining (47) and (48) enables the release velocities to be expressed as
vC,−=vC,−
nn+vC,−
t;vC,+=vC,+
nn+vC,+
t.(49)
It is worth noticing that all the velocity components described in (49) are generally distinct
and independent on opposing sides of the contact surface.
4. Second law of thermodynamics
In order to pave the way for the proof of entropy production presented in a subsequent
section, it is useful to introduce the notion of Hamiltonian H(X, t) [61, 63]. For the isothermal
case, this indeed can be understood as a generalised convex entropy function of the system of
conservation equations (1), coinciding with the definition of total energy per unit of undeformed
volume. Specifically, the Hamiltonian His defined by
H(X, t) = ˆ
H(p,F,H, J, α) = 1
2ρR
p·p+E(F,H, J, α),(50)
which represents the summation of the kinetic energy per unit of undeformed volume (i.e.
the first term on the right hand side of (50)) and the internal energy Eexpressed in terms
of the triplet deformation measures {F,H, J}and a set of state variables [64–66] (such as
plastic deformation or similar) collected in the form of a tensor α. Note here that H(X, t) and
ˆ
H(p,F,H, J, α) represent alternative functional representations of the same quantity.
It is instructive to revisit the second law of thermodynamics when written in terms of the
Hamiltonian. Taking the derivatives of ˆ
H(50) with respect to its arguments, the time rate of
the Hamiltonian for one of the bodies involving contact is obtained via the chain rule as follows
d
dt ZΩVHdΩV=ZΩV
∂ˆ
H(p,F,H, J, α)
∂t dΩV
=ZΩV ∂ˆ
H
∂p·∂p
∂t +∂ˆ
H
∂F:∂F
∂t +∂ˆ
H
∂H:∂H
∂t +∂ˆ
H
∂J
∂J
∂t +∂ˆ
H
∂α:∂α
∂t !dΩV
=ZΩVv·∂p
∂t +ΣF:∂F
∂t +ΣH:∂H
∂t + ΣJ
∂J
∂t +∂E
∂α:∂α
∂t dΩV
=ZΩVv·∂p
∂t + (ΣF+ΣHF+ ΣJH) : ∇0v+∂E
∂α:∂α
∂t dΩV
=ZΩVv·∂p
∂t +P:∇0v+∂E
∂α:∂α
∂t dΩV,
(51)
15
where, equations (5) and (10) have been substituted in the third and fifth lines of (51), respec-
tively. Subsequently, we can substitute the linear momentum conservation equation (1) into
(51) to give
d
dt ZΩVHdΩV=ZΩVv·fR+v·DIVP+P:∇0v+∂E
∂α:∂α
∂t dΩV.(52)
Recalling that v·DIVP+P:∇0v= DIV PTv, above equation reduces to
d
dt ZΩVHdΩV=ZΩVv·fR+ DIV(PTv) + ∂E
∂α:∂α
∂t dΩV.(53)
By performing integration by parts of the DIV term in expression (53), and after some re-
arrangement, it renders d
dt ZΩVHdΩV−˙
Πext =−˙
D, (54)
where ˙
Πext denotes the power introduced by external forces, defined as
˙
Πext =ZΩV
v·fRdΩV+Z∂ΩV\Γ
vB·tBdA +ZΓ
vC·tCdA. (55)
Here, Γ represents the boundary faces on contact region and ∂ΩV\Γ represents the remaining
boundary faces that are not in contact. In the above expression, the first term on the right hand
side represents external force acting on a body, the second term represents the non-contact
boundary forces obtained via the enforcement of standard Neumann or Dirichlet boundary
conditions, and the third term represents the contact boundary forces describing appropriately
the contact-impact phenomenon. Such contact boundary contributions are suitably enforced
by solving a Riemann-like problem as already presented in Section 3.
Consider the case of elasto-plasticity [47, 52, 58] where the elastic energy is expressed in
terms of elastic left Cauchy-Green tensor be=F C−1
pFT, thus in this case the internal state
variable is indeed the inverse of the plastic right Cauchy Green tensor, that is α=C−1
p. With
this, the rate of plastic dissipation ˙
Ddescribed in (54) becomes
˙
D=−ZΩV
∂E
∂C−1
p
:∂C−1
p
∂t dΩV.(56)
In view of the fact that the rate of plastic strain ˙
¯εphas been defined as the work conjugate to
the von Mises equivalent stress ¯τ[58], equation above can be alternatively expressed as [58]
˙
D=ZΩV
˙
¯εp¯τ dΩV; ¯τ=r3
2(τ′:τ′),(57)
where τ′represents the deviatoric component of the Kirchhoff stress. Noticing that in the
above expression the rate of dissipation is always non-negative, that is ˙
D≥0, equation (54)
can be transformed into the following inequality
d
dt ZΩVHdΩV−˙
Πext ≤0,(58)
which represents a valid expression for the second law of thermodynamics [59]. Satisfaction of
inequality (58) is a necessary ab initio condition to ensure stability, otherwise referred to as
the Coleman–Noll procedure [34]. This key concept will be further exploited in Section 5.2 at
a semi-discrete level.
16
5. Vertex centred finite volume method
5.1. Semi-discrete formulation for dynamic contact
(a) Interior node (b) Boundary node
Figure 3: Dual mesh of (a) an interior node and (b) a boundary node using the medial dual approach in two
dimensional triangular mesh.
The vertex centred finite volume spatial discretisation presented in this work requires the
generation of a median dual mesh [34, 35, 50] for the definition of control volumes (see Figure
3). With this in mind, expression (1) can now be spatially discretised over an undeformed
control volume Ωa
V, to give
Ωa
V
dUa
dt =−Z∂Ωa
V
FNdA + Ωa
VSa.(59)
Here, Uaand Saare the average values of both the conservation variables and source term
vector within the control volume, respectively.
Moreover, the surface integral of (59) is approximated by means of appropriate numerical
fluxes, resulting in5
Ωa
V
dUa
dt =−
X
b∈Λa
FI
Nab ||Cab|| +X
γ∈ΛB
a
FB
aCγ+X
β∈ΛC
a
FC
aCβ
+ Ωa
VSa,(61)
where b∈Λarepresents the set of neighbouring control volumes bassociated with the control
volume aand Cγ,β =Aγ,β
3Nγ,β represents the (tributary) boundary area vector. For a given
edge connecting nodes aand b, the mean undeformed area vector Cab satisfies the reciprocal
relation, that is Cab =−Cba. The terms within the parenthesis in (61) correspond to the
5In the second and third terms in the parenthesis of (61), the weighted average stencil proposed by L¨ohner
and co-authors [67] is used by computing the boundary flux over a boundary face γ(and β) in three dimensions
as
FB
a=6FB
a+FB
b+FB
c
8;FC
a=6FC
a+FC
b+FC
c
8(60)
where b,care the other two nodes that together with node adefine boundary face γ(and β).
17
evaluation of the control volume internal interface flux FI
Nab , non-contact boundary fluxes FB
a
and contact boundary fluxes FC
a. This evaluation is comprised of a summation over edges
(first term in the parenthesis), a summation over non-contact boundary faces (second term in
the parenthesis) and a summation over contact boundary faces (third term in the parenthesis).
The internal interface flux FI
Nab =FI
Nab (U−
ab,U+
ab,Nab) must be evaluated on the basis of
the contact-stick condition (see Section 3.2.1), which depends on the reconstructed states at
both sides of the mid-edge of ab, namely U−
ab and U+
ab. The non-contact boundary flux FB
ais
enforced through either Neumann or Dirichlet boundaries and the contact flux FC
ais determined
following strictly the contact procedure presented in Algorithm 1 (obeying appropriate contact-
impact physics). Notice that in (61), Cβ=0when the boundary face βis not in contact.
It is worth noticing that equation (61) would only lead to a first order solution in space
[45] provided that U−
ab and U+
ab are modelled following a piecewise constant representation.
For instance, U−
ab =Uaand U+
ab =Ub, thus leading to excessive numerical dissipation in the
solution. The physics of the problem can no longer be captured accurately unless excessively
fine meshes are used, which is clearly undesirable especially for large scale problems in practice.
To overcome this drawback, and to guarantee second order accuracy in space, a suitable linear
reconstruction procedure is used. A detailed discussion of this reconstruction procedure has
already been presented in [45, 47, 48].
Expression (61) is now particularised for each individual component of U, yielding
Ωa
V
dpa
dt =X
b∈Λa
tI||Cab|| +X
γ∈ΛB
a
tB
a||Cγ|| +X
β∈ΛC
a
tC
a||Cβ|| + Ωa
Vfa
R; (62a)
Ωa
V
dFa
dt =X
b∈Λa
vI⊗Cab +X
γ∈ΛB
a
vB
a⊗Cγ+X
β∈ΛC
a
vC
a⊗Cβ; (62b)
Ωa
V
dHa
dt =X
b∈Λa
FAve vI⊗Cab+X
γ∈ΛB
a
FavB
a⊗Cγ+X
β∈ΛC
a
FavC
a⊗Cβ; (62c)
Ωa
V
dJa
dt =X
b∈Λa
vI·HAveCab+X
γ∈ΛB
a
vB
a·(HaCγ) + X
β∈ΛC
a
vC
a·(HaCβ).(62d)
Here, [•]Ave =1
2([•]a+ [•]b). It is worth re-emphasising that the determination of internal
fluxes {tI,vI}is based on contact-stick mode (refer to equations (30), (33) and (34)) and the
non-contact boundary fluxes {tB
a,vB
a}are evaluated respecting the physical boundaries (i.e.
Neumann or Dirichlet). On the other hand, the evaluation of contact boundary fluxes {vC
a,tC
a}
must satisfy the Karush-Kuhn-Tucker condition for stick-slip-separation transition. The reader
can refer to equations {(34), (30), (33)}for contact-stick mode, equations {(40), (41), (43)}for
contact-slip mode, and equations {(44), (49), (47), (48)}for separation mode.
In ensuring discrete satisfaction of the involutions (4), and following the work of [34], one
viable option is to approximate the updates of F(62b) and H(62c) using central difference
approximations by neglecting the jump in traction in (30) and (33) (or jump in ΣJ(38a)) for
vIin (62b) and (62c). Additionally, in order to ensure the triplet deformation measures to
be solved in a consistent manner, the average strain variables FAve and HAve appearing in
expressions (62c) and (62d) will be replaced by Faand Ha. With this, and assuming the
jump in traction is dominated by jump in pressure (see Remark 4), the geometric conservation
18
equations (62b)-(62d) reduce to
Ωa
V
dFa
dt =X
b∈Λa
vWAvg ⊗Cab +X
γ∈ΛB
a
vB
a⊗Cγ+X
β∈ΛC
a
vC
a⊗Cβ; (63a)
Ωa
V
dHa
dt =Fa
X
b∈Λa
vWAvg ⊗Cab +X
γ∈ΛB
a
vB
a⊗Cγ+X
β∈ΛC
a
vC
a⊗Cβ
; (63b)
Ωa
V
dJa
dt =Ha:
X
b∈Λa
vWAvg ⊗Cab +X
γ∈ΛB
a
vB
a⊗Cγ+X
β∈ΛC
a
vC
a⊗Cβ
+X
b∈Λa
SΣJ
ab Σ+
J−Σ−
J.
(63c)
Here, the strictly positive parameter SΣJ
ab and the weighted average velocity field are defined as
SΣJ
ab =1
2ρRcp
cab ·cab
∥Cab∥;vWAvg =vWAvg
nn+vWAvg
t,(64)
with their components being described as
vWAvg
n=c−
pρ−
Rv−
n+c+
pρ+
Rv+
n
c−
pρ−
R+c+
pρ+
R
;vWAvg
t=c−
sρ−
Rv−
t+c+
sρ+
Rv+
t
c−
sρ−
R+c+
sρ+
R
;cab =HAveCab.(65)
It is important to emphasise that strong compatibility between the different kinematic fields
{F,H, J}is lost at the semi-discrete level. However, weak compatibility is maintained due to
the coupled nature of the semi-discrete system of conservation equations.
For visualisation purposes, the current deformed geometry is recovered by integrating in
time the discrete nodal velocity field obtained using (62a)
dxa
dt =va.(66)
With respect to the time integration of the above system (62a, 63a, 63b, 63c) along with the
geometry x(66), and keeping in mind a fast and efficient algorithm, we advocate for an explicit
one-step two-stage Total Variation Diminishing Runge–Kutta (TVD-RK) method, thoroughly
reported by the authors in [48] and references therein.
5.2. Entropy production
In this section, inequality (58) is assessed for the above set of semi-discrete equations (62a,
63a, 63b, 63c). For illustrative purposes, the body under consideration is said to be homo-
geneous. Additionally, and in line with the Godunov’s theorem [56, 57], we assume piecewise
constant approximation (first order in space) for variables across each control volume. Making
use of expression (57), the semi-discrete counterpart of (51) is
X
a
Ωa
V
dHa
dt =X
a
Ωa
Vva·dpa
dt +Σa
F:dFa
dt +Σa
H:dHa
dt + Σa
J
dJa
dt −˙
Da(67a)
=X
a
Ωa
Vva·dpa
dt + (Σa
F+Σa
HFa+ Σa
JHa) : dFa
dt −˙
Da+X
aX
b∈Λa
Σa
JSΣJ
ab (Σb
J−Σa
J)
(67b)
=X
a
Ωa
Vva·dpa
dt +Pa:dFa
dt +X
aX
b∈Λa
Σa
JSΣJ
ab (Σb
J−Σa
J)−X
a
Ωa
V˙
Da,(67c)
19
where, equations (63a)-(63c) and (10) have been substituted in the second and third lines of
(67), respectively. Subsequently, we can substitute the linear momentum conservation equation
(62a), the deformation gradient conservation equation (63a) and, after some algebra, gives
X
a
Ωa
V
dHa
dt =X
aX
b∈Λa
1
2[ta·vb−tb·va]∥Cab∥(68a)
+X
aX
b∈Λa
va·Sv
ab (vb−va) + X
aX
b∈Λa
Σa
JSΣJ
ab (Σb
J−Σa
J)−X
a
Ωa
V˙
Da+˙
Πext.
(68b)
Here, ˙
Πext denotes the semi-discrete power contribution, expressed as
˙
Πext =X
a
Ωa
Vva·fa
R+X
γ
Aγ
RtB
a·vB
a+X
β
Aβ
RtC
a·vC
a,(69)
and the positive definite matrices are
Sv
ab =ρRcp
2(n⊗n) + ρRcs
2(I−n⊗n).(70)
Noticing that the nested summation is carried out over control volumes in (68) and the anti-
symmetric nature of the first line of the right hand side, we can conclude that these terms
cancel and thus (68) reduces to
X
a
Ωa
V
dHa
dt −˙
Πext =X
aX
b∈Λa
va·(Sv
ab (vb−va)) + X
aX
b∈Λa
Σa
JSΣJ
ab (Σb
J−Σa
J)−X
a
Ωa
V˙
Da
(71a)
=X
aX
b∈Λa
vb·(Sv
ba (va−vb)) + X
aX
b∈Λa
Σb
JSΣJ
ba (Σa
J−Σb
J)−X
a
Ωa
V˙
Da.
(71b)
It is worth pointing out that the first two terms on the right hand side can be equivalently
written by swapping indices aand b. Simple averaging the first line and the second line of the
equation above, and noticing that Sv
ab =Sv
ba and SΣJ
ab =SΣJ
ba , an alternative expression is
X
a
Ωa
V
dHa
dt −˙
Πext =−"1
2X
aX
b∈Λa(vb−va)·Sv
ab (vb−va) + SΣJ
ab Σb
J−Σa
J2+X
a
Ωa
V˙
¯εa
p¯τa#.
(72)
Indeed, the first two terms in the square bracket of (72) are always non-negative. Moreover, in
the case of elasto-plasticity, the third term representing the rate of plastic dissipation is also
non-negative.
6. Algorithmic description
For ease of understanding, Algorithm 2 summarises the complete algorithmic description of
the proposed finite volume methodology for large strain contact dynamics. This algorithm6is
6Moreover, we have also implemented the algorithm using an in-house software for the proof-of-concept
one-dimensional and two-dimensional examples presented in the paper.
20
implemented in modern CFD code “OpenFOAM”, with an eye on large scale contact simulation
in future works.
Algorithm 2: Vertex centred finite volume algorithm for contact dynamics
(1) Initialise median dual mesh and solid dynamic variables for all bodies,i
(2) Initialise contact pairs
- Identify contact pairs and initialise contact variables
- Compute face area projection weights (see Remark 6)
while t<tend do
(3) Calculate allowable time step: ∆t
(4) Store all conserved variables: Uold
a,i =Un
a,i
for Runge-Kutta Stage = 1 to 2do
(5) Update contact pairs through two-way mapping (see Algorithm 3)
forall bodies do
(6) Apply linear reconstruction for interior fluxes (refer to Section 3.3 in [35])
(7) Compute numerical fluxes via Riemann solver:
- Interior fluxes as contact-stick (see Section 3.2.1): FI
Nab (U−
ab,U+
ab,Nab )
- Boundary fluxes (see Section 4.2 in [45]): FB
a
- Contact fluxes (see Algorithm 1): FC
a
(8) Update conservation variables: Ua=Ua+ ∆t˙
Ua
(9) Compute first Piola-Kirchhoff stress (see Remark 2): Pa
end
end
(10) Update conservation variables: Un+1
a,i =1
2Ua,i +Uold
a,i
(11) Compute first Piola-Kirchhoff stress (see Remark 2): Pa,i
end
Remark 7: Numerical simulations of contact problems often involve modelling the interaction of
multiple bodies across a non-conforming (or non-matching) interface mesh. In order to address
these scenarios, a pre-existing OpenFOAM library was employed by the proposed method. This
OpenFOAM library, known as Arbitrary Mesh Interface (AMI), is based on the conservative
local Galerkin projection procedure presented by Farrell and Maddison in [68]. By harnessing
this library a projection weighting is calculated based on the overlapping face area of the
two contact surfaces in the reference configuration. Since this AMI procedure is based on a
surface to surface projection, additional piecewise surface-to-vertex reconstruction algorithm is
then required. For illustrative purposes, Algorithm 3 summarises the non-conforming mapping
procedure in two dimensions, projecting variables from the “−” contact surface to the “+”
contact surface. Its graphical representation is also depicted in Figure 4.
Algorithm 3: The non-conforming mapping procedure in two dimensions.
(1) Obtain averaged variables at face centroid f:{t−
f,v−
f}←{t−
a,v−
a}
(2) Map face variables from “−” to “+” surface using AMI face area projection
weighting {tMap
f,vMap
f}
(3) Reconstruct face nodal variables {tMap
af ,vMap
af }via piecewise linear reconstruction
21
Figure 4: Two-dimensional vertex based mapping algorithm to project {ta,va}from “−” surface to “+” surface.
Table 1: Two-bar impact: material parameters used in the simulation for bar 1 and bar 2.
Young’s modulus E100 N m−2
Material density ρR0.01 kg m−3
Poisson’s ratio ν0.0
Shock wave speed cp100 m s−1
7. Numerical examples
In this section, a wide variety of numerical examples are presented in order to assess the
robustness, applicability and performance of the proposed formulation presented above. In the
following sections, it is important to demonstrate the overall algorithm
•ensures consistency and accuracy of the field variables at the contact interface for both
conforming and non-conforming interface meshes,
•guarantees long term stability by satisfying the discrete version of the second law of
thermodynamics (72), and
•circumvents hour-glassing and pressure instabilities even in the case of nearly incompress-
ible material and elasto-plasticity.
In the following numerical computations, we consider only the frictionless contact where the
value of friction coefficient kin (41) is set to zero. We also assume that for simplicity the
contact points between potential contacting interfaces are known a priori. In terms of the
temporal stability of the algorithm, the Courant-Friedrichs-Lewy number of αCFL = 0.3 has
been chosen. In addition, comparisons are also carried out against the results simulated using
the commercial software package Abaqus [69]. From the spatial discretisation standpoint, the
standard linear finite element method (triangular element in two dimension and tetrahedral
element in three dimension) and mean dilatation approach (quadrilateral element in two di-
mension and hexahedral element in three dimension) are employed, in conjunction with a set
of built-in artificial viscosity parameters in order to dissipate high frequency oscillations.
22
Figure 5: Two-bar impact: geometry and problem setup. The bar on the left is named as bar 1 and the bar on
the right is named as bar 2.
(a) (b)
(c) (d)
Figure 6: Two-bar impact: time evolution of (a) global total energy, (b) global numerical dissipation, (c)
different energy measures for bar 1 (via Mesh IV) and (d) different energy measures for bar 2 (via Mesh IV).
A neo-Hookean constitutive model as described in (9) is used. Their corresponding material parameters are
summarised in Table 1.
23
(a) (b)
Figure 7: Two-bar impact: time evolution of (a) velocity vxand (b) stress σxx monitored at point A in bar
1. Point A refers to position X= 10 m. A neo-Hookean constitutive model as described in (9) is used. Their
corresponding material parameters are summarised in Table 1.
7.1. Objective 1: Consistency and accuracy
7.1.1. One-dimensional two-bar impact for similar bars
The first example corresponds to the impact of two bars having equal length with an initial
gap of δ= 0.01 m, as shown in Figure 5. Bar 1 (on the left), is travelling with a given velocity
v1
0= 0.1 m/s towards bar 2 (on the right) which is at rest. Material properties for both bars
are exactly the same and are summarised in Table 1. The main objective of this classical
benchmarked problem is to examine the robustness and reliability of the proposed algorithm in
capturing contact mode transition. As reported in literature [6, 70–75], most of the methods
still exhibit severe non-physical oscillations in the velocity resolution throughout the duration of
contact and also post separation. Specific ad-hoc regularisation procedure is generally required
to limit these numerical artefacts.
In this example, a linear elastic model is considered and four different levels of mesh refine-
ments are used. For instance, {Mesh I,Mesh II,Mesh III,Mesh IV}comprise {128,256,512,1024}
number of elements, respectively. Both bars make first contact at time timpact =δ/v1
0= 0.1 s.
Such impact generates shock waves of speed cp=qE
ρR= 100 m/s, and they travel in opposite
directions along each bar and reflect back to the contact point at time t=timpact + 2L/cp= 0.3
s.
First, we demonstrate the proposed algorithm is capable of satisfying the second law of
thermodynamics, and hence ensuring long-term stability. Figure 6a shows the time history of
global total energy of the two bars. Its resolution after the contact at time t= 0.1 s is better
represented by refining the mesh. Another interesting variable of interest is the accumulated
numerical entropy (dissipation) present in the algorithm. This is achieved by integrating the
Hamiltonian energy of the system described in (72) over time7, which decreases over time for
the entire simulation. This is seen in Figure 6b. The total numerical dissipation is reduced
7Insofar as the linear elastic model is used in this case, we thus neglect the physical dissipation introduced
by the model, that is the rate of plastic dissipation.
24
(a) (b)
(c) (d)
(e) (f)
Figure 8: Two-bar impact: time evolution of (first row) velocity vx, (second row) stress σxx and (third row)
displacement ux. Results in the first column are monitored at point A in bar 1 and results in the second column
are monitored at point B in bar 2. Point A refers to position X= 10 m and point B refers to X= 10.01 m.
Comparison is carried out between the proposed algorithm (via Mesh IV) and the exact solutions for similar
bars. A neo-Hookean constitutive model as described in (9) is used. Their corresponding material parameters
are summarised in Table 1. 25
Table 2: Two-bar impact: material parameters used in the simulation.
Bar 1 Bar 2
Young’s modulus E49 100 N m−2
Material density ρR0.01 0.01 kg m−3
Poisson’s ratio ν0.0 0.0
Shock wave speed cp70 100 m s−1
when successively increasing the mesh density. In addition, Figures (6c) and 6d illustrate the
time histories of different forms of energy for bar 1 and bar 2, respectively. These include
kinetic energy Ktotal =RΩV
1
2ρRp·pdΩV, elastic strain energy ψtotal =RΩVEdΩV, and the total
energy being defined as the summation of kinetic energy and elastic strain energy. At time
t= 0, bar 1 (travelling at a given velocity v1
0) has the total energy fully dominated by kinetic
energy, whereas bar 2 has zero total energy (no movement and stress-free) since it is at rest.
When impact takes place at time t= 0.1 s, some of the total energy of bar 1 is transferred
into kinetic energy of bar 2 (both bars travel together whilst in contact) and some into elastic
strain energy of bar 2 (as both bar elastically deform during the period of contact). The elastic
strain energy of the bars peak at time t= 0.2 s due to the fact that the compressive stress wave
arrives at the free end of the bars. The stress wave is then reflected back from the free end (now
the stress wave becomes tensile stress wave) into the contact point which result in separation
at time t= 0.3 s. After separation, as expected, nearly all total energy is transferred from
bar 1 to bar 2, with only approximately 0.4 % of the total energy (via Mesh IV) dissipated
numerically due to the use of a Riemann-based algorithm (refer to the first two terms on the
right hand side of (72)).
Second, we highlight the importance of incorporating a slope limiter into the proposed
algorithm in order to improve the resolution of field variables in the vicinity of shocks. Figure
7 shows the time histories of velocity and stress for bar 1. The exact (analytical) solution
is also provided for verification purposes. As it can be observed, using piecewise constant
representation (first order accurate in space) for field interpolation, the solutions are fairly
dissipative over time for both velocity vxand axial stress σxx unless excessively fine mesh is used.
To enhance the accuracy, we introduce a piecewise linear reconstruction. Such enhancement,
as seen in Figure 7b, gives much better resolution in stress but fails prior to separation, where
non-physical oscillations are generated. In order to control these spurious modes, the classical
Barth and Jespersen limiter [57] is implemented. A great improvement is observed in Figure
7b. As compared to the classical finite element method, no post-separation velocity oscillations
are observed in the proposed algorithm.
Finally, for qualitative comparison purposes, we monitor the velocity vx, displacement ux
and stress σxx evolutions at two locations, namely point A in bar 1 and point B in bar 2.
Our results are in very good agreement with the given exact solutions (see Figure 8), without
showing undershoots/overshoots near a discontinuity.
7.1.2. One-dimensional two-bar impact for dissimilar bars
We next analyse the same impact problem but now considering a softer material (bar 1)
impacts against a stiffer material (bar 2). This is achieved by reducing the value of Young’s
modulus Eof bar 1 from 100 N/m2to 49 N/m2. The remaining material properties for bars
1 and 2 are exactly the same as those reported in Section 7.1.1, and are summarised again
in Table 2 for completeness. The purpose of this test case is to examine the applicability of
26
(a) (b)
(c) (d)
(e) (f)
Figure 9: Two-bar impact: time evolution of (first row) velocity vx, (second row) stress σxx and (third row)
displacement ux. Results in the first column are monitored at point A in bar 1 and results in the second column
are monitored at point B in bar 2. Point A refers to position X= 10 m and point B refers to X= 10.01
m. Comparison is carried out between the proposed algorithm (via Mesh IV) and the exact solutions for
dissimilar bars. A neo-Hookean model as described in (9) is used. Their corresponding material parameters are
summarised in Table 1. 27
the algorithm in addressing impact between two different materials. Figure 9 illustrates the
time evolutions of velocity vx, displacement uxand stress σxx monitored at points A and B.
The proposed algorithm can accurately capture the solutions during impact process and release
process, displaying extremely good agreement with the closed-form solutions. No specific ad-
hoc regularisation procedure is required.
28
Figure 10: Collision of rubber ring: geometry and problem setup. The rubber ring on the left is named as ring
1 and the rubber ring on the right is named as ring 2.
Table 3: Collision of rubber ring: material parameters used in the simulation for ring 1 and ring 2.
Young’s modulus E1×106N m−2
Material density ρR1000 kg m−3
Poisson’s ratio ν0.4
Lam´e parameters µ0.35714 MN m−2
λ1.42857 MN m−2
7.2. Objective 2: Spurious mechanism
7.2.1. Two dimensional compressible ring collision
As previously explored in References [76, 77], the main aim of this classical benchmark
problem is to examine the capability of the proposed algorithm in alleviating unwanted spurious
modes that may potentially arise in the contact (or shock) interface. The problem consists of
simulating the collision of two rubber rings, with an initial gap of 8mm, coming together at
a relative speed of 1.18m/s. The geometry of the problem is displayed in Figure 10. In this
example, a neo-Hookean model presented in (9) is considered. The values of all the relevant
material parameters used can be found in Table 3.
(a) (b) (c) (d)
Figure 11: Each ring domain is discretised with (a) Mesh I (2,480 linear triangle with 1,364 nodes), (b)
Mesh II (10,080 linear triangle with 5,292 nodes), Mesh I II (81,280 linear triangle with 20,828 nodes) and
Mesh IV (163,200 linear triangle with 82,620 nodes). To avoid repetition of image, only one rubber ring per
discretisation is shown above.
29
(a) (b)
(c) (d)
Figure 12: Collision of rubber ring: time evolution of (a) global total energy, (b) global numerical dissipation, (c)
different energy measures for ring 1 (via Mesh IV) and (d) different energy measures for ring 2 (via Mesh IV).
A neo-Hookean constitutive model as described in (9) is used. Their corresponding material parameters are
summarised in Table 3.
30
(a) (b)
Figure 13: Collision of rubber ring: comparison of deformed shapes at time t={10,20,30,40}ms (from top
to bottom) using (a) proposed algorithm (linear triangular mesh) with Mesh IV and (b) mean dilatation
technique (bilinear quadrilateral mesh). The colour contour plot indicates pressure field. A neo-Hookean model
as described in (9) is used. Their corresponding material parameters are summarised in Table 3.
31
(a) (b)
Figure 14: Collision of rubber ring: time evolution of (a) component of velocity vxand (b) component of stress
σxx at point A in ring 1. Point A refers to position X= [40,0]Tmm. A comparison is carried out between
the proposed algorithm with four different meshes and the mean dilatation approach. A neo-Hookean model as
described in (9) is used. Their corresponding material parameters are summarised in Table 3.
Aiming to show mesh convergence8, four successively refined meshes (see Figure 11) are
used. These include (Mesh I) 2,480, (Mesh II) 10,080, (Mesh III) 81,280 and (Mesh IV)
163,200 number of linear triangular elements for each ring. In order to ensure the algorithm
correctly reproduces the second law of thermodynamics, the global entropy and total energy are
monitored (see Figure 12a and 12b). Indeed, for all four meshes, both the global entropy and
total energy of the system decrease over time, whereby the irreversibility is caused by numerical
stabilisations introduced into the algorithm (which is precisely the square bracket term of (72)).
Before contact takes place, the total energy of the system is completely dominated by kinetic
energy. When time t > 6.8 ms, that is after collision takes place, the kinetic energy of the
system decreases and transforms into elastic strain energy. Additionally, a very small amount
of the kinetic energy also converts to monotonic decreasing numerical dissipation during a
deformation process. This is seen in Figures 12c and 12d.
For comparison purposes, we also simulate the same problem discretised using the stan-
dard linear finite element method (using 163,840 number of linear triangular meshes with
82,944 nodes) and the mean dilatation approach (using 81,920 number of bilinear quadrilateral
meshes with 82,944 nodes). Figure 13 illustrates the deformation process of the two rings at
time t={10,20,30,40}ms, displaying how the two rings collide, bounce off and then oscillate.
No spurious modes are observed. In comparison to the mean dilatation technique, very similar
results in terms of deformed shape and pressure field are observed. For completeness, the time
evolutions of the components of velocity vx, displacement uxand stress σxx are also monitored.
As shown in Figure 14, the obtained solutions converge to the results of mean dilatation tech-
nique by refining the mesh. In comparison to the standard linear finite element method, the
proposed algorithm clearly outperforms it by accurately capturing stress discontinuities (in this
8It is important to emphasise that the simulation of rubber ring collision could have been performed with
only one rubber ring via appropriate symmetry condition. However, in our case, we decided to consider the
complete two-ring impact in order to check the resolution at shock interface between two bodies.
32
(a) (b)
(c) (d)
(e) (f)
Figure 15: Collision of rubber ring: time evolution of (first row) velocity vx, (second row) stress σxx and (third
row) displacement ux. Results in the first column are monitored at point A in ring 1 and results in the second
column are monitored at point B in ring 2. Point A refers to position X= [40,0]Tmm and point B refers
to X= [48,0]Tmm. Comparison is carried out between the proposed algorithm (via Mesh IV), the linear
triangular finite element method and the mean dilatation approach. A neo-Hookean model described in (9) is
used. Their corresponding material parameters are summarised in Table 3.
33
Figure 16: Nearly incompressible bar impact: geometry and problem setup. The bar on the left is named as
bar 1 and the bar on the right is named as bar 2.
Table 4: Nearly incompressible bar impact: material parameters used in the simulation for bar 1 and bar 2.
Young’s modulus E5.85 ×108N m−2
Material density ρR8930 kg m−3
Poisson’s ratio ν0.495
Lam´e parameters µ0.19565 GN m−2
λ19.3696 GN m−2
case, σxx) without any spurious oscillations. This is seen in Figures 15c and 15d.
7.2.2. Two dimensional nearly incompressible bar impact
Similar to the objectives described in Section 7.2.1, another standard benchmark problem
previously adopted in [47] is considered. As shown in Figure 16, the example presents the
impact of two nearly incompressible rectangular bars travelling at each other with a relative
velocity of v0= [100,0]Tm/s. For each bar, its width is of w= 6.4 mm and its length is of
L= 32.4 mm. The normal separation between two bars is 8 mm. A neo-Hookean model is
used. The values of all the simulation parameters are summarised in Table 4. For completeness,
we discretise the problem using four different levels of mesh refinement, including (Mesh I)
640, (Mesh II) 2,560, (Mesh III) 10,240 and (Mesh IV) 40,960 number of linear triangular
elements per bar.
First, a mesh refinement study for the proposed algorithm is carried out. In Figure 17,
the deformation pattern of the structure predicted using a small number of elements (Mesh I)
agrees well with the results obtained using finer discretisations (Mesh II to Mesh IV). As
for the latter, improved pressure resolution is observed. For qualitative comparison purposes,
time evolution of velocity vxand displacement uxare monitored and compared in Figure 18.
Interestingly, double contact occurs between 80 µs and 100 µs. It is well-known that pressure
checker-boarding is commonly encountered in standard linear finite elements when attempting
to model materials with predominant nearly incompressible behaviour. This numerical artefact
can be completely resolved by the algorithm proposed in the current paper, without resorting
to any ad-hoc regularisation procedure. Comparing with mean dilatation technique (see Figure
19), smoother version in pressure profile is observed. Figure 20 shows the time evolution of the
deformation behaviour of two bars come into contact. Again, very smooth pressure profile is
seen throughout the entire contact-impact process. Neither hour-glassing nor pressure checker-
boarding are observed.
34
(a) Mesh I: 640 number of linear triangles with 369 number of nodes per bar
(b) Mesh II: 2,560 number of linear triangles with 1,377 number of nodes per bar
(c) Mesh III: 10,240 number of linear triangles with 5,313 number of nodes per bar
(d) Mesh IV: 40,960 number of linear triangles with 20,865 number of nodes per bar
Figure 17: Nearly incompressible bar impact: comparison of bar impact at time t= 90 µs using various mesh
refinements. In each subfigure, the first row depicts the current deformed state discretised with linear triangular
mesh and the second row illustrates pressure contour. A neo-Hookean model described in (9) is used. Their
corresponding material parameters are summarised in Table 4.
35
(a) (b)
(c) (d)
Figure 18: Nearly incompressible bar impact: time evolution of (first row) velocity vxand (second row) dis-
placement ux. Results in the first column are monitored at position X= [32.4,0]Tmm in bar 1 and results in
the second column are monitored at position X= [40.4,0]Tmm in bar 2. Comparison is carried out between
the proposed algorithm with Mesh IV, the linear triangular finite element method (40,960 number of linear
triangles with 20,865 number of nodes per bar) and the mean dilatation technique (20,480 number of bilinear
quadrilaterals with 20,865 number of nodes per bar). A neo-Hookean model described in (9) is used and the
material parameters are summarised in Table 4.
36
(a)
Step: FinalTim Frame: 125
Total Time: 0.000125
(b)
Step: FinalTim Frame: 2361
Total Time: 0.000100
(c)
Figure 19: Nearly incompressible bar impact: comparison of deformed shapes at time t= 100 µs using (a)
proposed algorithm with Mesh IV (40,960 number of linear triangles with 20,865 number of nodes per bar),
(b) linear triangle finite element method (40,960 number of linear triangles with 20,865 number of nodes per
bar) and (c) mean dilatation technique (20,480 number of bilinear quadrilaterals with 20,865 number of nodes
per bar). The colour contour plot indicates pressure field. A neo-Hookean model described in (9) is used. Their
corresponding material parameters are summarised in Table 4.
37
Figure 20: Nearly incompressible bar impact: a sequence of deformed structures with pressure resolution at
times t={50,75,100,125,150,200,250,300,325}µs (from top to bottom). Results obtained via Mesh IV. A
neo-Hookean model is used and the corresponding material parameters are summarised in Table 4.
38
Figure 21: Impact with non-matching interface mesh: geometry and problem setup. The bar on the left is
named as bar 1 and the bar on the right is named as bar 2. Both bars have width Wand height Hof 3.2 mm
with a length Lof 32.4 mm. The initial gap δbetween two bars is 0.4 mm.
Table 5: Impact with non-matching interface mesh: the number of linear tetrahedra for bar1 and bar2.
Bar 1 (elements; nodes) Bar 2 (elements; nodes)
Mesh I 30,000; 6,171 1,920; 525
Mesh II 30,000; 6,171 6,480; 1,519
Mesh III 30,000; 6,171 15,360; 3,321
Mesh IV 30,000; 6,171 51,840; 10,309
7.3. Objective 3: Non-matching contact interface
We now extend the above two dimensional bar impact problem to three dimension as dis-
played in Figure 21. This example serves the purpose to examine the accuracy and reliability
of the proposed algorithm when considering non-matching meshes at the contact interface. A
neo-Hookean model is chosen and the corresponding material properties remain exactly the
same as the one listed in Table 4.
Aiming to show mesh independent convergence for this problem, we begin this example by
performing a series of non-conforming mesh refinement analysis. The (bulk) mesh information
for the two bodies are presented in Table 5 and their respective non-matching interface meshes
are depicted in Figure 22. As shown in Figures 23 and 24, it is remarkable that the deformation
pattern together with pressure profile converge even with the use of a relatively coarse mesh
(Mesh I). Additionally, in Figure 25, we also monitor the time history of velocity component
vx, displacement component ux, stress component σxx and pressure at the contact point X=
[32.8,3.2,3.2]Tmm. Their corresponding spatial distributions at time t= 260 µs between
positions X= [32.8,1.6,1.6]Tmm and X= [65.2,1.6,1.6]Tmm are illustrated in Figure 26.
The solutions indeed converge with a progressive level of mesh refinement. In order for the
overall algorithm to ensure long term stability, the global entropy is monitored (see Figure
27). As expected, the global total entropy of the system decreases over time throughout the
entire simulation duration. Observe that our solution is slightly more dissipative than that of
the mean dilatation technique via tri-linear hexahedral elements (see Figure 28). No spurious
modes are seen comparing with the linear tetrahedral finite element method.
39
(a)
(b)
Figure 22: Impact with non-matching interface mesh: for ease of explication, we choose to display a series of non-
corforming mesh refinements on X-Y plane view. In (a), the first four rows represent the non-conforming mesh
discretisations for both bars (Mesh I to Mesh IV). Their respective close-up view of the interface between
two bodies can be seen in the first four columns of (b). For completeness, we also discretise the problem using
conforming mesh. This is illustrated in the last row of (a) and its associated close-up view is displayed in the
last column of (b).
40
Figure 23: Impact with non-matching interface mesh: comparison of three-dimensional bar impact at time
t= 120 µs using (a) non-conforming mesh discretisations (Mesh I to Mesh IV, from first row to fourth row)
and (b) conforming mesh discretisation (fifth row). Colour contour plot indicates the pressure profile. A neo-
Hookean model (9) is used. Their corresponding material parameters are summarised in Table 4.
41
Figure 24: Impact with non-matching interface mesh: comparison of three-dimensional bar impact at time
t= 260 µs using (a) non-conforming mesh discretisations (Mesh I to Mesh IV, from first row to fourth row)
and (b) conforming mesh discretisation (fifth row). Colour contour plot indicates the pressure profile. A neo-
Hookean model (9) is used. Their corresponding material parameters are summarised in Table 4.
42
(a) (b)
(c) (d)
Figure 25: Impact with non-matching interface mesh: time evolution of (a) velocity component vx, (b) dis-
placement component ux, (c) stress component σxx and (d) pressure. Results are monitored at position
X= [32.8,3.2,3.2]Tmm in bar 2. A neo-Hookean model (9) is used. Their corresponding material parameters
are summarised in Table 4.
43
(a) (b)
(c) (d)
Figure 26: Impact with non-matching interface mesh: spatial distribution at time t= 260 µs of (a) velocity
component vx, (b) displacement component ux, (c) stress component σxx and (d) pressure along the line in
bar 2 from X= [32.8,1.6,1.6]Tmm to X= [65.2,1.6,1.6]Tmm. A neo-Hookean model (9) is used. Their
corresponding material parameters are summarised in Table 4.
44
Figure 27: Impact with non-matching interface mesh: time evolution of (a) global total energy, (b) global
numerical dissipation, (c) different energy measures for bar 1 (via Mesh IV) and (d) different energy measures
for bar 2 (via Mesh IV). A neo-Hookean constitutive model as described in (9) is used. Their corresponding
material parameters are summarised in Table 4.
45
Figure 28: Impact with non-matching interface mesh: time evolution of global total energy using the proposed
method (Mesh IV), mean dilatation and linear finite element method. Comparison of deformed shapes at time
t= 205 µs is shown, where the colour plot indicates pressure distribution. A neo-Hookean constitutive model
as described in (9) is used. Their corresponding material parameters are summarised in Table 4. The results
of mean dilatation is obtained using 5,000 number of trilinear hexahedra with 6,171 number of nodes for bar
1 and 8,640 number of trilinear hexahedra with 10,309 number of nodes for bar 2. The results of linear finite
element method is obtained using 30,778 number of linear tetrahedra with 6,267 number of nodes for bar 1
and 50,261 number of linear tetrahedra with 9,976 number of nodes for bar 2.
46
Figure 29: Torus impact: geometry and problem setup.
Table 6: Torus impact: material parameters used in the simulation.
neo-Hookean von-Mises plasticity
Young’s modulus E1×1061×106N m−2
Material density ρR8,930 8,930 kg m−3
Poisson’s ratio ν0.45 0.45
Lam´e parameters µ0.34483 0.34483 MN m−2
λ3.10345 3.10345 MN m−2
Initial yield stress τ0
y- 1 ×104N m−2
Hardening Parameter H- 10 N m−2
7.4. Objective 4: Highly nonlinear impact
In the last example, we consider the rebound of a torus of outer radius Ro= 40mm, inner
radius ri= 30mm and diameter d0= 1mm. The torus impacts against a rigid frictionless wall
with an initial velocity of v0= [1.18,0,0]Tm/s where the separation distance between the torus
and the wall is of δ= 4mm. This is illustrated in Figure 29. A neo-Hookean model is first
chosen with the material properties summarised in the third column of Table 6.
Aiming to demonstrate the consistency of the algorithm, the domain is discretised using four
different levels of refinement, namely (Mesh I) 4,643, (Mesh II) 12,439, (Mesh III) 29,748
and (Mesh IV) 56,955 number of unstructured linear tetrahedral meshes. Figure 30c shows the
reduction of total numerical dissipation when successively increasing the mesh density. More
crucially, the global (numerical) entropy is non-positive and reduces over time for the entire
simulation. In Figure 30e, the evolution in time of the kinetic energy, elastic strain energy and
total energy (being the summation of both kinetic and elastic strain energy) are monitored.
At the start of the simulation, the total energy of the system is completely dominated by
kinetic energy as the torus is moving with the initial velocity until approximately 3.4 ms where
47
(a) (b)
(c) (d)
(e) (f)
Figure 30: Torus impact: time evolution of (first row) global total energy, (second row) global numerical
dissipation and (third row) different energy measures (via Mesh IV). The solution in the first column are
obtained via neo-Hookean model and the solution of the second column are obtained via a Hencky-based von
Mises plasticity model. Their corresponding material parameters are summarised in the third and fourth column
of Table 6, respectively.
48
Figure 31: Elastic torus impact: comparison of deformed shapes at time t={0.5,1,1.5,2}ms (from top
to bottom) using (left column) proposed algorithm with Mesh IV, (centre column) linear tetrahedral finite
element method (101,045 number of elements and 20,427 number of nodes) and (right column) mean dilatation
technique (29,441 number of hexahedra and 33,793 number of nodes). The colour contour plot indicates pressure
field. A neo-Hookean model described in (9) is used and the material parameters are summarised in third column
of Table 6.
49
Figure 32: Elastic torus impact: a sequence of deformed structures with pressure resolution at times t=
{2.5,5,7.5,. . . ,50}ms (from left to right and top to bottom). Results obtained using the proposed algorithm
discretised with linear tetrahedra (Mesh IV). A neo-Hookean model is used and the corresponding material
parameters are summarised in third column of Table 6.
50
Figure 33: Elasto-plastic torus impact: comparison of deformed shapes at time t={0.5,1,1.5,2}ms (from top
to bottom) using (left column) proposed algorithm with Mesh IV, (centre column) linear tetrahedral finite
element method (101,045 number of elements with 20,427 nodes) and (right column) mean dilatation technique
(29,441 number of hexahedra with 33,793 nodes). The colour contour indicates pressure field. A Hencky-based
von Mises plasticity model is used and the material parameters are summarised in fourth column of Table 6.
51
(a) (b)
(c) (d)
Figure 34: Elasto-plastic torus impact: time evolution of (a) velocity vx, (b) displacement ux, (c) stress σxx
and (d) pressure. Results are monitored at position X= [40,0,0]Tmm. Comparison is carried out between the
proposed algorithm with Mesh IV, the linear tetrahedral finite element method (101,045 number of elements
with 20,427 nodes) and the mean dilatation technique (29,441 number of hexahedra with 33,793 nodes). A
Hencky-based von Mises plasticity model is used and the material parameters are summarised in fourth column
of Table 6.
52
Figure 35: Elasto-plastic torus impact: a sequence of deformed structures experiencing plasticity with pressure
resolution at times t={2.5,5,7.5,. . . ,50}ms (from left to right and top to bottom). Results obtained using
the proposed algorithm discretised with linear tetrahedra (Mesh IV). A Hencky-based von Mises plasticity
model (with isotropic hardening) is used and the corresponding material parameters are summarised in fourth
column of Table 6.
53
impact occurs. In the elastic case, the kinetic energy is mostly transferred into elastic strain
energy. When time is approximately 30 ms, that is when the separation begins to occur, the
elastic strain energy is then converted back to kinetic energy as the torus bounces off the
rigid surface. Comparing with the standard linear finite element method (101,045 number of
linear tetrahedra and 20,427 number of nodes), the proposed algorithm can be used without
experiencing any spurious mechanism (see Figure 31). By making use of the proposed method,
a series of deformed states are shown in Figure 32, where the colour contour plot indicates the
pressure distribution.
Moreover, the same problem is further analysed by employing a Hencky-based von Mises
plasticity (with isotropic hardening) model. Its associated materials properties are summarised
in the fourth column of Table 6. When the torus impacts against a frictionless wall, the total
kinetic energy is partially converted into elastic strain energy whilst most of the kinetic energy
in this case is transferred into irrecoverable plastic energy dissipation. Indeed, the amount of
physical plastic dissipation introduced into this model can then be monitored by integrating
in time the term related to the internal dissipation ˙
Dappearing in the Hamiltonian energy
of the system described in (72). Again, as seen in Figures 33 and 34, the proposed algorithm
can effectively alleviate non-physical pressure instabilities. Our solutions match well with the
results obtained via mean dilatation technique (which is discretised with 29,441 number of
tri-linear hexahedra and 33,793 number of nodes). For visualisation purposes, Figure 35 shows
the time evolution of the plastic deformation of a torus with very smooth pressure field.
8. Conclusions
The paper presents an explicit vertex centred finite volume method for the dynamic solution
of non-smooth contact problems, where a mixed system of first order conservation equations
together with the associated jump conditions is used. Using the specific jump equation for
the conservation of linear momentum, several dynamic contact models are derived ensuring the
preservation of hyperbolic characteristic structure across contact interface. The formulation
has been implemented within the modern CFD code “OpenFOAM”, aiming to bridge the gap
between CFD and Computational Solid Dynamics. Through the examples presented in this
paper, the proposed algorithm proves to perform extremely well in dynamic contact-impact
problems without resorting to ad-hoc algorithmic regularisation correction. Specifically, the
proposed algorithm by construction overcomes a number of persistent numerical drawbacks
commonly found in the literature. No spurious hour-glassing is observed and correct (smooth)
pressure pattern are obtained in contrast to alternative finite element approaches, such as the
well known linear tetrahedral element technology. Crucially, the overall algorithm ensures long-
term stability by monitoring the global entropy production via the Hamiltonian energy of the
system. The consideration of nonlinear shock wave speeds in the contact-impact conditions
within the current computational framework is the next step of our work.
Acknowledgements
Runcie and Lee gratefully acknowledge the support provided by the EPSRC Strategic Support
Package: Engineering of Active Materials by Multiscale/Multiphysics Computational Mechan-
ics - EP/R008531/1. Gil and Lee would like to acknowledge the financial support received
through the project Marie Sk lodowska-Curie ITN-EJD ProTechTion, funded by the European
Union Horizon 2020 research and innovation program with grant number 764636. Runcie and
Lee would also like to acknowledge the many useful discussions with Dr. Peter Grassl from
54
University of Glasgow.
55
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