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A Total Lagrangian upwind Smooth Particle Hydrodynamics
algorithm for large strain explicit solid dynamics
Chun Hean Lee a,1, Antonio J. Gil b, 2, Ataollah Ghavamianb, Javier Bonetc
(a) Glasgow Computational Engineering Centre, School of Engineering
University of Glasgow, Glasgow, G12 8QQ, United Kingdom
(b) Zienkiewicz Centre for Computational Engineering, College of Engineering
Swansea University, Bay Campus, SA1 8EN, United Kingdom
(c) University of Greenwich, London, SE10 9LS, United Kingdom
Abstract
In previous work [1, 2], Lee et al. introduced a new Smooth Particle Hydrodynamics (SPH)
computational framework for large strain explicit solid dynamics with special emphasis on the
treatment of near incompressibility. A first order system of hyperbolic equations was presented
expressed in terms of the linear momentum and the minors of the deformation, namely the
deformation gradient, its co-factor and its Jacobian. Taking advantage of this representation,
the suppression of numerical deficiencies (e.g. spurious pressure, long term instability and/or
consistency issues) was addressed through well-established stabilisation procedures. In Refe-
rence [1], the adaptation of the very efficient Jameson-Schmidt-Turkel algorithm was presented.
Reference [2] introduced an adapted variationally consistent Streamline Upwind Petrov Galer-
kin methodology. In this paper, we now introduce a third alternative stabilisation strategy,
extremely competitive, and which does not require the selection of any user-defined artificial
stabilisation parameter. Specifically, a characteristic-based Riemann solver in conjunction with
a linear reconstruction procedure is used, with the aim to guarantee both consistency and con-
servation of the overall algorithm. We show that the proposed SPH formulation is very similar
in nature to that of the upwind vertex centred Finite Volume Method presented in [3]. In order
to extend the application range towards the incompressibility limit, an artificial compressibi-
lity algorithm is also developed. Finally, an extensive set of challenging numerical examples is
analysed. The new SPH algorithm shows excellent behaviour in compressible, nearly incom-
pressible and truly incompressible scenarios, yielding second order of convergence for velocities,
deviatoric and volumetric components of the stress.
Keywords: Conservation laws, SPH, Upwind, Riemann Solver, Explicit dynamics,
Incompressibility
1. Introduction
The classical (displacement-based) Smooth Particle Hydrodynamics (SPH) Lagrangian for-
malism [4–10] is well-known to suffer from a number of severe drawbacks, namely: (1) tensile
1Corresponding author: chunhean.lee@glasgow.ac.uk. This work was completed whilst at the previous af-
filiation: Zienkiewicz Centre for Computational Engineering, College of Engineering, Swansea University, Bay
Campus, SA1 8EN, United Kingdom
2Corresponding author: a.j.gil@swansea.ac.uk
Preprint submitted to Computer Methods in Applied Mechanics and Engineering September 27, 2018
instability, spurious pressure and hour-glassing [11, 12]; (2) numerical issues associated with
conservation, consistency, long term stability and convergence [13, 14] and (3) the reduced order
of convergence for derived variables (e.g. stresses and strains) [15–18].
Over the last two decades, significant effort has been devoted to enhance the robustness of
classical SPH algorithms. The numerical deficiencies described above can be partially alleviated
through the use of corrected kernel gradient approximations, combined with so-called non-
consistent stabilisation strategies [8] (i.e. artificial viscous fluxes [4, 19, 20] and conservative
strain smoothing regularisation [15, 21]). As reported in References [6, 8], the above enhanced
SPH methodologies ensure the reproducibility of complete polynomial basis in finite domains
[5, 22–24]), but still suffer from persistent artificial mechanisms similar to hour-glassing when
attempting to model problems with predominant nearly incompressible behaviour.
In very recent works [1, 2], some of the authors of the present manuscript have successfully
introduced a mixed-based SPH computational framework for explicit fast solid dynamics, where
the conservation of linear momentum pis solved along with conservation equations for the
deformation gradient F, its co-factor Hand its Jacobian J. Specifically, the SPH discretisation
of the new system of conservation laws {p,F,H, J }was introduced through a family of well-
established stabilisation strategies, namely a Jameson-Schmidt-Turkel (JST) algorithm [25, 26]
and a variationally consistent Streamline Upwind Petrov Galerkin (SUPG) algorithm [2, 27].
Both computational methodologies were capable of eliminating spurious hourglass-like modes
and spurious pressure oscillations in nearly incompressible scenarios.
The main objective of the current paper is to explore a third alternative SPH spatial discre-
tisation of the same system {p,F,H, J}by means of a very competitive Riemann solver based
stabilisation methodology, without the need to introduce any artificial stabilisation parameters.
The mixed-based set of equations is advanced in time by means of an explicit time integrator,
where the time step is controlled through the Courant-Friedrichs-Lewy number [28] dependent
on the volumetric wave speed cp. We also show that the new SPH algorithm is very similar
to an upwind vertex centred FVM [3], where the latter requires the generation of a dual mesh
using the medial dual approach [29]. In the near (or full) incompressibility limit, the wave
speed cpcan reach very high (even infinite in the degenerate case) values potentially leading to
a very inefficient algorithm [26, 30]. To address this issue, we then present an adapted artifi-
cial compressibility algorithm [31–33], supplemented with the use of a (conservative) Laplacian
viscosity term, if necessary, to allow for the explicit simulation of (nearly) incompressible solids.
The paper is organised as follows. Section 2 briefly summarises the system of first or-
der conservation laws {p,F,H, J }for large strain solid dynamics. Section 3 describes the
computational methodology of the Upwind SPH framework. The variational statement of the
{p,F,H, J}system, the SPH approximation, the Riemann solver based SPH discretisation and
the linear reconstruction procedure are presented. For clarity, the complete explicit Upwind
SPH algorithm is summarised in Section 4. In addition, an artificial compressibility algorithm
is also presented in Section 5 to account for truly and nearly incompressible solids. Section 6
summarises the flowchart of the proposed SPH methodology. In Section 7, an extensive set of
challenging numerical examples is examined to assess the performance of the proposed algo-
rithm. Finally, Section 8 presents some concluding remarks and current directions of research.
2. Reversible elastodynamics
Consider the three dimensional deformation of an isothermal body of material density ρ0
moving from its initial undeformed configuration occupying a volume V, of boundary ∂V , to
2
Figure 1: Motion of a deformable continuum domain
a current deformed configuration at time toccupying a volume v, of boundary ∂v (see Figure
1). The time dependent motion is defined through a deformation mapping x=φ(X, t) which
satisfies the following set of Total Lagrangian first order conservation laws [1, 2, 27, 34–38]
∂p
∂t −DIV P=f0; (1a)
∂F
∂t −∇0p
ρ0=0; (1b)
∂H
∂t −CURL p
ρ0
F=0; (1c)
∂J
∂t −DIV 1
ρ0
HTp= 0.(1d)
Here, p:= ρ0vis the linear momentum per unit of undeformed volume, vis the velocity field,
Fis the deformation gradient (or fibre map), His the co-factor of the deformation (or area
map), Jis the Jacobian of the deformation (or volume map), Pis the first Piola-Kirchhoff
stress tensor and f0is a body force term per unit of undeformed volume. The symbol
represents the tensor cross product between vectors and/or second order tensors in the sense of
[25, 30, 39, 40], DIV and CURL represent the material divergence and material curl operators
as defined in expressions (5) and (7) of Reference [25], respectively, and ∇0represents the
material gradient operator defined as ∇0:= ∂
∂X.
Above equation (1a) represents the conservation of linear momentum, whilst the rest of the
system equations (1b-1d) represent a supplementary set of conservation laws for the geometric
strain measures {F,H, J}. Additionally, appropriate involutions [1, 26] must be satisfied by
some of the strain variables {F,H}[41] of the system as
CURLF=0; DIVH=0.(2)
Notice that, if necessary, the use of these involutions enables the conservation equations for
3
the area and volume maps in (1c) and (1d) to be re-written as [1]
∂H
∂t −F∇0p
ρ0=0;∂J
∂t −H:∇0p
ρ0= 0.(3)
More generally, the above system of conservation laws (1) can be summarised in a concise
manner as ∂U
∂t +∂FI
∂XI
=S;∀I= 1,2,3,(4)
where Udenotes the set of conservation variables, Sthe source term and FIthe flux vector
in the material Cartesian direction I, as follows
U=
p
F
H
J
;FI=−
P EI
1
ρ0p⊗EI
F1
ρ0p⊗EI
H:1
ρ0p⊗EI
;S=
f0
0
0
0
,(5)
with EIis the I-th unit vector of the Cartesian basis defined as
E1=
1
0
0
;E2=
0
1
0
;E3=
0
0
1
.(6)
Furthermore, the corresponding flux vector (5b) associated with the material unit outward
normal Ncan be expressed as
FN=FINI=−
P N
1
ρ0p⊗N
F1
ρ0p⊗N
H:1
ρ0p⊗N
.(7)
Crucially, in the presence of non-smooth solutions, all of the above conservation laws (1a-1d)
are accompanied by appropriate jump conditions [34, 42] defined as
cJpK=−JPKN; (8a)
cJFK=−1
ρ0
JpK⊗N; (8b)
cJHK=−FAve 1
ρ0
JpK⊗N; (8c)
cJJK=−HAve :1
ρ0
JpK⊗N,(8d)
where J·Kdenotes the jump operator across a discontinuity surface with normal Nmoving with
speed cin the reference space and {FAve,HAve}being defined as an average of the geometric
strain measures between the left and right states of a discontinuity surface.
For the particular case of a reversible process, the closure of the system described in (1)
requires the introduction of a suitable constitutive law relating the stress tensor Pwith the
4
strain measures {F,H, J}, obeying the principle of objectivity [43] and thermodynamic consis-
tency (via the Colemann-Noll procedure) [44]. In this work, a polyconvex nearly incompressible
constitutive model is used (see Section 2.1 of Reference [1] on pg. 75). Taking advantage of the
complete set of conservation laws described in (1), the polyconvex nature of the constitutive
model (i.e. a guarantor of material stability [45, 46]) facilitates the transformation of the sy-
stem of conservation laws into a symmetric set of hyperbolic equations expressed in terms of the
entropy conjugates of the conservation variables [25, 30]. Finally, for the complete definition
of the initial boundary value problem, initial and boundary (essential and natural) conditions
must also be specified as appropriate.
Remark 1:
In the present manuscript, consideration of irreversible processes is restricted to the case of
an isothermal elasto-plastic model typically used in metal forming applications [24]. Particu-
larly, thermal effects will be neglected. In order to model irrecoverable plastic behaviour, the
standard rate-independent von Mises plasticity model [43] with isotropic hardening is used and
the basic structure is summarised for completeness in Appendix A.
3. Spatial discretisation
The set of local conservation equations described in (1a-1d) has recently been spatially
discretised by the authors using two well-established (and consistent) stabilisation strategies,
leading to the Jameson-Schmidt-Turkel Smooth Particle Hydrodynamics (JST-SPH) algorithm
[1] and the Streamline Upwind Petrov-Galerkin Smooth Particle Hydrodynamics (SUPG-SPH)
algorithm [2]. These numerical techniques resulted in being very efficient when attempting to
solve problems with predominant nearly incompressible behaviour. JST-SPH requires the eva-
luation of a blend of Laplacian and bi-Laplacian operators for JST stabilisation and SUPG-SPH
requires the generation of a secondary set of particles for the computation of a residual-based
SUPG stabilisation. In this paper, we explore a third and extremely competitive stabilised
SPH method. The new methodology is known as Upwind Smooth Particle Hydrodynamics
(Upwind-SPH) and will be discussed in the following section.
3.1. General remark
In general, a standard weak variational statement for the mixed-based system {p,F,H, J}
(1a-1d) (known as Bubnov-Galerkin contribution AGal) is established by multiplying the local
form of the conservation laws (4) by appropriate work conjugate virtual fields δVand integra-
ting over the volume Vof the body, to give
0 = AGal(U, δV) := ZV
δV•∂U
∂t dV −ZV
δV•SdV +ZV
δV•∂FI
∂XI
dV. (9)
The symbol •is used to denote the inner (dual) product of work conjugate pairs, δV:=
{δv, δΣF, δΣH, δΣJ}represent the virtual work conjugates of the conservation variables U,
δvis the virtual velocity field and {δΣF, δΣH, δΣJ}are appropriate conjugate stresses to
{F,H, J}respectively.
5
Figure 2: Particle approximation
Application of the Green-Gauss divergence theorem on the last term of (9) results in
0 = AGal(U, δV) := ZV
δV•∂U
∂t dV −ZV
δV•SdV −ZV
FI•∂δV
∂XI
dV +Z∂V
δV•FNdA. (10)
Above Galerkin representation (10) can be particularised to the case of the linear momentum
pand the extended set of geometric strain measures {F,H, J}
0 = Ap
Gal := ZV
δv·∂p
∂t dV +ZV
P:∇0δvdV −ZV
δv·f0dV −Z∂V
δv·tBdA; (11a)
0 = AF
Gal := ZV
δΣF:∂F
∂t −∇0p
ρ0dV ; (11b)
0 = AH
Gal := ZV
δΣH:∂H
∂t −F∇0p
ρ0dV ; (11c)
0 = AJ
Gal := ZV
δΣJ∂J
∂t −H:∇0p
ρ0dV, (11d)
with tBbeing a possible boundary traction vector. Notice that only the linear momentum
conservation equation (11a) has been integrated by parts in order to enable the imposition of
boundary tractions.
3.2. Corrected SPH approximation
For evaluation of the material gradient of any arbitrary vector function f, we employ the
following particle approximation via the use of the “Corrected Gradient of a Corrected Kernel
˜
∇0˜
W” [24, 47] described as follows
∇0f(Xa)≈X
b∈Λb
a
fb⊗Gb(Xa); Gb(Xa) := Vb˜
∇0˜
Wb(Xa).(12)
Here, Λb
arepresents the set of neighbouring particles bthat lie inside a sphere of a given radius
2haround Xa(see Figure 2), Vband fbrepresent the volume and time varying vector function f
6
stored at particle b. The kernel approximation W being employed in this paper is the standard
quadratic kernel function introduced in [48] (see equation (5) of Section 2.1 on pg. 350).
As shown in Reference [47] (see Section 4.3 on pg. 106-107), the above kernel gradient
evaluation Gb(Xa) reproduces exactly the gradient of any constant and/or linear function. For
this reason, equation (12), if necessary, can also be alternatively expressed by including the
redundant term fain (12) as
∇0f(Xa)≈X
b∈Λb
a
21
2(fa+fb)⊗Gb(Xa).(13)
Remark 2:
Another useful representation for the evaluation of a kernel gradient can be formulated by
artificially adding the redundant term −fato (12), resulting in
∇0f(Xa)≈X
b∈Λb
a
21
2(fb−fa)⊗Gb(Xa).(14)
One of the key contributions of this paper is to show a relationship between the SPH particle
approximation and the vertex centred Finite Volume Method (FVM) approximation [3], where
the latter requires the definition of a dual mesh3which is constructed using the median dual
approach [29]. This will enable a well-established upwinding finite volume spatial discretisation
[49, 50] commonly used in the CFD community to be adapted to a SPH mesh-free method.
To achieve this, we first recall the discrete gradient evaluation used within the control volume
associated with node aby utilising a discrete version of the Green-Gauss divergence theorem
∇0f(Xa)≈1
VaZ∂Va
f⊗NdA, (15)
with Nbeing the outward unit normal vector on the boundary of the control volume ∂Va.
Above discrete boundary integral (15) can then be discretised by means of the FVM where
a (second order) central difference approximation is used for the discretisation of the vector
function f, to give
∇0f(Xa)≈1
VaX
b∈Λb
a1
2(fa+fb)⊗Cab,(16)
where for a given edge connecting nodes aand b(see Figure (3a)), the material area vector Cab
is defined as4
Cab := X
k∈Λk
ab
NkAk,(17)
where Λk
ab is the set of facets belonging to edge ab,Akis the area of a given facet kand
Nkits outward normal. These material area vectors enable a substantial reduction in the
computational cost when computing the boundary integral used in the Green-Gauss divergence
theorem, since they save an additional loop on facets per edge ab [29, 38].
3The dual mesh is constructed by connecting edge midpoints with element centroids in two dimensions (see
Figure 3a) and edge midpoints with face centroids and element centroids in three dimensions.
4Due to the definition of a dual mesh, the area vectors satisfy Cab = -Cba.
7
(a) (b)
Figure 3: Control volume for (a) vertex centred FVM and (b) SPH mesh free approach in two dimensions. The
red shaded area is the control volume associated to node a. The blue lines are the edges connecting node a
to its neighbouring nodes bi. The vertex centred FVM requires the outward unit normal Nab to be defined
using the material outward normal Nkof every facet kbelonging to edge ab. However, for the SPH mesh free
approach, the normal Nab is defined by a direction vector between particles aand b.
Comparing equation (13) with (16), a useful relationship arises relating the (mesh-based)
material outward area vector with the (mesh-free) SPH kernel gradient evaluation, defined as
Cab := 2VaGb(Xa).(18)
Its reciprocal relationship can also be defined as Cba := 2VbGa(Xb). Due to the anti-symmetric
nature of the SPH gradient correction [7, 47], notice here that Gb(Xa)6=Ga(Xb) which in
turn lead to Cab 6=−Cba.
3.3. SPH spatial discretisation
Typically, in the context of Element Free Galerkin schemes [51, 52], the Galerkin weak
statements described in (11a-11d) are accurately evaluated using the necessary distribution of
(Gauss) quadrature points in order to avoid spurious hourglass (or zero-energy) modes [12,
15, 53]. In general, the positions of these quadrature points do not coincide with those of the
material particles [51, 54]. This is however not the case when employing a mesh-free SPH
discretisation. In the latter case, above integrands (11a-11d) are under-integrated at the cloud
of particles regarded as quadrature points [1, 4, 6, 8, 55, 56].
3.3.1. Bubnov-Galerkin contribution
Using a corrected kernel interpolation ˜
W (see Section 3.2 on pg. 77 in Reference [1]) along
with the corresponding discrete gradient evaluation of ∇0δvdescribed in (14), term Ap
Gal (11a)
results in the following semi-discrete expression
dpa
dt =Ea−Ta.(19a)
8
Here, the nodal external Eaand internal Taforce vectors are defined as
Ea=Aa
Va
ta
B+fa
0;Ta=X
b∈Λb
a
2VbhPb˜
∇0˜
Wb(Xa)iAve ,(20)
where hPb˜
∇0˜
Wb(Xa)iAve := 1
2hPb˜
∇0˜
Wa(Xb)−Pa˜
∇0˜
Wb(Xa)i.(21)
Here, Pa,b := P(Fa,b,Ha,b , Ja,b). Aaand ta
Bare the tributary area and traction vector of those
particles placed on the boundary, the latter being computed directly from the given traction
boundary conditions [3, 34, 42]. The internal force representation Tadescribed in (20) satisfies
the global conservation of linear momentum, that is PaVaTa=0.
Analogously to the above discretisation procedure, the extended set of geometric conserva-
tion laws AF,H,J
Gal (11b-11d) can now follow, but this time using the gradient approximation of
∇0p
ρ0presented in (13). This will yield
dFa
dt =X
b∈Λb
a
2pAve
ρ0⊗Gb(Xa); (22a)
dHa
dt =FaX
b∈Λb
a
2pAve
ρ0⊗Gb(Xa); (22b)
dJa
dt =Ha:X
b∈Λb
a
2pAve
ρ0⊗Gb(Xa),(22c)
where the average linear momentum pAve := 1
2(pa+pb).
Remark 3: It is easy to show that the vertex-centred FVM presented in Reference [3] is a parti-
cular case of the above representation (19a, 22a-22c). To show this, we utilise the relationship
described in (18) and replacing the kernel gradient Gb(Xa) in (19a, 22a-22c) with the material
area vector Cab, which gives
dpa
dt =1
Va
Ea−1
VaX
b∈Λb
a
[PbCab]Ave ; (23a)
dFa
dt =1
VaX
b∈Λb
apAve
ρ0⊗Cab; (23b)
dHa
dt =1
Va
FaX
b∈Λb
apAve
ρ0⊗Cab; (23c)
dJa
dt =1
Va
Ha:X
b∈Λb
apAve
ρ0⊗Cab; (23d)
with [PbCab]Ave := 1
2[PbCba −PaCab]. However, for the case of the dual mesh used in FVM,
the area vectors become Cab =Cab, and thus satisfying Cba =−Cab. As a result, the linear
momentum evolution (23a) reduces to the following expression
dpa
dt =1
Va
Ea+1
VaX
b∈Λb
a
PAveCab;PAve =1
2(Pa+Pb),(24)
9
X,x
Y,y
Z,z
N+
N−
n−
n+
n−
n+
c−
sc+
s
c+
p
c−
p
Time t=0
Time t
Figure 3: Contact mechanics (To be replaced!!!)
(23a) can be reduced to
dpa
dt =1
Va
Ea+1
VaXb
a
4.3.2. Riemann-based upwiding stabilisation
In this paper, by taking advantage of the conservation structure of the mixed-based set of
equations (1a-1d) to be solved, a well-established numerical stabilisation algorithm is introduced
via a characteristic-based Riemann solver previously explored in the context of finite volume
method [33, 41, 45]. In what follows, some of the results presented in Reference [41] (see Section
4.3 on pages. 417-420) will be employed.
10
However, the resulting system, either SPH discretisation described in (19a, 22a-22c) or FVM
discretisation described in (24, 23b-23d), suffers from severe numerical instabilities (e.g. hour-
glassing, pressure and tensile instabilities) [12, 18, 24, 52] and requires some form of numerical
viscosity to counterbalance the negative diffusion introduced by the classical Bubnov-Galerkin
approximation [25–27, 29, 33, 34, 37, 41, 45]. In this work, an adapted (particle) Riemann-
based upwinding stabilisation strategy is introduced, crucial to circumvent this shortcoming.
This will be addressed in the next section.
a
1
PAveCab;PAve =
b
V(t)
V(t)
a
b
V
a
φ (X, t)
φ (X, t)
Vb
Figure 4: Contact mechanics
and the rest of the evolution equations for the geometric strain measures (23b-23d) remain
exactly the same but replacing Cab with Cab. Notice that expressions (24, 23b-23d) are exactly
the semi-discrete equations presented for a vertex centred FVM [3].
However, the resulting spatially discretised systems, either the SPH discretisation described
in (19a, 22a-22c) or the FVM discretisation in (24, 23b-23d), suffer from severe numerical
instabilities (e.g. hour-glassing, pressure and tensile instabilities) [12, 18, 24, 57] and require
some form of numerical dissipation [3, 25–27, 30, 34, 35, 38, 42]. In this work, an adapted
(particle) Riemann-based upwinding stabilisation strategy is introduced to circumvent this
shortcoming. This will be presented in the next section.
3.3.2. Riemann-based upwinding stabilisation
By taking advantage of the conservation structure of the mixed-based set of equations (1a-
1d), numerical stabilisation is introduced via a characteristic-based Riemann solver previously
explored in the context of finite volume methods [3, 34, 42]. In what follows, some of the results
presented in Reference [42] (see Section 4.3 on pages. 417-420) will be employed.
In Lagrangian dynamics, it is often possible for two domains apart (i.e. Vaand Vb) to
come into contact with each other after some time t. The impact would generate two types of
shock waves (p-wave cpand s-wave cs) travelling within the domains, as illustrated in Figure
4. Similar waves propagate in the fictitious interface between neighbouring particles when
considering piecewise discontinuous solutions.
Referring to Section 4.3 of [42], the numerical (interface) fluxes FC
Nab across a discontinuity
10
surface defined by the outward unit normal vector Nab can be read as follows
FC
Nab := −
PCNab
1
ρ0pC⊗Nab
Fa1
ρ0pC⊗Nab
Ha:1
ρ0pC⊗Nab
.(25)
These numerical fluxes can be additively decomposed into the summation of an average state
and an upwinding numerical stabilisation term, expressed as [·]C:= [·]Ave + [·]stab.
In order to ensure the discrete satisfaction of the involutions described by equation (2), we
must not introduce any numerical dissipation into (25b) and (25c) by setting the values of pC=
pAve. This implies that the update for {F,H}are naturally curl- and divergence-free as their
semi-discrete equations are formulated in terms of a material discrete gradient of a continuous
velocity field [3, 30, 38]. As a result, above flux approximations (25) now reduce to
FC
Nab =−
PAveNab
1
ρ0pAve ⊗Nab
Fa1
ρ0pAve ⊗Nab
Ha:1
ρ0pAve ⊗Nab
| {z }
Average
−
PstabNab
0
0
Ha:1
ρ0pstab ⊗Nab
| {z }
Upwinding stabilisation
.(26)
The first term on the right hand side of (26) represents the average (Galerkin) state, whereas
the second term (being high order corrections) can be interpreted as an upwinding numerical
stabilisation that damps the possible instabilities arising from the first term [34]. A detailed
derivation of the upwinding stabilisation term described in (26) can be found in Reference [42].
For completeness, the expressions for the stabilised first Piola Kirchhoff stresses and linear
momentum are [42]
Pstab := Sp
ab p+
f−p−
f⊗Nab;pstab := SP
ab P+
f−P−
fNab,(27)
with the (acoustic) stabilisation matrices {Sp
ab,SP
ab}being defined as
Sp
ab =1
2[cp(nab ⊗nab) + cs(I−nab ⊗nab )] ; SP
ab =1
2cp
(nab ⊗nab).(28)
Here, the outward spatial normal is defined by the current direction vector between particles
aand bsuch as nab := xb−xa
kxb−xakand {p−,+
f,P−,+
f}denote the left and right states of {p,P}at
fictitious interface f. The procedure of evaluating the reconstructed states {p−,+
f,P−,+
f}will
be discussed in Section 3.3.3.
It is now possible to introduce Riemann based stabilisation terms to the Galerkin contribu-
tions described in (24) and (23d). To achieve this, we need to replace {PAve,pAve}of (24) and
(23d) with {PC,pC}via (26), and after some simple algebraic manipulation gives
dpa
dt =
1
Va
Ea+1
VaX
b∈Λb
a
1
2(PbCba −PaCab)
+D(pa); (29a)
dJa
dt =
1
Va
Ha:X
b∈Λb
apAve
ρ0⊗Cab
+D(Ja),(29b)
11
where the Riemann based dissipative terms are
D(pa) = 1
VaX
b∈Λb
a
PstabCab ;D(Ja) = 1
VaX
b∈Λb
a
1
ρ0
pstab ·[HaCab].(30)
Substitution of the fundamental relation (18) described by Cab = 2VaGb(Xa) (and its
reciprocal relation Cba = 2VbGa(Xb)) into (29) and (30), the squared bracket terms in (29a,b)
recover the Galerkin discrete expressions for the conservation equations of linear momentum
(19a) and Jacobian (22c), whereas the particle upwinding stabilisation terms can be found as
D(pa) = X
b∈Λb
a
2VbPstab ˜
∇0˜
Wb(Xa); D(Ja) = X
b∈Λb
a
2Vb
pstab
ρ0·hHa˜
∇0˜
Wb(Xa)i.(31)
Since in this case ˜
∇0˜
Wb(Xa)6=˜
∇0˜
Wa(Xb), the dissipation term D(pa) presented in (31a)
will not automatically satisfy the global conservation requirement [47]. However, this can be
easily fulfilled by replacing ˜
∇0˜
Wb(Xa) with its average counterpart ˜
∇0˜
WAve
b(Xa) defined as
˜
∇0˜
WAve
b(Xa) := 1
2h˜
∇0˜
Wb(Xa)−˜
∇0˜
Wa(Xb)i.(32)
Similarly, in order to ensure the stabilisation term D(Ja) is globally conservative, the term
hHa˜
∇0˜
Wb(Xa)ipresented in (31b) must also be replaced by hHa˜
∇0˜
Wb(Xa)iAve defined by
hHa˜
∇0˜
Wb(Xa)iAve := 1
2hHb˜
∇0˜
Wa(Xb)−Ha˜
∇0˜
Wb(Xa)i.(33)
3.3.3. Linear reconstruction procedure
In order to guarantee second order accuracy in space, a linear reconstruction procedure
[34] is employed for the evaluation of the neighbouring states of the Riemann values, that is
{p−,+
f,P−,+
f}.
The reconstruction procedure can be achieved using two different strategies. The first
standard approach is to reconstruct both the linear momentum and the deformation gradient
tensor (e.g. {p−
f,F−
f}and {p+
f,F+
f}) at the mid-edge fconnecting particles aand b, followed
by the computation of the corresponding stresses {P(F−
f),P(F+
f)}. The second strategy is to
first compute the stresses at each particle (i.e. P(Fa) and P(Fb)) and then reconstruct the
computed stresses to the interface f, namely {P−
f,P+
f}. The latter is of particular interest in
this paper since it requires fewer evaluations of the stresses and, as complex constitutive models
are typically used in solid mechanics, this can result in a faster algorithm.
For any individual component Uof the variables pand P, the linear reconstructed value
at any position Xis in the form of
U(X) = Ua+Ga·(X−Xa).(34)
To achieve this, it is necessary to obtain an appropriate (particle) gradient operator Gathrough
the least squares minimisation process [34, 42].
Introducing first the objective functional Π given as
Π(Ga) = 1
2X
b∈Λb
a
1
kdabk2[Ub−(Ua+Ga·dab )]2;dab =Xb−Xa.(35)
12
Here, b∈Λb
arepresents a set of neighbouring (particle) values bassociated with particle a, and
dab represents a material vector measured from position ato b. Taking directional derivative
of (35) with respect to Ga, yields
0=DΠ[Ga] = X
b∈Λb
a
1
kdabk2[Ub−(Ua+Ga·dab )] dab.(36)
Rearranging equation (36) gives the following expression for Ga
Ga=
X
b∈Λb
a
Nab ⊗Nab
−1
X
b∈Λb
aUb− Ua
kdabkNab ,(37)
where Nab is the unit vector in the direction of dab described in (C.3) (see Figure 3b).
4. Complete Upwind-SPH algorithm
Addition of the discrete upwinding stabilisation terms (31) to the Bubnov-Galerkin discrete
expressions (see (19a) and (22a-22c)), finally yields the complete semi-discrete set of governing
equations for {p,F,H, J}as
dpa
dt =Ea−Ta+D(pa); (38a)
dFa
dt =X
b∈Λb
a
2pAve
ρ0⊗Gb(Xa); (38b)
dHa
dt =FaX
b∈Λb
a
2pAve
ρ0⊗Gb(Xa); (38c)
dJa
dt =Ha:X
b∈Λb
a
2pAve
ρ0⊗Gb(Xa) + D(Ja).(38d)
The Riemann-based stabilising terms {D(pa),D(Ja)}and the internal Taand external Ea
force vectors are already defined in Section 3.3. Notice here that the upwinding stabilisation
is only applied to the linear momentum evolution D(pa) (38a) and the volume map evolution
D(Ja) (38d). The former alleviates the appearance of spurious zero-energy (hourglass-like [8])
modes due to rank deficiency inherent to the use of Galerkin particle integration, whereas the
latter removes pressure instabilities in near incompressibility [26].
Remark 4:
Alternatively, it is also possible to show that the (particle) discrete system (38a-38d) can
13
be re-written as
dpa
dt =Ea−X
b∈Λb
a
Vb
Va
PbGa(Xb) + D(pa); (39a)
dFa
dt =X
b∈Λb
a
pb
ρ0⊗Gb(Xa); (39b)
dHa
dt =FaX
b∈Λb
a
pb
ρ0⊗Gb(Xa); (39c)
dJa
dt =Ha:X
b∈Λb
a
pb
ρ0⊗Gb(Xa) + D(Ja).(39d)
Insofar as the mixed-based system {p,F,H, J}, either (38) or (39), is rather large, it will
only be suitable to employ an explicit type of time integrator. For simplicity, an explicit one-
step two-stage Total Variation Diminishing Runge-Kutta (TVD-RK) scheme has been used
[1–3, 25–27, 34, 38, 42]. This is described by the following time update equations from time
step tnto tn+1:
U?
a=Un
a+ ∆t˙
Un
a(Un
a, tn);
U??
a=U?
a+ ∆t˙
U?
a(U?
a, tn+1);
Un+1
a=1
2(Un
a+U??
a).
(40)
In this manuscript, the geometry is also updated through the above TVD-RK algorithm
[1, 2, 42]. This results in a monolithic time integration procedure where the unknowns U=
(p,F,H, J, )Talong with the geometry xare all updated via (40).
The maximum time step ∆t:= tn+1 −tnis governed by a standard Courant-Friedrichs-Lewy
(CFL) condition [28] given as
∆t=αCF L
hmin
cp,max
,(41)
where cp,max is the maximum p-wave speed, hmin is the characteristic particle spacing within the
computational domain and αCF L is the CFL stability number. For the numerical computations
presented in this paper, a value of αCF L = 0.3, unless otherwise stated, has been chosen to
ensure both accuracy and stability [38] of the algorithm.
It is important to point out that the resulting Upwind-SPH algorithm (either (38) or
(39)) does not intrinsically fulfil conservation of angular momentum, since the strain measures
{F,H, J}are no longer exclusively computed from the current geometry [42]. For instance,
the deformation gradient F6=Fx:= ∇0x, its co-factor H6=Hx:= 1
2FxFxand its Jacobian
J6=Jx:= det Fx. In this paper, a monolithic discrete angular momentum projection algorithm
presented in Lee et al. [1] is employed and is summarised in Appendix B. Specifically, both the
internal nodal force Taand the particle-based upwinding stabilisation term D(pa), described
in (38a), are modified in a least-square sense in order to preserve the total angular momentum,
whilst still ensuring the global conservation of linear momentum.
14
5. Artificial compressibility
5.1. General remark
As it is well known, in the case of truly incompressible or nearly incompressible materials,
the volumetric wave speed cpcan reach very large values leading to prohibitively small time
steps [28]. This can have a very negative effect on the computational efficiency of any time-
explicit algorithm. One of the most popular approaches to address this issue is the artificial
compressibility method, originally developed for the Navier-Stokes equations [31–33]. Taking
inspiration from the fractional step algorithm presented in Gil et al. [26, 30], the artificial
compressibility approach is here adapted to the set of first order conservation law equations
presented in (1).
To achieve this, it is first important to re-write the volume map conservation law (3b) in
terms of the pressure unknown by utilising the volumetric constitutive law used in this paper,
namely p=κ(J−1) (refer to equation (11) on pg. 75 in Reference [1]). This equation now
reads 1
κ
∂p
∂t =H:∇0p
ρ0.(42)
In this case, the new unknowns of the problem are {p,F,H, p}(with Jbeing replaced by the
pressure variable p). For a truly incompressible material, κ≈ ∞ and the left hand side of
equation (42) vanishes, resulting in an incompressibility constraint [31].
In this approach, we first discretise the continuum equations (1a, 1b, 1c, 42) in time and
then proceed to discretise in space via a Riemann-based upwinding stabilisation as presented in
Section 3.3.2. For satisfaction of the incompressibility constraint, we can update the evolution
equations of the linear momentum (1a) and pressure (42) as follows [26, 30]
pn+1 −pn
∆t−DIVP(Fn,Hn, pn+1)−fn
0=0; (43a)
pn+1 −pn
κ∆t−Hn:∇0pn+1
ρ0= 0.(43b)
Both the pressure and linear momentum conservation variables are solved implicitly in tn+1,
assuming the deformation gradient Fand its cofactor Hto be frozen at time tn.
In order to solve equations (43a,b), a predictor-corrector algorithm is used. The algorithm
is first advanced explicitly yielding intermediate variables {pint, pint}which are then projected
after iteratively solving an implicit system known as pressure correction. Therefore, the first
(predictor or intermediate) step of the scheme over a time step ∆tis defined as
pint −pn
∆t−DIVP(Fn,Hn, pn)−fn
0=0; (44a)
Fn+1 −Fn
∆t−∇0pn
ρ0=0; (44b)
Hn+1 −Hn
∆t−Fn∇0pn
ρ0=0; (44c)
1
κ
pint −pn
∆t−Hn:∇0pn
ρ0= 0.(44d)
15
The second (corrector or projection) step becomes
pn+1 −pint
∆t−DIV pn+1 −pnHn=0; (45a)
1
κpn+1 −pint
∆t−Hn:∇0pn+1
ρ0−pn
ρ0= 0.(45b)
Here, the maximum time step ∆tof the predictor-corrector algorithm described above is go-
verned by the maximum p-wave speed dependent on the bulk modulus κof a material.
For nearly (and truly) incompressible materials, the bulk modulus present in the first term
of equation (44d) can potentially reach very high values (even infinite), leading to extremely
small time steps. This will then destroy the explicit nature of the predictor step of the scheme.
It is for this reason that a fictitious bulk modulus ˜κis used in its place, yielding
1
˜κ
pint −pn
∆t−Hn:∇0pn
ρ0= 0.(46)
As a consequence, the projection step of the pressure equation (45b) now becomes
1
κ
(pn+1 −pn)
∆t−1
˜κ
pint −pn
∆t−Hn:∇0pn+1
ρ0−pn
ρ0= 0.(47)
Notice that the summation of both equations (46) and (47) recovers the original assumption in
which the linear momentum variable is treated implicitly in the formulation.
Finally, in order to solve the implicit system for the correction (refer to (45a) and (47)),
one approach is to employ an iterative artificial compressibility algorithm [29]. This can be
achieved through the introduction of a pseudo time derivative term ∂
∂τ to expressions (45a) and
(47), and after rearranging gives
∂p
∂τ = DIV pn+1 −pnHn−pn+1 −pint
∆t; (48a)
1
γ
∂p
∂τ =1
˜κ
pint −pn
∆t+Hn:∇0pn+1
ρ0−pn
ρ0−1
κ
(pn+1 −pn)
∆t,(48b)
where γrepresents the artificial compressibility parameter. In this work, the pseudo terms
are also advanced in time using exactly the same time integrator described in (40). The SPH
spatial discretisation for the predictor-corrector system (44a, 44b, 44c, 46, 48a, 48b) will be
presented in the following section.
Remark 5:
In contrast to the artificial compressibility algorithm used in system (48a-48b), an alterna-
tive approach based on a pressure Poisson equation [26, 30] is also presented here for comple-
teness. In this approach, the pressure correction can be re-formulated by combining (46) into
(47), to give
1
κ
(pn+1 −pn)
∆t−Hn:∇0pn+1
ρ0.(49)
Substitution of (45a) into (49) for pn+1 yields
1
κ
(pn+1 −pn)
∆t−∆tHn:∇0pint
ρ0−Hn:∇0DIV 1
ρ0pn+1 −pnHn= 0.(50)
16
For evaluation of the pressure increment, equation above however requires the solution of a
system of nonlinear equations at each time step. This is indeed not computationally efficient
when employing a mesh-free SPH discretisation, hence not pursued in this work.
5.2. SPH artificial compressibility algorithm
Following the SPH discretisation procedure presented in Section 3, the predictor step of the
mixed-based system {p,F,H, p}is
pint
a−pn
a
∆t=En
a−Tn
a+D(pn
a); (51a)
Fn+1
a−Fn
a
∆t=X
b∈Λb
apn
b
ρ0⊗Gb(Xa); (51b)
Hn+1
a−Hn
a
∆t=Fn
aX
b∈Λb
apn
b
ρ0⊗Gb(Xa); (51c)
1
˜κ
pint
a−pn
a
∆t=Hn
a:X
b∈Λb
apn
b
ρ0⊗Gb(Xa) + D(Jn
a),(51d)
with the stabilising terms {D(pa),D(Ja)}and the internal and external force vectors {Ta,Ea}
defined in Section 3.3.
In addition, the corrector step of the discrete system {p,F,H, p}becomes
dpa
dτ =X
b∈Λb
apn+1
b−pn
bHn
bGb(Xa)−pn+1
a−pint
a
∆t; (52a)
1
γ
dpa
dτ =1
˜κ
pint
a−pn
a
∆t−1
κ
(pn+1
a−pn
a)
∆t+Hn
a:X
b∈Λb
apn+1
b
ρ0−pn
b
ρ0⊗Gb(Xa).(52b)
At each time step ∆t, above system (52) is iteratively solved for the linear momentum and
pressure within the pseudo time integration, aims at obtaining convergence to a pseudo steady
state (e.g. dpa
dτ ≈0and dpa
dτ ≈0).
5.3. Iteration speed-up procedure
To accelerate the speed of convergence within the iterative process, we can incorporate an
additional Laplacian (or harmonic) dissipative operator to equation (52b), which results in
1
γ
dpa
dτ =1
˜κ
pint
a−pn
a
∆t−1
κ
(pn+1
a−pn
a)
∆t+Hn
a:X
b∈Λb
apn+1
b
ρ0−pn
b
ρ0⊗Gb(Xa) + Dpseudo(pa).(53)
The discrete pseudo viscosity term at particle acan be written as
Dpseudo(pa) := ηL[p(Xa)] ; η=αc2
s∆t
µ,(54)
where αis a dimensionless user-defined parameter in the range of [0,1]. A challenging aspect
remains as to how to approximate the (particle based) Laplacian dissipative operator to ensure
fulfillment of the global conservation requirement, that is PaVaDpseudo(pa) = 0.
17
Referring to Appendix C, the discrete Laplacian viscosity operator (C.4) used in this work
is approximated as [64]
L[p(Xa)] ≈2X
b∈Λb
a
Vbpb−pa
kXb−XakNab·˜
∇0˜
Wb(Xa).(55)
Unfortunately, the evaluation of Laplacian operator prevents the exact global conservation due
to the lack of symmetry of the kernel gradient correction, that is ˜
∇0˜
Wb(Xa)6=˜
∇0˜
Wa(Xb).
This issue can also be addressed by replacing ˜
∇0˜
Wb(Xa) with ˜
∇0˜
WAve
b(Xa) (32).
Remark 6: Following [1], an alternative approach to approximate the (globally conservative)
Laplacian evaluation of the pressure variable pcan be expressed as
L[p(Xa)] ≈X
b∈Λb
a
Vb[pb−pa]˜
∆0WAve
b(Xa),(56)
where ˜
∆0WAve
b(Xa) is defined as
˜
∆0WAve
b(Xa) := 1
2h˜
∆0Wb(Xa) + ˜
∆0Wa(Xb)i.(57)
Here, ˜
∆0is a corrected Laplacian approximation which can be obtained through a least-square
minimisation procedure (see Appendix A in [1] for further details). This requires the solution
of a system of equations and for this reason not pursued in this paper.
Thus, the discrete pseudo viscosity operator can be described as
Dpseudo(pa)≈2ηX
b∈Λb
a
Vbpb−pa
kXb−XakNab·˜
∇0˜
WAve
b(Xa).(58)
Notice here that the only purpose of adding viscosity operator Dpseudo(pa) to (53) is to accelerate
the speed of convergence within the pseudo time integration process when iteratively solving
the implicit system for pressure correction (52a, 53). This is in clear contrast to the upwinding
stabilisation terms {D(pa),D(Ja)}introduced in the predictor step (51), crucial to ensure the
robustness (stability) of the algorithm.
6. Algorithmic description
For ease of understanding, Algorithm 1 summarises the complete algorithmic description of
the mixed-based {p,F,H, J}Upwind Smooth Particle Hydrodynamics (Upwind-SPH) metho-
dology, with all the necessary numerical ingredients. Notice that simpler {p,F}and {p,F, J}
versions of the algorithm can be easily obtained by neglecting the relevant geometric conserva-
tion laws (i.e. (39b) and/or (39c)).
7. Numerical examples
An ample spectrum of numerical examples is presented in order to examine the performance
of the proposed SPH methodology in compressible, nearly incompressible and truly incompres-
sible scenarios. Specifically, three stabilised mixed-based SPH methodologies are analysed,
namely {p,F},{p,F, J}and {p,F,H, J }Upwind-SPH. For validation purposes, some of
the results are benchmarked against other in-house mixed-based numerical schemes, including:
Finite Element Method [25–27, 30, 35], Finite Volume Method [3, 34, 38, 42] and SPH [1, 2].
18
Algorithm 1: Complete stabilised Upwind-SPH mixed methodology
Input : Un
awhere U= [p F H J]T
Output: Un+1
a,Pn+1
a,xn+1
a
(1) ASSIGN old primary variables: Uold
a=Un
aand xold
a=xn
a
(2) EVALUATE p-wave speed: cp(see References [30, 42])
(3) COMPUTE time increment: ∆t
for TVD-RK time integrator = 1 to 2do
(4) COMPUTE right-hand-side of the mixed-based system:
˙
pa(38a), ˙
Fa(38b), ˙
Ha(38c) and ˙
Ja(38d)
(5) APPLY discrete angular momentum preserving algorithm (see Section 6 of [1])
(6) EVOLVE {Ua,xa}via TVD-RK (40)
(7) COMPUTE first Piola Pa(see Eqn. (12) on pg. 75 in Reference [1])
end
(8) UPDATE {Un+1
a,xn+1
a}(see 40)
(9) COMPUTE first Piola Pn+1
a(see Eqn. (12) on pg. 75 in Reference [1])
Table 1: Numerical values for the relative error of the components of linear momentum and stress as compa-
red to the exact solution, measured with the L2norm. Comparison between the {p,F,H, J }Upwind-SPH,
{p,F,H, J }SUPG-SPH and {p,F,H, J}JST-SPH. Convergence rate calculated using the results of the two
finest meshes.
Error values px
h{p,F,H, J}Upwind {p,F,H, J}SUPG-SPH {p,F,H, J }JST-SPH
1/4 2.490 ×10−32.690 ×10−36.635 ×10−3
1/8 8.123 ×10−48.987 ×10−41.986 ×10−3
1/16 2.198 ×10−42.578 ×10−45.697 ×10−4
1/24 9.342 ×10−51.177 ×10−42.510 ×10−4
conv. rate 2.110 1.934 2.022
Error values PxX
h{p,F,H, J}Upwind {p,F,H, J}SUPG-SPH {p,F,H, J }JST-SPH
1/4 4.422 ×10−33.964 ×10−39.976 ×10−3
1/8 2.269 ×10−32.051 ×10−34.156 ×10−3
1/16 5.554 ×10−45.805 ×10−41.184 ×10−3
1/24 2.513 ×10−42.456 ×10−45.248 ×10−4
conv. rate 1.956 2.122 2.007
19
(a) (b)
Figure 5: Swinging cube. L2norm convergence of components of (a) Linear momentum; and (b) Stresses at a
particular time t= 0.002 s. Results obtained using a neo-Hookean constitutive model with A=B=C= 1
and U0= 5 ×10−4using the proposed SPH methodologies. The material properties used are Young’s modulus
E= 17 MPa, density ρ0= 1100 kg/m3, Poisson’s ratio ν= 0.3 and αCFL = 0.3.
7.1. Conservation, consistency and convergence
7.1.1. Swinging cube
As already explored in References [2, 3, 25–27, 30, 34, 38, 42, 58], the first example shows
a cube of unit side length with symmetric boundary conditions (e.g. restricted normal dis-
placement) at faces X= 0, Y= 0 and Z= 0 and skew-symmetric boundary conditions (i.e.
restricted tangential displacement) at faces X= 1m, Y= 1m and Z= 1m (see Figure 3 on
pg. 526 in [2]). The main objective of this example is to show the convergence behaviour of
the proposed {p,F},{p,F, J }and {p,F,H, J}Upwind-SPH methodologies5. A linear elastic
material with Young’s modulus E= 17 MPa and Poisson’s ratio ν= 0.3 is considered. The den-
sity of the material is ρ0= 1100 kg/m3and the Courant-Friedrich-Levy number is αC F L = 0.3.
Following Section 8.2 on pg. 85 of Reference [1], the convergence analysis is carried out by
computing the L2norm of the error between the analytical solution of this problem and the
numerical solution obtained for different values of the particle spacing. Table 1 shows a global
L2convergence analysis of the linear momentum pand the first Piola Kirchhoff stress tensor P
simulated using the {p,F,H, J}Upwind-SPH method6, as compared to the analytical solution
described in equation (52) of [1] (see Section 8.2 on pg. 85). Their corresponding graphical
representations are depicted in Figure 5. As expected, all of the SPH schemes show equal
second order convergence for both linear momentum and the components of the stress tensor.
Both Upwind-SPH and SUPG-SPH methodologies show very similar convergence behaviour,
as the stabilisation terms introduced in both methodologies are very well designed from a mat-
hematical standpoint (e.g. Riemann solver and residual based stabilisation). However, a slight
disadvantage of the SUPG-SPH method is the need to establish a secondary cloud of particles
(e.g. stress points) for the evaluation of the residual based stabilisation term, which would
5Nearly identical results have been obtained for {p,F}and {p,F, J}Upwind-SPH methodologies, hence
not presented.
6Given the fact that A=B=C, all the three components of linear momentum and stresses are of the same
magnitude, namely px=py=pzand PxX =PyY =PzZ .
20
Figure 6: Satellite problem setup
require a non-trivial intervention of the user prior to the analysis run. More interestingly, the
proposed Upwind-SPH method shows better accuracy than the JST-SPH algorithm previously
reported in [1], with the same slope but with a lower translation error (see Figure 5). This is
due to the fact that the Upwind-SPH method requires careful design (via an acoustic Riemann
solver) of the stabilisation terms, without relying on the use of any user defined stabilisation
parameters.
7.1.2. Satellite-like structure
In order to demonstrate the momentum conservation characteristics of the proposed al-
gorithm, the motion of a satellite-like structure originally proposed by [59] is studied. The
structure of a truncated cone of base radius 4 m, of top radius 2 m and of height 3 m, along
with four attached arms of unit cross-section that extend 6.5 m from the centre of the structure
(see Figure 6). The satellite is released without any initial deformation but with an initial
angular velocity of Ω = 1 rad/s about the centre of mass. The velocity field relative to the
centre of mass Xcm is given as
v0(X) = ω×(X−Xcm) ; ω= (0,0,Ω)T;X= (X, Y , 0) .(59)
A neo-Hookean model is chosen with the material parameters ρ0= 1000 kg/m3, Young’s
modulus E= 50.05 kPa and Poisson’s ratio ν= 0.3. For visualisation purposes, a sequence of
snapshots capturing the deformation of the satellite-like structure is shown in Figure 7. Figure
8 shows the time history of the components of the global angular and linear momenta, where
both fields can be seen to remain constant whilst the structure deforms free from external
effects. Activation of the angular momentum preserving algorithm enables the global angular
momentum to remain constant. We show how the new Riemann-based {p,F}Upwind-SPH
algorithm fulfill global conservation, namely PaVaD(pa) = 0and PaVaTa=0(see Figure
9).
7.2. Spurious pressure
7.2.1. Tensile test
A perforated block is clamped on its bottom surface and is left free on the rest of the
boundaries (see Figure 10a). The main aim of this example is to illustrate the capability of
the proposed Total Lagrangian Upwind-SPH algorithm in surpressing spurious pressure modes.
The block is initially pulled with a linear variation in initial velocity field v0=V[0,0,(Z/H)]T
where V= 100 m/s and H= 0.5 m. This problem is solved using a nearly incompressible
21
(a) t = 0 s (b) t = 1.66 s
(c) t = 5.94 s (d) t = 11.11 s
(e) t = 16 s (f) t = 23.33 s
Figure 7: Satellite-like structure. Time evolution of the deformation along with the pressure distribution using
the mixed-based {p,F}Upwind-SPH. Results obtained with an angular velocity of ω= [0,0,1]Trad/s. A
neo-Hookean constitutive model is employed with Young’s modulus E= 50046 Pa, density ρ0= 1000 kg/m3,
Poisson’s ratio ν= 0.3 and αCFL = 0.3. 22
(a) (b)
Figure 8: Satellite-like structure. Time evolution of the components of (a) Global angular momentum; and (b)
Global linear momentum using the mixed-based {p,F}Upwind-SPH. Results obtained with an angular velocity
of ω= [0,0,1]Trad/s. A neo-Hookean constitutive model is employed with Young’s modulus E= 50046 Pa,
density ρ0= 1000 kg/m3, Poisson’s ratio ν= 0.3 and αCFL = 0.3.
(a) (b)
Figure 9: Satellite-like structure. Time evolution of the components of (a) Global internal force PaVaTa≈0
and (b) Global Riemann-based dissipation PaVaD(pa)≈0of linear momentum evolution using the mixed-
based {p,F}Upwind-SPH algorithm. Results obtained with an angular velocity of ω= [0,0,1]Trad/s. A
neo-Hookean constitutive model is employed with Young’s modulus E= 50046 Pa, density ρ0= 1000 kg/m3,
Poisson’s ratio ν= 0.3 and αCFL = 0.3.
23
Z, z
v0=−V[0,0,(Z/H)]Tm/s
(b)
Table 3: Final radii of copper bar at t= 80µs. Results obtained using the proposed {p,F}Upwind-SPH
mesh-free algorithm, benchmarked against other published methodologies.
Method Final radius (cm)
Standard 4-Noded Tetrahedra [15] 0.555
Under-integrated 8-Noded Hexahedra [15] 0.695
Average Nodal Pressure 4-Noded Tetrahedra [15] 0.699
Jameson-Schmidt-Turkel Vertex Centred FVM [37] 0.698
Petrov-Galerkin FEM [27] 0.700
Upwind Cell Centred FVM [41] 0.700
Upwind-SPH mesh-free method 0.689
8.2.3. Taylor impact plasticity
Previously explored in References [27, 37, 41], a copper bar of initial length L= 0.0324
m and of initial radius r= 0.0032 m impacts against a rigid wall with a dropping velocity
of 227 m/s, as shown in Figure 17. The main objective of this benchmark problem is to
assess the performance of the proposed mesh-free algorithm in capturing plastic deformation
with application to metal forming. A von Mises hyperelastic-plastic material with isotropic
hardening is chosen to simulate the material. The material parameters are Young’s modulus
E= 117 GPa, density ρ0= 8.930 ×103kg/m3, Poisson’s ratio ν= 0.35, yield stress ¯τ0
y= 0.4
GPa and hardening modulus H= 0.1 GPa. Figure 18 shows the contours of both plastic strain
and pressure at eight different time instants. Extremely smooth pressure field is observed. For
comparison purposes, the final radius of copper bar at time t= 80µs predicted by the {p,F}
Upwind-SPH algorithm is shown in Table 3, benchmarked against other published numerical
results [15, 27, 37]. As reported in References [15, 16], the solutions obtained using the linear 4-
noded tetrahedra suffers from volumetric locking and pressure instabilities. The new mesh-free
method clearly circumvents these issues.
24
Y, y
D= 0.2m
H= 0.5m
Z, z
v0Tm/s
Y, y
D= 0.2m
H= 0.5m
(−0.5,−0.5,0.5) m(−0.5,−0.5,0.5) m
X, x (0.5,0.5,0) mX, x (0.5,0.5,0) m
(a)
=V[0,0,(Z/H)]
(a)
Z, z
v0=−V[0,0,(Z/H)]Tm/s
(b)
Table 3: Final radii of copper bar at t= 80µs. Results obtained using the proposed {p,F}Upwind-SPH
mesh-free algorithm, benchmarked against other published methodologies.
Method Final radius (cm)
Standard 4-Noded Tetrahedra [15] 0.555
Under-integrated 8-Noded Hexahedra [15] 0.695
Average Nodal Pressure 4-Noded Tetrahedra [15] 0.699
Jameson-Schmidt-Turkel Vertex Centred FVM [37] 0.698
Petrov-Galerkin FEM [27] 0.700
Upwind Cell Centred FVM [41] 0.700
Upwind-SPH mesh-free method 0.689
8.2.3. Taylor impact plasticity
Previously explored in References [27, 37, 41], a copper bar of initial length L= 0.0324
m and of initial radius r= 0.0032 m impacts against a rigid wall with a dropping velocity
of 227 m/s, as shown in Figure 17. The main objective of this benchmark problem is to
assess the performance of the proposed mesh-free algorithm in capturing plastic deformation
with application to metal forming. A von Mises hyperelastic-plastic material with isotropic
hardening is chosen to simulate the material. The material parameters are Young’s modulus
E= 117 GPa, density ρ0= 8.930 ×103kg/m3, Poisson’s ratio ν= 0.35, yield stress ¯τ0
y= 0.4
GPa and hardening modulus H= 0.1 GPa. Figure 18 shows the contours of both plastic strain
and pressure at eight different time instants. Extremely smooth pressure field is observed. For
comparison purposes, the final radius of copper bar at time t= 80µs predicted by the {p,F}
Upwind-SPH algorithm is shown in Table 3, benchmarked against other published numerical
results [15, 27, 37]. As reported in References [15, 16], the solutions obtained using the linear 4-
noded tetrahedra suffers from volumetric locking and pressure instabilities. The new mesh-free
method clearly circumvents these issues.
24
Y, y
D= 0.2m
H= 0.5m
Z, z
v0=−V[0,0,(Z/H)]Tm/s
Y, y
D= 0.2m
H= 0.5m
(−0.5,−0.5,0.5) m(−0.5,−0.5,0.5) m
X, x (0.5,0.5,0) mX, x (0.5,0.5,0) m
(a)
(b)
Figure 10: Problem setup: (a) Tensile test and (b) Punching test
neo-Hookean material with Young’s modulus E= 1.7×107Pa, density ρ0= 1.1×103kg/m3
and Poisson’s ratio ν= 0.45.
As illustrated in Figure 11, all of the proposed Total Lagrangian SPH methodologies, namely
{p,F},{p,F,J}and {p,F,H,J}Upwind-SPH, are capable of eliminating tensile instability
without showing pressure oscillations. Figure 12 shows the time evolution of the deformation
pattern along with its pressure contour plot with three different views: isometric view, front
view and top view. In this problem, extremely large distortions around the holes of the block
are displayed7. Notice that in this type of problems, mesh-based methods as presented in
[3, 25–27, 30, 34, 38, 42] are not suitable unless adaptive mesh refinement [60] is carried out.
To further examine the efficiency of the algorithm, the same problem is now assessed using
a larger value of Poisson’s ratio ν= 0.499. As presented in Figure 13, the {p,F}Upwind-
SPH methodology shows excessive pressure fluctuations which eventually lead to an incorrect
deformation path. For this reason, the conservation law for the Jacobian Jis incorporated in
order to stabilise the formulation, as already explored in previous publications [1, 2]. A smooth
pressure spatial representation can then be observed when using {p,F, J}and {p,F,H, J}
Upwind-SPH methodologies. For completeness, Figure 14 depicts the time evolution of the
deformation of the problem simulated using the {p,F,H, J}algorithm, displaying a smooth
pressure contour.
7.2.2. Punch test
Similar to the tensile test described in Section 7.2.1, the perforated block is now left free
on its top face and constrained with roller supports (i.e. symmetric boundary conditions) on
the rest of the boundaries (see Figure 10b). The main objective of this example is to show
the ability of the proposed methodology in alleviating severe pressure oscillations in highly
constrained problems. The block is punched on a quarter of the domain with a linear variation
of a velocity field v0=−V[0,0,(Z/H)]Twhere V= 250 m/s and H= 0.5 m. A nearly
incompressible neo-Hookean material is used where Young’s modulus E= 1.7×107Pa, density
ρ0= 1.1×103kg/m3and Poisson’s ratio ν= 0.499.
As shown in [1, 2], addition of the volume map conservation law to the {p,F}mixed-based
system seems to be very efficient when solving problems with predominant nearly incompressible
7For simplicity, the inter-particle contact algorithm [56] is not considered in the present manuscript.
24
t = 0.0058 s t = 0.0171 s t = 0.0211 s
(a) Mixed-based {p,F}Upwind-SPH
(b) Mixed-based {p,F,J}Upwind-SPH
(c) Mixed-based {p,F,H,J}Upwind-SPH
Figure 11: Tensile cube. A sequence of deformed configurations using the mixed-based (a) {p,F}, (b) {p,F,
J}and (c) {p,F,H,J}Upwind-SPH. Results obtained with velocity field v0=V[0,0,(Z/H)]Twhere V= 100
m/s and H= 0.5 m. A neo-Hookean constitutive model is employed with Young’s modulus E= 1.7×107Pa,
density ρ0= 1.1×103kg/m3, Poisson’s ratio ν= 0.45 and αCFL = 0.3.
25
t = 0.0266 s t = 0.0330 s t = 0.0417 s
(a) Isometric view
(b) Front view
(c) Top view
Figure 12: Tensile cube. A sequence of deformed configurations in three different views via the {p,F}Upwind-
SPH, namely: (a) Isometric view, (b) Front view, and (c) Top view. Results obtained with velocity field
v0=V[0,0,(Z/H)]Twhere V= 100 m/s and H= 0.5 m. A neo-Hookean constitutive model is employed with
Young’s modulus E= 1.7×107Pa, density ρ0= 1.1×103kg/m3, Poisson’s ratio ν= 0.45 and αCFL = 0.3.
26
t = 0.0058 s t = 0.0171 s t = 0.0211 s
(a) Mixed-based {p,F}Upwind-SPH
(b) Mixed-based {p,F,J}Upwind-SPH
(c) Mixed-based {p,F,H,J}Upwind-SPH
Figure 13: Tensile cube. A sequence of deformed configurations using the mixed-based (a) {p,F}, (b) {p,
F,J}and (c) {p,F,H,J}Upwind-SPH. Results obtained with velocity field v0=V[0,0,(Z/H)]Twhere
V= 100 m/s and H= 0.5 m. A nearly incompressible neo-Hookean constitutive model is employed with
Young’s modulus E= 1.7×107Pa, density ρ0= 1.1×103kg/m3, Poisson’s ratio ν= 0.499 and αCFL = 0.3.
27
t = 0.0266 s t = 0.0330 s t = 0.0417 s
(a) Isometric view
(b) Front view
(c) Top view
Figure 14: Tensile cube. A sequence of deformed configurations in three different views via the {p,F,H,
J}Upwind-SPH: (a) Isometric view, (b) Front view, and (c) Top view. Results obtained with velocity field
v0=V[0,0,(Z/H)]Twhere V= 100 m/s and H= 0.5 m. A nearly incompressible neo-Hookean model is
used with Young’s modulus E= 1.7×107Pa, density ρ0= 1.1×103kg/m3, Poisson’s ratio ν= 0.499 and
αCFL = 0.3. 28
behaviours. In Figure 15, it is clear that the {p,F}Upwind-SPH introduces spurious pressure
modes. These non-physical pressure instabilities can be effectively removed by using alternative
{p,F, J}and {p,F,H, J }stabilised schemes. For completeness, Figures 16 and 17 show a
sequence of deformed states of the problem for a relatively long period of time. Very smooth
pressure field is observed surrounding the holes of the perforated block.
7.2.3. Taylor impact plasticity
Previously presented in References [27, 38, 42], a copper bar of initial length L= 0.0324
m and of initial radius r= 0.0032 m impacts against a rigid wall with a dropping velocity of
227 m/s, as shown in Figure 18. The main objective of this benchmark problem is to assess
the performance of the proposed mesh-free algorithm in capturing large plastic deformations
with application to metal forming [43]. A von Mises hyperelastic-plastic material with isotropic
hardening is chosen for the simulation of this problem. The material parameters are Young’s
modulus E= 117 GPa, density ρ0= 8.930 ×103kg/m3, Poisson’s ratio ν= 0.35, yield stress
¯τ0
y= 0.4 GPa and hardening modulus H= 0.1 GPa. Figure 19 shows the contours of both
plastic strain and pressure at eight different time instants. Extremely smooth pressure field is
observed. For verification purposes, the final radius of the copper bar at time t= 80µs predicted
by the {p,F}Upwind-SPH algorithm is shown in Table 2, benchmarked against other published
numerical results [15, 27, 38]. As shown in References [15, 16], the solutions obtained using
the standard linear 4-noded tetrahedra (being widely used in commercial software) typically
suffers from volumetric locking and pressure instabilities. The new mesh-free method clearly
circumvents these issues.
Table 2: Final radii of copper bar at t= 80µs. Results obtained using the proposed {p,F}Upwind-SPH
mesh-free algorithm, benchmarked against other published methodologies.
Method Final radius (cm)
Standard 4-Noded Tetrahedra [15] 0.555
Under-integrated 8-Noded Hexahedra [15] 0.695
Average Nodal Pressure 4-Noded Tetrahedra [15] 0.699
Jameson-Schmidt-Turkel Vertex Centred FVM [38] 0.698
Petrov-Galerkin FEM [27] 0.700
Upwind Cell Centred FVM [42] 0.700
Upwind-SPH mesh-free method 0.689
7.3. Robustness
7.3.1. Twisting column
In order to examine the robustness of the algorithm, the challenging twisting column ex-
ample previously explored in References [1, 2, 42] is now considered (see Figure 20). The
problem is initialised with a sinusoidal angular velocity field relative to the origin given by
ω0= [0,Ω sin(πY /2L),0]T, where Ω = 105 rad/s and L= 6 m is the length of the column.
A nearly incompressible neo-Hookean material is considered and the material parameters are
Young’s modulus E= 1.7×107Pa, density ρ0= 1.1×103kg/m3and Poisson’s ratio ν= 0.4995.
A particle refinement study is shown in Figure 21. Interestingly, both the deformation
and pressure resolution obtained are practically identical, showing optimal convergence for the
29
t = 0.0018 s t = 0.0066 s t = 0.0113 s
(a) Mixed-based {p,F}Upwind-SPH
(b) Mixed-based {p,F,J}Upwind-SPH
(c) Mixed-based {p,F,H,J}Upwind-SPH
Figure 15: Punch test. A sequence of deformed configurations using the mixed-based (a) {p,F}, (b) {p,F,
J}and (c) {p,F,H,J}Upwind-SPH. Results obtained with velocity field v0=−V[0,0,(Z/H)]Twhere
V= 250 m/s and H= 0.5 m. A nearly incompressible neo-Hookean constitutive model is employed with
Young’s modulus E= 1.7×107Pa, density ρ0= 1.1×103kg/m3, Poisson’s ratio ν= 0.499 and αCFL = 0.3.
30
(a) t = 0.0016 s (b) t = 0.0023 s (c) t = 0.0030 s
(d) t = 0.0046 s (e) t = 0.0063 s (f) t = 0.0073 s
(g) t = 0.0089 s (h) t = 0.0100 s (i) t = 0.0113 s
Figure 16: Punch test. A sequence of deformed configurations using the mixed-based {p,F,H, J}Upwind-
SPH. Results obtained with velocity field v0=−V[0,0,(Z/H)]Twhere V= 250 m/s and H= 0.5 m. A nearly
incompressible neo-Hookean constitutive model is employed with Young’s modulus E= 1.7×107Pa, density
ρ0= 1.1×103kg/m3, Poisson’s ratio ν= 0.499 and αCFL = 0.3.
31
(a) t = 0.0011 s (b) t = 0.0021 s (c) t = 0.0032 s
(d) t = 0.0041 s (e) t = 0.0054 s (f) t = 0.0068 s
(g) t = 0.0096 s (h) t = 0.0104 s (i) t = 0.0113 s
Figure 17: Punch test. A sequence of deformed configurations (emphasising bottom view) using the mixed-
based {p,F,H, J }Upwind-SPH. Results obtained with velocity field v0=−V[0,0,(Z/H)]Twhere V= 250
m/s and H= 0.5 m. A nearly incompressible neo-Hookean constitutive model is employed with Young’s
modulus E= 1.7×107Pa, density ρ0= 1.1×103kg/m3, Poisson’s ratio ν= 0.499 and αCFL = 0.3.
32
X,x
Y,y
v0
(−0.0032, 0, 0)
(0.0032, 0.0324, 0)
Z,z
r0
Figure 18: Taylor bar configuration
proposed method. It is also interesting to notice how the methodology preserves perfect axial
rotation without introducing out-of-axis characteristics. This can be easily shown in Figure 22
by monitoring the evolution of the horizontal velocity and horizontal displacement components
at point X= [0.5,0.5,6]Tm, both within zero machine accuracy. For benchmarking purposes,
Figure 23 depicts a comparison of the new Upwind-SPH methodology against the recently
proposed JST-SPH [1], SUPG-SPH [2] mesh-free methods and other in-house mixed-based
methodologies. All of the schemes described above produce very similar results in terms of
deformed shape and pressure field.
In comparison to the Upwind-SPH and SUPG-SPH schemes, it is clear that the JST-SPH
(see Figure 23c) requires a considerably larger number of particles in order to capture the
correct deformation pattern of the column. The results simulated using the SUPG-SPH (see
Figure 23b) are in good agreement with the Upwind-SPH scheme (see Figure 23a). However,
the SUPG-SPH method requires the generation of a secondary set of particles for the evaluation
of the SUPG stabilisation, thus resulting in a greater computational cost.
The problem becomes significantly more challenging by increasing the initial angular velocity
now to Ω = 200 rad/s with a Poisson’s ratio of ν= 0.4995. A refinement study is also carried
out in Figure 24 using a sequentially refined number of particles of 5 ×5×30, 6 ×6×36 and
7×7×42. Notably, the number of twists shown in the column is captured extremely well even
with a small number of particles. As illustrated in the top view of the simulation of Figure 24,
no out-of-plane deformation can be observed.
Finally, we can further examine the robustness of the algorithm by increasing the value of
Poisson’s ratio to the limit such that ν≈0.5. Its main objective is to show the efficiency of the
artificial compressibility approach (see Section 5) in comparison to the explicit Upwind-SPH,
especially in problems characterised by nearly or truly incompressible behaviours. Figure 25
shows the qualitative comparison between those two approaches. As expected, the artificial
compressibility algorithm introduces a larger numerical dissipation due to the use of the (con-
33
(a) t= 10µs (b) t= 20µs (c) t= 30µs (d) t= 40µs
(e) t= 50µs (f) t= 60µs (g) t= 70µs (h) t= 80µs
Figure 19: Taylor impact. Time evolution of plastic strain in the left quarter and pressure distribution in the
right quarter of the domain along with the deformation. Results obtained using the proposed {p,F}Upwind-
SPH algorithm with dropping velocity v0= (0,−227,0)Tm/s. A hyperelastic-plastic constitutive model is
employed with Young’s modulus E= 117 GPa, density ρ0= 8930 kg/m3, Poisson’s ratio ν= 0.35, yield stress
¯τ0
y= 0.4 GPa, hardening modulus H= 0.1 GPa and αCFL = 0.3.
34
X, x
Y, y
(−0.5,0,0.5)m
(0.5,6,−0.5)m
Z, z
ω0= Ω [0,sin(πY /2H),0]Trad/s
H= 6 m
Figure 20: Twisting column problem setup
servative) Laplacian viscosity term described in (58). A particle refinement study is carried
out in Figure 26, with a progressive level of refinement. It is remarkable that the deformation
pattern predicted with a small number of particles agrees extremely well with those results
obtained using a fine discretisation. The accuracy of the pressure contour is clearly enhanced
as we increase the number of particles. Figure 27 shows the importance of adding (particle ba-
sed) pseudo viscosity term to the artificial compressibility approach, with the primary aim to
speed up the iterative process of the algorithm. As presented in Figure 27, the use of viscosity
term dramatically reduces the number of iterations required for convergence within the pseudo
time integration when solving truly incompressible solids.
7.3.2. Complex geometry
In the last example of this paper, we demonstrate the robustness of the proposed SPH
algorithm on a complex geometry displayed in Figure 28. As reported in Reference [61], the
geometry used in this example is a simplified version of a cardiovascular stent used in biomedical
applications. The structure has an initial outer diameter of 20 mm, a thickness of 0.5 mm and
a total length of 20 mm. The diameter of every hole is 2 mm. In this problem, we study
the deformation pattern of this stent-like structure by applying a velocity field at the top and
bottom of the structure, described as follows
v0=(h0,0,−abγ
(γ+1)2iTif t≤0.03 s
0otherwise ;γ= exp [a(c−t)],(60)
where a= 800, b= 0.006 and c= 0.015. Due to the presence of three symmetry planes, only
one eighth of the problem is solved with appropriate boundary conditions. The structure is
made of a nearly incompressible neo-Hookean material with density ρ= 1100 kg/m3, Young’s
Modulus E= 17 MPa and Poisson’s ratio ν= 0.45. Figure 29 shows the overall deformation of
a structure at time t= 0.0216 s. It is remarkable that the pressure is extremely well captured
35
Top view
(a) 4 ×4×24 (b) 5 ×5×30 (c) 6 ×6×36
Figure 21: Twisting column. A sequence of particle refinement using the mixed-based {p,F,H, J }Upwind-
SPH at time step t= 0.1s: (a) 4 ×4×24, (b) 5 ×5×30, and (c) 6 ×6×36 particles. Results obtained with an
angular velocity field ω0= [0,Ω sin (πY /2L),0] where Ω = 105 rad/s and L= 6 m. A nearly incompressible
neo-Hookean constitutive model is employed with Young’s modulus E= 1.7×107Pa, density ρ0= 1.1×103
kg/m3, Poisson’s ratio ν= 0.4995 and αCFL = 0.3.
36
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7
-5
510-10
(a)
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7
-1
0
110-10
(b)
Figure 22: Twisting column. Time evolution of the components of (a) Velocity and (b) Displacement at point
X= [0.5,0.5,6]Tm via the mixed-based {p,F,H, J }Upwind-SPH algorithm. A nearly incompressible neo-
Hookean constitutive model is employed with Young’s modulus E= 1.7×107Pa, density ρ0= 1.1×103kg/m3,
Poisson’s ratio ν= 0.4995 and αCFL = 0.3.
even with a minimum of two particles across the thickness. As can be seen in Figure 30, the
mixed-based {p,F,H, J}Upwind-SPH method produces reliable results that are freed from
zero energy modes. This will open up interesting possibilities for modelling in the field of
biomechanics [62, 63], where this consideration is very relevant.
8. Conclusions
This paper presents a new Upwind Smooth Particle Hydrodynamics (Upwind-SPH) com-
putational methodology for the analysis of nearly and truly incompressible large deformations
in explicit fast solid dynamics. The methodology is established starting from a mixed-based
system of Total Lagrangian first order conservation laws, where the linear momentum pconser-
vation law is solved along with conservation equations for the extended set of geometric strain
measures {F,H, J}.
In this work, a Riemann solver based spatial discretisation has been employed for stabi-
lisation, offering a series of advantages over the recently proposed JST-SPH method [1] and
SUPG-SPH method [2]. First, the nature of the Riemann based upwinding stabilisation does
not explicitly rely on the use of any artificial stabilisation parameters. This differs from the
JST-SPH and SUPG-SPH methodologies, as the former requires the evaluation of user-defined
harmonic and bi-harmonic operators and the latter requires the selection of a number of user-
defined stabilisation parameters for the evaluation of the SUPG stabilisation. Second, the
Upwind-SPH framework is more accurate than its JST-SPH counterpart, and its accuracy is
comparable to that of the SUPG-SPH but with a reduced computational cost.
It has also been shown that the resulting Upwind-SPH algorithm overcomes a number of
numerical difficulties posed by the classical SPH method, namely non-physical hydrostatic pres-
sure fluctuations, hour-glassing and numerical issues associated with conservation, consistency,
long term instability and convergence. As a result, the new algorithm provides a good balance
37
(a) (b) (c) (d) (e)
Pressure (Pa)
Figure 23: Twisting column. Comparison of deformed shapes plotted with pressures at time t= 0.1 s using:
(a) Mixed-based {p,F}Upwind-SPH; (b) Mixed-based {p,F,H, J}SUPG-SPH-H1 (τF= ∆t,ξF= 0.2,
τp= 0.1∆t); (c) Mixed-based {p,F}JST-SPH (ε(2)
p= 0 and ε(4)
p=1
8); (d) PG-FEM [26]; and (e) Constrained-
TOUCH [42]. Results obtained with an angular velocity field ω0= [0,Ω sin(πY/2L),0] where Ω = 105 rad/s
and L= 6 m. A neo-Hookean material is used with density ρ0= 1100 kg/m3, Young’s modulus E= 17 MPa
and Poisson’s ratio ν= 0.495.
38
Top view
(a) 5 ×5×30 (b) 6 ×6×36 (c) 7 ×7×42
Figure 24: Twisting column. A sequence of particle refinement using the mixed-based {p,F,H, J }Upwind-
SPH at time step t= 0.1 s: (a) 5 ×5×30, (b) 6 ×6×36, and (c) 7 ×7×42 particles. Results obtained with an
angular velocity field ω0= [0,Ω sin (πY /2L),0] where Ω = 200 rad/s and L= 6 m. A nearly incompressible
neo-Hookean constitutive model is employed with Young’s modulus E= 1.7×107Pa, density ρ0= 1.1×103
kg/m3, Poisson’s ratio ν= 0.4995 and αCFL = 0.3.
39
Top view
(a) Artificial compressibility approach (b) Explicit Upwind-SPH
Figure 25: Twisting column. A comparison between (a) {p,F,H, p}artificial compressibility approach and
(b) Explicit {p,F,H, J }Upwind-SPH at time step t= 0.1s. Results obtained with an angular velocity
field ω0= [0,Ω sin (πY /2L),0] where Ω = 105 rad/s and L= 6 m. A nearly incompressible neo-Hookean
constitutive model is employed with Young’s modulus E= 1.7×107Pa, density ρ0= 1.1×103kg/m3,
Poisson’s ratio ν= 0.4999 and αCFL = 0.3.
40
Top view
(a) 4 ×4×24 (b) 6 ×6×36 (c) 8 ×8×48
Figure 26: Twisting column. A sequence of particle refinement using the mixed-based {p,F}Upwind-SPH
through artificial compressibility at time step t= 0.1s: (a) 4 ×4×24, (b) 6 ×6×36, and (c) 8 ×8×48 particles.
Results obtained with an angular velocity field ω0= [0,Ω sin (πY /2L),0] where Ω = 105 rad/s and L= 6 m.
A truly incompressible neo-Hookean constitutive model is employed with Young’s modulus E= 1.7×107Pa,
density ρ0= 1.1×103kg/m3, Poisson’s ratio ν= 0.5 and αCFL = 0.3.
41
0 0.1 0.2 0.3 0.4 0.5
8
8.2
8.4
8.6
8.8
9
0 2 4 6 8
4
4.5
5
5.5
6106
(a)
22
24
26
28
30
32
1
1.5 107
0.5
0 10 20 30
20
0 0.1 0.2 0.3 0.4 0.5
(b)
0 0.1 0.2 0.3 0.4 0.5
20
40
60
80
100
120
140
160
180
0 10 20 30
0.5
1
1.5
2107
(c)
0 0.1 0.2 0.3 0.4 0.5
0
100
200
300
400
500
600
700
800
900
0.1 0.2 0.3 0.4 0.5
20
30
40
0 10 20 30
0.5
1
1.5
2107
(d)
Figure 27: Twisting column. The effect of incorporating Laplacian viscosity (54) to the artificial compressibility
approach with the primary aim to enhance the iterative process of the algorithm in various scenarios: (a)
ν= 0.499, (b) ν= 0.4999, (c) ν= 0.49999 and (d) ν= 0.5. Results obtained for the first step of Runge-Kutta
time integrator with an angular velocity field ω0= [0,Ω sin (πY /2L),0] where Ω = 105 rad/s and L= 6 m.
The green curve represents the number of iterations required to achieve steady state convergence in pseudo-
time for a fixed Laplacian viscosity α. A neo-Hookean constitutive model is employed with Young’s modulus
E= 1.7×107Pa, density ρ0= 1.1×103kg/m3and αCFL = 0.3.
42
Figure 28: Stent-like structure [61]. The Computer Aided Design (CAD) file can be downloaded from GrabCAD.
(a) Front view (b) Top view (c) Interior view
Figure 29: Stent-like structure. Deformed configuration of the structure at t= 0.0216 s simulated using the
mixed-based {p,F,H,J}Upwind-SPH algorithm. A nearly incompressible neo-Hookean constitutive model
is employed with Young’s modulus E= 0.9 MPa, density ρ0= 1000 kg/m3, Poisson’s ratio ν= 0.45 and
αCFL = 0.3.
43
t = 0 s t = 0.0128 s t = 0.0148 s
t = 0.0168 s t = 0.0189 s t = 0.0208 s
t = 0.0233 s t = 0.0248 s t = 0.0026 s
Figure 30: Stent-like structure. A sequence of deformed configurations using the mixed-based {p,F,H,J}
Upwind-SPH algorithm. A nearly incompressible neo-Hookean constitutive model is employed with Young’s
modulus E= 0.9 MPa, density ρ0= 1000 kg/m3, Poisson’s ratio ν= 0.45 and αCFL = 0.3.
44
between accuracy and speed of computation. In terms of computational efficiency, an adapted
artificial compressibility approach has been formulated particularly useful when dealing with
extremely large (or even infinite) wave speeds in the incompressible limit.
Finally, a comprehensive set of numerical examples has been presented in order to ben-
chmark the results obtained against other numerical methodologies, including Finite Element
Method [25–27, 30, 35], Finite Volume Method [3, 34, 38, 42] and Smooth Particle Hydrody-
namics Method [1, 2]. Both velocities and (volumetric and deviatoric) stresses display equal
second order of convergence, advantageous in those scenarios where stresses are of primary
interest. The consideration of thermoelasticity within the current mesh-free computational fra-
mework is the next step of our work.
Acknowledgements
The authors gratefully acknowledge the financial support provided by the Sˆer Cymru Natio-
nal Research Network for Advanced Engineering and Materials, United Kingdom. The third
author acknowledges the financial support received through the European Commission EA-
CEA Agency, Framework Partnership Agreement 2013-0043 Erasmus Mundus Action 1b, as a
part of the EM Joint Doctorate Simulation in Engineering and Entrepreneurship Development
(SEED).
Appendix A. von Mises plasticity model
The standard algorithm to ensure that the Kirchhoff stress satisfies a von Mises type of
plastic constraint is summarised here for completeness in Algorithm 2.
Appendix B. Discrete angular momentum preserving algorithm
The resulting Upwind-SPH algorithm does not intrinsically fulfil conservation of angular
momentum, since the strain measures {F,H, J}are no longer exclusively obtained from the
current geometry [42]. Specifically, the deformation gradient F6=Fx:= ∇0x, its co-factor
H6=Hx:= 1
2FxFxand its Jacobian J6=Jx:= det Fx.
In this paper, and following the work of Jibran et al. [42], a monolithic discrete angular
momentum projection algorithm is presented. Specifically, the local internal nodal force Tais
modified (in a least-square sense) in order to preserve the total angular momentum, whilst still
ensuring the global conservation of linear momentum.
Following Reference [42], sufficient conditions for the global preservation of the discrete
linear and angular momentum within a time step are enforced at each stage of the one-step
two-stage Runge Kutta time integrator (40) described as:
X
a
VaTχ
a=0;X
a
VaXa×Tχ
a=0;Xa=xn
a, χ =n
xn
a+∆t
2ρ0(pn
a+p?
a), χ =?; (B.1)
where χ={n, ?}.
A least-square minimisation procedure is used to obtain a modified set of internal nodal
forces ˆ
Tathat satisfy the above conditions (B.1). This can be achieved by computing the
45
Algorithm 2: Time update of first Piola Kirchoff stress tensor
Input : Fn+1,C−1
pn, εn
p
Output: Pn+1
(1) Obtain Jacobian of deformation: Jn+1
F= det Fn+1
(2) Evaluate pressure: pn+1 =κln (Jn+1
F)
Jn+1
F
(3) Compute trial elastic left strain tensor: ¯
bn+1
e=Fn+1 C−1
pnFTn+1
(4) Spectral decomposition of ¯
bn+1
e:¯
λe,i ,¯ni←− ¯
bn+1
e=P3
i=1 ¯
λe,i2(¯ni⊗¯ni)
(5) Obtain trial deviatoric Kirchoff stress tensor:
¯τ0=P3
i=1 ¯τ0
ii (¯ni⊗¯ni),¯τ0
ii = 2µln ¯
λe,i−2
3µln Jn+1
F
(6) Obtain yield criterion: f¯τ0, εn
p=q3
2(¯τ0:¯τ0)−τ0
y+Hεn
p
(7) Compute direction vector and plastic multiplier:
if f¯τ0, εn
p>0then
Direction vector: υn+1
i=¯τ0
ii
√2
3(¯τ0:¯τ0)
Plastic multiplier: ∆γ=f(¯τ0,εn
p)
3µ+H
else
υn+1
i= ∆γ= 0
end
(8) Evaluate elastic stretch: λn+1
e,i = exp ln(¯
λe,i)−∆γυn+1
i
(9) Set spatial normals: nn+1
i=¯ni
(10) Compute Kirchoff stresss tensor:
τn+1 =P3
i=1 τii nn+1
i⊗nn+1
i
τii =τ0
ii +Jn+1
Fpn+1, τ 0
ii =1−2µ∆γ
√2
3(¯τ0:¯τ0)¯τ0
ii
(11) Evaluate first Piola Kirchoff stress tensor: Pn+1 =τn+1 F−Tn+1
(12) Update elastic left strain tensor: bn+1
e=P3
i=1 λn+1
e,i 2nn+1
i⊗nn+1
i
(13) Compute plastic right Cauchy Green tensor: C−1
pn+1 =F−1n+1 bn+1
eF−Tn+1
(14) Update plastic strain: εn+1
p=εn
p+ ∆γ
46
minimum of the following functional [38, 42] (ignoring time arguments for brevity):
ΠT(ˆ
Ta,λang,λlin) = 1
2X
a
Va(ˆ
Ta−Ta)·(ˆ
Ta−Ta)
−λang · X
a
VaXa׈
Ta!−λlin · X
a
Vaˆ
Ta!.
(B.2)
After some simple algebra, a modified set of internal nodal forces ˆ
Taarise:
ˆ
Ta=Ta+λang ×Xa+λlin.(B.3)
The Lagrange multipliers {λang,λlin}are the solutions of the following system of equations
PaVa[(Xa·Xa)I−Xa⊗Xa]PaVaˆ
Xa
PaVaˆ
Xa−PaVa λang
λlin =−PaVaXa×Ta
PaVaTa,(B.4)
with the indicial notation hˆ
Xaiik =Eijk [Xa]j.
In addition, an extra condition must be added for the satisfaction of the global angular
momentum preservation, namely
X
a
VaD(pa) = 0;X
a
VaXa×D(pa) = 0.(B.5)
Similarly to the least-square minimisation procedure described above, a set of modified upwin-
ding dissipation terms can now be obtained by replacing {ˆ
Ta,Ta,λlin,λang}in (B.3) and (B.4)
with {ˆ
D(pa),D(pa),µDp,λDp}.
Appendix C. Laplacian evaluation
The artificial compressibility procedure to be employed in this paper to effectively handle
near incompressibility regimes may require the evaluation of the Laplacian (harmonic operator
[1]) of a solution function. For this purpose, consider the Laplacian of any arbitrary scalar
function gto be numerically approximated as
L[g(Xa)] := ∇0·[∇0g(Xa)] ≈X
b∈Λb
a
∇0g(Xb)·Gb(Xa).(C.1a)
Addition of the redundant term ∇0g(Xa) to (C.1a) yields an alternative expression as
L[g(Xa)] ≈X
b∈Λb
a
2∇0g(Xa) + ∇0g(Xb)
2·Gb(Xa).(C.2)
It is worth pointing out that the squared bracket term shown in (C.2) represents the gradient
approximation at the mid-edge connecting particles aand b. This can be further approximated
via a second order central difference scheme as [64]
∇0g(Xa) + ∇0g(Xb)
2≈gb−ga
kXb−XakNab,Nab := Xb−Xa
kXb−Xak.(C.3)
47
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