![a) illustrates computational setup of the considered benchmark test. The free span length and thickness of the plate are respectively set as L = 0.2 m and hs = 0.02 m, that corresponds to the geometry in the setup of Gray et al. [73]. The plate is r S = 1000 kg/m 3 in density and n S = 0.4 in Poisson's ratio. The plate begins to oscillate with an initial velocity distribution of uy(x). By referring to the work of Diani [72], the theoretical time history of a viscoelastic plate's deflection η for first (fundamental) mode of vibration with an initial condition of η (L,0) = Δ0](https://www.researchgate.net/profile/Abbas-Khayyer/publication/361975553/figure/fig2/AS:1178728088518656@1658042084026/a-illustrates-computational-setup-of-the-considered-benchmark-test-The-free-span_Q320.jpg)
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1
On systematic development of FSI solvers in the context of particle methods
Abbas Khayyer
1
, Hitoshi Gotoh, Yuma Shimizu
Department of Civil and Earth Resources Engineering, Kyoto University,
Kyoto, Kyoto, Japan
ABSTRACT
The paper presents a concise review on state-of-the-art of developments corresponding to FSI
solvers developed within the context of particle methods. The paper reviews and highlights the
potential robustness of entirely Lagrangian meshfree FSI solvers in reproducing FSI
corresponding to extreme events and portrays the future perspectives for systematic
developments towards reliable engineering applications with respect to rapid advances in
technology and emergence of so-called advanced materials that can result in complex and
highly non-linear structural responses. Accordingly, the paper highlights the necessity for
reproduction of comprehensive structural responses, including viscoelastic, elastoplastic and
progressive damages/failures, by the advanced FSI solvers developed within the context of
particle methods. In this regard, extensions of the structure model are suggested to be
conducted in a variationally consistent framework to ensure stability, accuracy and physical
reliability including thermodynamic consistency. The paper reviews basics of mathematical
and numerical modelling for entirely Lagrangian meshfree hydroelastic FSI solvers and
presents a brief background on extensions of such solvers towards reproducing viscoelastic
structural responses. Some preliminary numerical results on structural viscoelasticity achieved
by an extended Hamiltonian SPH model are presented. This vision paper also concisely
portrays the future perspectives for systematic development of particle-based FSI solvers.
KEYWORDS: Fluid-Structure Interaction; entirely Lagrangian meshfree, SPH, Hamiltonian
SPH, viscoelasticity
1
Corresponding author, Email: khayyer@particle.kuciv.kyoto-u.ac.jp
2
1. INTRODUCTION
Extreme environmental conditions, e.g. tsunamis, rogue waves, typhoons or storm surges, may
result in devastating consequences in coastal and offshore regions. In the past decade, there has
been an increasing interest in construction of “resilient” structures, i.e. “tenacious” and
“sustainable” structures that keep withstanding external forces with easy maintenances and can
preserve their functionality even in extreme events. To achieve a resilient structure design, an
in-depth understanding of the complex Fluid-Structure Interaction (FSI) system is of
substantial importance. On the other hand, considering the coastal/ocean structures against
extreme waves or severe slamming, precise realization of coastal/offshore fluid-structure
systems would be significantly challenging on account of their large scale and limitations of
experimental or field measurements. Therefore, computational modelling would be considered
as a robust and potentially reliable approach. Meanwhile, presence of violent fluid flow fields,
complex moving boundaries and topological changes, complex fluid-structure interactions and
intense non-linear structural responses would result in distinct challenges for reliable
computational modelling of FSI in coastal/ocean engineering.
Challenges in computational modelling of FSI in coastal/ocean engineering are primarily
linked to presence of violent flows and associated complex structural deformations that have
been conventionally considered to be addressed within the context of hydroelastic FSI. Such
challenges would be more pronounced in cases with viscoelastic effects, especially with respect
to emergence of advanced engineered structural elements such as viscoelastic layered
composites that would benefit from viscoelastic materials [1], e.g., for impact energy
absorption and reduction of slamming-induced impact pressures [2] or enhanced service life
and noise/vibration control [3]. In addition, especially in extreme events, the maximum stresses
in structures can exceed the material yield stress, causing large plastic flow and permanent
plastic deformations that may result in damage accumulation, material properties degradation,
fracture, fatigue damage and progressive failures [4] leading to structural functionality
disruptions or even structural collapse.
In light of rapid advancements of computational technology, advanced computational methods
have been continuously developed in testing the applicability or design of advanced engineered
materials [5]. In addition, reflecting on the aforementioned challenges corresponding to critical
FSI phenomena encountered in ocean/coastal engineering, e.g., violent free-surface flows,
3
large/abrupt hydrodynamic loads and large structural deformations, Lagrangian meshfree or
particle methods (e.g., SPH, Smoothed Particle Hydrodynamics [6]; ISPH, Incompressible
SPH [7]; MPS, Moving Particle Semi-implicit [8]) are advantageous for reliable computational
modelling of hydroelastic FSI. In specific, Entirely Lagrangian Meshfree hydroelastic FSI
solvers (i.e., FSI solvers utilizing particle methods for discretizations of both fluid and structure
domains) have provided substantial potential as reliable computational tools for hydroelastic
FSI [9-11] due to their flexibility, robust extendibility and precise impositions of interface
boundary conditions.
Fig. 1. Three important aspects of reliability, adaptivity and generality in development of FSI
solvers – Parts (a) [12], (b) [10], (c) [9], (d) [13], (e) [14] present representative results by a set
of novel computational methods developed by the authors.
In development of FSI solvers including the entirely Lagrangian meshfree ones, several key
aspects need to be thoroughly considered. These aspects can be categorized into three groups
of reliability, adaptivity and generality, as portrayed in a recent review article by Gotoh et al.
[11]. Since 2013, our research team has continuously and coherently worked on development
of particle-based FSI solvers with respect to these three important aspects as portrayed in Fig.
1. The presented figure also includes representative results by a set of novel computational
methods developed by our research group. These methods include (i) 2D ISPH-SPH [12], as
the first entirely Lagrangian meshfree hydroelastic FSI solver with projection-based fluid
model; (ii) 3D ISPH-SPH (Multi-Resolution) [10], characterised by refined adaptive schemes;
4
(iii) 3D MPS-Hamiltonian MPS [9], as the first 3D entirely Lagrangian meshfree projection-
based hydroelastic FSI solver; (iv) 2D ISPH for fluid-porous media interactions [13] with
precise satisfaction of interface boundary conditions; (v) and 2D ISPH-Hamiltonian SPH [14]
as the first entirely Lagrangian meshfree hydroelastic FSI solver for composite structures.
Despite developments of particle methods towards hydroelastic FSI simulations, particle-based
structure models have not yet been thoroughly developed for precise and comprehensive
reproductions of structural responses including elasto-visco-plastic deformations (Fig. 1)
through precise considerations of viscoelasticity and elastoplasticity. There have been a few
studies on reproduction of elasto-plasticity in the context of SPH, e.g. [15-19], however, these
studies have been performed in the context of Newtonian mechanics and thus, variational
consistency of formulations is not maintained after discretizations. In addition, most of these
models [15-18] are not applicable to large-strain elasto-plasticity as encountered in
coastal/ocean engineering. In view of the variationally-consistent framework of the HSPH
structure model, this model serves as an excellent candidate to be coherently and effectively
further extended for simulations of comprehensive structural responses comprising elastic,
viscoelastic and plastic (elasto-visco-plastic) deformations. Accordingly, the coupled ISPH-
extended HSPH would serve as an excellent candidate for reproduction of FSI corresponding
to violent fluid flows and complex structural systems in a reliable and efficient manner. In
addition, in many FSIs encountered in coastal/ocean engineering, the effect of air phase cannot
be ignored (e.g., [20]). This includes wave impacts including air entrapment/entrainment or
water slamming involving entrapped air in between the marine structure and water surface [21].
Hence, the developed FSI solvers need to incorporate the air phase and serve as a multi-phase
air-water-structure FSI solver. Finally, with respect to achieving a precise, comprehensive
computational modelling, damage/fracture mechanics needs to be incorporated systematically
and in a variationally consistent manner, so that we can establish an “Advanced Multi-physics
Entirely Lagrangian Meshfree FSI solver” for design of resilient coastal/ocean structures.
This brief review or vision paper aims at shedding light on the state-of-the-art of FSI modelling
in the context of particle methods with specific focus on Entirely Lagrangian Meshfree FSI
solvers in which particle methods are incorporated to discretize both fluid and structure sub-
domains. The paper is organised in the following manner. First, the most recently developed
5
particle-based hydroelastic FSI solvers and their distinct features are briefly reviewed in section
2. Section 3 presents the mathematical & numerical modelling corresponding to entirely
Lagrangian meshfree hydroelastic FSI solvers, along with highlighting some of the key features
that are crucial for reliable modelling of FSI corresponding to design of resilient structures in
ocean and coastal engineering. In section 4, a brief background and some preliminary results
corresponding to extended HSPH towards hydro-viscoelastic FSI modelling are presented.
Finally, section 5 presents a set of concluding remarks and concisely portrays the future
perspectives corresponding to systematic development of FSI solvers in the context of particle
methods.
2. RECENTLY DEVELOPED PARTICLE-BASED FSI SOLVERS
In our recent review article [11], the state-of-the-art of particle-based FSI solvers is portrayed.
Hence, here only the most recently developed particle-based FSI solvers are concisely reviewed.
Li et al. [22] have recently presented a partitioned IFEMP (immersed finite element material
point) method consisting of an improved incompressible material point method or iMPM (as
fluid model) coupled with finite element method (as structure model) with the interactions
between the iMPM and FEM being handled by a sharp immersed interface approach. Zhang et
al. [23] also incorporated a partitioned coupling strategy to couple MPS with FEM for
hydroelastic FSI corresponding to 3D dam-break flows with elastic gate structures. Previously
Zhang et al. [24] investigated different strategies for partitioned MPS-FEM coupling. McLoone
and Quinlan [25] presented a novel and robust hydroelastic solver through coupling Finite
Volume Particle Method (FVPM) for fluid with FEM for structure. The coupled solver was
effectively applied to reproduce fluid flows interacting with very thin elastic structures. Long
et al. [26] presented a coupled Edge-based Smoothed Finite Element Method-SPH for
hyrdroelastic FSI with coupling being achieved through a virtual particle scheme with virtual
particles being dynamically and automatically generated for each interaction pair of ES-FEM
segment and SPH particle. Zhang et al. [27] presented a coupled SPH-Smoothed FEM (SFEM)
for fluid flow interactions with rigid or deformable structures. The implementation of SFEM
facilitated the reproduction of highly stiff structures. Recently, Shimizu et al. [28] presented an
advanced entirely Lagrangian meshfree FSI solver with an implicit structure model. The key
feature of this developed solver corresponded to consistent time coupling between fluid and
6
structure and its high applicability in reproducing fluid flow interactions with highly stiff
structures.
Ng et al. [29] presented a multi-resolution Smoothed Particle Hydrodynamics-Volume
Compensated Particle Method (SPH-VCPM) for reproduction of hydroelastic FSI. Ng et al.
[30] also extended their model for complex fluid-solid mixture flow problems through coupling
SPH for fluid and VCPM for elastic solid and Discrete Element Method (DEM) for modelling
the contact force between colliding elastic solid bodies. Meng et al. [31] developed a novel
hydroelastic FSI solver entirely based on the Riemann SPH method, where Riemann SPH was
incorporated for both fluid and elastic structure models for 2D and 3D hydroelastic FSI
simulations. Peng et al. [32] presented a coupled hydroelastic FSI solver through coupling
WCSPH with Reproducing Kernel Particle Method (RKPM) beam formulation based on
degenerated continuum approach.
Sun et al. [33] developed a hydroelastic FSI solver by incorporating a so-called Modified
Sequential Staggered (MSS) coupling algorithm to couple
d
+-SPH fluid model with a Total
Lagrangian SPH structure model. The solver could accurately reproduce several 2D/3D
challenging hydroelastic FSI problems. Lyu et al. [34] highlighted the necessity for
simultaneous implementation of TIC (Tensile Instability Control) and PST (Particle Shifting
Technique) for reliable (WCSPH-TLSPH) simulations of general hydroelastic problems
encountered in ocean engineering. Zhang et al. [35] coupled a
d
-SPH fluid model with a SPIM
(Smoothed Point Interpolation Method) structure model for hydroelastic FSI. Monteleone et al.
[36] proposed an FSI solver comprising of SPH for fluid and simplified particle-spring model
for structure with a coupling scheme imposing the coupling conditions through additional
boundary particles. Xie et al. [37] presented a coupled Improved MPS-DEM coupled method
for liquid-solid multiphase flow simulations. Harada et al. [38] presented a three-dimensional
MPS-DEM coupled method for simulation of a specific type of fluid-soil interactions, namely,
coastal morphodynamics including swash ripple formation. Sizkow and El Shamy [39] applied
a multiphase mixture SPH-DEM coupled method to study the seismic response of shallow
foundations resting on liquefiable soil.
Yan and Oterkus [40] proposed a peridynamics based methodology for fluid-structure
interaction problems where the structure model was established through using the ordinary
7
state-based peridynamics theory while the fluid model was developed by incorporating the
peridynamics differential operator. Rahimi et al. [41] proposed a coupled SPH-Peridynamics
(PD) to simulate fluid-structure interactions that may involve failure propagations. The
structure was considered as linear-elastic and isotropic, yet, said to be straightforwardly
extendable to reproduce responses and possible failure propagations corresponding to
composite and anisotropic structures.
There have been impressive efforts in extending/developing open-source SPH software capable
of handling hydroelastic FSI. For instance, Zhang et al. [42] presented detailed features of an
open-source SPH software referred to as SPHinXsys including hydroelastic FSI simulations.
The embedded hydroelastic FSI scheme of SPHinXsys also accommodates multi-resolution
simulations [43]. O’Connor and Rogers [44] presented a robust SPH formulation for
hydroelastic FSI within the open-source DualSPHysics code. Incorporation of a TLSPH
formalism along with kernel correction and zero-energy mode suppression were considered to
avoid potential deficiencies corresponding to linear inconsistency, tensile instability and rank
deficiency. Yilmaz et al. [45] presented the application of DualSPHysics-Project Chrono
coupling for hydroelasticity problems encountered in ocean engineering. In their coupled FSI
solver, Yilmaz et al. [45] modelled the fluid sub-domain by the WCSPH formulation of
DualSPHyscis while the solid sub-domain was considered as rigid body elements attached by
hinges with torsional and damping stiffness.
Despite great efforts devoted to development of FSI solvers in the context of particle methods,
most of these solvers are capable of only reproducing linearly elastic material responses and
have been configured in a thoroughly Newtonian framework. Certainly, with respect to
advances made in engineering design through incorporation of so-called “advanced materials”,
such as viscoelastic layered composites [2], rubber-like polymeric materials such as dielectric
elastomers [46,47] or advanced fibre-reinforced composites [48-51], the developed FSI solvers
must be capable of reproducing comprehensive structural responses including non-linear and
large-strain elastic, viscoelastic and elastoplastic responses. In addition to the complexities
associated with reliable reproduction of comprehensive structural responses, another challenge
corresponding to composite structures would be linked to presence of large material
discontinuities and thus, discontinuities in stress field. In this regard, implementation of a
variationally consistent Hamiltonian formalism for the structure model would be of great
advantage. In Hamiltonian structure models, dynamics and kinematics of the structure system
8
would be variationally linked with respect to a predefined energy potential. Therefore,
stability/accuracy of the structure model would be ensured without the need for artificial
numerical stabilizers. In addition, variational consistency would bring about excellent physical
reliability in terms of conservation properties and thermodynamical consistency. Another
advantage of Hamiltonian formalism would be straightforward and rigorous extensions to
model linear & non-linear viscoelasticity and elastoplasticity including large-strain responses.
3. ENTIRELY LAGRANGIAN MESHFREE HYDROELASTIC FSI SOLVERS
This section presents a concise description of mathematical & numerical modelling
corresponding to entirely Lagrangian meshfree hydroelastic FSI solvers.
3.1 Fluid Models
In principle, the fluid models are founded on the continuity and Navier-Stokes equations (Eqs.
1 and 2) corresponding to an incompressible fluid.
(1)
(2)
where, D/Dt stands for Lagrangian time derivative, ρ represents density, t stands for time, u
denotes particle velocity vector, p symbolizes particle's pressure, g signifies gravitational
acceleration vector and ν represents laminar kinematic viscosity; aSF corresponds to the
acceleration imposed on a target fluid particle (F) due to the presence of its neighbouring
structure particles (S); the superscript F denotes the physical quantity of fluid particle. The
continuity equation (Eq. 1) is imposed by projecting an intermediate velocity field into a
velocity divergence-free space through solving a Poisson Pressure Equation (PPE). This
projection is made on the basis of Helmholtz-Leray projection theorem that guarantees
exactness and uniqueness of solutions to continuity/Navier-Stokes on the condition that
boundary conditions are well imposed [52].
In order to augment the accuracy of projection particle methods several refined differential
operator models and error mitigating terms can be incorporated to minimize numerical errors
and result in a more accurate projection. Accordingly, the SPH or MPS fluid models may
0
F
Ñ× =u
2
F
SF
FF
F
Dp
Dt
n
r
Ñ
=-+Ñ+ +
uuga
9
benefit from a set of enhanced schemes, i.e. the so-called Higher-order Laplacian (HL), Higher-
order Source term (HS), Gradient Correction (GC), Error Compensating Source (ECS) and
Dynamic Stabilization (DS). The aforementioned enhanced schemes are comprehensively
described in several papers including those by Khayyer et al. [53,54] and Gotoh and Khayyer
[55].
3.2 Structure Models
Lagrangian meshfree structure models are configured either in the frameworks of Newtonian
mechanics (so-called MPS or SPH structure models) or Hamiltonian one (so-called HMPS or
HSPH).
3.2.1 Newtonian formulation of structural dynamics
In the context of Newtonian mechanics, the MPS or SPH structure models are established based
on equations for conservation of linear and angular momenta (Eqs. 3 and 4) in a Lagrangian
form.
(3)
(4)
where σS represents the stress tensor of a structure particle, r denotes position vector, aFS
corresponds to the acceleration imposed on a target structure particle (S) due to the presence of
its neighbouring fluid particles (F), where ρSaFS = -ρFaSF, i.e., f
FS = -f
SF, for the whole fluid
and structure subdomains, i.e., the force from fluid to structure would be equal in magnitude
and opposite in direction with respect to that from structure to fluid; I represents the moment
of inertia, ω signifies the angular velocity vector and m denotes the mass of a particle.
The stress tensor σS for a typical structure particle can be expressed according to Hooke’s law,
i.e., a linear elastic constitutive law, as follows:
(5)
1
S
S FS
S
D
Dt ρ
=- Ñ× + +
uσga
( ) ( )
SS
DD
Im
Dt Dt
=´ωru
() 2
SSS SS
λtr μ=+σ ε Iε
10
where, εS symbolizes the strain tensor; λS and μS are the Lame’s constants, i.e., mechanical
properties of the material calculated from Young’s modulus, ES, and the Poisson's ratio, υS, and
I signifies the unit tensor. The strain tensor εS is expressed as:
(6)
where, s represents the pure elastic deformation vector, which is described as follows:
, (7)
where R signifies the rigid body rotation tensor, while θ (rotation angle) is updated at each time
step from the solution of the equation of angular momentum conservation (Eq. 4). In Eq. 7,
subscripts i and j represent target particle and its neighbouring structure particles, respectively;
and superscript “0” represents the values evaluated with respect to the reference configuration.
3.2.2 Hamiltonian description of structural dynamics
Theoretically, Newtonian and Hamiltonian frameworks are equivalent in continuous form.
However, in discrete form, different numerical results would be achieved due to incorporation
of different differential operator models. In specific, Newtonian structure models need to
provide precise approximations of divergence of stress tensor to reproduce the kinematics of a
system. While in Hamiltonian structure models, dynamics and kinematics of the structural
system are variationally linked with respect to a predefined energy potential that corresponds
to the Helmholtz free energy of the system. The variational feature of Hamiltonian structure
models would also lead to advantages in terms of stability, accuracy and conservation features.
A distinct advantage of Hamiltonian formalism would be convenient and rigorous extensions
to model non-linear and large-strain elastic, viscoelastic and elastoplastic structural responses,
in a mathematically-physically consistent variational manner.
In Hamiltonian structure models, the general form of momentum equation for nonlinear
elastodynamics is derived based on minimization of the action of structure system with respect
to Hamilton's principle of least action as [14]:
( )
2
T
SÑ+Ñ
=-
ss
ε
( )
00
0.5
ij ij j ij i ij
=- ×+×sr RrRr
cos sin
sin cos
ii
i
ii
qq
qq
-
æö
=ç÷
èø
R
11
(8)
where, represents the strain energy density function, i.e., an energy potential. For a
hyperelastic material, spatial derivative of strain energy density function in Eq. (8) can be
written based on the first Piola-Kirchhoff stress tensor PS and the deformation gradient tensor
FS as:
(9)
where, “:” denotes double dot product of two tensors, FS represents deformation gradient tensor
and PS signifies the first Piola-Kirchhoff stress tensor which is obtained from Eq. (10).
(10)
In Eq. (10), SS represents the second Piola-Kirchhoff stress tensor which is obtained from
derivative of the strain energy density function (defined with respect to the Saint Venant-
Kirchhoff hyper-elasticity model) with respect to the Green-Lagrange strain tensor (ES) as:
, , (11)
Considering Eqs. (8) and (11), it is clear that the kinematics and dynamics of the structural
system are variationally linked with respect to variations of a strain energy density function.
As previously stated, an outcome of this important variational consistency would be precise
modelling of FSI corresponding to composite structures. Fig. 2 portrays representative
snapshots corresponding to ISPH-HSPH (Incompressible SPH-Hamiltonian SPH) modelling
of water slamming on sandwich composites in 2D & 3D. The presented figure presents
qualitatively accurate results of pressure and stress fields for a sandwich hull comprising large
discontinuities in Young’s modulus and density. As a one-step advancement, the 3D ISPH-
HSPH treats the face sheets as transversely isotropic, more consistent with derived theoretical
1
SS
FS
S
D
Dt
y
r
¶
=-+ +
¶
uga
r
S
y
::
SSS S
S
S
yy
¶¶¶ ¶
==
¶¶ ¶ ¶
FF
P
rF r r
SSS
=×PFS
S
S
S
y
¶
=
¶
S
E
( )
2
SS S SS
tr
lµ
=+SEE
( )
1
2
T
=×-EFFI
12
solution by Qin and Batra [56]. Future advancements correspond to rigorous treatment of finite-
strain viscoelastic as well as elastoplastic structural responses within the variationally
consistent Hamiltonian framework for design of resilient structures that would likely benefit
from advanced materials characterized by complex and highly non-linear responses, especially
under extreme loading conditions, as in case of extreme sea states.
3.2.3 Fluid-structure coupling scheme
Two-way coupling of fluid-elastic structure models can be established mainly through two
general coupling schemes referred to as FSA (Fluid Structure Acceleration-based) [12] and PI
(Pressure Integration) [57,12] schemes. The advantage of the FSA scheme corresponds to
perfect satisfaction of normal stress continuity in both continuum and discrete domains [12].
However, this scheme would not be preferable in FSI simulations corresponding to laminated
composites comprising large discontinuities in density field [14]. The PI scheme satisfies the
normal stress continuity to a certain extent and has shown to be effective for reproduction of
FSI corresponding to laminated composites [14]. In partitioned FSI coupling in the context of
particle methods with projection-based MPS or ISPH fluid models, an important matter
corresponds to satisfaction of boundary conditions corresponding to Helmholtz-Leray
decomposition (e.g., no-slip boundary condition at the fluid-structure interface) with respect to
uniqueness, exactness of solutions to continuity/Navier-Stokes equations [52]. In this regard,
robust algorithms can be developed and implemented to further enhance the accuracy and
reliability of fluid-structure coupling without compromising the computational efficiency.
13
Fig. 2. Representative snapshots corresponding to ISPH-HSPH (Incompressible SPH-
Hamiltonian SPH) modelling of water slamming on sandwich composites in 2D (a) & 3D (b)
14
4. TOWARDS HYDRO-VISCOELASTICITY IN PARTICLE-BASED FSI
For precise reproduction of FSI corresponding to viscoelastic structures, careful establishment
of a rigorous viscoelastic structure model is essential. The proposed viscoelastic structure
models should be able to produce precise viscoelastic structural responses in a physically
consistent manner. For this reason, variational and thermodynamic consistencies of proposed
viscoelastic structure model would be of prime importance. In this brief paper, we present a
general framework for establishment of a variationally & thermodynamically consistent
viscoelastic structure model in the context of SPH (section 4.1). Some preliminary numerical
results are also presented in section 4.2.
To our best knowledge, modelling of structural viscoelasticity has not been previously targeted
in the context of SPH. This paper presents preliminary results of structural viscoelasticity
corresponding to an extended Hamiltonian SPH, in a variationally and thermodynamically
consistent form.
4.1. Modelling of Viscoelastic Structural Responses
A general constitutive law corresponding to viscoelastic materials can be expressed as follows
[58]:
! " !!
#
$
%
&
'(
) *"#$
%
+
,
%
& - .
'/
$
%
&
'0
) 1"#%
&
+
,
%
& - .
'/
$
%
&
'0 (12)
2
where G(t - s) = C(t - s) - C(t); C is right Cauchy-Green strain tensor and δ corresponds to
an infinitesimal value. In the above equation, the second Piola-Kirchhoff stress tensor S is
decomposed into elastic, short-time and long-time memory viscous parts (first, second and
third terms on the right-hand side of Eq. 12, respectively). The short-time part includes rate-
dependent material responses corresponding to rapid material loading which is of importance
in extreme events and FSIs involving impacts, slamming, etc. This short-term viscous response
would be a function of external variables corresponding to right Cauchy-Green strain tensor C
and its time-rate [59]. The long-term response corresponds to stress relaxation and creep
[59,60] and would be a function of the history of right Cauchy-Green strain tensor C.
Accordingly, Eq. (12) would be written in the following form:
! " !!%$%&'' ) !'%$
3
%&'/ $%&'' )
4
5
&
%6%,%& - .'7 ./ $%&''8.
(13)
C
!
15
As for modelling of second term on right hand side of Eq. (13), i.e., short-time response, proper
modelling can be made through definition of a thermodynamically admissible [58,61] viscous
potential
y
v [58,59] as a function of strain and strain-rate invariants (I1-I3, J1-J7) that can well
describe non-linear short-time responses [61]. In that case:
9'" 9'
%
:(7 :)7:*7;(7<7 ;+
'
/22222!," =-.!
-𝑪
0
" !'
>
$
3%
&
'
/$
%
&
'? (14)
In this paper, as a preliminary work for viscoelastic modelling in the SPH framework, the
following simple equation is considered:
!'
>
$
3%
&
'
/$
%
&
'?
"2@"$
3
" @"
#
A
3
1B A ) A1B A
3( (15)
where the short-time stress response is simply considered to be a function of time-rate of right-
Cauchy-Green strain tensor and the overall viscous response of the system would be obtained
as time integration of short-time response calculated by Eq. (15); hence, this approach can be
simply referred to as Time-integrated Short-time Response (TSR). It should be noted that in
Eq. (15), πS stands for a dissipation coefficient (unit: Pa･s). Note that Zhang et al. [62]
developed simple artificial damping method for their TLSPH structure model based on a
similar form of Eq. (15). Certainly, this equation is one of the simplest forms of short-time
response and several other relevant terms (Eq. 33 in [61]) are not included here for the sake of
simplicity. Another approach for short-term viscous responses can be based on a multiplicative
decomposition of deformation gradient into elastic and viscous parts [63,64] in a manner
similar to the Kröner-Lee decomposition applied in finite elastoplasticity.
As for long-term memory viscoelastic response, a common way is to formulate the stress
response by convolution integrals and through Generalized relaxation (Maxwell) Models [65],
for this reason this general approach is referred to as Convolution Integral Models [66]. In case
of the GMM, the viscoelastic model comprises a single spring element (an equilibrium
modulus of ) and N sets of parallelly connected maxwell elements (consisting of a series
of spring and dashpot elements with an elastic modulus of and a dashpot with a relaxation
E∞
i
E
16
time of ). In the GMM, the elastic modulus containing relaxation effect is typically
approximated by Prony series as follows:
(16)
In non-linear visco-hyperelasticity, the constitutive equation can be formulated in a
convolution form as follows [67]:
(17)
The function is a non-dimensional relative modulus defined through the Prony series
using viscoelastic parameters of GMM [65]. If the strain history C(t) is known, the stress
would be a function of only time, i.e. . Accordingly, we have the following
equation:
(18)
where stress is decomposed into a long-term hyperelastic response and a visco-hyperelastic
contribution. Following the derivation presented by Goh et al. [68], Eq. (18) would be written
as follows:
(19)
where and k represents the time step number.
i
t
( )
1
exp
N
i
ii
t
Et E E
t
=
æö
=+ -
ç÷
èø
å
∞
( ) ( ) ( ) ( ) ( )
e
e0
,t
tgtgts ds
s
¶¶
=*=-
¶¶
òSC C
SC S C
C
( )
gt
( )
,tSC
( )
tS
( ) ( ) ( )
( ) ( )
e
e0
1
e
1
exp
Nt
i
ii
N
i
i
s
ts
tg t g ds
s
gt t
t
=
=
¶
æö
-
=+ -
ç÷
¶
èø
=+
åò
å
S
SS
Sh
∞
∞
( ) ( ) ( )
1
ee
1
1exp
exp
N
i
kk k
ii
ii
i
t
t
tg t t g t
t
t
t
-
=
æö
æö
D
--
ç÷
ç÷
æö
Dèø
ç÷
=+ -+D
ç÷
ç÷
D
èø
ç÷
ç÷
èø
å
SS h S
∞
( ) ( )
1
ee e
kk
tt
-
D=-SS S
17
It must be highlighted here that another approach to model viscoelastic response is the so-called
Internal Variable Model where differential-type models are applied as functions of tensorial
internal variables A [69,70]. The proper approach should depend on the target problem and
materials involved [71].
4.2. Preliminary Numerical Investigations for Reproduction of Viscoelastic Structural
Responses
In this section, two types of viscoelastic HSPH structure models, namely, TSR-type and GMM-
type are considered for preliminary investigations. The considered numerical test corresponds
to free vibration of a 2D linear viscoelastic cantilever plate, where the theoretical solution can
be obtained based on the consideration of the Euler-Bernoulli theory [72].
Fig. 3(a) illustrates computational setup of the considered benchmark test. The free span length
and thickness of the plate are respectively set as L = 0.2 m and hs = 0.02 m, that corresponds to
the geometry in the setup of Gray et al. [73]. The plate is
r
S = 1000 kg/m3 in density and
n
S =
0.4 in Poisson’s ratio. The plate begins to oscillate with an initial velocity distribution of uy(x).
By referring to the work of Diani [72], the theoretical time history of a viscoelastic plate’s
deflection η for first (fundamental) mode of vibration with an initial condition of η (L,0) = Δ0
and ∂η/∂t |t=0 = 0 is obtained as:
(20)
where δ1 is wave number of fundamental vibration mode (δ1L = 1.875) and ω1 stands for
frequency. For simplicity of calculation and with respect to the considered Total Lagrangian
framework, instead of providing an initial displacement to the plate, the plate is initially placed
horizontally along the x axis and an initial velocity field is imposed to the plate at t’ =
s which corresponds to the quarter of the oscillation period corresponding to a zero deflection
( ) ( )
( )
( )
( )
( )
( )
( ) ( )
1
01theo
1
11 1
11
1
1
,cos;
cosh cos
cosh cos
exp Im R
sin sinh sinh si
e
n
fx
xt t fL
xx
fx x x x x xx
t
h
d
w
dd
dd
w
dd d
=
+
-+
D-
=+-
( )
1
1
4
2
Re
p
w
18
instant. This initial velocity field can be simply obtained by taking the time derivative of Eq.
(20), i.e., ∂ηtheo/∂t |t= t’.
The simulation is performed by both TSR and GMM viscoelastic models. Figs. 3(b) and 3(c)
correspond to the results by TSR and GMM models, respectively. For the TSR model, the
Young’s modulus of the plate is set as ES = 2.0E+6 Pa and the dissipation coefficient πS is tuned.
For the GMM, equilibrium modulus is set as = 2.0E+6 Pa, and the number of Maxwell
elements as well as their elastic modulus and relaxation times are adjusted. Two types of
combinations of the Maxwell elements are considered, i.e. single Maxwell element case: N =
1; =(1.9E+7 Pa, 1.0E-4 s), and triple Maxwell element case: N = 3; = (1.9E+7
Pa, 1.0E-4 s), = (1.0E+6 Pa, 1.0E-3 s), = (1.0E+5 Pa, 1.0E-2 s).
Fig. 3(b1) presents snapshots of particles illustrating stress field at t = 0.84 s reproduced by the
TSR extended HSPH model with dissipation coefficient of πS = 500 and 5000 Pa･s with particle
diameter of d0 = 1.0E-3 m and maximum deflection of Δ0 = -0.03 m. From the presented figures,
it is found that smooth stress fields are achieved by the TSR extended HSPH model for two
presented dissipation coefficients, with clear reduction of deflection amplitude for the larger
dissipation coefficient, i.e. πS = 5000 Pa･s. Fig. 3(b2) depicts time variations of deflection at
the plate’s free end with a set of different dissipation coefficients. From the presented figure,
it is evident that the gradual increase of the dissipation coefficient from πS = 0 to πS = 5000,
has consistently resulted in increase of deflection attenuation.
Fig. 3(c1) portrays typical snapshots of particles including stress distributions at t = 0.84 s
reproduced by the GMM extended HSPH model corresponding to particle diameter of d0 =
1.0E-3 m. According to this figure, the GMM extended HSPH model is shown to provide stable
and smooth stress fields for both single and triple Maxwell element cases, i.e. N = 1 and N = 3.
The figure also portrays the qualitative accuracy of GMM extended HSPH for a case of large
deformation, i.e., Δ0 = -0.06 m. Fig. 3(c2) presents deflection time histories at the plate’s free
end for the case of N = 3 and Δ0 = -0.03 m for a set of different particle diameters along with
the corresponding theoretical solution derived for the present triple Maxwell element case [72].
From this figure, by refining the spatial resolution, the accuracy of GMM extended HSPH
model is clearly improved, indicating the convergence property of the GMM extended HSPH
E∞
i
E
i
t
( )
11
,E
t
( )
11
,E
t
( )
22
,E
t
( )
33
,E
t
19
model. The preliminary results presented in this article, portray the potential robustness of
variationally consistent HSPH to be extended towards reproduction of viscoelastic structural
responses in a precise and thermodynamically consistent manner.
5. CONCLUDING REMARKS
This review article portrays the latest advances corresponding to particle-based FSI solvers
with the main focus on entirely Lagrangian meshfree FSI solvers in which particle methods are
applied as both fluid and structure models. The paper highlights the potential robustness of
these advanced solvers in reproducing FSI corresponding to extreme events and portrays the
future perspectives for continued developments towards reliable practical engineering
applications, especially with respect to rapid advances made in the field of material engineering
and emergence of so-called advanced materials, characterized by complex non-linear and
potentially viscoelastic and/or elastoplastic structural responses.
Developments made in the context of particle methods corresponding to FSI solvers, mainly
focus on reproducing linearly elastic material responses and have been mainly configured in a
thoroughly Newtonian framework. The paper highlights the need for development of advanced
structure models in the context of particle methods being capable of reproducing
comprehensive structural responses corresponding to material non-linearities as well as
viscoelasticity and elastoplasticity. Such developments need to be made coherently and
preferably in a variationally consistent framework, where dynamics and kinematics of the
structural system would be variationally linked. Such variational consistency would be
important to ensure stability, accuracy and physical reliability including conservation
properties and thermodynamical consistency.
In this paper, followed by a brief review on latest advances corresponding to particle-based FSI
solvers, basics of mathematical and numerical modelling for entirely Lagrangian meshfree
hydroelastic FSI solvers are briefly presented, along with highlighting some of the important
features that are crucial for reliable modelling of FSI corresponding to design of resilient
structures in extreme events in ocean/coastal engineering. As an important step for coherent
extension of particle-based structure models towards reproducing viscoelastic structural
responses, a brief background of corresponding mathematical & numerical modelling is
20
presented along with some preliminary numerical results achieved by an extended Hamiltonian
SPH model.
Followed by rigorous establishment of viscoelastic structure models and corresponding FSI
solvers in the context of particle methods, further academic research should be devoted to
development of variationally consistent elastoplastic structure models capable of reproducing
finite strain elastoplasticity [74,75]. To ensure comprehensive modelling of structural
responses, in addition to viscoelasticity and elastoplasticity, damage and fracture modelling
needs to be considered as well. In this regard, accumulated structural damage needs to be
calculated in a precise, reliable and physically consistent manner. For this purpose, the
variationally consistent elastoplastic SPH model can be coupled with a Continuum Damage
Mechanics (CDM) model [76] that links microscale damage mechanisms to macroscale failures.
To ensure general applicability of the developed solvers for FSI characterized by air
entrapment/entrainment, the fluid model must be extended to include the air phase, resulting in
a multi-phase & multi-physics air-water-structure FSI solver. Certainly, each step in such
coherent developments need to be accompanied by scrupulous validations through
consideration of rigorous reference solutions including classical and experimental test cases.
21
Fig. 3. Preliminary results by TSR & GMM extended HSPH structure models in reproducing
free vibration of a 2D linear viscoelastic cantilever plate
6. ACKNOWLEDGEMENT
The authors would like to express their gratitude to Professor Decheng Wan at Shanghai Jiao
Tong University for invitation for this vision paper. The first author, A. Khayyer, would like
to express his sincere appreciation to Professor Antonio J. Gil at Swansea University and Dr
Chun Hean Lee at the University of Glasgow for discussions regarding viscoelastic and
elastoplastic modelling. The authors appreciate the research grants by JSPS (Japan Society for
the Promotion of Science), grants number JP18K04368, JP21H01433 and JP21K14250.
22
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