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REVISED PROOF
Journal : Large 11831 Article No : 9438 Pages : 32 MS Code : 9438 Dispatch : 2372020
Vol.:(0123456789)
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Archives of Computational Methods in Engineering
https://doi.org/10.1007/s1183102009438w
ORIGINAL PAPER
Stochastic Oblique Impact onComposite Laminates: AConcise Review
andCharacterization oftheEssence ofHybrid Machine Learning
Algorithms
T.Mukhopadhyay1 · S.Naskar2 · S.Chakraborty3· P.K.Karsh4· R.Choudhury5· S.Dey5
Received: 21 September 2019 / Accepted: 29 April 2020
© CIMNE, Barcelona, Spain 2020
Abstract
Due to the absence of adequate control at diﬀerent stages of complex manufacturing process, material and geometric proper
ties of composite structures are often uncertain. For a secure and safe design, tracking the impact of these uncertainties on
the structural responses is of utmost signiﬁcance. Composite materials, commonly adopted in various modern aerospace,
marine, automobile and civil structures, are often susceptible to lowvelocity impact caused by various external agents. Here,
along with a critical review, we present machine learning based probabilistic and nonprobabilistic (fuzzy) low–velocity
impact analyses of composite laminates including a detailed deterministic characterization to systematically investigate the
consequences of source uncertainty. While probabilistic analysis can be performed only when complete statistical descrip
tion about the input variables are available, the nonprobabilistic analysis can be executed even in the presence of incom
plete statistical input descriptions with sparse data. In this study, the stochastic eﬀects of stacking sequence, twist angle,
oblique impact, plate thickness, velocity of impactor and density of impactor are investigated on the crucial impact response
parameters such as contact force, plate displacement, and impactor displacement. For eﬃcient and accurate computation, a
hybrid polynomial chaos based Kriging (PCKriging) approach is coupled with inhouse ﬁnite element codes for uncertainty
propagation in both the probabilistic and non probabilistic analyses. The essence of this paper is a critical review on the
hybrid machine learning algorithms followed by detailed numerical investigation in the probabilistic and nonprobabilistic
regimes to access the performance of such hybrid algorithms in comparison to individual algorithms from the viewpoint of
accuracy and computational eﬃciency.
1 Introduction
Due to the high speciﬁc strength, stiﬀness, rigidity, fatigue,
corrosion resistance and other outstanding mechanical
characteristics (with tunable characteristics) compared to
standard metallic structural materials, laminated composite
plates have a broad application in the spacecraft, marine,
automotive, mechanical and civil sectors. Composite struc
tures are often susceptible to lowvelocity impact caused
by various external agents, leading to a signiﬁcant inﬂu
ence on the intended performance of the system. Therefore,
investigating the behaviour of composite structures sub
jected to impact load is of utmost importance. On the other
hand, uncertainties in a composite material may arise due
to presence of voids in between the laminate, incomplete
knowledge about the ﬁbre parameters, porosity, alternation
in ply thickness and various other inevitable issues involved
in the complex manufacturing process. Quite naturally, the
lowvelocity impact responses are aﬀected by the presence
T. Mukhopadhyay, S. Naskar, S. Chakraborty, P. K. Karsh, R.
Choudhury and S. Dey have contributed equally to this work.
* T. Mukhopadhyay
tanmoy@iitk.ac.in
S. Naskar
susmitanaskar@iitb.ac.in
1 Department ofAerospace Engineering, Indian Institute
ofTechnology Kanpur, Kanpur, India
2 Department ofAerospace Engineering, Indian Institute
ofTechnology Bombay, Mumbai, India
3 Department ofApplied Mechanics, Indian Institute
ofTechnology Delhi, NewDelhi, India
4 Department ofMechanical Engineering, Parul Institute
ofEngineering andTechnology, Parul University, Vadodara,
India
5 Mechanical Engineering Department, National Institute
ofTechnology Silchar, Silchar, India
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Journal : Large 11831 Article No : 9438 Pages : 32 MS Code : 9438 Dispatch : 2372020
T.Mukhopadhyay et al.
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of these uncertainties. A general overview of the sources
of uncertainty in the computational framework of a struc
tural system is presented in Fig.1 [1]. One adhoc way to
deal with these uncertainties is to introduce the so called
partial safety factors at the design stage. However, a more
rigorous method will demand quantiﬁcation of the eﬀect of
the material and the geometric uncertainties on the output
responses. To this end, we would pursue both probabilistic
and nonprobabilistic lowvelocity impact assessment of
composite laminates to cover two possible instances of get
ting an adequate statistical report on the input parameters,
or unavailability of the same owing to restrictions on per
forming experiments involving a large number of samples.
Researchers, over the years, have studied the behaviour of
composite structures under the action of impact load. While
Xu and Chen [2] conducted lowvelocity impact analysis
of carbon epoxy laminates for damage detection, Liu etal.
[3] studied the influence of shape of impactor (such as
conical, hemispherical and ﬂat) on the lowvelocity impact
responses of sandwich plate. In both cases, experimental
as well as numerical analyses were performed. Jagtap etal.
[4] carried out ﬁnite element (FE) simulation for damage
identiﬁcation of laminated plates due to impact loading.
The eﬀect of boundary condition and velocity of impactor
were determined. Similarly, Balasubramani etal. [5] per
formed numerical investigation to determine the eﬀect of
boundary conditions, the thickness of laminate, impactor’s
mass and velocity on transverse and longitudinal stress of
the composite laminate due to lowvelocity impact loading.
Tan and Sun [6] and Sun and Chen [7] also used the ﬁnite
element method with Newmark time integration scheme to
investigate lowvelocity impact on composite structures. A
comprehensive review on lowvelocity impact loading on
composite structures can be found in [8]. Ahmed and Wei
[9] also reviewed numerical and experimental methods for
computing dynamic and static responses of composite plates
subjected lowvelocity impact and quasistatic loads.
Several works dealing with failure mechanism of compos
ite plates subjected to lowvelocity impact load can be found
in the literature. While Yuan etal. [10] used an analytical
model based on the theory of ﬁrst order shear deformation
for the analysis of damage and deformation of laminated
glass under lowvelocity impact, Zhang and Zhang [11]
applied FE model for damage detection in composite struc
tures due to lowvelocity impact. Feng and Aymerich [12],
Maio etal. [13] and Kim etal. [14] developed and applied
Fig. 1 General overview of the sources of uncertainty in the computational framework of a structural system
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Journal : Large 11831 Article No : 9438 Pages : 32 MS Code : 9438 Dispatch : 2372020
Stochastic Oblique Impact onComposite Laminates: AConcise Review andCharacterization of…
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progressive damage models to investigate the failure mecha
nism of laminated composite due to the lowvelocity impact.
Lipeng etal. [15] investigated delamination failure due to
impact load by using a selfadapting delamination element
method. Johnson etal. [16] presented diﬀerent models for
failure analysis of composite plates by considering internal
damage and delamination due to impact loading. Coutellier
etal. [17] developed a model for delamination detection in
thin composite structures. Jih and Sun [18], on the other
hand, investigated experimentally the delamination in lami
nated composite plates due to lowvelocity impact.
Despite the vast literature on lowvelocity impact analysis
of composite structures, none of these studies consider the
presence of uncertainties in the system. Due to the complexity
of manufacturing, accurate design speciﬁcations of composite
structures cannot be achieved in real life. As a consequence,
uncertainties in a composite structure are unavoidable. In
composite material, the main sources of uncertainties are due
to variation in material properties and inaccurate geometrical
properties. Such uncertainties are introduced in the elementary
input level (elemental mass and stiﬀness matrix), and propa
gate to the global level (global mass and stiﬀness matrix) of
composite structures and hence, leads to a signiﬁcant devia
tion from the deterministic value of impact responses. In the
present paper, the eﬀects such sourceuncertainties on the low
velocity oblique impact (refer to Fig.2a) response of compos
ite plates are aimed to be addressed. The analysis is divided
into three sections namely deterministic, probabilistic and non
probabilistic, the later two sections being dedicated to stochas
tic analysis and uncertainty quantiﬁcation (UQ). Only when
the probabilistic distributions of uncertain input parameters
are accessible can the probabilistic analyses be performed. In
many instances though, it is not possible to obtain the complete
probabilistic distributions of the input variables. In such cases,
nonprobabilistic fuzzy analysis can be employed to portray
the eﬀects of uncertainty. It is to be noted that both conven
tional probabilistic and nonprobabilistic analysis techniques
involve signiﬁcant computational eﬀorts due to the require
ment of performing thousands of expensive ﬁnite element
simulations. One way to circumvent this issue is to develop a
machine learning model on the basis of representative origi
nal ﬁnite element simulations. It is worthy to note here that
machine learning is a broad domain. A schematic diagram
showcasing the various aspects of machine learning techniques
and its relationship with data science is shown in Fig.3. In this
work, we are only interested in supervised learning techniques.
Popular supervised learning techniques include Gaussian pro
cess or Kriging [19–22], Polynomial chaos expansion (PCE)
[23–25], analysisofvariance decomposition [26–29], Polyno
mial chaos based Kriging (PC Kriging) [30–33] etc. In this
work, we review three machine learning techniques in the con
text of stochastic lowvelocity impact analysis. The machine
learning techniques reviewed here are polynomial chaos
expansion, Kriging and polynomial chaos based Kriging.
This paper is composed of six sections in the order of
chronological interdependence including the current intro
duction section. Section2 describes governing equations for
the analysis of the transient lowvelocity oblique impact of
composite plates that includes the descriptions of dynamic
equations, contact law and Newmark’s integration scheme. In
Sect.3, detailed description of the surrogate model based on
PCKriging is provided. Section4 provides both probabilis
tic and nonprobabilistic stochastic approaches for the impact
analysis of lowvelocity. The numerical results are presented
in Sect.5 (deterministic, probabilistic and fuzzy based non
probabilistic results including the comparative performance
of three diﬀerent surrogate models i.e. PCE, Kriging and PC
Kriging). Finally, in Sect.6, major observations and conclu
sion are provided along with an overview of the current level
of development in relevant research ﬁelds.
2 Review oftheGoverning Equations
forLowVelocity Impact onLaminated
Composites
A laminated composite plate is considered with length L,
width b, and thickness t subjected to normal and oblique
impact loading (as shown in Fig.2). The dynamic equation
[34] of such system can be expressed as
where
M(̃
𝜍)
,
(̃
𝜍)
,
𝛿
and
̈
𝛿
are the randomized mass matrix,
randomized stiﬀness matrix, displacement vector and accel
eration vector, respectively, while {F} is externally applied
force vector. Here,
(̃
𝜍)
indicates the degree of randomization.
The force vector including the contact force
(
F
C)
in case of
impact can be expressed as
The equation of motion for the rigid impactor is given by
where
mimp(̃
𝜍)
is the mass of impactor while
̈
𝛿imp
is the accel
eration of impactor.
2.1 Contact Law
Modiﬁed Hertzian contact law can be utilized to calculate
the contact force between impactor and the composite plate
[35]. The impactor is assumed as a spherical elastic solid
body.
The contact force can be obtained during loading as
(1)
[M(̃
𝜍)]{
̈
𝛿}+[K(̃
𝜍)]{𝛿}={F(̃
𝜍)}
(2)
F
(̃
𝜍)=
{
000F
C
(̃
𝜍)000
}
(3)
mimp
(̃
𝜍)
̈
𝛿
imp
+F
c
(̃
𝜍)=
0
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REVISED PROOF
Journal : Large 11831 Article No : 9438 Pages : 32 MS Code : 9438 Dispatch : 2372020
T.Mukhopadhyay et al.
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Fig. 2 a Laminated composite plate subjected to normal and oblique impact load by a spherical mass. b A typical example of twisted plate. c
Geometric details of twist in the plate
REVISED PROOF
Journal : Large 11831 Article No : 9438 Pages : 32 MS Code : 9438 Dispatch : 2372020
Stochastic Oblique Impact onComposite Laminates: AConcise Review andCharacterization of…
1 3
where
𝛾
denotes the local indentation and k is the modiﬁed
contact stiﬀness [36] which can be expressed by contact
theory as
where
Ei
is the elastic modulus of the impactor,
Eyy
is the
elastic modulus of laminated composite plate of the upper
most ply in the transverse direction, while
Rimp
and
𝜐
are the
radius and Poisson’s ratio of impactor, respectively. At the
time of loading and unloading the contact force
(
F
C)
can be
estimated as
where
Fm
and
𝛾m
are the maximum contact force and maxi
mum indentation, respectively. The permanent indentation
(𝛾0)
in loading and unloading cylce is given by
where
𝛽c
is the constant, while
𝛾Cr
is the critical identation.
For the oblique impact, the local indentaion is given by
where
𝛾imp
and
𝛾plt
are impactor’s displacement and targeted
plate displacements, respectively, while
𝛽and 𝜓
are the
oblique impact angle and twist angle, respectively, along the
(4)
Fc
(̃
𝜍)=k(̃
𝜍)𝛾(̃
𝜍)
1.5
0
≤
𝛾
≤
𝛾
m
(5)
k
(̃
𝜍)=
4
3
√
Rimp
1
[1−𝜐2
i(̃
𝜍)]
E
i
(̃
𝜍)+1
E
yy
(̃
𝜍)
(6)
F
c(̃
𝜍)=Fm
[
𝛾(̃
𝜍)−𝛾0
𝛾
m
−𝛾
0]5∕2
and Fc(̃
𝜍)=Fm
[
𝛾(̃
𝜍)−𝛾0
𝛾
m
−𝛾
0]3∕2
(7)
𝛾
0
=0when 𝛾
m
<𝛾
Cr
𝛾0
=𝛽
c(
𝛾
m
−𝛾
Cr)
when 𝛾
m≥
𝛾
Cr
(8)
𝛾(t)(
̃
𝜍)=𝛾imp(t)(
̃
𝜍)cos 𝛽+𝛾plt(xc,yc,t)(
̃
𝜍)cos 𝜓
global zdirection, respectively. The contact force elements
at the global direction of contact point can be described as
2.2 Newmark’s Time Integration Scheme
The contact force involved in the equilibrium Eqs.(1) and
(3) is generally transient in nature for the dynamic response
of a laminated composite plate under the impact by a spheri
cal impactor. The time integration scheme of Newmark [37]
is used to solve the equations that depend on time. Use of
above scheme with time interval
t
gives the subsequent
relations at the time
t+Δt
where
[̄
K]
and
[̄
k]
are the eﬀective stiﬀness matrix of the
plate and impactor, respectively, and given by
Eﬀective contact forces at time
t+Δt
can be derived as
(9)
Fix =0, Fiy =Fc(̃
𝜍)sin 𝜓,Fiz =Fc(̃
𝜍)cos 𝜓.
(10)
[̄
K]𝛿(t+t)={
̄
F}(t+t)
(11)
k
imp𝛿
(t+Δt)
imp
={FC}(t+Δt
)
(12)
[̄
K]=K+a0M
(13)
[̄
k]=a
0
m
imp
(14)
{
F}
(t+Δt)
={F}
(t+𝛿t)
+[M]
(
a
0
𝛿
(t)
+a
1̇
𝛿
(t)
+a
2⃛
𝛿
(t))
(15)
̄
F
(t+Δt)
C
e
=F(t+Δt)
C+mimp
(
a0𝛿(t)
imp +a1̇
𝛿(t)
imp +a2̈
𝛿(t)
imp
)
Fig. 3 Diﬀerent facets of
machine learning techniques
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REVISED PROOF
Journal : Large 11831 Article No : 9438 Pages : 32 MS Code : 9438 Dispatch : 2372020
T.Mukhopadhyay et al.
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The acceleration and velocity can be derived from displace
ment at time
t+Δt
as
The initial boundary condition considered as
where
V0
is the initial velocity of the impactor. The time
integration constants can be expressed as
For the present study, the value of
′′
and
𝛽′′
are considered
as 0.5 and 0.25, respectively.
3 Hybrid Machine Learning Based
onKriging andPCE
Let, x=
{
x
1
,…,x
N}
∈ℝN to be the input variables and
y
ℝ
O
to be the output responses. We also assume
M(⋅)
to be the computational model (FE model in present case)
such that
For impact analysis, the model
M(
⋅
)
is computationally
expensive to evaluate and hence, the task of quantifying the
uncertainties in the output response
y
becomes diﬃcult. One
way to deal with this issue is to replace the computationally
expensive ﬁnite element model
M(
⋅
)
with a surrogate
⌢
M(
⋅
)
.
It can be noted here that we have used the words surrogate
modelling and machine learning in identical sense keeping
in mind its purpose in the context of this article. We would
review three methods of machine learning in this section that
can be used as a surrogate of the original simulation model.
3.1 Polynomial Chaos Expansion
Polynomial chaos expansion (PCE) is one of the most pop
ular methods available in literature. This was ﬁrst imple
mented by Wiener [38] and hence, is also known as ‘Wiener
Chaos expansion’. Xiu and Karniadakis [23] subsequently
generalized the technique and proved its eﬀectiveness for
diﬀerent continuous and discrete systems from the so called
(16)
{̈
𝛿}
(t+Δt)
=a0({𝛿}
(t+Δt)
−{𝛿}
(t)
)−a1{
̇
𝛿}
(t)
−a2{
̈
𝛿}
(t)
̈
𝛿
(t+Δt)
imp =a0(𝛿(t+Δt)
imp −𝛿(t)
imp)−a1̇
𝛿(t)
imp −a2̈
𝛿(t)
imp
{
̇
𝛿}(t+Δt)={̇
𝛿}(t)+a3{̈
𝛿}(t)+a4{̈
𝛿}(t+Δt)
̇
𝛿
(t+Δt)
imp
=̇
𝛿(t)
imp
+a3̈
𝛿(t)
imp
+a4̈
𝛿(t+Δt)
imp
(17)
𝛿
=
̇
𝛿=
̈
𝛿=0, 𝛿
imp
=
̈
𝛿
imp
=0and
̇
𝛿
imp
=V
0
(18)
a
0=
1
𝛽��𝛿t2,a1=
1
𝛽��Δt
,a2=
1
2𝛽�� −
1,
a3
=
(
1−��
)
Δtand a
4
=��Δt
(19)
y=M(x)
Askeyscheme,
L2
convergence in the corresponding Hilbert
space.
Assuming
𝐢
=
(
i
1
,i
2
,…,i
N)
∈ℕ
N
0
to be a multiindex
with
i=i1+i2+
⋯
+iN,
and let n ≥ 0 be an integer. The
nth order PCE of g(X) is given as:
where {ai} are unknown coeﬃcients that must be deter
mined. Φi(X) are Ndimensional orthogonal polynomials
with maximum order of and satisﬁes
Here, δij denotes the multivariate kronecker delta function.
It is to be noted that the orthogonal polynomials are depend
ent on the PDF ϖ(x) of input variables. Table1 presents the
orthogonal polynomial type and the random variable type
correspondence [23].
Over last two decades, researchers have developed and
utilized diﬀerent variants of PCE. Xiu and Karniadakis
[23] proposed the Wiener–Askey PCE where the unknown
coeﬃcients associated with the coeﬃcients were deter
mined by using either collocation method or the Galerkin
projection. With this method, it is possible to solve sto
chastic partial diﬀerential equations in an eﬃcient way.
However, Wiener–Askey PCE is intrusive in nature and
hence, knowledge about the governing partial diﬀerential
equation of the system is required. As a consequence, this
method is not applicable to cases where the user only have
some data and no knowledge about the process from which
the data is generated.
To tackle the abovementioned problem, researchers
focused on developing nonintrusive (datadriven) PCE.
The easiest and most popular way to train a datadriven
PCE is by minimizing the least square error of the system
(20)
̂
g
(X)=
n
∑
i=0
ai𝛷i(X
)
(21)
E
(𝛷i(X)𝛷j(X)) =
�
𝛺
𝛷i(X)𝛷j(X)𝜛(x)=𝛿ij,0
≤
i

,

j
≤N
Table 1 The Correspondence of the type of orthogonal polynomial
with distribution pattern
Type Random variables Type of orthogo
nal polynomial
Support
Continuous Gaussian Hermite (− ∞,∞)
Gamma Laguerre [0,∞)
Beta Jacobi [a, b]
Uniform Legendre [a, b]
Discrete Poisson Charlier
{0, 1, …}
Binomial Krawtchouk
{0, 1, …,N}
Negative binomial Meixner
{0, 1, …}
Hypergeometric Hahn
{0, 1, …,N}
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Stochastic Oblique Impact onComposite Laminates: AConcise Review andCharacterization of…
1 3
[39, 40]. However, this method is susceptible to overﬁtting
and as a result often performs poorly. Methods for train
ing a PCE model by using the quadrature rule can also be
found in the literature [41, 42]. However, both these train
ing algorithms suﬀer from the curse of dimensionality and
hence, are only applicable to smallscale problems with
limited number of input variables.
To address the curse of dimensionality associated with
leastsquare and quadrature based training algorithms,
Blatman and Sudret [24, 43] two adaptive sparse PCEs
which can used for solving problems having hundreds of
input variables. Both the methods proposed follow similar
ﬂow where an iterative algorithm is used to determine the
importance of the terms involved in PCE and the lesser
important terms are removed. In the ﬁrst method, the
important terms in PCE are determined by tracking the
change in coeﬃcient of determination 2 (due to addition/
removal of a term). In the second approach, a more rigor
ous framework, referred to as the leastangle regression
is used to determine the important terms of PCE. With
both these approaches, there is a signiﬁcant reduction in
the number of unknown coeﬃcients associated with PCE
and thereby, issues with hundreds of input variables can
be solved.
Jacquelin etal. [44] identiﬁed that for lightly damped sys
tems, the convergence of PCE is very poor. It was proposed
that integrating Aitken’s transformation into the framework
of PCE can improve its convergence signiﬁcantly. Pascual
and Adhikari [45] hybridized the basic formulation of PCE
by coupling it with perturbation method. Four variants of the
hybrid perturbationPCE was proposed and reduced spectral
method was used to identify unknown coeﬃcients associ
ated with the bases. The proposed approaches were utilized
to solve the stochastic eigenvalue problem. It was observed
that the approaches proposed lead to a better approximation
of larger eigenvalues.
3.2 Kriging
In today’s time, one of the most popular machine learning
technique is perhaps the Gaussian process, a.k.a. Krig
ing is a Bayesian machine learning technique where we
assume that the response y, conditioned on input x is a
sample from a Gaussian process.
where
𝜇(
⋅
;𝐁)
is the mean function and
R(
⋅
,
⋅
;𝜽)
is the cor
relation kernel.
𝐁,𝜎
and
𝜽
are the hyperparameters of the
Gaussian process respectively, denotes the unknown coef
ﬁcients related to the mean function, the process variance
and the lengthscale parameter associated with the correla
tion kernel. In order to use Gaussian process as a machine
(22)
y
x;𝐁,𝜎,𝜽∼GP
(
𝜇(x;𝐁),𝜎
2
R
(
x
1
,x
1
;𝜽
))
learning technique, the hyperparameters needs to be esti
mated based on some training data. This can either be
achieved by maximizing the likelihood [21] or by using the
Bayes rule [46–49].
The most popular form of Gaussian process is the zero
mean Gaussian process or the simple Kriging. In this vari
ant, we assume
𝜇(
⋅
;𝐁)=0
. As a consequence, only
𝜎
and
𝜽
are the only hyperparameters associated with the system.
An improvement to the simple Kriging is the ordinary
Kriging where we assume the mean function is assumed
to be constant,
𝜇(
⋅
;𝐁)=a0
where
a0
is a constant. Unfor
tunately, the fact that the mean function is modelled as a
constant often results in erroneous models.
To enhance the Kriging model’s precision, universal
Kriging was developed [50, 51]. In universal Kriging, the
mean function represented as a linear regression model by
using multivariate polynomials
where
bi(x)
represents the ith basis function and ai denotes
the coeﬃcient associated with the ith basis function. With
this setup, the mean function captures the largest variance
in the data and the correlation function interpolates the
residual. Considering,
x
=
{
x
1
,x
2
,…,x
n}
to be input sam
ples and
g={g1,g2,…,gn}
to be the responses, the design
matrix and the correlation matrix can be represented. The
regression portion can be written as a n × p model matrix F,
whereas the stochastic process is deﬁned using a n × n cor
relation matrix Ψ
where ψ(∙,∙) is a correlation function, parameterised by a
set of hyperparameters θ. As already stated, the hyperpa
rameters are identiﬁed either by using maximum likelihood
estimation (MLE) or by using the Bayes rule.
Similar to PCE discussed in previous section, univer
sal Kriging also suﬀers from the curse of dimensional
ity. To address this issue, blind Kriging was proposed in
[51–54]. In blind Kriging, the polynomial order used to
represent the mean function of the Gaussian process is
selected in an adaptive manner. Bayes rule is used to train
ing the blind Kriging model. It is worthwhile to mention
that blind Kriging satisﬁes both the hierarchy criterion
and the heredity criterion. As per the hierarchy criterion,
(23)
𝜇
(⋅,𝐁)=
P
∑
i=1
aibi(x
)
(24)
F
=
b1
x
1
…bp
x
1
⋮⋱⋮
b
1
(xn)⋯b
p
(xn)
(25)
𝜓
=
𝜓
x1,x1
…𝜓
x1,xn
⋮⋱⋮
𝜓
xn,x1
⋯𝜓(xn,xn)
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REVISED PROOF
Journal : Large 11831 Article No : 9438 Pages : 32 MS Code : 9438 Dispatch : 2372020
T.Mukhopadhyay et al.
1 3
lower order eﬀects in the mean function are selected before
the higher order eﬀects. Whereas, as per the heredity cri
terion, an eﬀect can only be important if its parent eﬀects
are already important. Other variants of Kriging includes
CoKriging [55] and stochastic Kriging [56–58]. A com
parative assessment from the viewpoint of accuracy and
computational eﬃciency can be found in ref [19], where
both high and low dimensional input parameter space was
considered for a comprehensive analysis.
The choice of suitable correlation function is a crucial
element for all the Kriging variants [59–61]. Correlation
function that are commonly used with Gaussian process
are mostly stationary and hence,
With such as correlation function, it is possible to represent
multivariate functions as product of onedimensional corre
lations. Popular stationary correlation functions includes: (a)
exponential correlation function (b) generalised exponential
correlation function (c) Gaussian correlation function (d)
linear correlation function (e) spherical correlation function
(f) cubic correlation function and (g) spline correlation func
tion. The mathematical forms of all the correlation functions
are provided below:
1. Exponential correlation function:
2. Generalised exponential correlation function:
3. Gaussian correlation function:
4. Linear correlation function:
5. Spherical correlation function:
6. Cubic correlation function:
7. Spline correlation function:
(26)
𝜓
(x,x
�
)=
∏
j
𝜓j(𝜃,xi−x
�
i
)
(27)
𝜓j(𝜃;dj)=exp(−𝜃jdj)
(28)
𝜓j
(𝜃;d
j
)=exp(−𝜃
j
d
j𝜃
n+1),0<𝜃
n+1≤2
(29)
𝜓
j(𝜃;dj)=exp(−𝜃jd
2
j)
(30)
𝜓j(𝜃;dj)=max{0, 1 −𝜃jdj}
(31)
𝜓
j(𝜃;dj)=1−1.5𝜉j+0.5𝜉
2
j
,𝜉j=min{0, 𝜃j

dj
}
(32)
𝜓
j(𝜃;dj)=1−3𝜉
2
j
+2𝜉
3
j
,𝜉j=min{1, 𝜃j

dj
}
where
𝜉j=𝜃jdj
For all the correlation functions described above,
dj
=x
i
−x
�
i
. The hyperparameters associated with the
covariance functions are determined either by using the
maximum likelihood estimate (MLE) or by using the
Bayes rule. A detailed account of MLE in the context of
Kriging is given in [21].
3.3 Polynomial Chaos Based Kriging (PCKriging)
Finally, we discuss about a hybrid machine learning tech
nique, referred to as the polynomial chaos based Kriging
(PCKriging) [30–32]. PCKriging is a novel surrogate
model that combine two wellknown surrogates, namely,
polynomial chaos expansion (PCE) [23, 25] and Kriging [19,
20]. PCKriging can be viewed as a Kriging model where
the mean/trend function is modelled by using PCE. With
this setup, it is possible to achieve a higher order accuracy
as compared to PCE and Kriging.
PCKriging is a special kind of Kriging where the mean
function of the Gaussian process is modelled by using poly
nomial chaos expansion. More speciﬁcally,
𝝁(⋅)
in Eq.(23)
is represented by using Eq.19. Under limiting condition,
PCKriging converges either to PCE or to Kriging. Similar
to Kriging, the hyperparameters in PCKriging are learned
by maximizing the likelihood. For further details, interested
readers may refer [19, 62].
Despite PCKriging’s beneﬁt over its individual PCE
and Kriging, the hybrid metamodel suﬀers from the curse
of dimensionality due to the factorial growth of unknown
coeﬃcients with a rise in the number of input parameters
N. This limitation originates from the PCE component
of PCKriging. To address this problem, a variant of PC
Kriging, referred to as Optimal PC Kriging (OPCKriging)
[31] was proposed. In OPCKriging, least angle regression
(LAR) is used to only retain the important components of
PCE. The OPCKriging follows an iterative algorithm where
each polynomial can be added to the trend part onebyone.
Figure4 presents a ﬂowchart depicting the algorithm of
OPCKriging.
(33)
𝜓
j(𝜃;dj)=
1−5𝜉2
j+30𝜉3
j,0
≤
𝜉j
≤0.2
1.251−𝜉3
j, 0.2 ≤𝜉j≤
1
0, 𝜉j>1
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Journal : Large 11831 Article No : 9438 Pages : 32 MS Code : 9438 Dispatch : 2372020
Stochastic Oblique Impact onComposite Laminates: AConcise Review andCharacterization of…
1 3
4 Machine Learning Based Stochastic
Impact Analysis
A major objective of the present study is to determine sto
chastic response of lowvelocity impact loading on com
posite plates following probabilistic and nonprobabilistic
frameworks. Both geometric and material uncertainties are
considered in this work. To be speciﬁc, uncertainties in the
composite plate stem from variation in the material prop
erties, ﬁbre orientation angle, twist angle, oblique impact
angle, initial velocity of impactor, mass density of impac
tor, and thickness of target plates, inclusion of which in
the analysis (following probabilistic and nonprobabilistic
approaches) is discussed here.
4.1 Probabilistic Impact Analysis
For probabilistic impact analysis, statistical descriptions of
the stochastic inputs are necessary. To that end, the machine
learning techniques discussed in previous section have been
coupled with our inhouse FE code for lowvelocity impact
analysis. For quantifying the uncertainty in the output
responses, ﬁrst input training samples are obtained using
an appropriate design of experiment (sampling) scheme.
Due to its simplicity and already proven superior perfor
mance, Sobol sequence [63, 64] has been used in this study.
In the next step, the training outputs are obtained by using
the actual FE solver. In the third step, the machine learning
models are trained and the hyperparameters associated with
the models are computed. Finally, Monte Carlo simulation
is carried out based on the trained ML model to compute
the probability density function of the output responses. A
ﬂowchart depicting the ML based probabilistic uncertainty
quantiﬁcation algorithm is presented in Fig.5. For the cur
rent study, the following cases of uncertainties are consid
ered at each lamina level (layerwise uncertainty modelling)
1. Variation of ﬁbreorientation angle:
2. Variation of twist angle:
3. Variation of oblique impact angle:
4. Variation of initial velocity of impactor:
5. Variation of mass density of impactor:
6. Variation of thickness of the plate:
7. Variation in location of loading point:
𝜓1{
𝜃
,
E
,
G
,
𝜐
,
𝜌
}=
𝛩
[{
𝜃
(̃
𝜍
)},{
E
(̃
𝜍
)},{
G
(̃
𝜍
)},{
𝜐
(̃
𝜍
)},{
𝜌
(̃
𝜍
)}]
𝜓2{𝜓,𝜃,E,G,𝜐,𝜌}=𝛩[𝜓,{𝜃(̃
𝜍)},{E(̃
𝜍)},{G(̃
𝜍)},{𝜐(̃
𝜍)},{𝜌(̃
𝜍)}]
𝜓3{,𝜃,E,G,𝜐,𝜌}=𝛩[,{𝜃(̃
𝜍)},{E(̃
𝜍)},{G(̃
𝜍)},{𝜐(̃
𝜍)},{𝜌(̃
𝜍)}]
𝜓4{V,𝜃,E,G,𝜐,𝜌}=𝛩[V,{𝜃(̃
𝜍)},{E(̃
𝜍)},{G(̃
𝜍)},{𝜐(̃
𝜍)},{𝜌(̃
𝜍)}]
𝜓5{𝜌imp,𝜃,E,G,𝜐,𝜌}=𝛩[𝜌imp ,{𝜃(̃
𝜍)},{E(̃
𝜍)},{G(̃
𝜍)},{𝜐(̃
𝜍)},{𝜌(̃
𝜍)}]
𝜓
6
{tplt,𝜃,E,G,𝜐,𝜌}=𝛩[tplt ,{𝜃(̃
𝜍)},{E(̃
𝜍)},{G(̃
𝜍)},{𝜐(̃
𝜍)},{𝜌(̃
𝜍)}]
Fig. 4 Flowchart for OPC
Kriging
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Journal : Large 11831 Article No : 9438 Pages : 32 MS Code : 9438 Dispatch : 2372020
T.Mukhopadhyay et al.
1 3
Here
̃
𝜁
is used to denote the stochastic representation
of the system parameters. The parameters
E,G,𝜐,𝜌
are
the set of Young’s moduli, shear moduli, mass den
sity and Poisson’s ratio in different directions, where
the entire set of stochastic material properties is
{
E
1,
E
2,
E
3,
G
12,
G
13,
G
23,
𝜐
12,
𝜐
13,
𝜐
23,
𝜐
32,
𝜐
21,
𝜐
31,
𝜌
}
. Unless
otherwise mentioned, the degree of stochasticity from the
respective deterministic values is taken as
±10%
(as per
standard design practice) for each of the components in
the set of material properties.
𝜓7{Lp,𝜃,E,G,𝜐,𝜌}=𝛩[Lp,{𝜃(̃
𝜍)},{E(̃
𝜍)},{G(̃
𝜍)},{𝜐(̃
𝜍)},{𝜌(̃
𝜍)}]
4.2 Fuzzy Impact Analysis
Although probabilistic analysis is more rigorous as it pro
vides the probability distribution of the output responses,
it is limited by the fact that we require probability distribu
tion of the input variables for carrying out such analysis.
In the reallife scenario, we may not have knowledge about
the probability distribution of the input variables due to
the requirement of extensive experimental characterization
of the materials involving thousands of physical samples.
Under such circumstances of sparse data availability, we
have to opt for non probabilistic analysis. Out of diﬀer
ent non probabilistic analysis methods available in litera
ture, fuzzy based nonprobabilistic analysis is employed
Fig. 5 Flowchart for probabilistic impact analysis based on hybrid machine learning models coupled with FE simulations
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REVISED PROOF
Journal : Large 11831 Article No : 9438 Pages : 32 MS Code : 9438 Dispatch : 2372020
Stochastic Oblique Impact onComposite Laminates: AConcise Review andCharacterization of…
1 3
for uncertainty quantiﬁcation and propagation in low
velocity impact analysis of the laminated composite plate.
The fuzzy theory is employed in the intermediate stage
between nonmembers and members known as member
ship function
[
𝜇
pi]
that signiﬁes the degree to which each
component in the territory leads to the fuzzy set [65]. The
triangular membership function is employed for the fuzzy
number
[Pi(̃
𝜍𝛼)]
and expressed as
where
PM
i
,
PU
i
,
PL
i
denotes the mean value, upper bound and
lower bound, respectively. Here
̃
𝜍𝛼
indicates the fuzziﬁed
variations corresponding to each αcut, where α is known
as the degree of fuzziness or membership grade ranging
from 0 to 1. As an example, the Gaussian distribution can
be approximated by using the triangle as shown in Fig.6a,
where the area under the Gaussian distribution is equal to
the area under the triangular function [66]. The triangular
fuzzy membership function is written as
where
𝜆=(2
𝜋𝜎
x)1∕2
, Xi and
𝜎x
represents the mean and
standard deviation (S.D.) of the Gaussian distribution. In
the present study, triangular membership function [µP(i)] is
employed as
By applying the αcut method, the fuzzy input number Pi
can be grouped into the set
̄
Pi
of (n + 1) intervals Pi
(j)
where n is the number of αcut levels. The interval of jth
level of ith fuzzy number can be expressed
where Pi
(j,U) and Pi
(j,L) represent the upper and lower bound
of the interval at the jth level, respectively. At j = n,
P(n,U)
i
=P
(n,L)
i
=P
(n,M)
i
. The superscript U represents the upper
bound, while L denotes lower bound. The fuzzy input num
bers are considered as the uncertain model parameters for
the uncertainty analysis and an interval analysis is carried
out at diﬀerent αlevels [67].
Even though in the present study we have considered
triangular membership functions for the input parameters,
(34)
Pi
(̃
𝜍
𝛼
)=[P
U
i
,P
M
i
,P
L
i]
(35)
𝜇
P(i)=max
⎡
⎢
⎢
⎣
0, 1 −
�
�
�
X(j)
i−Xi
�
�
�
𝜆
⎤
⎥
⎥
⎦
(36)
𝜇
P(i)=1−
(
P
M
i−Pi
)/(
P
M
i−P
L
i
)
,for P
L
i
≤
Pi
≤
P
M
i
𝜇
P(i)=1−(Pi−PM
i)/(PU
i−PM
i),for PM
i≤Pi≤P
U
i
𝜇P(i)
=0, otherwise
(37)
̄
Pi
(̃
𝜍
𝛼
)=[P
(0)
i
,P
(1)
i
,P
(2)
i
,P
(3)
i
,…,P
(j)
i
,…,P
(n)
i]
(38)
P
(j)
i=
[
P(j,L)
i,P(j,U)
i
]
the input membership functions can be augmented further
depending on the availability of limited number of input
dataset. In this work, we start by evaluating the deterministic
solution at
=1
level ﬁrst and continue towards the lower
−
cut levels using an interval analysis. As a special case,
if the input–output relation of the problem in hand is mono
tonic in nature, computing the bounds of the fuzzy outputs
becomes trivial. Unfortunately, for most reallife problems,
the input–output relation is not monotonic in nature. Under
such circumstances, a maximization and minimization algo
rithm involving multiple simulations is necessary. In this
work, we proceed by ﬁrst formulating the machine learn
ing models as a surrogate to the actual FE code. Then we
perform MCS on the trained machine learning models to
compute the maximum and minimum values of the response
quantities of interest for a particular αcut level. It is to be
noted that only a single machine learning model is required
in this case corresponding to
=0
as the same model can
be reused for other αcut levels. The number of actual FE
simulations required in this study is therefore equal to the
number of training samples needed to train the models of
machine learning. The procedure of the present fuzzy impact
approach is summarized in Figs.6b and 7.
5 Numerical Investigation andDiscussion
In this work a glass–epoxy laminated composite plate
having dimensions
L=1m, b=1m
and
t=0.002 m
is considered. Unless otherwise mentioned, the plate is
considered to be subjected to normal and oblique impact
loadings at the centre of the plate. The deterministic
material properties of glass–epoxy are
E1
=38.6 ×10
9
Pa,
E2=8.27 ×109
Pa,
G12 =G13 =4.144 ×109
Pa,
G23
=1.657 ×10
9
Pa,
𝜌=2600 kg/m3
,
𝜐=0.26
[68]. The
diameter of spherical steel ball (impactor) is considered
as
0.0127 m
. It is assumed that the ﬁbre orientation angle
may have a variation of 5% and the material properties
may have a variation of
10%
with respect to the determin
istic values. Such variations are considered as per stand
ard industrial practices; however, the current analysis can
be extended to other percentages of variation, if required.
Contact force (CF), impactor displacement (ID) and
plate displacement (PD) are considered to be the output
response variables. The inhouse deterministic ﬁnite ele
ment code for impact analysis is validated with results of
Sun and Chen [7] (refer to Fig.8), wherein it is observed
that the current results are extremely close to the results
of literature.
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REVISED PROOF
Journal : Large 11831 Article No : 9438 Pages : 32 MS Code : 9438 Dispatch : 2372020
T.Mukhopadhyay et al.
1 3
5.1 Deterministic Impact Analysis
Deterministic numerical results of the lowvelocity impact
are discussed in this subsection (Tables2, 3, 4, 5, 6, 7, 8)
to study the basic and fundamental inﬂuence of diﬀerent
system parameters such as ﬁbreorientation angle, oblique
impact angle, twist angle, initial velocity of impactor, mass
density of impactor, thickness of plate and location of
impact loading. Here we study four diﬀerent crucial stacking
sequences of the composite laminate: bending stiﬀ laminate
([0°/0°/30°/− 30°]s), cross ply laminate ([90°/0°/90°/0°]s),
torsion stiﬀ laminate). The eﬀects of stacking sequence on
Fig. 6 a Triangular membership function approximated from Gaussian distribution. b Fuzzy analysis for diﬀerent value of αcuts
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REVISED PROOF
Journal : Large 11831 Article No : 9438 Pages : 32 MS Code : 9438 Dispatch : 2372020
Stochastic Oblique Impact onComposite Laminates: AConcise Review andCharacterization of…
1 3
lowvelocity impact responses are furnished in Table2. It is
observed that the peak CF is highest for the torsion stiﬀ lam
inates. On the other hand, peak ID and peak PD are found
to be minimum for torsion stiﬀ laminates and maximum for
bending stiﬀ laminates. Table3 shows the variation of peak
impact responses with the change in twist angle. The peak
CF is found to increase with increase in twist angle. On the
contrary, peak ID and peak PD decrease with the increase
in twist angle. The inﬂuence of oblique impact angle on the
responses is shown in Table4. While peak CF and peak PD
Fig. 7 Flowchart for nonprobabilistic impact analysis based on fuzzy approach (Machine learning models are used instead of direct FE model,
as indicated using a blue colour box)
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decrease with increase in the impact angle, peak ID is found
to follow a reverse trend. All the peak responses are found
to increase with increase in the initial velocity, as shown
in Table5. Eﬀect of mass density on the impact response
is shown in Table6. In this case, increase in mass density
raises the peak responses. The eﬀect of plate thickness, as
presented in Table7, reveals that peak CF increases with the
increase in plate thickness, while peak PD and peak ID show
an opposite trend. The eﬀect of impact point on the critical
impact responses is shown in Table8, where it is found that
peak CF, PD and ID are maximum at point 2, point 3 and
point 3, respectively.
Fig. 8 Time histories of a contact force and b deﬂection of glass
epoxy composite plates considering a centrally impacted bending
stiﬀ laminated composite plate
(±0
◦
∕±30
◦
)
with dimension L = 1m,
b = 1 m, and t = 0.002 m, ѱ = 0°, β = 0°, initial velocity of impac
tor = 5m/s, diameter of spherical steel ball = 0.0127 m, mass density
of impactor
(𝜌)=0.0085
N−s
cm
4 [7]
Table 2 Eﬀect of stacking sequence (quasiisotropic stiﬀ, tor
sion stiﬀ, cross ply and bending stiﬀ laminates on lowvelocity
impact responses considering t = 0.002 m, ѱ = 0°, β = 0°, V = 5 m/s,
ρ = 0.0085
N−s
cm
4
Stacking sequence Impact responses (maximum value)
CF (N) ID (m) PD (m)
Bending stiﬀ 744.7855 0.000225 0.090134
Quasiisotropic stiﬀ 770.0546 0.000221 0.0854
Cross ply 770.45 0.000219 0.08794
Torsion stiﬀ 773.31 0.000219 0.08548
Table 3 Eﬀect of twist angle
(𝜓)
on lowvelocity impact responses
with considering t = 0.002m, β = 0°, V = 5 m/s, ρ = 0.0085
N−s
cm
4 , bend
ing stiﬀ laminate (02°/± 30°)s
Twist angle Impact responses (maximum value)
CF (N) ID (m) PD (m)
𝜓=0◦
744.7855 0.000227 0.090134
𝜓=15◦
776.3958 0.000221 0.0882
𝜓=30◦
874.1484 0.000206 0.0833
𝜓=45◦
1053.9 0.000188 0.07413
Table 4 Eﬀect of oblique impact angle
(𝛽)
on lowvelocity impact
responses with considering t = 0.002 m, ѱ = 0°, V = 5m/s, ρ = 0.0085
N−s
cm4
, bending stiﬀ laminate (02°/± 30°) s
Oblique impact
angle
Impact responses (maximum value)
CF (N) ID (m) PD (m)
𝛽=0◦
744.7855 0.000225 0.090134
𝛽=15◦
724.1631 0.000232 0.08965
𝛽=30◦
661.4398 0.000251 0.087791
𝛽=45◦
553.4121 0.00029 0.084967
Table 5 Eﬀect of initial velocity of impactor on lowvelocity impact
responses with considering t = 0.002m, ѱ = 0°, β = 0°, ρ = 0.0085
N−s
cm
4 ,
bending stiﬀ laminate (02°/± 30°)s
Initial velocity of
impactor (m/s)
Impact responses (maximum value)
CF (N) ID (m) PD (m)
V = 5 744.7855 0.000227 0.090134
V = 10 1549.402 0.00042 0.177863
V = 15 2365.073 0.000606 0.263738
V = 20 3193.182 0.000789 0.349809
Table 6 Eﬀect of mass density of impactor (ρ in
N−s
cm
4 ) on lowveloc
ity impact responses with considering t = 0.002 m, ѱ = 0°, β = 0°,
V = 5m/s, bending stiﬀ laminate (02°/± 30°)s
Mass density of
impactor
Impact responses (maximum value)
CF (N) ID (m) PD (m)
𝜌
= 75 × 10−4 719.9314 0.00021 0.08149
𝜌
= 80 × 10−4 733.6016 0.000219 0.085852
𝜌
= 85 × 10−4 744.7855 0.000227 0.090134
𝜌
= 90 × 10−4 755.3778 0.000235 0.094109
𝜌
= 95 × 10−4 766.9816 0.000242 0.098161
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5.2 Stochastic Impact Analysis
In this section, results corresponding to the probabilistic
and nonprobabilistic impact analysis are presented. The
formation of surrogate models based on PCEKriging is
discussed ﬁrst including comparative assessment of other
related surrogates. After validating the accuracy of the sur
rogate models, detailed stochastic analyses are carried out
in the subsequent subsections.
5.2.1 Surrogate Modelling andValidation
In this section, ﬁrst we discuss about training the machine
learning models. To be speciﬁc, convergence studies to
determine the optimal number of training samples are pre
sented. Second, we perform a comparative assessment of
PCE, Kriging and PCKriging.
5.2.1.1 Design ofExperiments One important task in sur
rogate modelling is to generate suitable training samples
for training the surrogate model. As already stated in the
preceding section, Sobol sequence is adopted in this study
to generate samples for training ML model. However, the
optimal number of training samples required still needs to
be determined. To that end, a study by varying the number
of training samples has been carried out. Figure9 shows the
PDF of responses (for direct MCS and PCEKriging based
MCS) with respect to training sample size of 32, 64, 128,
256, 512 and 1024. For all the three output responses, the
results obtained using 512 training samples are almost iden
tical to those obtained using 1024 samples. Based on this
observation, we conclude that 512 is the optimal number
of training samples. Note that all the subsequent results are
obtained by training the surrogate with 512 training sam
ples.
5.2.1.2 PCE Versus Kriging Versus PCKriging: A Compara
tive Study The surrogate PCKriging is developed by
combine PCE and Kriging. In this section, we examine the
performance of the three surrogate models (PCE, Kriging
and PCKriging) in the context of probabilistic lowvelocity
impact analysis. To that end, coeﬃcient of determination
(
R2
)
and root mean square error (RMSE) have been com
puted corresponding to training sample size of 32, 64, 128,
256, 512 and 1024. Figure10 shows the
R2
and RMSE cor
responding to the diﬀerent training sample size and the three
surrogate models.
It is observed that PCKriging consistently outperforms
PCE and Kriging; although the results obtained using PCE
are found to be extremely close to the PCKriging results.
Moreover, similar to the observations in previous section,
the results obtained corresponding to sample size of 512
and 1024 are almost identical (with
R2
close to 1), indi
cating that the surrogate models converge at 512 training
samples. Figure11 shows the probability density functions
obtained using PCE, Kriging and PCKriging, wherein the
results are compared with benchmark Monte Carlo simula
tion results. For all the three cases, PCKriging is found to
yield best results followed by PCE, establishing the superior
ity of PCKriging over PCE and Kriging. All the subsequent
results in this paper are obtained using PCKriging trained
with 512 training samples. It can be noted in this context
that stochastic analysis of composite structures leading to
the uncertainty quantiﬁcation of diﬀerent global responses
have recently received signiﬁcant attention from the scien
tiﬁc community [69–79]. However, most of these studies
consider a single machine learning algorithm to map the
stochastic input–output domain. The current investigation
is the ﬁrst attempt to investigate the performance of hybrid
machine learning algorithms for any structural response of
composite structures.
Table 7 Eﬀect of thickness of plate (t) on lowvelocity impact
responses with considering ѱ = 0°, β = 0°, V = 5 m/s, ρ = 0.0085
N−s
cm
4 ,
bending stiﬀ laminate (02°/± 30°) s
Thickness of plate
(m)
Impact responses (maximum value)
CF (N) ID (m) PD (m)
t = 0.002 322.9597 0.000548 0.266644
t = 0.004 744.7855 0.000225 0.090134
t = 0.006 1054.777 0.000188 0.050085
t = 0.008 1248.632 0.000176 0.033335
Table 8 Eﬀect of location of impactor contacting point on low
velocity impact responses with dimension t = 0.002m, ѱ = 0°, β = 0°,
V = 5 m/s, ρ = 0.0085
N−s
cm
4 , bending stiﬀ laminate (02°/± 30°)s (loca
tion of impact points on the laminated composite plate is indicated in
the inset of Fig.19a)
Location of impactor Impact responses (maximum value)
CF (N) ID (m) PD (m)
Location 1 731.8873 0.000228 0.075777
Location 2 744.7855 0.000225 0.090134
Location 3 735.177 0.000229 0.124411
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5.2.2 Probabilistic Impact Analysis
Having established the superiority of PCKriging over
PCE and Kriging, we present results for probabilistic
impact analysis in this subsection based on the PCKriging
assisted approach. The results presented here correspond to
the impact location at the centre of the plate, unless other
wise mentioned. Figures12 shows the variation of contact
force, displacements of impactor and plate, and velocity of
impactor with respect to time history for diﬀerent stacking
sequences. The ﬁgure also shows the corresponding stochas
tic response bounds arising due to the source uncertainties.
It is found that contact force initially increases at a signiﬁ
cant rate with time and then decreases up to zero gradually.
Impactor and plate displacements are noticed to gradually
increase to a peak value and then reduce with the elapse of
time. The velocity of the impactor reduces gradually over
time and becomes constant after a certain duration.
The inﬂuence of ﬁbre orientation angle in composite
laminates is shown in Fig.13. It is observed that the peak
CF occurs for the torsion stiﬀ laminates. The eﬀects of
twist angle on the critical impact responses are furnished in
Fig.14. In this case, the CF increases with the increase in
twist angle, while peak ID and peak PD have a reverse trend.
In case of impact loading, impact angle has a signiﬁcant
eﬀect on the critical impact responses as shown in Fig.15.
The peak CF and peak PD decreases with the increase in
impact angle from 0° to 45° while peak PD is found to have
a reverse trend. All the impact responses increase with the
increase in the initial velocity of the impactor as shown in
Fig. 9 Convergence study for PCKriging with respect to the number of training samples. For all the three responses, PCKriging converges at
512 training samples
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Fig.16 due to the increase in kinetic energy. The standard
deviation of the response parameters is also found to follow
a similar trend for initial velocity of impactor. The increase
in impactor mass density also leads to an increase of all
impact responses for the same reason as shown in Fig.17.
The eﬀect of plate thickness on the impact responses are
shown in Fig.18, wherein contact force is found to increase
with the increase in plate thickness. On the other hand, the
displacement of the impactor and plate displacement reduce
as the plate thickness increases. The standard deviation of
the response parameters is also found to follow a simi
lar trend for thickness. The eﬀect of location of impactor
Fig. 10 PCE vs Kriging versus
PCKriging (PCKriging is
found to yield the best results)
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contacting point on the impact responses is shown in Fig.19.
It is observed that contact force is maximum at the loca
tion 2 i.e. centre of the plate, while plate displacement and
impactor displacement are maximum at location 3. The rela
tive coeﬃcient of variation is shown for various inﬂuencing
system parameters in Fig.20 to understand about their rela
tive degree of inﬂuence on the impact response parameters.
The coeﬃcient of variation (COV) is obtained by taking the
ratio of standard deviation to mean of the responses. Here
the relative coeﬃcient of variation (RCOV) is computed by
Fig. 11 Comparison of PCE, Kriging and PCKriging results. All the three models are trained with 512 training samples
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Fig. 12 Stochastic variation of the time history of lowvelocity impact responses for diﬀerent stacking sequences of the composite plate a–d
for torsion stiﬀ laminate (45°, − 45°, 45°, − 45°)s, e–h for bending stiﬀ laminate (0°, 0°, 30°, − 30°)s considering t = 0.002m, ѱ = 0°, β = 0°,
V = 5m/s, ρ = 0.0085
N−s
cm
4 , and Δt = 1 microsecond. Stochastic variation of the time history of lowvelocity impact responses for diﬀerent stack
ing sequences a–d for cross ply laminate (90°, 0°, 90°, 0°)s, e–h for quasiIsotropic stiﬀ laminate (0°, 45°, − 45°, 90°)s considering t = 0.002m,
ѱ = 0°, β = 0°, V = 5m/s, ρ = 0.0085
N−s
cm4
, and Δt = 1 microsecond
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normalizing the COVs with respect to the sum of all COVs.
The relative sensitivity [67] of critical impact responses for
the six cases indicated in Sect.4.1 (considering impact at
the centre of the plate) can be clearly understood from this
analysis.
5.2.3 Fuzzy Based Nonprobabilistic Impact Analysis
In this subsection, we present numerical results correspond
ing to the nonprobabilistic assessment based on fuzzy
analysis, which is beneﬁcial if the complete description of
the probability distribution of the input variables is not avail
able. In this paper, the fuzzy approach is used to ﬁnd out the
Fig. 13 Eﬀect of variation of stacking sequence (quasiisotropic stiﬀ
laminate (0°, 45°, − 45°, 90°)s, torsion stiﬀ laminate (45°, − 45°, 45°,
− 45°)s, cross ply laminate (90°, 0°, 90°, 0°)s and bending stiﬀ lami
nate (0°, 0°, 30°, − 30°)s on lowvelocity impact responses consider
ing t = 0.002m, ѱ = 0°, β = 0°, V = 5m/s, ρ = 0.0085
N−s
cm
4
Fig. 14 Eﬀect of variation of twist angle (
𝜓
) on PDF plots of low
velocity impact responses considering t = 0.002m, β = 0°, V = 5 m/s,
ρ = 0.0085
N−s
cm
4 , bending stiﬀ laminate (02°/± 30°)s
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nonprobabilistic responses by means of a predeﬁned inter
val of input parameters. The membership grade is considered
0–1 at a level of 0.25. Similar to probabilistic analysis, PC
Kriging models trained with 512 training samples are used.
Fuzzy triangular membership function of the stochastic input
parameters is formed to address the variation of contact force,
plate displacement, and impactor displacement corresponding
to each level of α cut. It is found that the resulting output
membership functions show a deviation from the triangular
distribution of input membership functions.
Similar to the probabilistic analysis, Figs.21, 22, 23, 24,
25, 26 and 27 show the inﬂuence of diﬀerent input vari
ables on the lowvelocity impact responses following the
fuzzy based approach. In Fig.21, inﬂuence of plyangle on
low velocity impact responses are shown. For torsion stiﬀ
Fig. 15 Eﬀect of variation of impact angle (β) on PDF plots of low
velocity impact responses considering t = 0.002m,
𝜓=0◦
, V = 5m/s,
ρ = 0.0085
N−s
cm
4 , bending stiﬀ laminate (02°/± 30°)s
Fig. 16 Eﬀect of variation of initial velocity of impactor (V) on
PDF plots of lowvelocity impact responses considering t = 0.002m,
ѱ = 0°, β = 0°, ρ = 0.0085
N−s
cm4
, bending stiﬀ laminate (02°/± 30°)s
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Fig. 17 Eﬀect of variation of mass density of impactor (ρ in
N−s
cm
4 ) on
PDF plots of lowvelocity impact responses considering t = 0.002m,
ѱ = 0°, β = 0°, V = 5m/s, bending stiﬀ laminate (02°/± 30°)s
Fig. 18 Eﬀect of variation of thickness of plate (m) on PDF of low
velocity impact responses considering ѱ = 0°, β = 0°, V = 5 m/s,
ρ = 0.0085
N−s
cm
4 , bending stiﬀ laminate (02°/± 30°)s
and bending stiﬀ laminate conﬁgurations, the maximum and
minimum values of contact forces are identiﬁed respectively.
On the other hand, maximum plate displacement and impac
tor displacement are observed for bending stiﬀ laminate.
Figure22 shows the eﬀect of the twist angle on fuzzy low
velocity impact response behaviour of laminated composite
plates. The contact force peak value is noticed to increase
as the angle of twist increases. On the other hand, as the
angle of twist increases the plate displacement is found to
reduce. The inﬂuence of oblique impact angle is shown in
Fig.23. The contact force and plate displacement decrease
with the increase in the oblique impact angle; impact dis
placement is found to have a reverse trend. Figures24 and
25 show the eﬀect of the mass and initial velocity of the
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impactor on the transient impact responses, respectively. All
the critical responses increase with increase in the impac
tor mass and initial impactor velocity. The inﬂuence of the
thickness of plate is shown in Fig.26, where the contact
force increases with the increase in thickness of plate. The
plate displacement and impactor displacement are found
to decrease with the increase in the thickness of laminate.
Finally, Fig.27 shows the eﬀect of location of impactor on
the fuzzy responses considering three diﬀerent points on
the plate surface. The maximum value of contact force is
observed when the impact occurs at the centre of the plate.
On the other hand, both platedisplacement and impactor
displacement are observed to have maximum value when
the impact is on location 3.
6 Remarks andPerspective onHybrid
Machine Learning Models
In this paper, we reviewed the possibility of using a hybrid
machine learning technique (PCKriging) for stochastic
computational mechanics considering a critical impact
problem. Note that the concept of hybrid machine learn
ing approaches is not new; in fact, there exist a plethora of
hybrid machine learning approaches in the literature. The
primary idea of these methods is to combine more than one
machine learning models so as to exploit the advantages of
both (or, all of them). The ﬁrst use of hybrid machine learn
ing model is perhaps the ‘ensemble method’ proposed in
[80, 81]. The primary premise of this work was to represent
the response as a weighted combination of more than one
machine learning techniques. The ‘ensemble of surrogate’
method has gained signiﬁcant attention and its applica
tion can be found in diﬀerent domain [82–83]. Analysis of
variance (ANOVA) decomposition [84], also known as the
highdimensional model representation (HDMR) [85], is a
popular choice among researchers for hybridization. Over
the years, researchers have come up with diﬀerent variants
of HDMR/ANOVA by combining it with other machine
learning techniques. For example, Shan and Wang [86,
87] combined radial basis function (RBF) with cutHDMR
(aka anchored ANOVA) to formulate RBFHDMR. Within
this framework, the basis functions in cut HDMR are rep
resented by using RBF. In an independent study, Chowd
hury etal. [88–90] formulated moving least square based
cutHDMR (MLSHDMR) for solving structural reliability
analysis problems. The formulation for MLSHDMR and
RBFHDMR are similar; the only diﬀerence resides in the
Fig. 19 Eﬀect of variation of impactor contacting point on PDF plots
of lowvelocity impact responses considering t = 0.002 m, ѱ = 0°,
β = 0°, V = 5 m/s, ρ = 0.0085
N−s
cm
4 , bending stiﬀ laminate (02°/± 30°) s
(Location of impact points on the laminated composite plate is indi
cated in the inset of Fig.19a)
Fig. 20 Relative coeﬃcient of variation (RCOV) for peak con
tact force, plate displacement, and impactor displacement of cen
trally impacted glass–epoxy laminated composite plates consider
ing bending stiﬀ laminate (02°/± 30°)s, ѱ = 45°, β = 30°, V = 10 m/s,
ρ = 0.0090
N−s
cm
4 , t = 0.004m
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Fig. 21 Eﬀect of variation of stacking sequence (quasiisotropic stiﬀ
laminate (0°, 45°, − 45°, 90°)s, torsion stiﬀ laminate (45°, − 45°, 45°,
− 45°)s, cross ply laminate (90°, 0°, 90°, 0°)s and bending stiﬀ lami
nate (0°, 0°, 30°, − 30°)s on lowvelocity impact responses consider
ing t = 0.002m, ѱ = 0°, β = 0°, V = 5m/s, ρ = 0.0085
N−s
cm
4
Fig. 22 Eﬀect of variation of twist angle (ѱ) on lowvelocity impact
responses considering t = 0.002 m, β = 0°, V = 5 m/s, ρ = 0.0085
N−s
cm
4 ,
bending stiﬀ laminate (02°/± 30°)s
fact that the basis functions for MLSHDMR are represented
by using MLS based regression. As an improvement over
MLSHDMR and RBFHDMR, Huang etal. [91] proposed
support vector regression HDMR (SVRHDMR) in 2015. It
was illustrated that the performance of SVRHDMR outper
forms RBFHDMR. Note that all the HDMR based hybrid
machine learning algorithms discussed above are based on
cutHDMR.
Hybrid machine learning approaches based on random
sampling HDMR, also known as the ANOVA decomposi
tion, can also be found in the literature. Chakraborty and
Chowdhury [92] developed a sequential experimental design
based generalized ANOVA by coupling polynomial chaos
expansion [23, 25, 43] with RSHDMR [93, 94]. An adap
tive version of this algorithm was also proposed [95]. Later
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generalized ANOVA was further hybridized by coupling
Gaussian process [96–98] with it. This was referred to as the
hybrid polynomial correlated function expansion (HPCFE).
In essence, HPCFE is a fusion of three machine learning
algorithms, namely PCE, RSHDMR and Gaussian process
[22, 46, 47, 99]. The primary idea of HPCFE is to represent
the mean function of Gaussian process by using general
ized ANOVA. Adaptive variants of HPCFE was proposed
in [100, 101]. It was illustrated that with hybridization (or
fusion), the accuracy of the machine learning algorithm
improves.
Apart from HDMR, hybrid machine learning algorithms
based on Gaussian process, also known as the Kriging [20,
21, 62] is also popular in the literature. In [102], a new
hybrid machine learning algorithm was developed by com
bining fuzzy logic, artiﬁcial neural network and Kriging.
In another work, Pang etal. [103] combined Gaussian pro
cess with neural network. The two methods diﬀer in how
Fig. 23 Eﬀect of variation of impact angle (β) on lowvelocity impact
responses considering t = 0.002 m, ѱ = 0°, V = 5 m/s, ρ = 0.0085
N−s
cm
4 ,
bending stiﬀ laminate (02°/± 30°)s
Fig. 24 Eﬀect of variation of mass density of impactor (ρ in
N−s
cm
4 ) on
lowvelocity impact responses considering t = 0.002m, ѱ = 0°, β = 0°,
V = 5m/s, bending stiﬀ laminate (02°/± 30°)s
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Journal : Large 11831 Article No : 9438 Pages : 32 MS Code : 9438 Dispatch : 2372020
Stochastic Oblique Impact onComposite Laminates: AConcise Review andCharacterization of…
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the neural network and Gaussian processes are combined.
While the former uses neural network and Gaussian process
sequentially, separated by fuzzy logic, the latter uses neural
network to represent the covariance function of the Gauss
ian process. The method used in the current paper is also a
hybrid machine learning technique, referred to as the poly
nomial chaos based Kriging (PCKriging). This method was
Fig. 25 Eﬀect of variation of initial velocity of impactor (V in m/s) on
lowvelocity impact responses considering t = 0.002m, ѱ = 0°, β = 0°,
ρ = 0.0085
N−s
cm
4 , bending stiﬀ laminate (02°/± 30°)s
Fig. 26 Eﬀect of variation of plate thickness (t) on lowvelocity
impact responses considering ѱ = 0°, β = 0°, V = 5 m/s, ρ = 0.0085
N−s
cm4
, bending stiﬀ laminate (02°/± 30°)s
ﬁrst proposed in [30] and was then further improved in [31].
In this method, the mean function of Kriging is represented
by using polynomial chaos expansion. It is argued that poly
nomial chaos expansion performs a global approximation by
using basis function and Kriging performs local approxima
tion by using the covariance kernel.
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T.Mukhopadhyay et al.
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Based on the literature and the results presented in this
paper, it is safe to conclude that hybrid machine learning
approaches are generally more accurate as compared to a
single machine learning approach (note that such single
machine learning approaches have been shown to predict
accurately in various engineering problems [104–133]).
However, there is no free lunch and this enhancement in
the accuracy normally comes at a cost of the eﬃciency. For
instance, HPCFE discussed above is more accurate but less
eﬃcient as compared to the generalized ANOVA. Similarly,
polynomial chaos based Kriging used in this paper is more
accurate than polynomial chaos and Kriging; however, the
computational time necessary for training a polynomial
chaos based Kriging is more. To address this issue, research
ers over the last few years have developed diﬀerent adaptive
algorithms. Having said that, there is still a signiﬁcant scope
for further developments when it comes to hybrid machine
learning algorithms.
7 Conclusions
This paper deals with the eﬀects of inputuncertainty on
lowvelocity impact responses of composite laminates,
which is investigated based on an eﬃcient machine learning
algorithm. The Newmark’s time integration scheme is imple
mented to solve time–histories of transient responses, while
the modiﬁed Hertzian contact law is employed to obtain
the contact force and other parameters. First, a determinis
tic analysis is carried to investigate the eﬀects of diﬀerent
system parameters (such as stacking sequence, twist angle,
impact angle, initial velocity of impactor, mass density of
impactor and thickness of plate). Subsequently a proba
bilistic analysis is presented to characterize the complete
probabilistic descriptions of lowvelocity impact responses.
Finally, to address the scenario where complete statistical
descriptions of the input data are not available (sparse input
data), a fuzzy based nonprobabilistic approach is presented
for lowvelocity impact analysis of composites. Since con
ventional methods for probabilistic and nonprobabilistic
analyses are exorbitantly computationally expensive, we
integrated a hybrid polynomial chaos based kriging (PC
Kriging) approach with the conventional framework to
obtain a high level of computational eﬃciency. By hybridiz
ing the two powerful metamodelling techniques, polynomial
chaos and kriging (to capture the global and local behaviour
of a system, respectively), it is possible to exploit the com
plementary advantages of these two models in a single com
putational framework. In essence, here we have presented
a numerical demonstration of the superiority of hybrid
machine learning algorithms over individual models in a
systematic way including a critical review of the algorithms.
The novelty of this paper lies in characterizing the eﬀect
of sourceuncertainty on low velocity impact of compos
ite plates as well as development of the hybrid simulation
approach based on PCKriging coupled with the finite
element model of composite laminates to achieve compu
tational eﬃciency. We have presented a comprehensive
study following both the probabilistic and nonprobabilistic
approaches that covers every possible scenario of the avail
ability or unavailability of the statistical distributions of the
input parameters. The stochastic results (both probabilistic
and nonprobabilistic) in this paper show that the inevita
ble eﬀect of uncertainty has signiﬁcant eﬀect on the critical
impact responses of composite laminates. Thus it is impor
tant to adopt an inclusive design approach by quantifying
the stochastic variation of the global responses to ensure
adequate safety and serviceability of the structure under low
velocity impact. Besides that, the hybrid PCKriging based
approach adopted in this study to achieve computational eﬃ
ciency in the expensive and timeconsuming process of mod
elling impact in complex structural forms like composite
Fig. 27 Eﬀect of variation of impactor contacting point on low
velocity impact responses considering t = 0.002 m, ѱ = 0°, β = 0°,
V = 5m/s, ρ = 0.0085
N−s
cm
4 , bending stiﬀ laminate (02°/± 30°) s (Loca
tion of impact points on the laminated composite plate is indicated in
the inset of Fig.19a)
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Stochastic Oblique Impact onComposite Laminates: AConcise Review andCharacterization of…
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structures can be useful for other computationally intensive
problems of structural analyses and mechanics.
Acknowledgements TM and SN acknowledge the initiation grants
received from IIT Kanpur and IIT Bombay, respectively. PKK and RC
are grateful for the ﬁnancial support from MHRD, India during the
research work. SC acknowledges the support of XSEDE and Center for
Research Computing, University of Notre Dame for providing compu
tational resources required for carrying out this work.
Data availability All data used to generate these results is available in
the main paper. Further details could be obtained from the correspond
ing author(s) upon request.
Compliance with Ethical Standards
Conflict of interest The authors declare that they have no conﬂict of
interest
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