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Due to the absence of adequate control at different stages of complex manufacturing process, material and geometric properties 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 low-velocity impact caused by various external agents. Here, along with a critical review, we present machine learning based probabilistic and non-probabilistic (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 description about the input variables are available, the non-probabilistic analysis can be executed even in the presence of incomplete 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 (PC-Kriging) approach is coupled with in-house 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 non-probabilistic regimes to access the performance of such hybrid algorithms in comparison to individual algorithms from the viewpoint of accuracy and computational efficiency.
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Journal : Large 11831 Article No : 9438 Pages : 32 MS Code : 9438 Dispatch : 23-7-2020
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Archives of Computational Methods in Engineering
https://doi.org/10.1007/s11831-020-09438-w
ORIGINAL PAPER
Stochastic Oblique Impact onComposite Laminates: AConcise Review
andCharacterization oftheEssence ofHybrid 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 low-velocity impact caused by various external agents. Here,
along with a critical review, we present machine learning based probabilistic and non-probabilistic (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 non-probabilistic 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 (PC-Kriging) approach is coupled with in-house 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 non-probabilistic
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 low-velocity 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
low-velocity 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 ofAerospace Engineering, Indian Institute
ofTechnology Kanpur, Kanpur, India
2 Department ofAerospace Engineering, Indian Institute
ofTechnology Bombay, Mumbai, India
3 Department ofApplied Mechanics, Indian Institute
ofTechnology Delhi, NewDelhi, India
4 Department ofMechanical Engineering, Parul Institute
ofEngineering andTechnology, Parul University, Vadodara,
India
5 Mechanical Engineering Department, National Institute
ofTechnology Silchar, Silchar, India
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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 ad-hoc 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 non-probabilistic low-velocity 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 low-velocity impact analysis
of carbon- epoxy laminates for damage detection, Liu etal.
[3] studied the influence of shape of impactor (such as
conical, hemispherical and flat) on the low-velocity impact
responses of sandwich plate. In both cases, experimental
as well as numerical analyses were performed. Jagtap etal.
[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 etal. [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 low-velocity impact loading.
Tan and Sun [6] and Sun and Chen [7] also used the finite
element method with Newmark time integration scheme to
investigate low-velocity impact on composite structures. A
comprehensive review on low-velocity 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 low-velocity impact and quasi-static loads.
Several works dealing with failure mechanism of compos-
ite plates subjected to low-velocity impact load can be found
in the literature. While Yuan etal. [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 low-velocity impact, Zhang and Zhang [11]
applied FE model for damage detection in composite struc-
tures due to low-velocity impact. Feng and Aymerich [12],
Maio etal. [13] and Kim etal. [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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Stochastic Oblique Impact onComposite Laminates: AConcise Review andCharacterization of…
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progressive damage models to investigate the failure mecha-
nism of laminated composite due to the low-velocity impact.
Lipeng etal. [15] investigated delamination failure due to
impact load by using a self-adapting delamination element
method. Johnson etal. [16] presented different models for
failure analysis of composite plates by considering internal
damage and delamination due to impact loading. Coutellier
etal. [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 low-velocity impact.
Despite the vast literature on low-velocity 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 source-uncertainties 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,
non-probabilistic fuzzy analysis can be employed to portray
the effects of uncertainty. It is to be noted that both conven-
tional probabilistic and non-probabilistic 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 [1922], Polynomial chaos expansion (PCE)
[2325], analysis-of-variance decomposition [2629], Polyno-
mial chaos based Kriging (PC- Kriging) [3033] etc. In this
work, we review three machine learning techniques in the con-
text of stochastic low-velocity 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 inter-dependence including the current intro-
duction section. Section2 describes governing equations for
the analysis of the transient low-velocity 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
PC-Kriging is provided. Section4 provides both probabilis-
tic and non-probabilistic stochastic approaches for the impact
analysis of low-velocity. 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 oftheGoverning Equations
forLow-Velocity Impact onLaminated
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)
(̃
𝜍)=
000F
(̃
𝜍)000
(3)
mimp
(̃
𝜍)
̈
𝛿
imp
+F
c
(̃
𝜍)=
0
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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
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Stochastic Oblique Impact onComposite Laminates: AConcise Review andCharacterization of…
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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
[1𝜐2
i(̃
𝜍)]
E
i
(̃
𝜍)+1
E
yy
(̃
𝜍)
(6)
F
c(̃
𝜍)=Fm
[
𝛾(̃
𝜍)−𝛾0
𝛾
m
𝛾
0]52
and Fc(̃
𝜍)=Fm
[
𝛾(̃
𝜍)−𝛾0
𝛾
m
𝛾
0]32
(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 z-direction, 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
tt
where
[̄
K]
and
[̄
k]
are the effective stiffness matrix of the
plate and impactor, respectively, and given by
Effective contact forces at time
tt
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
(tt)
imp
={FC}(tt
)
(12)
[̄
K]=K+a0M
(13)
[̄
k]=a
0
m
imp
(14)
{
F}
(tt)
={F}
(t+𝛿t)
+[M]
(
a
0
𝛿
(t)
+a
1̇
𝛿
(t)
+a
2
𝛿
(t))
(15)
̄
F
(tt)
C
e
=F(tt)
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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The acceleration and velocity can be derived from displace-
ment at time
tt
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
onKriging andPCE
Let, x=
{
x
1
,,x
N}
N to be the input variables and
y
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}
(tt)
=a0({𝛿}
(tt)
−{𝛿}
(t)
)−a1{
̇
𝛿}
(t)
a2{
̈
𝛿}
(t)
̈
𝛿
(tt)
imp =a0(𝛿(tt)
imp 𝛿(t)
imp)a1̇
𝛿(t)
imp a2̈
𝛿(t)
imp
{
̇
𝛿}(tt)={̇
𝛿}(t)+a3{̈
𝛿}(t)+a4{̈
𝛿}(tt)
̇
𝛿
(tt)
imp
=̇
𝛿(t)
imp
+a3̈
𝛿(t)
imp
+a4̈
𝛿(tt)
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)
Askey-scheme,
L2
convergence in the corresponding Hilbert
space.
Assuming
𝐢
=
(
i
1
,i
2
,,i
N)
N
0
to be a multi-index
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 N-dimensional 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. Table1 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 above-mentioned problem, researchers
focused on developing nonintrusive (data-driven) PCE.
The easiest and most popular way to train a data-driven
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 onComposite Laminates: AConcise Review andCharacterization of…
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[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 small-scale problems with
limited number of input variables.
To address the curse of dimensionality associated with
least-square 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 least-angle 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 etal. [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 perturbation-PCE 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 length-scale parameter associated with the correla-
tion kernel. In order to use Gaussian process as a machine
(22)
y|
x;𝐁,𝜎,𝜽GP
(
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 [4649].
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
[5154]. 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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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
Co-Kriging [55] and stochastic Kriging [5658]. 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 [5961]. 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 one-dimensional 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,xix
i
)
(27)
𝜓j(𝜃;dj)=exp(−𝜃j|dj|)
(28)
𝜓j
(𝜃;d
j
)=exp(−𝜃
j|
d
j|𝜃
n+1),0<𝜃
n+12
(29)
𝜓
j(𝜃;dj)=exp(−𝜃jd
2
j)
(30)
𝜓j(𝜃;dj)=max{0, 1 𝜃j|dj|}
(31)
𝜓
j(𝜃;dj)=11.5𝜉j+0.5𝜉
2
j
,𝜉j=min{0, 𝜃j
|
dj
|}
(32)
𝜓
j(𝜃;dj)=13𝜉
2
j
+2𝜉
3
j
,𝜉j=min{1, 𝜃j
|
dj
|}
where
𝜉j=𝜃j|dj|
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 (PC-Kriging)
Finally, we discuss about a hybrid machine learning tech-
nique, referred to as the polynomial chaos based Kriging
(PC-Kriging) [3032]. PC-Kriging is a novel surrogate
model that combine two well-known surrogates, namely,
polynomial chaos expansion (PCE) [23, 25] and Kriging [19,
20]. PC-Kriging 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.
PC-Kriging 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,
PC-Kriging converges either to PCE or to Kriging. Similar
to Kriging, the hyperparameters in PC-Kriging are learned
by maximizing the likelihood. For further details, interested
readers may refer [19, 62].
Despite PC-Kriging’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 PC-Kriging. To address this problem, a variant of PC-
Kriging, referred to as Optimal PC- Kriging (OPC-Kriging)
[31] was proposed. In OPC-Kriging, least angle regression
(LAR) is used to only retain the important components of
PCE. The OPC-Kriging follows an iterative algorithm where
each polynomial can be added to the trend part one-by-one.
Figure4 presents a flowchart depicting the algorithm of
OPC-Kriging.
(33)
𝜓
j(𝜃;dj)=
15𝜉2
j+30𝜉3
j,0
𝜉j
0.2
1.251𝜉3
j, 0.2 𝜉j
1
0, 𝜉j>1
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Stochastic Oblique Impact onComposite Laminates: AConcise Review andCharacterization of…
1 3
4 Machine Learning Based Stochastic
Impact Analysis
A major objective of the present study is to determine sto-
chastic response of low-velocity impact loading on com-
posite plates following probabilistic and non-probabilistic
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 non-probabilistic
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 in-house FE code for low-velocity 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 (layer-wise uncertainty modelling)
1. Variation of fibre-orientation 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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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 real-life 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 non-probabilistic analysis is employed
Fig. 5 Flowchart for probabilistic impact analysis based on hybrid machine learning models coupled with FE simulations
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Stochastic Oblique Impact onComposite Laminates: AConcise Review andCharacterization 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 non-members 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 α-cut, 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)12
, 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 α-cut 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 α-cut levels. The interval of j-th
level of i-th fuzzy number can be expressed
where Pi
(j,U) and Pi
(j,L) represent the upper and lower bound
of the interval at the j-th 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 α-levels [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)
iXi
𝜆
(36)
𝜇
P(i)=1
(
P
M
iPi
)/(
P
M
iP
L
i
)
,for P
L
i
Pi
P
M
i
𝜇
P(i)=1(PiPM
i)/(PU
iPM
i),for PM
iPiP
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 real-life 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 α-cut 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 α-cut 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 andDiscussion
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 in-house 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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1 3
5.1 Deterministic Impact Analysis
Deterministic numerical results of the low-velocity impact
are discussed in this subsection (Tables2, 3, 4, 5, 6, 7, 8)
to study the basic and fundamental influence of different
system parameters such as fibre-orientation 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 α-cuts
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Journal : Large 11831 Article No : 9438 Pages : 32 MS Code : 9438 Dispatch : 23-7-2020
Stochastic Oblique Impact onComposite Laminates: AConcise Review andCharacterization of…
1 3
low-velocity impact responses are furnished in Table2. 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. Table3 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 Table4. While peak CF and peak PD
Fig. 7 Flowchart for non-probabilistic 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 Table5. Effect of mass density on the impact response
is shown in Table6. In this case, increase in mass density
raises the peak responses. The effect of plate thickness, as
presented in Table7, 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 Table8, 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 = 1m,
b = 1 m, and t = 0.002 m, ѱ = 0°, β = 0°, initial velocity of impac-
tor = 5m/s, diameter of spherical steel ball = 0.0127 m, mass density
of impactor
(𝜌)=0.0085
Ns
cm
4 [7]
Table 2 Effect of stacking sequence (quasi-isotropic stiff, tor-
sion stiff, cross ply and bending stiff laminates on low-velocity
impact responses considering t = 0.002 m, ѱ = 0°, β = 0°, V = 5 m/s,
ρ = 0.0085
Ns
cm
4
Stacking sequence Impact responses (maximum value)
CF (N) ID (m) PD (m)
Bending stiff 744.7855 0.000225 0.090134
Quasi-isotropic 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 low-velocity impact responses
with considering t = 0.002m, β = 0°, V = 5 m/s, ρ = 0.0085
Ns
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 low-velocity impact
responses with considering t = 0.002 m, ѱ = 0°, V = 5m/s, ρ = 0.0085
Ns
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 low-velocity impact
responses with considering t = 0.002m, ѱ = 0°, β = 0°, ρ = 0.0085
Ns
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
Ns
cm
4 ) on low-veloc-
ity impact responses with considering t = 0.002 m, ѱ = 0°, β = 0°,
V = 5m/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 non-probabilistic impact analysis are presented. The
formation of surrogate models based on PCE-Kriging 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 andValidation
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 PC-Kriging.
5.2.1.1 Design ofExperiments 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. Figure9 shows the
PDF of responses (for direct MCS and PCE-Kriging 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 PC-Kriging: A Compara-
tive Study The surrogate PC-Kriging is developed by
combine PCE and Kriging. In this section, we examine the
performance of the three surrogate models (PCE, Kriging
and PC-Kriging) in the context of probabilistic low-velocity
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. Figure10 shows the
R2
and RMSE cor-
responding to the different training sample size and the three
surrogate models.
It is observed that PC-Kriging consistently outperforms
PCE and Kriging; although the results obtained using PCE
are found to be extremely close to the PC-Kriging 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. Figure11 shows the probability density functions
obtained using PCE, Kriging and PC-Kriging, wherein the
results are compared with benchmark Monte Carlo simula-
tion results. For all the three cases, PC-Kriging is found to
yield best results followed by PCE, establishing the superior-
ity of PC-Kriging over PCE and Kriging. All the subsequent
results in this paper are obtained using PC-Kriging 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 [6979]. 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 low-velocity impact
responses with considering ѱ = 0°, β = 0°, V = 5 m/s, ρ = 0.0085
Ns
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 = 0.002m, ѱ = 0°, β = 0°,
V = 5 m/s, ρ = 0.0085
Ns
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 PC-Kriging over
PCE and Kriging, we present results for probabilistic
impact analysis in this subsection based on the PC-Kriging
assisted approach. The results presented here correspond to
the impact location at the centre of the plate, unless other-
wise mentioned. Figures12 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 PC-Kriging with respect to the number of training samples. For all the three responses, PC-Kriging 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
PC-Kriging (PC-Kriging 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 PC-Kriging results. All the three models are trained with 512 training samples
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Fig. 12 Stochastic variation of the time history of low-velocity impact responses for different stacking sequences of the composite plate ad
for torsion stiff laminate (45°, − 45°, 45°, − 45°)s, eh for bending stiff laminate (0°, 0°, 30°, − 30°)s considering t = 0.002m, ѱ = 0°, β = 0°,
V = 5m/s, ρ = 0.0085
Ns
cm
4 , and Δt = 1 micro-second. Stochastic variation of the time history of low-velocity impact responses for different stack-
ing sequences ad for cross ply laminate (90°, 0°, 90°, 0°)s, eh for quasi-Isotropic stiff laminate (0°, 45°, − 45°, 90°)s considering t = 0.002m,
ѱ = 0°, β = 0°, V = 5m/s, ρ = 0.0085
Ns
cm4
, and Δt = 1 micro-second
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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 Non-probabilistic Impact Analysis
In this sub-section, we present numerical results correspond-
ing to the non-probabilistic 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 (quasi-isotropic 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 low-velocity impact responses consider-
ing t = 0.002m, ѱ = 0°, β = 0°, V = 5m/s, ρ = 0.0085
Ns
cm
4
Fig. 14 Effect of variation of twist angle (
𝜓
) on PDF plots of low-
velocity impact responses considering t = 0.002m, β = 0°, V = 5 m/s,
ρ = 0.0085
Ns
cm
4 , bending stiff laminate (02°/± 30°)s
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non-probabilistic 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 low-velocity impact responses following the
fuzzy based approach. In Fig.21, influence of ply-angle 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 = 0.002m,
𝜓=0
, V = 5m/s,
ρ = 0.0085
Ns
cm
4 , bending stiff laminate (02°/± 30°)s
Fig. 16 Effect of variation of initial velocity of impactor (V) on
PDF plots of low-velocity impact responses considering t = 0.002m,
ѱ = 0°, β = 0°, ρ = 0.0085
Ns
cm4
, bending stiff laminate (02°/± 30°)s
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Fig. 17 Effect of variation of mass density of impactor (ρ in
Ns
cm
4 ) on
PDF plots of low-velocity impact responses considering t = 0.002m,
ѱ = 0°, β = 0°, V = 5m/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
Ns
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.
Figure22 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. Figures24 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 plate-displacement and impactor
displacement are observed to have maximum value when
the impact is on location 3.
6 Remarks andPerspective onHybrid
Machine Learning Models
In this paper, we reviewed the possibility of using a hybrid
machine learning technique (PC-Kriging) 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 [8283]. Analysis of
variance (ANOVA) decomposition [84], also known as the
high-dimensional 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 cut-HDMR
(aka anchored ANOVA) to formulate RBF-HDMR. Within
this framework, the basis functions in cut HDMR are rep-
resented by using RBF. In an independent study, Chowd-
hury etal. [8890] formulated moving least square based
cut-HDMR (MLS-HDMR) for solving structural reliability
analysis problems. The formulation for MLS-HDMR and
RBF-HDMR are similar; the only difference resides in the
Fig. 19 Effect of variation of impactor contacting point on PDF plots
of low-velocity impact responses considering t = 0.002 m, ѱ = 0°,
β = 0°, V = 5 m/s, ρ = 0.0085
Ns
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
Ns
cm
4 , t = 0.004m
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Fig. 21 Effect of variation of stacking sequence (quasi-isotropic 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 low-velocity impact responses consider-
ing t = 0.002m, ѱ = 0°, β = 0°, V = 5m/s, ρ = 0.0085
Ns
cm
4
Fig. 22 Effect of variation of twist angle (ѱ) on low-velocity impact
responses considering t = 0.002 m, β = 0°, V = 5 m/s, ρ = 0.0085
Ns
cm
4 ,
bending stiff laminate (02°/± 30°)s
fact that the basis functions for MLS-HDMR are represented
by using MLS based regression. As an improvement over
MLS-HDMR and RBF-HDMR, Huang etal. [91] proposed
support vector regression HDMR (SVR-HDMR) in 2015. It
was illustrated that the performance of SVR-HDMR outper-
forms RBF-HDMR. Note that all the HDMR based hybrid
machine learning algorithms discussed above are based on
cut-HDMR.
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 RS-HDMR [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 [9698] with it. This was referred to as the
hybrid polynomial correlated function expansion (H-PCFE).
In essence, H-PCFE is a fusion of three machine learning
algorithms, namely PCE, RS-HDMR and Gaussian process
[22, 46, 47, 99]. The primary idea of H-PCFE is to represent
the mean function of Gaussian process by using general-
ized ANOVA. Adaptive variants of H-PCFE 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 etal. [103] combined Gaussian pro-
cess with neural network. The two methods differ in how
Fig. 23 Effect of variation of impact angle (β) on low-velocity impact
responses considering t = 0.002 m, ѱ = 0°, V = 5 m/s, ρ = 0.0085
Ns
cm
4 ,
bending stiff laminate (02°/± 30°)s
Fig. 24 Effect of variation of mass density of impactor (ρ in
Ns
cm
4 ) on
low-velocity impact responses considering t = 0.002m, ѱ = 0°, β = 0°,
V = 5m/s, bending stiff laminate (02°/± 30°)s
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Stochastic Oblique Impact onComposite Laminates: AConcise Review andCharacterization 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 (PC-Kriging). This method was
Fig. 25 Effect of variation of initial velocity of impactor (V in m/s) on
low-velocity impact responses considering t = 0.002m, ѱ = 0°, β = 0°,
ρ = 0.0085
Ns
cm
4 , bending stiff laminate (02°/± 30°)s
Fig. 26 Effect of variation of plate thickness (t) on low-velocity
impact responses considering ѱ = 0°, β = 0°, V = 5 m/s, ρ = 0.0085
Ns
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 [104133]).
However, there is no free lunch and this enhancement in
the accuracy normally comes at a cost of the efficiency. For
instance, H-PCFE 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 input-uncertainty on
low-velocity 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 low-velocity impact responses.
Finally, to address the scenario where complete statistical
descriptions of the input data are not available (sparse input
data), a fuzzy based non-probabilistic approach is presented
for low-velocity impact analysis of composites. Since con-
ventional methods for probabilistic and non-probabilistic
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 source-uncertainty on low- velocity impact of compos-
ite plates as well as development of the hybrid simulation
approach based on PC-Kriging 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 non-probabilistic
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 non-probabilistic) 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 PC-Kriging based
approach adopted in this study to achieve computational effi-
ciency in the expensive and time-consuming 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 = 5m/s, ρ = 0.0085
Ns
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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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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