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Abstract

This paper deals with the stochastic sensitivity analysis of functionally graded material (FGM) plates subjected to free vibration and low-velocity impact to identify the most influential parameters in the respective analyses. A hybrid moment-independent sensitivity analysis is proposed coupled with the least angle regression based adaptive sparse polynomial chaos expansion. Here the surrogate model is integrated in the sensitivity analysis framework to achieve computational efficiency. The current paper is concentrated on the relative sensitivity of material properties in the free vibration (first three natural frequencies) and low-velocity impact responses of FGM plates. Typical functionally graded materials are made of two different components, where a continuous and inhomogeneous mixture of these materials is distributed across the thickness of the plate based on certain distribution laws. Thus, besides the overall sensitivity analysis of the material properties, a unique spatial sensitivity analysis is also presented here along the thickness of the plate to provide a comprehensive view. The results presented in this paper would help to identify the most important material properties along with their depth-wise spatial sensitivity for low-frequency vibration and low-velocity impact analysis of FGM plates. This is the first attempt to carry out an efficient adaptive sparse PCE based moment-independent sensitivity analysis (depth-wise and collectively) of FGM plates under the simultaneous susceptibility of vibration and impact. Such simultaneous multi-objective sensitivity analysis can identify the important system parameters and their relative degree of importance in advanced multi-functional structural systems.
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A hybrid stochastic sensitivity analysis for low-frequency vibration and low-
velocity impact of functionally graded plates
P.K. Karsha,b,1, T. Mukhopadhyayc, 1, *, S. Chakrabortyd,e,1, S. Naskarf,1, S. Deya,1
aMechanical Engineering Department, National Institute of Technology Silchar, India
bDepartment of Mechanical Engineering, Parul Institute of Engineering & Technology, Vadodara, India
cDepartment of Aerospace Engineering, Indian Institute of Technology Kanpur, Kanpur, India
dCenter for Informatics and Computational Science, University of Notre Dame, USA
eDepartment of Aerospace and Mechanical Engineering, University of Notre Dame, USA
fWhiting School of Engineering, Johns Hopkins University, Baltimore, USA
*Corresponding author’s email: tanmoy@iitk.ac.in; tanmoy.mukhopadhyay@eng.ox.ac.uk
1All authors have contributed equally
Abstract
This paper deals with the stochastic sensitivity analysis of functionally graded material (FGM)
plates subjected to free vibration and low-velocity impact to identify the most influential
parameters in the respective analyses. A hybrid moment-independent sensitivity analysis is
proposed coupled with the least angle regression based adaptive sparse polynomial chaos
expansion. Here the surrogate model is integrated in the sensitivity analysis framework to
achieve computational efficiency. The current paper is concentrated on the relative sensitivity
of material properties in the free vibration (first three natural frequencies) and low-velocity
impact responses of FGM plates. Typical functionally graded materials are made of two
different components, where a continuous and inhomogeneous mixture of these materials is
distributed across the thickness of the plate based on certain distribution laws. Thus, besides
the overall sensitivity analysis of the material properties, a unique spatial sensitivity analysis is
also presented here along the thickness of the plate to provide a comprehensive view. The
results presented in this paper would help to identify the most important material properties
along with their depth-wise spatial sensitivity for low-frequency vibration and low-velocity
impact analysis of FGM plates. This is the first attempt to carry out an efficient adaptive sparse
PCE based moment-independent sensitivity analysis (depth-wise and collectively) of FGM
plates under the simultaneous susceptibility of vibration and impact. Such simultaneous multi-
objective sensitivity analysis can identify the important system parameters and their relative
degree of importance in advanced multi-functional structural systems.
Keywords: Moment-independent sensitivity analysis; Functionally graded plate; Adaptive
Sparse polynomial chaos expansion; Free vibration; Low-velocity impact
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1. Introduction
Typical functionally graded plates are composed of two materials such as metal and
ceramic wherein continuous and inhomogeneous mixture of these component materials are
distributed across the thickness of plate by using certain laws such as, power law, sigmoid law,
exponential law. The function of ceramic is to provide the thermal resistance while the metal
provides the mechanical load carrying capacity. Unlike conventional materials, functionally
graded plates can withstand higher thermal and mechanical load. Hence, a wide range of
applications of functionally graded materials (FGMs) can be found in aircraft, medical
equipments and implants, sports accessories, automobile sectors, etc. Additionally, unlike
composite materials, the fear of failure due to delamination is not present in the FGMs, as it is
not a layered structure in the true sense.
Many researchers carried out free vibration and impact analysis of FGM structures such
as Chen et al. [1] conducted free vibration analysis on three different configurations of FGM
sandwich shallow shell by using the shear deformation theory considering stretching effects,
while Abrate [2] studied on the behaviour of FGM plates which behave like homogeneous
material. Civalek and Baltacıoglu [3] performed a frequency analysis of FGM and laminated
composites by applying the harmonic differential quadrature and discrete singular convolution
solving approach while Tian et al. [4] developed a new model for free vibration analysis of
FGM beam with porosity. Akbari et al. [5] carried out free vibration analysis of FGM conical
panels using the first order shear deformation theory considering various boundary conditions.
Mirsalehi et al. [6] performed buckling and free vibration analysis of FGM microplate by
employing the modified strain gradient theory. Sofiyev and Hui [7] conducted the stability and
vibration analysis of FGM shells under extreme pressure condition by using the FOSDT
considering mixed boundary conditions. Fu et al. [8] numerically solved non-linear dynamic
responses of a shallow spherical shell in the thermal environment by Chebyshev collocation
method and Newmark scheme while Cui and Kiernan [9] developed graded foam composed of
microscopic polymer cellular structures to improve energy absorption capacity. Gunes and
3
Aydin [10] investigated the impact behaviour of FGM plate by drop weight and found the
effects of the velocity of impactor, power law exponent and plate radius on the impact
responses. After that Gunes et al. [11] determined elasto-plastic impact behaviour of FGM
plates under impact loading. Smahat and Megueni [12] carried out numerical modelling for the
impact analysis of FGM plate under thermal environment. Larson et al. [13] conducted an
impact analysis of FGM composite plate using both experimental and numerical technique
while Asemi and Salami [14] applied finite element method for the low-velocity impact
responses of FGM plate and found the effects of property gradation index, projectile velocity
and density on the normal stress, contact force, and lateral deflections. Mao et al. [15]
determined the nonlinear impact responses of FGM shells subjected to impact loading in the
thermal environment. Over the last decade, multiple deterministic and stochastic vibration and
low-velocity impact analysis are reported for laminated composites, sandwich and FG structure
[16-34].
Despite the vast literature on FGM as discussed above, most of the studies are restricted
to the deterministic domain and it is found that no particular attention is provided on the
relative importance of the influencing parameters for various global structural responses. It is
important to quantify such relative sensitivity to decide the degree of quality control needed in
an inherently uncertain system as well as to facilitate the process of achieving multi-objective
design goals effectively. In this paper it is attempted to present an efficient sensitivity analysis
for FGM plates considering the free vibration and low-velocity impact responses.
The sensitivity analysis methods available in literature can be broadly classified into
two categories, namely local sensitivity analysis and global sensitivity analysis. In local
sensitivity analysis, sensitivity indices are computed based on the derivates of the response
function. Although easy to compute, these methods only quantify the sensitivity with respect to
a base point and hence, the overall behaviour cannot be tracked. Popular local sensitivity
analysis methods include the score-function based approach, finite difference based approach
and filter based approach. On the contrary, in global sensitivity analysis, the sensitivity indices
4
are actually representative of the overall behaviour. It is worthwhile to mention that in some
literatures, global sensitivity analysis are also referred to as stochastic sensitivity analysis as it
deals to stochastic systems. Popular global sensitivity analysis methods include variance based
sensitivity analysis, moment independent sensitivity analysis, Fourier amplitude sensitivity test
and others. Note that global sensitivity analysis methods are computationally expensive as it
requires several function evaluations. To address these issues, surrogate based global
sensitivity methods are proposed in literature. The primary idea behind these methods is
depicted to employ the surrogate models by replacing the computationally expensive simulator.
Popular surrogate models available in literature includes polynomial chaos expansion, analysis
of variance decomposition, Kriging, support vector machine, neural network. A brief review on
sensitivity analysis in the field of engineering and science is provided in the following
paragraph.
Hamdia et al. [35] applied three methods Morris One-At-a-Time, PCE-Sobol and
Extended Fourier amplitude sensitivity for determining the input (material properties)
sensitivity of the energy conversion factor of flexoelectric materials. Antonio and Hoffbauer
[36] conducted the global sensitivity analysis of composite structure to determine the effect of
uncertain material properties on the reliability of the structure by using the artificial neural
network and Monte Carlo simulation while Zhang et al. [37] applied the inverse algorithm for
damage detection of composite beam with considering the noises and sensitivity analysis is
also performed to determine the effect of noise on the damage parameters. De Sousa et al. [38]
performed sensitivity analysis of laminated composite to obtain the optimal design by using the
topological derivative mapping methodology while Plischke et al. [39] introduced global
sensitivity analysis for the data having a large number of factors and input-output set. Bishay
and Sofi [40] carried out the sensitivity analysis of smart soft composite which are used as
robotic finger and Mandal et al. [41] carried out moment independent sensitivity analysis of
fibre reinforced plastic composite joints. Bodjona and Lessard [42] conducted the variance
based sensitivity analysis of single-lap bolted composite and determine the relative effects of
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different parameters on the load sharing while similarly, extended their investigation on
Borgonovo MIS analysis by using the Gaussian radial basis function based surrogate model
and found that this approach is cost-effective [43]. Zadeh et al. [44] performed a comparative
study between two global sensitivity analysis methods such as variance-based method and
moment independent method. Zhao and Bu [45] investigated on hierarchical sparse
metamodelling method for global sensitivity analysis. Vu-Bac et al. [46] developed a software
framework for probabilistic sensitivity analysis for computationally expensive models.
Despite all the works discussed above, sensitivity analysis of functionally graded
material plates under uncertainty is not attempted. To fill this apparent void, the present paper
focuses on a novel hybrid moment-independent sensitivity analysis of functionally graded
plates subjected to low-frequency free vibration and low-velocity impact. It can be noted that
such multi-dimensional sensitivity analysis considering multiple forms of analysis is crucial for
advanced structures like FGM owing to their susceptibility in simultaneous occurrence of
vibration and impact in many applications. The advantage of the adopted moment independent
sensitivity analysis resides in its robustness. To be specific, unlike variance based sensitivity
analysis, moment independent sensitivity analysis yields accurate results for skewed
distributions as well. Having said that, moment independent sensitivity analysis is
computationally expensive. To address this issue, least angle regression based adaptive sparse
polynomial chaos expansion is coupled within the moment-independent sensitivity analysis
framework. Using this method, relative effects of uncertainty in elastic modulus, shear
modulus, Poisson’s ratio and mass density on the responses are determined. The novelty of this
work includes least angle regression based adaptive sparse PCE is adopted for an efficient
moment independent sensitivity analysis. It is portrayed that unlike other surrogate based
sensitivity analysis framework, the proposed framework can actually be employed for realistic
systems having over hundred stochastic variables. To the best of authors’ knowledge, moment-
independent sensitivity analysis is firstly attempted in engineering problem (FGM plates). As
the material properties in FGM structures vary across the depth, besides the conventional
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(a)
(b)
Figure 1. (a) Geometric view of FGM plate (b) View of FGM plate subjected to impact
loading
sensitivity analysis, it is also aimed to present the depth-wise sensitivity of the material
properties. Such comprehensive results on quantifying the relative importance of the
influencing system parameters will be extremely useful in designing FGM structures and
understanding the effect of input uncertainties on the structural responses. Hereafter, the paper
is organised into five sections; section 2 includes the mathematical formulation for material
modelling, low-frequency free vibration and low-velocity impact analysis; section 3 discusses
7
the surrogate based moment independent sensitivity analysis algorithm; section 4 focuses on
the results and discussion, and section 5 provides concluding remarks and future scope study.
2. Mathematical formulations of FGM plates
2.1. Materials modelling
FGM is an inhomogeneous mixture of two materials, where material properties are
varying along one dimension i.e. thickness. If ‘f’ denotes a function of material properties then,
=
=n
iiiVff 1
(1)
where
and
i
V
represent the material property and volume fraction of constituent material ‘i,
respectively. The material properties (f) of FGM can also be expressed as [47]
3
3
2
21
1
10 1TfTfTfTfff +++++=
(2)
where f0, f-1, f1, f2 and f3 are the coefficients of temperature (T) in Kelvin. The material
properties of FGM plate varies continuously and smoothly throughout the depth [48]. The
effective material properties of FGM plate can be obtained by using power law distribution,
1
()
2
p
m c m y
E E E E t

= + +


(3)
1
()
2
p
m c m y
t
 

= + +


(4)
1
()
2
p
m c m y
E E E t

= + +


(5)
where E, µ and ρ represent the Young’s modulus, Poisson’s ratio and mass density,
respectively. The subscripts c and m denote the properties associated with ceramic and metal
respectively. In Eqs. (3) (5), t represents thickness of the plate, and y = (-t/2) for top surface
and y = (t/2) for bottom surface. The parameter p denotes the power law exponent (index).
2.2. Low-frequency free vibration analysis
In the present study, first order shear deformation theory is employed. In an orthogonal
coordinate system (x, y, z), considering (x, y) as the mid-plane of the reference plane, the
displacement can be expressed as [26]
8
u (x, y, z) = u0 (x, y) z θx (x, y)
v (x, y, z) = v0 (x, y) z θy (x, y)
w (x, y, z) = w0 (x, y) = w (x, y)
(6)
where u, v and w are the displacement components in the x, y and z direction, respectively.
0 0 0
, , andu v w
are the displacement at the mid plane and θx, θy are the rotations of cross sections
along the x and y axes. For a particular case, the transformed stress-strain relation for a lamina
with respect to (x-y-z) coordinate system can be expressed as
11 12 16
12 22 26
16 26 66
44 45
45 55
00
00
00
0 0 0
0 0 0
xx xx
yy yy
xy xy
yz yz
xz xz
Q Q Q
Q Q Q
Q Q Q
QQ
QQ






 

 

 

 
=
 

 

 

 

 

(7)
where
11
Q
=Q11Cos4 + 2(Q12 +2Q66)Sin2 Cos2 + Q22Sin4,
2
1
Q
= (Q11Q22-4Q66)Sin2Cos2 + Q12(Sin4+Cos4),
22
Q
= Q11Sin4+2(Q12+2Q66) Sin2Cos2+Q22Cos4,
16
Q
= (Q11-Q12-2Q66)SinCos3 + (Q12-Q22+2Q66)Sin3Cos,
26
Q
= (Q11-Q12-2Q66)Sin3Cos + (Q12-Q22+2Q66)SinCos3,
66
Q
= (Q11+Q22-2Q12-2Q66)Sin2Cos2+Q66(Sin4+COs4),
44
Q
=Q44Cos2 + Q55Sin2,
45
Q
= (Q55-Q44)SinCos,
55
Q
=Q55Cos2 + Q44Sin2
(8)
Here Qij denotes the in-plane elements of the stiffness matrix. In general, the force and moment
resultants are obtained from stresses as
 
/2
/2
{}
{}
T
x y xy x y xy x y
hT
x y xy x y xy xz yz
h
F N N N M M M Q Q
z z z dz
 
=
=
(9)
9
In matrix form, the in-plane force resultant {N}, the moment resultant {M}, and the transverse
shear resultants {Q} can be expressed as
 
0
[ ]{ } [ ]{ }N A B k
=+
(10)
 
0
[ ]{ } [ ]{ }M B D k
=+
 
[ *]{ }QA
=
Thus,
 
 
 
*.FD
=
(11)
where
11 12 16 11 12 16
12 22 26 12 22 26
16 26 66 16 26 66
11 12 16 11 12 16
12 22 26 12 22 26
16 26 66 16 26 66
44 45
45 55
00
00
00
00
[] 00
00
0 0 0 0 0 0
0 0 0 0 0 0
A A A B B B
A A A B B B
A A A B B B
B B B D D D
DB B B D D D
B B B D D D
SS
SS






=






(12)
where Aij , Bij and Dij represent the in-plane stiffness, in-plane-out of plane coupling stiffness
and the bending stiffness, respectively. Considering Hamilton’s principle by employing
Lagrange’s equation of motion, the dynamic equilibrium equation at element level can be
expressed by
 
 
 
 
   
 
Re
()
e e e e Ee e e Ce e
M X C X K K K X F F
+ + + − = +
(13)
After assembling all the element matrices and the force vectors (the subscript e is used to
denote elementary level matrices) with respect to the common global coordinates, the resulting
equilibrium equation of the structure becomes
 
 
 
 
   
 
 
 
2
( ) ( )
ER
M X C X K K K X F F
+ + + =  +
(14)
where [M] ϵ
nn
R
, [C] ϵ
nn
R
, [KE] ϵ
nn
R
, [Kσ] ϵ
nn
R
and [KR] ϵ
nn
R
are the global
matrices such as, mass matrix, damping matrix, elastic stiffness matrix, geometric stiffness
matrix (depends on initial stress distribution) and rotational stiffness matrix, respectively.
10
{F2)} is the vector of nodal equivalent centrifugal forces, {F} is the global vector of
externally applied force and {X} is the global displacement vector. In the above equation, [K]
can be defined as the global stiffness matrix wherein
[ ] ( [ ] [ ] [ ])
ER
K K K K
= + −
ϵ
nn
R
. The
dynamic equilibrium equation in global form considering external forces and no damping
effect can be expressed as,
 
 
 
 
}
~
{)()( FXKXM =+
(15)
where
)(
denotes the degree of stochasticity and
 
}{)(}
~
{2FFF +=
. The simplified
equation of motion of free vibration system with n degrees of freedom can expressed as [26]
 
 
 
 
0)()( =+ XKXM
(16)
For free vibration, the natural frequencies [ωn] are determined from the standard eigenvalue
problem which is solved by the QR iteration algorithm,
}){(}{)]([ XXS
=
(17)
where
)]([)]([)]([ 1
MKS
=
2
)}({ 1
)(
n
=
(18)
2.3. Low-velocity impact analysis
From equation (15), the dynamic equilibrium equation can also be idealized as
)}({})]{([})]{([
impact
FXKXM =+
(19)
Here
)(
denotes the degree of stochasticity. For the impact loading, the externally applied load
vector Fimpact =
F
can be expressed as,
T
C
FF }000......)(.....000{)(
=
(20)
where
C
F
is the contact force. The dynamic equation of motion for the rigid impactor can be
expressed as
0)()( =+
Cimpimp FXm
(21)
11
where
imp
m
and
imp
X
are the impactor’s mass and acceleration, respectively. The contact force
between the elastic spherical impactor and FGM plate can be determined by using the modified
Hertzian contact law which is based on permanent indentation. The contact force during impact
loading can be expressed as [51-53, 80],
mc kF
= 0)()( 5.1
(22)
3
][ 1
3
16 B
C
kk
k
pimp
+
=
(23)
where
k
denotes the modified contact stiffness and
is the change in the distance between
plate and impactor called the local indentation [52, 53, 80]. The constant
B
depends on the
radii of curvature of the contacting surface of the impactor and plate. For the contact between
flat surface and spherical object,
2=B
is assumed [80]. The constant C depends on the
curvature of the impactor and plate and given by
1
1 2 1 2
1 1 1 1 1
i i p p
CR R R R

= + + +


(24)
where
12
and
ii
RR
radius of curvature of the impactor and
12
and
pp
RR
are the radius of
curvature of the target. For the flat plate and spherical impactor,
12ii
RR=
and
12pp
RR=
.
p
k
is the stiffness parameter related to the plate, while
imp
k
is the stiffness parameter related to the
impactor which can be expressed as,
( )
( )
 
( )
1
222
22 11 22 12
2
11 22 12
2
xy xy
p
xy
A A A G A G
kG A A A
+ + +
=
11 (1 )
yx
AE
= − 
2
22
(1 ( / ))
1
x xy x y
x
E E E
A
−
=+
12 x xy
AE
=
2
1
1 2 ( / )
x xy x y
EE

= −−
(25)
12
where, E, G, and µ (with respective subscripts) are effective plate material properties (namely,
elastic modulus, shear modulus and Poisson ratio) in x and y directions. The parameter
imp
k
can be obtained in a similar way as
p
k
. In case of rigid impactor
0
imp
k=
.
The local indentation
)(
can be expressed as
cos)(),,(cos)()()()( tyxtt ccpimp +=
(26)
where
imp
and
p
are impactor’s displacement and targeted plate displacements, respectively,
while
and

are the impact angle (measured from normal vertical) and twist angle along the
global z-direction, respectively. For the present study, impact angle and twist angle are
considered as zero. The components of force at the impact loading pointed at the centre of the
plate can be derived as
cos)(,sin)(,0 cizciyix FFFFF ===
(27)
For the untwisted plate, the forces in x and y direction become zero, and in z direction force
will be equal to the contact force (
C
F
) which is transient in nature. The equations such as (19)
and (21) are considered as the ordinary differential equations with constant coefficients and
made to be satisfied at discrete time intervals
t
apart. These time-dependent equations are
solved by using the Newmark’s integration approach [54]. By using Newmark’s integration
approach with time step
t
, it can be expressed as
tttt
pFK ++ =}{}]{[
(28)
tt
c
tt
ii Fk ++ =}{}]{[
(29)
where
[]
p
K
and
[]
i
k
represent the effective stiffness matrix of the plate and impactor, which
are expressed as
][][][ 0McKK +=
(30)
ii mck 0
][ =
(31)
Effective force at time
tt +
are calculated as
)}{}{}{]([}{)}({}{ 210
2t
p
t
p
t
p
tttttt cccMFFF
++++= +++
(32)
13
)}{}{}{(}{}{ 210 t
imp
t
imp
t
impi
tt
c
tt
ccccmFF
+++= ++
(33)
The acceleration and velocity of plate and impactor can be computed as
)}{}{)}{}({}{ 210 t
p
t
p
t
p
tt
p
tt
pccc
= ++
(34)
)}{}{)}{}({}{ 210 t
imp
t
imp
t
imp
tt
imp
tt
imp ccc
= ++
(35)
tt
p
t
p
t
p
tt
pcc ++ ++= }{}{}{}{ 43
(36)
tt
imp
t
imp
t
imp
tt
imp cc ++ ++= }{}{}{}{ 43
(37)
Considering the following initial boundary conditions,
0}{}{}{ === ppp
(38)
0}{}{ == impimp
and
V
imp =}{
(39)
where, V is the initial velocity of impactor. The integration constants are calculated as
02
0
1
cat
=
,
10
1
cat
=
,
20
11
2
ca
=−
,
3(1 )c D t= −
,
4
c D t=
(40)
The value of
0
a
and D are considered as 0.5 and 0.25, respectively [55].
In the current finite element modelling for all the analyses, an eight noded
isoparametric quadratic plate element with five degrees of freedom at each node (three
translation and two rotations) is considered, wherein the entire plate is discretised into 64
elements (8 x 8 mesh, finalized using a convergence study) with 225 nodes.
3. Surrogate based moment independent sensitivity analysis
After description of the mathematical formulation for vibration and impact analysis of
FGM plates in the previous section, it is needed to focus on the methodology for moment
independent sensitivity analysis. In the present study, it utilizes a surrogate based moment
independent sensitivity analysis framework wherein the least angle regression based adaptive
sparse polynomial chaos expansion (PCE) is used as a surrogate. It can be noted that there are
various other surrogate models available in literature and they have been widely used to deal
14
with computationally intensive problems across different fields [56 - 71]. In this section, we
will first discuss the mathematical formulation for the surrogate model and subsequently,
proceed to the moment independent sensitivity analysis framework.
3.1. Least angle regression based adaptive sparse PCE
It considers a set of input variables to be the input variables (e.g.,
material properties, load, boundary etc.) and to be the output (e.g.,
impact response, natural frequencies etc.). We also assume that we have access to a computer
code that maps from the inputs x to the response y. In practice, can be some finite
element code (e.g., the mathematical formulation discussed in Section 2), resulting .
In practice, the computer code is often expensive and hence, it becomes quite difficult to
perform uncertainty quantification, sensitivity analysis and design under uncertainty. One way
to address this issue is to replace the computationally expensive simulator with a surrogate,
such that . The popular surrogate models in literature includes analysis of
variance decomposition, Kriging linked Gaussian process, neural network, support vector
machine, polynomial chaos expansion (PCE). In this work, we utilize a variant of PCE,
referred to as least angle regression based PCE.
In PCE, the inputs x and the output yare mapped as [72]
( ) ( )
N
=
y x a xM

N
(41)
where the series in Eq. (41) converges in the
L2
norm and are the unknown coefficients and
y
’s are multivariate polynomials. In PCE, the orthogonal polynomial are used such that
(42)
where
d
ij
is the Kronecker delta function. In order to utilize Eq. (37), we need to truncate the
infinite series and compute the unknown coefficients . The infinite series in Eq. (41) is
15
generally truncated arbitrarily whereas, the unknown coefficients are computed either by using
quadrature methods or by using least square method. However, PCE quickly becomes
intractable as the number of input variables, N, increases.
In order to address the above mentioned issue of intractability, adaptive sparse PCE is
proposed in literature. The primary idea behind adaptive sparse PCE is to only retain the terms
that are significant. With this, the number of unknown coefficients associated with PCE can be
significantly reduced. Although, there exist several ways to formulate the adaptive sparse PCE,
in this work we utilize the least angle regression based adaptive sparse PCE as presented in
[72]. The least angle regression is an efficient method for variables selection. In the current
work, we utilize least angle regression to select the basis functions
y
that have the greatest
impact on the model response (among probable set of candidate solutions). In other words,
least angle regression provides a sparse polynomial chaos expansion where only certain
components of the full polynomial chaos expansion are retained. It is to be noted that least
angle regression does not provide a single surrogate but a collection of surrogates. Once all the
surrogates are obtained, we select the best surrogate model based on some scoring criterion.
The steps involved in least angle regression based polynomial chaos expansion are explained
below:
Generate the possible sparse polynomial chaos expansion representations by using least
angle regression algorithm.
Compute the unknown coefficients of all the sparse polynomial chaos expansion by using
least square regression.
Assess the surrogates based on some statistical test.
Retain the surrogate model with highest score.
From above discussion, it is obvious that there are two specific questions for the procedure to
work. Firstly, it is needed to formulate the sparse polynomial chaos expansions by using least
angle regression. Secondly, how to determine the best surrogates. To clarify the first query, it
is presented by the algorithm of least angle regression as furnished in Algorithm 1.
16
Algorithm 1: Least angle regression
Initialize: Initialize the unknown coefficients . Set the initial residual equal to
y
.
Find the vector which is most correlated to the current residual.
Move from 0 towards the least square solution of the current residual on ,
until some other predictor has as much correlation with the current residual as
does .
Augment and ,
ya
j,
ya
k
{ }
and move them towards the joint least square
solution until some other predictor has as much correlation with the current
residual.
Continue until
( )
min , 1m P N−
predictors have been entered where P is the
maximum polynomial order and N is the number of variables.
Using least angle regression, it is possible to select only the important components of
polynomial chaos expansion. However, to actually utilize this expansion in practice, it is
important to assess the accuracy of this method. To that end, we have utilized the leave-one-out
cross validation test, the steps of which is shown in Algorithm 2.
Algorithm 2: Leave-one-out cross validation
Initialize: Generate training samples . Select one of the PCE models.
Iterate
1,..., s
iN=
Formulate (training samples set with i-th sample excluded).
Train polynomial chaos expansion based on .
Predict at the i-th input and compute the error
e
i
.
Compute the leave-one-out-error for the current model as,
1
1s
N
oi
i
s
N

=
=
Once the optimal polynomial chaos expansion is selected by using algorithms 1 and 2, we can
use it to replace the actual simulator and make future predictions in an efficient manner.
3.2. Moment independent sensitivity analysis
After describing the least angle regression based polynomial chaos expansion in the
preceding subsection, the moment independent sensitivity analysis is explained here. Note that
the function of least angle regression based polynomial chaos expansion is to achieve
17
computational efficiency in moment independent sensitivity analysis without compromising
the accuracy of results. The moment independent sensitivity analysis is a popular technique for
the sensitivity analysis, in which the effects of input parameters are evaluated on the whole
distribution of output responses, instead of determining the mean or variance of output
response [73]. Sensitivity analysis can be divided into local sensitivity analysis and global
sensitivity analysis, respectively. In the moment independent sensitivity analysis method,
which is a form of global sensitivity analysis, the focus is on the finding those inputs
parameters that, if fixed at their distribution ranges, will have greatest effect on the probability
density function of the model responses on average [74]. Consider the input and the output
variables defined in Section 3.1. In moment independent sensitivity analysis, “We are asked to
bet on the factor that, if determined would lead to the greatest expected modification in the
distribution of y[75]. Considering to be sensitivity index, it can be expressed as
(43)
where
yÎWy
and
w
·
( )
is the probability density function. Pictorial representation of
moment-independent sensitivity analysis is show in Figure 2. Note that moment independent
sensitivity indices are individually and jointly normalized:
(44)
and
(45)
It is to be noted that practical implementation of equation 43 is computationally demanding.
This is because we have to solve two integrals for computing sensitivity index by using Eq.
(43) one for computing the area denoted in Fig. 2 and other for computing the expectation
with respect to variable
xi,"i
. If we employ Monte Carlo sampling with
Nsimu
samples for
computing the expectation and quadrature based approach with
Nquad
points for computing the
area between the curves, total number of actual function evaluations (
E
F
) in moment
independent sensitivity indices is [41]
18
Figure 2. Pictorial representation of equation 43. Sensitivity indices is given by the shaded
portion
(46)
Clearly, the computational effort associated with moment independent sensitivity analysis is
significant and hence, direct implementation of a high-fidelity model (e.g., a FE model of FGM
structures) is not an option. Hence, alternatives need to be explored.
To address the above mentioned issue, we propose to utilize least angle regression
based polynomial chaos expansion as a surrogate of the actual computationally expensive
simulator. Assuming
NS
to be number of training samples required for training the surrogate,
the computational cost for the proposed sensitivity analysis is
(47)
where we have performed the MCS based on the trained surrogate (represented as superscript
sur) and hence, time required for the MCS part is negligible. The steps involved for performing
moment independent sensitivity analysis using the least angle regression based polynomial
chaos expansion are shown in Algorithm 3.
19
Algorithm 3: Moment independent sensitivity analysis using LAR-PCE
Initialize: Generate training samples .
Train the PCE model using algorithms 1 and 2.
Generate
Nsimu
sur
samples of the inputs.
Obtain
y
at the simulation points by using the trained PCE.
Use kernel density estimator to obtain the unconditional probability density function
w
y
( )
( )
.
Generate
Nquad
sur
quadrature points for each of the inputs.
Iterate: For the k-th observation of the i-th variables
xi
k
Set
xi
1:Nsimu
sur =xi
k
Obtain
y
at the simulation points by using the trained PCE.
Use kernel density estimator to obtain the conditional probability density function
w
y|xi
k
( )
( )
Once the code has iterated over all the possible observations of
xi
, compute
sensitivity indices using Eq. (36)
Terminate: Terminate when sensitivity indices of all the variables have been computed.
4. Results and discussion
In this section, numerical results for global sensitivity analysis of a cantilever FGM
plate are presented using the computationally efficient approach proposed in the preceding
section considering low-frequency free vibration and low-velocity impact. In general,
cantilever boundary conditions are common in various idealized structural forms involving
advanced lightweight materials such as aircraft wings and wind turbine blades. Thus, to present
numerical results, we have adopted this boundary condition. It can be noted that even though
the sensitivity analyses results are presented considering cantilever boundary condition here,
the results can be extended for other boundary conditions following the efficient adaptive
sparse PCE based moment-independent sensitivity analysis framework.
The dimension of the FGM plate is considered as L = b = 1m, t = 0.002 m. In this
context it can be noted that the thickness considered in the current paper is quite normal for
thin plates and such values were considered in multiple previous studies (For example, refer to
reference [76]). The total thickness of the plate is divided into 16 layers with different material
20
properties obtained by using the power law distribution. So each layer has different material
properties such as the longitudinal elastic modulus
( )
1
E
, longitudinal shear modulus
( )
12
G
,
Poisson’s ratio
( )
, and mass density
( )
. The relative effects of these layer-wise material
properties on the first three natural frequencies and impact analysis parameters (such as
maximum contact force, maximum plate displacement, and maximum impactor displacement)
are presented in terms of sensitivity index. The FGM plate is composed of two materials
namely aluminium and zirconia and their material properties are given as [77]:
70GPa,
Al
E=
151GPa
Zr
E=
,
0.25
Al
=
,
0.3
Zr
=
,
3
2707Kg/m
Al
=
,
3
3000Kg/m
Zr
=
. In this paper,
cantilever plate is considered for the stochastic sensitivity analysis of FGM plates subjected to
free vibration and low-velocity impact. Using simple cantilever plate geometry, the system’s
total number of degrees of freedom can be significantly reduced leads to reduction in
computational time. Since stochastic finite element models take thousands of iterations, and
therefore many hours to complete the simulation, it is effective to use simple cantilever plate.
But the same approach can be applied in future for other geometries with different boundary
condition.
4.1. Validation and convergence study
Before carrying out the sensitivity analysis, two different forms of validation and
convergence studies are required to be performed to achieve adequate level of confidence in
the current analyses. The finite element models as well as the surrogate model are validated in
this section for the vibration and impact analyses. The results of the present FE model are
validated with the results from literature for both the free vibration and low-velocity impact
analysis as shown in Table 1 and Figure 3 respectively. A mesh convergence study was carried
out revealing that the mesh size of 8 x 8 is sufficient to obtain satisfactory level of accuracy.
The purpose of this paper is to propose an efficient multi-dimensional sensitivity analysis for
FGM structures under the simultaneous occurrence of vibration and impact. From the
validation results presented in Table 1 and Figure 3, it is apparent that the adopted structural
21
Table 1. Non-dimensional first natural frequency
( )
0.5
[ / ]
mm
E
 
=
of FG square plate.
Power
law
index
Thickness
Length
Ta and
Noh [78]
Baferani et al.
[79]
Present Study
0
0.1
0.1134
0.1134
0.1139
0.2
0.4152
0.4154
0.4159
1
0.1
0.0869
0.0891
0.0883
0.2
0.3205
0.3299
0.3261
2
0.1
0.0788
0.0819
0.0797
0.2
0.2897
0.3016
0.3004
Figure 3. Validation results (with respect to Kiani et al. [80]) for time histories of contact force
for Silicon Nitride and Stainless Steel FG beam clamped at both ends considering t1 =1.0 μ-sec,
L =135 mm, b = 15 mm, t =10 mm, mi = 0.01 Kg, Ri = 12.7 mm, V =1.0 m/s
analysis model can capture the underlying physics of the system adequately. As the basic
physics remains same for a fairly simplified structural analysis model and a more complicated
analysis model, results for relative sensitivity of different material properties are not expected
to vary significantly, albeit a more complicated model may lead to an improvement of the
accuracy slightly at the cost of increased computational intensiveness. It can be noted that the
22
(a)
(b)
(c)
Figure 4. Scatter plots between the original FE model and surrogate model for first three
natural frequencies with different sample size (N).
23
efficient adaptive sparse PCE based moment-independent sensitivity analysis framework can
be adopted in conjunction with a more complicated structural analysis model, if more accuracy
is desired in any future applications.
In the present study, the least angle regression based polynomial chaos expansion is
employed as the surrogate model. The validation of the surrogate model is carried out by using
scatter plots and probability density function (PDF) plots of percentage error. Figure 4
represents the scatter plots between the original FE model and surrogate model with different
sample sizes(N) for first three natural frequencies and it can be found that a sample size (N) =
512 is adequate to obtain sufficient level of accuracy. Figure 5 illustrates the probability
density function plots of percentage of error of surrogate model with different sample size for
the first three natural frequencies, revealing an interesting outcome regarding the probability of
occurrence of various level of errors including the bounds. Thus for the sensitivity analysis of
FGM plate subjected to free vibration, a sample size of 512 for the surrogate model is
considered. Similarly, Figure 6 represents the scatter plots between the surrogate model and
original FE model with different sample size for the low-velocity impact responses namely
maximum contact force, maximum plate displacement, and maximum impactor displacement
and reveals that the sample size (N) = 512 has adequately less scattered data as compared to the
lower sample sizes for all impact responses. Figure 7 illustrates the PDF plots of percentage of
error in the surrogate model with different sample sizes for the three impact responses
reaffirming the outcome of figure 6. Thus a sample size of 512 is employed for surrogate model
formation in both vibration and impact analyses.
4.2. Sensitivity analysis for low-frequency free vibration
In this section, the numerical results for stochastic sensitivity analysis of FGM plates
are presented using the least angle regression based polynomial chaos expansion assisted
moment independent sensitivity analysis method. The relative importance of input parameters
such as material properties (both depth-wise and collectively) on the first three natural
24
(a)
(b)
(c)
Figure 5. Probability density function plots of percentage of error of surrogate model with
different sample size for the first three natural frequencies.
25
(a)
(b)
(c)
Figure 6. Scatter plots between original FE model and surrogate model with different sample
size (N) for the maximum contact force, maximum plate displacement, and maximum impactor
displacement.
26
(a)
(b)
(c)
Figure 7. Probability density function plots of percentage of error of surrogate model with
different sample size for the maximum contact force, maximum plate displacement, and
maximum impactor displacement.
frequencies are quantified. Such in-depth sensitivity analysis of a complex structural form like
FGM under simultaneous consideration of multiple objectives, as presented in this article, is
27
normally not achievable using direct analytical methods. A finite element based approach is
suitable for analysing such complex systems. We have coupled the FE analysis with a
surrogate based approach to achieve computational efficiency.
In this paper we have concentrated on first three modes of vibration. To be more
specific about the content of the article, we have added ‘low-frequency vibration’ in the title
and elsewhere in the paper. However, it can be noted that the efficient adaptive sparse PCE
based moment-independent sensitivity analysis framework can be adopted for analysing higher
modes of vibration as well in future. There are 64 input parameters (16 values of each material
property) in the current analysis, which are obtained by using the power law distribution (refer
to Table S1- S4 in the supplementary material). Figure 8 represents the depth-wise and
collective sensitivity indices of all material properties for the first three natural frequencies.
The bar chart shows that the elastic modulus (E1) and mass density
( )
have highest sensitivity
index for all the three natural frequencies. The shear modulus has very less effect on the first
and third natural frequencies, but it has significant effect on the second natural frequency. The
sensitivity index of elastic modulus has larger effects at top and bottom face of the plate, while
it has lesser effect as it is approached towards the middle of the plate. Shear modulus and
Poisson’s ratio also have a similar trend, while mass density has almost constant sensitivity
index throughout the depth.
Based on the efficient surrogate models of the first three natural frequencies, which
are capable of readily predicting the natural frequencies corresponding to any set of input
material properties within the design domain, further deterministic plots are presented in figure
9 for different degree (%) of individual variation of the input material properties. For plotting
these variations of the natural frequencies, a single material property of each layer is varied
with respect to the corresponding deterministic value while keeping the other material
properties fixed to their respective deterministic values (refer to Table S9 - S12 in the
28
(a) First natural frequency
(b) Second natural frequency
(c) Third natural frequency
Figure 8. Sensitivity index of material properties for the first three natural frequencies.
29
(a) First natural frequency (FNF)
(b) Second natural frequency (SNF)
(c) Third natural frequency (TNF)
Figure 9. Variation of natural frequencies with individual deterministic deviation of the
material properties with respect to their respective nominal values
supplementary material). The figures clearly portray individual effects of the material
properties on the natural frequencies in a deterministic framework. Best-fit linear
approximations are presented along with the points corresponding to different percentage of
variations of the material properties. The relative slopes of the straight lines for the four
30
different material properties can give a preliminary estimate of relative sensitivity of these
input parameters, albeit it is not possible to quantify the sensitivity thoroughly using the slopes.
It is interesting to notice that the relative slopes for the three natural frequencies shown in
figure 9 are in good agreement with the relative trend of collective sensitivity presented in the
insets of figure 8. Such agreement in the trend of results provides a qualitative validation of the
sensitivity analysis presented in this article.
4.3. Sensitivity analysis for low-velocity impact
The results of the stochastic sensitivity analysis for FGM plates subjected to low-
velocity impact loading are presented in this section considering: initial velocity of impactor as
5 m/sec, impact angle as , plate thickness as 0.002 m, mass density of impactor as 85
10-4
24
N s cm
, power law index as 1 and twist angle as . Figure 10 represents the sensitivity
indices of material properties for the critical low-velocity impact responses namely maximum
contact force, maximum plate displacement, and maximum impactor displacement (refer to
Table S5- S8 in the supplementary material). For all the impact responses, mass density has the
highest sensitivity index, followed by elastic modulus, shear modulus and Poisson’s ratio. The
sensitivity indices of mass density are almost constant from top surface to bottom surface,
while for the other material properties, the sensitivity index reduces from the faces towards the
middle of the plate.
Based on the efficient surrogate models of the critical impact responses, which are
capable of readily predicting the output quantities corresponding to any set of input material
properties within the design domain, further deterministic plots are presented in figure 11 for
different degree (%) of individual variation of the input material properties. For plotting these
variations of the critical impact responses, a single material property of each layer is varied
with respect to the corresponding deterministic value while keeping the other material
properties fixed to their respective deterministic values(refer to Table S13- S16 in the
supplementary material). The figures clearly portray individual effects of the material
31
(a)Maximum contact force
(b)Maximum plate displacement
(c)Maximum impactor displacement
Figure 10. Sensitivity index of material properties for the maximum contact force, maximum
plate displacement, maximum impactor displacement
32
(a) Maximum contact force
(b) Maximum plate displacement
(c) Maximum impactor displacement
Figure 11. Variation of impact responses with individual deterministic deviation of the
material properties with respect to their respective nominal values
properties on the critical impact responses in a deterministic framework. Best-fit linear
approximations are presented along with the points corresponding to different percentage of
33
variations of the material properties. The relative slopes of the straight lines for the four
different material properties can give a preliminary estimate of relative sensitivity of these
input parameters, albeit it is not possible to quantify the sensitivity thoroughly using the slopes.
It is interesting to notice that the relative slopes for the critical impact responses shown in
figure 11 are in good agreement with the relative trend of collective sensitivity presented in the
insets of figure 10. Such agreement in the trend of results provides a qualitative validation of
the sensitivity analysis presented in this article.
It can be noted in this context that based on the surrogate models developed for each of
the critical output parameters related to low-frequency vibration and low-velocity impact,
several other analyses can be performed efficiently (involving negligible additional
computational effort) in the deterministic as well as stochastic domain. For example, using a
Monte Carlo simulation based approach of optimization, Table S17 of the supplementary
material presents the depth-wise deterministic material properties that lead to the maximum
values of the first three natural frequencies. Similarly, the optimum material properties for
maximizing the critical impact responses are presented in Table S18 S20 of the
supplementary material.
5. Summary and conclusions
A novel hybrid sensitivity analysis approach is presented in this paper for functionally
graded plates subjected to free vibration and low-velocity impact. The least angle regression
based polynomial chaos expansion is integrated with moment independent sensitivity analysis
approach to achieve computational efficiency. The power law distribution is applied for
gradation of material properties in the FGM plates across the depth. Relative importance of the
material properties are quantified in a depth-wise sense as well as collectively for the free
vibration and low-velocity impact responses. The sensitivity results reveal that for first and
third natural frequencies, elastic modulus and mass density are the most sensitive parameters;
while for the second natural frequency shear modulus and mass density are most sensitive. The
34
Poisson’s ratio has less sensitivity for all natural frequencies. In case of the low-velocity
impact loading, mass density is more sensitive than other material properties for all the impact
responses. Besides the collective sensitivity results, the depth-wise spatial sensitivity analysis
for both the analyses reveals that mass density has almost uniform sensitivity across the depth,
while the sensitivity of other material properties are highest at the faces and reduce towards the
middle of the plate. From a physical viewpoint, the depth-wise variation of sensitivity of the
elastic properties can be justified by their relative contribution to the stiffness of the structure
based on the distance of a particular layer from the neutral axis.
The novelty of this work is two-folded. Firstly, a least angle regression based adaptive
sparse PCE approach is coupled with the moment-independent sensitivity analysis algorithm to
achieve computational efficiency. Thus, a generic computationally efficient sensitivity method
is proposed to employ in multitude of problems. Secondly, to the best of authors’ knowledge,
this is the first attempt wherein a stochastic sensitivity analysis method is applied to an
important engineering problem concerning the vibration and impact analysis of FGM plates. It
can be noted that such multi-dimensional sensitivity analysis considering multiple forms of
analysis is crucial for advanced structures like FGM owing to their susceptibility in
simultaneous occurrence of vibration and impact in many applications. As the material
properties in a FGM structure vary across the depth, besides the conventional sensitivity
analysis, the depth-wise sensitivity analysis of the material properties carries a critical practical
significance. The results of depth-wise spatial sensitivity analysis and the collective sensitivity
analysis in general (simultaneously considering multiple objectives)will serve as important
guidelines for designing and quality control of functionally graded structures. Moreover, the
quantification of relative importance of the input parameters, as presented in this paper, would
be helpful in carrying out efficient optimization, model updating, uncertainty and reliability
analysis by neglecting the less important effects. In future, the proposed computationally
efficient approach of sensitivity analysis can be utilised for various other materials and
structural systems.
35
Acknowledgement
The first author would like to acknowledge the financial support received from Ministry of
Human Resource and Development (MHRD), Govt. of India during the period of this research
work.
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1
Supplementary material
A hybrid stochastic sensitivity analysis for low-frequency vibration and
low-velocity impact of functionally graded plates
P.K. Karsha,b,1, T. Mukhopadhyayc, 1, *, S. Chakrabortyd,e,1, S. Naskarf,1, S. Deya,1
aMechanical Engineering Department, National Institute of Technology Silchar, India
bDepartment of Mechanical Engineering, Parul Institute of Engineering & Technology, Vadodara, India
cDepartment of Aerospace Engineering, Indian Institute of Technology Kanpur, Kanpur, India
dCenter for Informatics and Computational Science, University of Notre Dame, USA
eDepartment of Aerospace and Mechanical Engineering, University of Notre Dame, USA
fWhiting School of Engineering, Johns Hopkins University, Baltimore, USA
*Corresponding author’s email: tanmoy@iitk.ac.in; tanmoy.mukhopadhyay@eng.ox.ac.uk
1All authors have contributed equally
In this supplementary material we have provided additional numerical data in a tabulated
form. These will be helpful for comprehensive understanding of this article and facilitate
prospective direct numerical validations in future studies.
Table S1. Sensitivity results of Young’s modulus for low-frequency vibration (Here Ei, i =
1, 2, … 16 represents the Young’s modulus of ith idealized layer in the FGM plate)
Material
Properties
First natural
frequency
Second natural
frequency
Third natural
frequency
E1
0.140827
0.028758
0.144147
E2
0.09167
0.003349
0.094046
E3
0.060517
0.016963
0.061275
E4
0.032522
0.001003
0.032246
E5
0.014692
0.007292
0.01868
E6
0.000815
0.002965
0.000656
E7
0
0.000476
0.000337
E8
0
0.000369
0
E9
0
0
0
E10
0.011142
0.003381
0.010655
E11
0.022792
0.001698
0.023627
E12
0.045217
0.00181
0.043981
E13
0.058221
0
0.05696
E14
0.07658
0
0.077121
E15
0.103864
0.026064
0.102117
E16
0.117365
0.00246
0.115712
2
Table S2. Sensitivity results of shear modulus for low-frequency vibration (Here Gi, i = 1,
2, … 16 represents the shear modulus of ith idealized layer in the FGM plate)
Material
Properties
First natural
frequency
Second natural
frequency
Third natural
frequency
G1
0.000206
0.134951
0.0000633
G2
0
0.073504
0
G3
0.00049
0.066707
0.000199
G4
0.000744
0.00141
0.000364
G5
0.000396
0.002418
0.000177
G6
0.000268
0.003145
0.000155
G7
0
0
0
G8
0.000787
0.00282
0.000786
G9
0.000655
0
0.00035
G10
0.000811
0.000549
0.000283
G11
0.000438
0
0.000167
G12
0.000929
0.039479
0.000329
G13
0.000836
0.048134
0.000665
G14
0.00000467
0.056596
0.000952
G15
0.001729
0.09582
0.000844
G16
0.000625
0.113024
0.000275
Table S3. Sensitivity results of Poisson’s ratio for low-frequency vibration (Here µi, i = 1,
2, … 16 represents the Poisson’s ratio of ith idealized layer in the FGM plate)
Material
Properties
First natural
frequency
Second natural
frequency
Third natural
frequency
µ1
0.023225
0.004352
0.013145
µ2
0.014158
0.002149
0.00358
µ3
0.001003
0.002588
0.000463
µ4
0
0.001451
0
µ5
0.000302
0.001366
0.0000813
µ6
0.000902
0.002986
0.000247
µ7
0
0.002889
0.000494
µ8
0
0
0
µ9
0.000612
0.000675
0.000214
µ10
0
0
0.000611
µ11
0.002988
0.017888
0.003613
µ12
0.001113
0.001819
0.000398
µ13
0.001415
0.001474
0.000545
µ14
0.003284
0.001903
0
µ15
0.000577
0
0.000579
µ16
0.009225
0.001087
0.003821
3
Table S4. Sensitivity results of mass density for low-frequency vibration (Here ρi, i = 1, 2,
… 16 represents the mass density of ith idealized layer in the FGM plate)
Material
Properties
First natural
frequency
Second natural
frequency
Third natural
frequency
ρ1
0.048181
0.054983
0.049308
ρ2
0.048233
0.056569
0.050392
ρ3
0.057494
0.058668
0.056572
ρ4
0.049918
0.057595
0.05178
ρ5
0.054862
0.064546
0.057069
ρ6
0.052424
0.059146
0.054605
ρ7
0.053749
0.067879
0.053124
ρ8
0.055624
0.06366
0.054238
ρ9
0.05447
0.062268
0.054717
ρ10
0.055848
0.061399
0.056356
ρ11
0.050775
0.046182
0.049408
ρ12
0.050579
0.044399
0.053193
ρ13
0.050512
0.042663
0.050042
ρ14
0.050986
0.060468
0.051688
ρ15
0.046783
0.047278
0.049169
ρ16
0.048863
0.05115
0.049054
Table S5. Sensitivity results of Young’s modulus for low-velocity impact responses (Here
Ei, i = 1, 2, … 16 represents the Young’s modulus of ith idealized layer in the FGM plate)
Material
Properties
Max. Contact
Force
Max. Plate
Displacement
Max. Impactor
Displacement
E1
0.055302
0.035043
0.04596
E2
0.038546
0.023909
0.032508
E3
0.024876
0.01546
0.020954
E4
0.015071
0.009719
0.012095
E5
0.000321
0.005049
0.005986
E6
0.000623
0.002355
0.001906
E7
0
0.000876
0.000103
E8
0.000479
0.0000283
0.000242
E9
0
0.001053
0.000212
E10
0
0.003642
0.004429
E11
0.00878
0.007988
0.009291
E12
0.016888
0.011564
0.014761
E13
0.025861
0.016476
0.020789
E14
0.03863
0.023455
0.030858
E15
0.048116
0.030638
0.03946
E16
0.062263
0.039522
0.051482
4
Table S6. Sensitivity results of shear modulus for low-velocity impact responses (Here Gi,
i = 1, 2, … 16 represents the shear modulus of ith idealized layer in the FGM plate)
Material
Properties
Max. Contact
Force
Max. Plate
Displacement
Max. Impactor
Displacement
G1
0.018931
0.020521
0.01903
G2
0.016139
0.013482
0.011962
G3
0.007231
0.009443
0.007309
G4
0.004879
0.005625
0.003942
G5
0
0.003827
0.002083
G6
0
0.002952
0.000855
G7
0.000272
0.004017
0.001056
G8
0.000703
0.005474
0.000213
G9
0.00043
0.007978
0.001925
G10
0.00039
0.011887
0.005804
G11
0.00915
0.016794
0.009058
G12
0.016054
0.02171
0.012067
G13
0.015374
0.027446
0.016654
G14
0.023323
0.034281
0.02166
G15
0.027865
0.043481
0.028712
G16
0.03575
0.053396
0.034853
Table S7. Sensitivity results of Poisson’s ratio for low-velocity impact responses (Here µi,
i = 1, 2, … 16 represents the Poisson’s ratio of ith idealized layer in the FGM plate)
Material
Properties
Max. Contact
Force
Max. Plate
Displacement
Max. Impactor
Displacement
µ1
0.06111
0.061207
0.053105
µ2
0.041292
0.044988
0.03946
µ3
0.029491
0.032468
0.025529
µ4
0.019677
0.021178
0.017034
µ5
0.009954
0.013492
0.009629
µ6
0.005485
0.007419
0.004752
µ7
0
0.003443
0.001901
µ8
0
0.000864
0.0000520
µ9
0
0.000167
0.0000266
µ10
0
0
0.0000719
µ11
0
0
0.001463
µ12
0
0.002648
0.003637
µ13
0.009969
0.006056
0.008505
µ14
0.012383
0.010115
0.012703
µ15
0.021683
0.013015
0.016835
µ16
0.021735
0.01839
0.023058
5
Table S8. Sensitivity results of mass density for low-velocity impact responses (Here ρi, i
= 1, 2, … 16 represents the mass density of ith idealized layer in the FGM plate)
Material
Properties
Max. Contact
Force
Max. Plate
Displacement
Max. Impactor
Displacement
ρ1
0.043139
0.050731
0.057128
ρ2
0.044889
0.049859
0.0566
ρ3
0.043989
0.048682
0.05532
ρ4
0.042277
0.048013
0.0548
ρ5
0.040221
0.048696
0.0559
ρ6
0.04466
0.047888
0.0549
ρ7
0.04691
0.047579
0.053647
ρ8
0.040162
0.047043
0.054061
ρ9
0.040206
0.046792
0.055082
ρ10
0.045423
0.048594
0.056059
ρ11
0.042843
0.046991
0.054799
ρ12
0.044309
0.047704
0.055029
ρ13
0.040636
0.048299
0.054634
ρ14
0.042058
0.047815
0.055602
ρ15
0.043803
0.047913
0.053436
ρ16
0.041461
0.047484
0.053283
Table S9. Individual effect of deterministic variation in elastic modulus on first three
natural frequencies (FNF First natural frequency; SNF Second natural frequency; TNF
Third natural frequency)
E
G
µ
ρ
FNF
SNF
TNF
0.85E
G
µ
ρ
11.756
30.838
72.661
0.88E
G
µ
ρ
11.961
30.980
73.897
0.91E
G
µ
ρ
12.162
31.119
75.112
0.94E
G
µ
ρ
12.360
31.256
76.305
0.97E
G
µ
ρ
12.555
31.390
77.478
1.0E
G
µ
ρ
12.747
31.522
78.633
1.03E
G
µ
ρ
12.935
31.652
79.769
1.06E
G
µ
ρ
13.122
31.780
80.888
1.09E
G
µ
ρ
13.305
31.906
81.990
1.12E
G
µ
ρ
13.486
32.031
83.077
1.15E
G
µ
ρ
13.665
32.153
84.148
6
Table S10. Individual effect of variation in shear modulus on first three natural
frequencies (FNF First natural frequency; SNF Second natural frequency; TNF Third
natural frequency)
E
G
µ
ρ
FNF
SNF
TNF
E
0.85G
µ
ρ
12.742
29.742
78.441
E
0.88G
µ
ρ
12.743
30.110
78.483
E
0.91G
µ
ρ
12.744
30.472
78.523
E
0.94G
µ
ρ
12.745
30.828
78.561
E
0.97G
µ
ρ
12.746
31.178
78.598
E
1.0G
µ
ρ
12.747
31.522
78.633
E
1.03G
µ
ρ
12.748
31.862
78.666
E
1.06G
µ
ρ
12.749
32.196
78.699
E
1.09G
µ
ρ
12.750
32.525
78.729
E
1.12G
µ
ρ
12.751
32.849
78.759
E
1.15G
µ
ρ
12.752
33.169
78.788
Table S11. Individual effect of Poisson’s ratio in elastic modulus on first three natural
frequencies (FNF First natural frequency; SNF Second natural frequency; TNF Third
natural frequency)
E
G
µ
ρ
FNF
SNF
TNF
E
G
0.85µ
ρ
12.632
31.420
78.302
E
G
0.88µ
ρ
12.653
31.439
78.364
E
G
0.91µ
ρ
12.675
31.459
78.428
E
G
0.94µ
ρ
12.698
31.479
78.494
E
G
0.97µ
ρ
12.722
31.501
78.562
E
G
1.0µ
ρ
12.747
31.522
78.633
E
G
1.03µ
ρ
12.772
31.545
78.706
E
G
1.06µ
ρ
12.799
31.568
78.781
E
G
1.09µ
ρ
12.826
31.592
78.859
E
G
1.12µ
ρ
12.855
31.617
78.939
E
G
1.15µ
ρ
12.884
31.643
79.022
7
Table S12. Individual effect of variation in density on first three natural frequencies (FNF
First natural frequency; SNF Second natural frequency; TNF Third natural
frequency)
E
G
µ
ρ
FNF
SNF
TNF
E
G
µ
0.85ρ
13.826
34.191
85.289
E
G
µ
0.88ρ
13.588
33.603
83.823
E
G
µ
0.91ρ
13.362
33.045
82.430
E
G
µ
0.94ρ
13.147
32.513
81.103
E
G
µ
0.97ρ
12.942
32.006
79.839
E
G
µ
1.0ρ
12.747
31.522
78.633
E
G
µ
1.03ρ
12.560
31.060
77.479
E
G
µ
1.06ρ
12.381
30.617
76.375
E
G
µ
1.09ρ
12.209
30.193
75.316
E
G
µ
1.12ρ
12.044
29.786
74.301
E
G
µ
1.15ρ
11.886
29.395
73.325
Table S13. Individual effect of variation in elastic modulus on impact responses (Max. CF
- maximum contact force; Max. PD - maximum plate displacement; Max. ID - maximum
impactor displacement)
E
G
µ
ρ
Max. CF
Max. PD
Max. ID
0.85E
G
µ
ρ
567.7461
0.1158
0.0002680
0.88E
G
µ
ρ
571.9020
0.1144
0.0002657
0.91E
G
µ
ρ
576.3519
0.1132
0.0002636
0.94E
G
µ
ρ
580.5395
0.1120
0.0002615
0.97E
G
µ
ρ
584.6342
0.1108
0.0002595
1.0E
G
µ
ρ
588.8216
0.1097
0.0002576
1.03E
G
µ
ρ
592.7098
0.1086
0.0002558
1.06E
G
µ
ρ
596.8470
0.1076
0.0002541
1.09E
G
µ
ρ
600.6416
0.1066
0.0002525
1.12E
G
µ
ρ
604.6926
0.1056
0.0002509
1.15E
G
µ
ρ
608.4800
0.1047
0.0002493
8
Table S14. Individual effect of variation in shear modulus on impact responses (Max. CF
- maximum contact force; Max. PD - maximum plate displacement; Max. ID - maximum
impactor displacement)
E
G
µ
ρ
Max. CF
Max. PD
Max. ID
E
0.85G
µ
ρ
578.5799
0.1138
0.0002630
E
0.88G
µ
ρ
579.9168
0.1133
0.0002624
E
0.91G
µ
ρ
581.0372
0.1129
0.0002619
E
0.94G
µ
ρ
582.2547
0.1126
0.0002613
E
0.97G
µ
ρ
583.3836
0.1122
0.0002608
E
1.0G
µ
ρ
588.8216
0.1097
0.0002576
E
1.03G
µ
ρ
585.1957
0.1115
0.0002598
E
1.06G
µ
ρ
585.9014
0.1112
0.0002593
E
1.09G
µ
ρ
586.7139
0.1108
0.0002588
E
1.12G
µ
ρ
587.5238
0.1105
0.0002583
E
1.15G
µ
ρ
588.2576
0.1101
0.0002579
Table S15. Individual effect of Poisson’s ratio in elastic modulus on impact responses
(Max. CF - maximum contact force; Max. PD - maximum plate displacement; Max. ID -
maximum impactor displacement)
E
G
µ
ρ
Max. CF
Max. PD
Max. ID
E
G
0.85µ
ρ
606.0869
0.1058
0.0002503
E
G
0.88µ
ρ
606.9264
0.1055
0.0002499
E
G
0.91µ
ρ
607.7786
0.1053
0.0002495
E
G
0.94µ
ρ
608.8342
0.1050
0.0002492
E
G
0.97µ
ρ
609.9050
0.1048