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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

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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

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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

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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

i

f

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]

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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)

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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

{F(Ω2)} 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

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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 0̊, 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 0̊. 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