Lixin Huang*, Ming Yang, Xiaojun Zhou, Qi Yao and Lin Wang ...

Sci Eng Compos Mater 2016; 23(6): 685–698

*Corresponding author: Lixin Huang, School of Civil Engineering, Guangxi University, Nanning 530004, China; and The Key Laboratory of Disaster Prevention and Structural Safety of the Education Ministry, Guangxi University, Nanning 530004, China, e-mail: [email protected] Yang: School of Civil Engineering, Guangxi University, Nanning 530004, ChinaXiaojun Zhou: Guangzhou Urban Planning and Design Survey Research Institute, Guangzhou 510060, ChinaQi Yao: Shazhou Professional Institute of Technology, Civil Engineering Department, Zhangjiagang 215600, ChinaLin Wang: Times Institute of Architectural Design (Fujian) Co. Ltd., Fuzhou 350003, China

Lixin Huang*, Ming Yang, Xiaojun Zhou, Qi Yao and Lin Wang

Material parameter identification in functionally graded structures using isoparametric graded finite element model

Abstract: An identification algorithm based on an isopara metric graded finite element model is developed to identify the material parameters of the plane structure of func tionally graded materials (FGMs). The material parameter identification problem is formulated as the problem of minimizing the objective function, which is defined as a square sum of differences between measured displacement and calculated displacement by the isoparametric graded finite element approach. The minimization problem is solved by using the LevenbergMarquardt method, in which the sensitivity calculation is based on the differentiation of the governing equations of the isoparametric graded finite element model. The validity of this algorithm is illustrated by some numerical experiments. The numerical results reveal that the proposed algorithm not only has high accuracy and stable convergence, but is also robust to the effects of measured displacement noise.

Keywords: functionally graded materials; isoparametric graded finite element; LevenbergMarquardt method; material parameter identification; sensitivity analysis.

DOI 10.1515/secm-2014-0289Received August 29, 2014; accepted January 1, 2015; previously published online April 17, 2015

1 Introduction

Functionally graded materials (FGMs) are a new kind of inhomogeneous composite materials with material properties that vary continuously in space. Due to the unique graded feature, FGMs possess some advantages over common homogeneous materials and traditional composites, such as improved residual and thermal stress distribution, reduced stress intensity factors, and higher fracture toughness and bonding strength [1]. By designing the material gradients, FGMs can be adapted to a broad range of applications in various fields, such as aerospace, automobile, chemical engineering, electronic packages, and so on. The analysis and design of FGM structures have received considerable attention in recent years. The fruitful research and application of FGMs require their accurate material properties.

Based on the measurements of standardized test samples with a welldefined geometry and loading, the frameworks of determining material properties have long been established for isotropic materials. The unique graded feature makes FGMs behave differently from isotropic materials; therefore, calibrating the material properties of FGMs by these frameworks becomes much more difficult. Due to the limitations of these frameworks, several advanced methods have been developed to identify the material parameters of FGMs. Giannakopoulos and Suresh [2, 3] developed analytical solutions for the evolution of stresses and deformation fields due to indentation from a rigid indenter on a graded substrate. Suresh et al. [4] proposed an analytical solutionbased method to estimate the Young’s modulus variations of graded materials by experimental measurements. Combining instrumented microindentation with inverse analysis and the Kalmen filter technique, some researchers also proposed a measurement procedure to determine the material properties of FGMs [5, 6]. A socalled mixed numericalexperimental method, which integrates experimental techniques, numerical method and optimization techniques, was first developed by Kavanagh and Clough [7–9] for characterizing material parameter of composite structures. In order

686 L. Huang et al.: Material parameter identification in FGMs using IGFEM

to improve the efficiency and accuracy of the proposed method, many efforts have still actively been devoted to developing the mixed numericalexperimental method for the material parameter identification of composite structures ever since [10–24].

For an FGM structure under given boundary and loading conditions, determining the material parameters by a set of measured structure behavior data defines an inverse problem. The corresponding direction problem is that of calculating the structure behavior for given material parameters. The material parameter identification is formulated as the minimization of the objective function defined as a square sum of differences between the experimental and calculated structure behavior data, which belongs to an inverse problem. Given that the material properties of FGMs vary continuously in space and are functions of the coordinates, numerical approaches are required to solve a series of direct problems for the complex FGM structures. As a powerful numerical method, the finite element method (FEM) offers great flexibility in terms of dealing with engineering structures of complex geometry; thus, FEM has been widely used to analyze the FGM structures. The solver of the direction problem is sometimes called hundreds of times in the inverse process [14]. Therefore, the efficiency and accuracy of solving the direct problem plays a very important role in the material parameter identification.

A kind of approach to solve the direct problem is the homogeneous layer element approach, in which the FGM structure is divided into a large number of homogeneous thin layer elements. Such a conventional homogeneous layer element attains a stepwise constant approximation to a gradual change material property. However, this approach is computationally expensive and cannot offer enough accuracy for the material parameter identification. Some researchers developed graded element models with better performance for the analysis of the FGM structures. By generalization of the isoparametric concept, Kim and Paulino [25] developed an isoparametric graded finite element approach (IGFEA), in which the material properties of the FGM structure can be interpolated from the element nodal values using isoparametric shape functions. Wang and Qin [26] presented a boundary integralbased graded element model for analyzing twodimensional functionally graded solids. In their model, the governing equations of the problem are analytically satisfied in the element, with the fundamental solutions approximating the intraelement fields. The natural variation of material definition can be retained in the graded element without any approximation. The fundamental solutions corresponding to FGMs with a certain material variation law are prerequisites for this model. If there are

y

x

-TiAl 100%Y-TZP 0%

-TiAl 0%Y-TZP 100%

Graded layer h

w

Figure 1: Metal-ceramic FGMs.

no fundamental solutions, the variation of material definition can be approximated by the isoparametric shape functions described in the model of Kim and Paulino. Both two models can ensure the accuracy and efficiency of modeling FGM structures than the homogeneous layer element approach.

In the current article, an algorithm is proposed to identify the material parameter of FGM structures. The displacement response is used as the structure behavior data for material parameter identification of FGM structures. The material properties are determined by minimizing the objective function, which is defined as a square sum of differences between the measured displacement response and the one calculated by the IGFEA. The LevenbergMarquardt method is employed to solve the minimization problem. In this method, the sensitivities of displacements with respect to the material parameters are computed by differentiating the governing equations, together with upper and lower bounds on the material parameters. The IGFEA, as the chosen direction problem solver, provides more accuracy and efficiency than other numerical methods, such as the homogeneous layer element approach, which results in reducing the iterative step and computational cost. As the solver of the inverse problem, the LevenbergMarquardt method used in this model belongs to the gradientbased search technique. Thus, the LevenbergMarquardt method has a higher probability to converge to optima for initial guesses than genetic algorithms and neural network when a priori information of parameters is used in the material parameter identification process. Numerical experiments have been tested to illustrate the validity of the proposed material parameter identification method.

2 Materials and methodsAs shown in Figure 1, the FGMs are microscopically heterogeneous and are composed of homogeneous isotropic

L. Huang et al.: Material parameter identification in FGMs using IGFEM 687

derivation of the IGFEA for FGM plane structure can be found in [25].

Consider a typical finite element e for plane problems. The displacements (u, υ) at any point within the element can be approximated as

1,

me

i ii

u N u=

=∑

(9)

1,

m

ii

eiNυ υ

=

=∑

(10)

where ()e indicates a quantity relating to the element, similarly hereinafter, Ni represents shape functions, e

iu and e

iυ denote the nodal displacements corresponding to node i, and m is the number of nodal points within the element e. In this study, as shown in Figure 2, the quadrilateral elements of eight nodes [27] are employed to mesh the FGM structures. The shape functions are given by

1 ( 1 )( 1 )( -1), 1, 2, 3, 44i i i i iN iξξ ηη ξξ ηη= + + + =

(11)

21 ( 1- )( 1 ), 5, 7

2i iN iξ ηη= + =

(12)

21 ( 1 )( 1- ), 6, 8

2i iN iξξ η= + =

(13)

where (ξ, η) denote local coordinates with their limit from 1 to +1, and (ξi, ηi) are the local coordinates of node i. The socalled isoparametric finite element is that the global Cartesian coordinates (x, y) of a point in the element are interpolated from nodal values by using the same shape functions as the displacements. Therefore, the global Cartesian coordinates (x, y) can be given as

1,

me

i ii

x N x=

=∑

(14)

1.

me

i ii

y N y=

=∑

(15)

metal and ceramic components, such as titanium aluminide (γTiAl) and yttriastabilized tetragonal zirconia polycrystal (YTZP) [4]. It is usually assumed that material properties are either a linear or an exponential function of a spatial variable. Thus, Young’s modulus and Poisson’s ratio can be expressed as functions of the Cartesian coordinate y with a linear or an exponential variation [25], i.e.,

0( ) EE y E yγ= + (1)

0( )y yν

ν ν γ= + (2)

for linear material variation and

0( ) exp( ),EE y E yβ= (3)

0( ) exp( ),y yν

ν ν β= (4)

for exponential material variation,where E0 = E(0), ν0 = ν(0) are the material properties at the y = 0 line; E1 = E(h), ν1 = ν(h) are the material properties at the y = h line; and γE and γ

ν are independent nonhomoge

neity parameters characterized by

1 0-( )- ( 0) ,E

E EE h Eh h

γ = =

(5)

1 0-( )- ( 0) .h

h hν

ν νν νγ = =

(6)

βE and βν are independent nonhomogeneity parameters

characterized by

1

0

1 ( ) 1ln ln ,(0)E

EE hh E h E

β

= =

(7)

1

0

1 ( ) 1ln ln .(0)h

h hν

ννβ

ν ν

= =

(8)

3 Isoparametric graded finite element approach for the direct problem

As mentioned in Section 1, numerical approaches are required to solve a series of direct problems for the complex FGM structures in material parameter identification. Due to its high accuracy and efficiency, the numerical approach for the direct problem in the work described here is based on the IGFEA. Only the concept and the equations necessary for the material parameter identification in this paper are summarized. The detailed

3

6

25

1

8

4 7 η

ξ

Figure 2: Quadrilateral element of eight nodes.


The concept of isoparametric finite elements in the finite element analysis was first introduced by Taig [28]. By generalization of the isoparametric concept, Kim and Paulino [25] proposed that material properties be interpolated from the nodal material properties of the element for the FGMs structures. Thus, the Young’s modulus and Poisson’s ratio are interpolated as

1,

me

i ii

E N E=

=∑

(16)

1.

me

i iiNν ν

=

=∑

(17)

In the finite element analysis, the variation of the material properties can be approximated by isoparametric shape functions defining displacements and geometry. Thus, the above framework offers the same procedures as used in the standard FEM [27] for the analysis of FGM structures.

4 Parameter identification algorithm for functionally graded structures

4.1 Objective function

Consider an FMG plane structure. The general stressstrain relations of the FMG plane structure can be given as

( ) ,x x

y y

xy xy

yσ ε

σ ε

τ γ

=

D

(18)

in which [D(y)] is an elasticity matrix containing the appropriate material properties defined by

2

1 ( ) 0( )( ) ( ) 1 0 .

1- ( ) 1- ( )0 02

yE yy y

y y

ν

νν

ν

=

D

(19)

The governing equation of the FEM for the FMGs plane structure can be presented as

{ } { }( ) ( ) ,y y = K D u D f

(20)

where [K[D(y)]] is the stiffness matrix of structure, {u[D(y)]} is the vector of nodal displacements, and { f } is the vector of nodal forces.

In the present study, the Poisson’s ratio ν is assumed constant. The material parameters to be identified include

two independent parameters, i.e., E0 and E1. It means that the design variables are E0 and E1 in the parameter identification process. Thus, the parameter identification problem based on the measured displacement can be formulated as an ordinary leastsquares problem expressed as

2

1

1( ) ( ), , ,Minim e2

izl

ni

iF r l n

=

= ∈ ≥∑p p p R

(21)

{ } { }( ) ( )Subje ,ct to = K p u p f (22)

where p is the vector of parameter to be identified, i.e., 1 2 0 1 ;

T Tp p E E = = p F(p) is the objective function;

l is the number of measured displacements; and n is the number of parameters to be identified. The function ri(p) is defined by

( ) ( )- ,i i ir u u∗=p p (23)

where ui(p) is the calculated displacement by the IGFEA, and iu

∗ is the measured displacement.The components ri(p)(i = 1, 2, …, l) from Eq. (23) are

assembled into the vector r(p), which is defined by

1 2( ) ( ) ( ) ( ) .T

lr r r = r p p p p� (24)

Substituting Eq. (23) into Eq. (24) yields

1 1 2 2( ) ( ) ( ) ( ) ( )- .T

l lu u u u u u∗ ∗ ∗ ∗ = − = − − r p u p u p p p… (25)

4.2 Sensitivity analysis

Minimizing the objective function F(p) defined in Eq. (21) is a difficult task because the objective function F(p) is nonlinear. It is necessary to analyze the sensitivity of F(p) or r(p) with respect to the parameters in order to implement the inverse analysis procedure. Given that the measured displacements ( )1, 2, , iu i l∗ = � in Eq. (25) are independent of the material parameters, the sensitivity of r(p) with respect to the material parameters, i.e., the Jacobian matrix of r(p), is defined by

1 1

0 1

2 2

0 12

0 1

( ) ( )

( ) ( )( )

( ) .

( ) ( )

i

j l

l l

u uE E

u uu

E Ep

u uE E

∂ ∂

∂ ∂

∂ ∂∂

∂ ∂∂

∂ ∂

∂ ∂

×

= =

p p

p pp

J p

p p� �

(26)


The Jacobian matrix J(p) can be calculated by differentiating the system of algebraic equations presented in Eq. (22). Given that {f} is independent of the material parameters to be identified, differentiating Eq. (22) with respect to p yields

{ }( ) ( )( ) ( ) 0.

j jp p∂ ∂

∂ ∂

+ =

K p u pu p K p

(27)

From Eqs. (26) and (27), the Jacobian matrix J(p) can be given as

{ }-1

2

( ) ( )( ) - ( ) ( ) .i

j jl

up p

∂ ∂∂ ∂

×

= =

p K pJ p K p u p

(28)

For the quadrilateral element of eight nodes, the element stiffness matrix is

[ ] [ ]16 3 3 1616 16 3 3( ) ( ) ,

T Te e e ey t dxdy× ×× × = ∫∫K p B D B

(29)

where t is the element thickness, [Be] is called the straindisplacement matrix [27]. Under the circumstances of constant Poisson’s ratio, the elasticity matrix [D(y)] can be given by using isoparametric shape functions, i.e.,

8

12

1 0( )( ) 1 0 .

1- 10 0 ( 1- )2

i i iiN E y

yν

νν

ν

=

=

∑D

(30)

Differentiating Eq. (29) with respect to the design variables yields

16 3 3 16

16 16 3 3

( ) ( ) ,e eT Te e

j j

y t dxdyp p

∂ ∂∂ ∂× ×

× ×

= ∫∫K p DB B

(31)

in which the derivatives of the elasticity matrix with respect to the design variables can be derived, respectively. For linear material variation, the elasticity matrix [D(y)] is given by

8 1 001

2

- 1 0( ) 1 0 .

1- 10 0 ( 1- )2

i ii

E EN E y

hyν

νν

ν

=

+ =

∑D

(32)


8

1

20

1 1 01-( ) 1 0 ,1- 10 0 ( 1- )

2

i iiN yy h

E

ν∂

ν∂ ν

ν

=

=

∑D

(33)

8

1

21

1 1 0( ) 1 0 .

1- 10 0 ( 1- )2

i iiN yy h

E

ν∂

ν∂ ν

ν

=

=

∑D

(34)

For exponential material variation, the elasticity matrix [D(y)] is expressed as

8 101

02

exp ln 1 0( ) 1 0 .

1- 10 0 ( 1- )2

iii

y EN E

h Ey

ν

νν

ν

=

=

∑D

(35)


8 11

02

0

1- exp ln 1 0( ) 1 0 ,

1- 10 0 ( 1- )2

i iii

y y EN

h h EyE

ν∂

ν∂ ν

ν

=

=

∑D

(36)

8 0 11

1 02

1

exp ln 1 0( ) 1 0 .

1- 10 0 ( 1- )2

i iii

E y y EN

E h h EyE

ν∂

ν∂ ν

ν

=

=

∑D

(37)

The stiffness matrix of structure can be assembled by

1( ) ( ) ,

numele

e=

= ∑K p K p

(38)

where numel is the number of element in the structure. The summation implies the assembly of element matrices by the addition of overlapping terms according to node numbers. Differentiating Eq. (38) with respect to the design variables gives

1

( ) ( ) .enumel

ej jp p∂ ∂

∂ ∂=

=

∑K p K p

(39)

All the derivatives of displacements with respect to the design variables can be determined by substituting Eq. (39) into Eq. (28).

4.3 Levenberg-Marquardt method

For the nonlinear leastsquares problems defined in Eq. (21), the LevenbergMarquardt method is convenient and effective in the small residual case. A sequence of steps to adjust the parameters can be obtained by using


the LevenbergMarquardt method until convergence is achieved according to the specified criteria. The adjusted parameters at iteration k are calculated from the equations given by

( ) ( ) ( ) ( ) ( ) ( )( ) ( ) - ( ) ( ),k T k k k k T kα + = J p J p I J p r pδ (40)

( 1) ( ) ( ) ,k k k+ = +p p δ (41)

where I is a unit matrix, and α(k) is the Levenberg Marquardt parameter, which is a non negative scalar.

The convergence criteria are defined as

( ) ( )

1( ) ( ) ,k T k ε<J p r p (42)

( 1) ( )

2( )1

-,

k kli i

ki i

p pp

ε+

=

<∑

(43)

where ε1 and ε2 are the specified accuracy requirements.

5 Numerical results and discussionAccording to the scheme described above, numerical experiments are presented to validate the proposed algorithm. Figures 3 and 7 show the experiment model problems as well as the square and rectangular plane structures made from a sintered γTiAl/YTZP functionally graded material [4], where loads, geometry dimensions, and boundary conditions are specified. The true material parameters are listed in Table 1.

y

x

Graded layer

-TiAl 100%Y-TZP 0%

-TiAl 0%Y-TZP 100%

E(y)=0.3

h=90 mm

q=100 N/mm

90 mm

Figure 3: Square FGM plane structure.

Table 1: The true values of the material parameter.

Material component γ-TiAl Y-TZP

Young’s modulus E (GPa) 186 209

In these numerical experiments, unless otherwise specified, the following values and conditions are used:

Poisson’s ratio ν = 0.3 Initial value of the LevenbergMarquardt parameter α(0) = 1018

Accuracy requirements of convergence criterions ε1 = 106 and ε2 = 103

Upper and lower bounds of the two parameters to be identified

0 147 GPa 372 GPa, 52 GPa 418 GPaE E≤ ≤ ≤ ≤

Both the linear material variation and exponential material variation are investigated. In the parameter identification process, the displacements computed by the IGFEA using the true values of material parameters replace the measured ones. In this study, both the noisefree and noisecontaminated displacements are considered for the parameter identification. The noisecontaminated displacements are generated by adding the Gauss noise of different levels to the IGFEAcomputed displacements with the true values of material parameters. A series of pseudorandom numbers, i.e., the Gauss noise, are produced from a Gauss distribution with mean μ = 0 and standard deviation ˆ.σ The standard deviation σ̂ is defined as [14, 29]

2

1

1ˆ ( ) ,n

me i

ip u

nσ

== × ∑

(44)

where miu denotes the IGFEAcomputed displacement at

the ith measurement point, n is the number of measured displacements, and pe is the level of the noise contamination, such as 1%, 2%, and 5%. Six experiment groups of noisecontaminated displacements for each noise level and one experiment group of noisefree displacements are employed to study the sensitivity and stability of the proposed identification algorithm to noise.

5.1 Square FGMs plane structure

As shown in Figure 3, a square FGM plane structure under the uniform tension of 100 N/mm on the right side is investigated. Only the material parameter E1 is to be identified. ABAQUS standard finite element program, in which the FGM plane structure is discretized by 100 isoparametric graded quadrilateral eightnode elements and 341 nodes, is used to analyze the deformation. Figure 4 shows the finite element mesh of the square FGM plane structure and vertical displacement at node m1; the horizontal displacement at node m2 are regarded as the measured data.


high accuracy and stable convergence. Furthermore, the parameter identification process can be successfully finished in a few iteration steps (no more than five steps).

Table 2 lists the initial values of parameter E1 and two cases are investigated.

5.1.1 Linear material variation

The identified results of case 1 and case 2 are listed in Tables 3 and 4, respectively. With noisefree displacements, the maximum error of the identified parameter is very small, i.e., 0.0083% in case 1. For the noise levels of 1%, 2% and 5%, the maximum errors of the identified parameter are 1.87% in group 3 of case 1, 4.42% in group 4 of case 1 and 4.78% in group 2 of case 2, respectively.

5.1.2 Exponential material variation

Tables 5 and 6 present the identified results of case 1 and case 2, respectively. For noisefree displacements, the maximum error of the identified parameter is 0.07% in case 1. For the noise levels of 1%, 2% and 5%, the maximum errors of the identified parameter are 2.44% in group 3 of case 1, 4.34% in group 3 of case 2 and 6.94% in group 6 of case 1, respectively.

For both linear material variation and exponential material variation, the identified results shown in Tables 3–6 illustrate that the proposed algorithm has

x

y

m1(72, 27)

m2(90, 9)m2

m1

Figure 4: Finite element mesh of square FGM plane structure (measurement locations are indicated by the heavy circle).

Table 2: Initial values of material parameter in a square FGM plane structure.

Cases Parameter Initial values (GPa)

True values (GPa)

Initial value/True

value

1 E1 105 209 0.52 E1 313 209 1.5

Table 3: Identified parameters of a square FGM plane structure with linear material variation (case 1: initial value E1 = 105 GPa).

Noise levels

Groups

E1 (GPa)

Iterations

True value Identified Errors (%)

Noise free 1 209.00 208.98 -0.0083 41% noise 1 209.00 209.44 0.21 4

2 209.00 207.21 -0.86 4 3 209.00 205.10 -1.87 4 4 209.00 207.97 -0.49 4 5 209.00 212.67 1.76 4 6 209.00 205.82 -1.52 4

2% noise 1 209.00 213.06 1.94 4 2 209.00 210.65 0.79 4 3 209.00 208.10 -0.43 4 4 209.00 199.76 -4.42 4 5 209.00 209.13 0.06 4 6 209.00 211.76 1.32 4

5% noise 1 209.00 210.76 0.84 4 2 209.00 199.80 -4.40 4 3 209.00 214.24 2.51 4 4 209.00 216.43 3.56 4 5 209.00 216.11 3.40 4 6 209.00 211.71 1.30 4

Table 4: Identified parameters of a square FGM plane structure with linear material variation (case 2: initial value E1 = 313 GPa).

Noise levels

Groups

E1 (GPa)

Iterations


Noise free 1 209.00 208.99 -0.0062 41% noise 1 209.00 208.03 -0.46 4

2 209.00 205.55 -1.65 4 3 209.00 208.12 -0.42 4 4 209.00 209.56 0.27 4 5 209.00 211.81 1.34 4 6 209.00 209.95 0.45 4

2% noise 1 209.00 202.67 -3.03 4 2 209.00 211.99 1.43 4 3 209.00 208.48 -0.25 4 4 209.00 203.84 -2.47 4 5 209.00 213.10 1.96 4 6 209.00 217.46 4.05 4

5% noise 1 209.00 202.28 -3.22 4 2 209.00 219.00 4.78 4 3 209.00 211.98 1.43 4 4 209.00 215.81 3.26 4 5 209.00 208.45 -0.26 4 6 209.00 203.72 -2.53 4


As further assessment, the iterative identification processes with the maximum errors are chosen to investigate the performance of convergence and stability of the

Table 5: Identified parameters of a square FGM plane structure with exponential material variation (case 1: initial value E1 = 105 GPa).

Noise levels

Groups

E1 (GPa)

Iterations


Noise free 1 209.00 208.85 -0.07 41% noise 1 209.00 210.44 0.69 4

2 209.00 208.13 -0.42 4 3 209.00 214.11 2.44 4 4 209.00 212.08 1.47 4 5 209.00 210.17 0.56 5 6 209.00 207.43 -0.75 4

2% noise 1 209.00 216.75 3.71 4 2 209.00 212.42 1.64 5 3 209.00 210.01 0.48 4 4 209.00 210.03 0.49 5 5 209.00 204.61 -2.10 4 6 209.00 214.28 2.53 4

5% noise 1 209.00 211.93 1.40 4 2 209.00 208.30 -0.33 4 3 209.00 208.17 -0.40 4 4 209.00 203.53 -2.62 4 5 209.00 205.76 -1.55 4 6 209.00 223.50 6.94 4

Table 6: Identified parameters of a square FGM plane structure with exponential material variation (case 2: initial value E1 = 313 GPa).

Noise levels

Groups

E1 (GPa)

Iterations


Noise free 1 209.00 208.98 -0.01 41% noise 1 209.00 210.91 0.91 4

2 209.00 206.68 -1.11 4 3 209.00 213.14 1.98 4 4 209.00 210.08 0.52 4 5 209.00 208.18 -0.39 4 6 209.00 210.80 0.86 4

2% noise 1 209.00 207.35 -0.79 4 2 209.00 212.08 1.47 4 3 209.00 218.08 4.34 4 4 209.00 212.50 1.67 4 5 209.00 211.00 0.96 4 6 209.00 211.79 1.33 4

5% noise 1 209.00 197.14 -5.67 4 2 209.00 203.44 -2.66 4 3 209.00 215.39 3.06 4 4 209.00 213.49 2.15 4 5 209.00 215.64 3.18 4 6 209.00 200.66 -3.99 5

0 1 2 3 40

5

10

15

20

25

30

35

40

45

50

Number of iteration

Err

(%

)

0% noice, case 1

1% noice, group 3 of case 1



Figure 5: Convergence rate in the identification process of a square FGM plane structure with linear material variation.

proposed model. Thus, the relative errors of the identified parameter E1 are given by

-Err 100%,I T

T

V VV

= ×

(45)

where VI and VT are the identified value and the true value of the identified parameter E1, respectively. Figures 5 and 6 show the iterative identification processes. As can be seen, the proposed model has higher convergence rate and the identified values finally approach the true ones within a stable range.

5.2 Rectangular FGM plane structure

As a further illustration, the proposed parameter identification algorithm is applied to a rectangular FGM plane structure, as shown in Figure 7. Two concentrated forces are applied on the upper edge of the structure. As shown in Figure 8, the rectangular FGM plane structure is meshed by 300 isoparametric graded quadrilateral eightnode elements and 971 nodes. The ABAQUS standard finite element program is used to analyze the deformation in the parameter identification process. Vertical displacement at node m1 and horizontal displacements at nodes m2, m3, and m4 are regarded as the measured data for the material parameter identification. Four cases of initial values of parameters listed in Table 7 are selected for the material parameter identification.


5.2.1 Linear material variation

For the four cases of the initial values, the identified results of the parameters are shown in Tables 8–11, respectively. As can be seen, the identification performance is satisfactory. For the computed displacements without adding Gauss noise, the maximum errors of the identified parameter E0 and E1 are 0.06% in case 4 and 0.05% in case 2, respectively. For the noisecontaminated displacements of 1%, 2% and 5% levels, the maximum errors of the identified parameter E0 are 1.44% in group 4 of case

0 1 2 3 40

5

10

15

20

25

30

35

40

45

50

Number of iteration

Err

(%

)

0% noice, case 1




Figure 6: Convergence rate in the identification process of a square FGM plane structure with exponential material variation.

Graded layer

-TiAl 100%Y-TZP 0%

-TiAl 0%Y-TZP 100%

E(y)=0.3

F=50 kN F=50 kNy

x

70 mm 70 mm 60 mm

h=150 mm

Figure 7: Rectangular FGM plane structure.

m1(20, 0)

y

x

m2

m1

m3

m4

m2(100, 90)

m3(110, 60)

m4(120, 90)

Figure 8: Finite element mesh of rectangular FGM plane structure (measurement locations are indicated by the heavy circle).

Table 7: Initial values of material parameter in a rectangular FGM plane structure.

Cases Parameters Initial values (GPa)

True values (GPa)

Initial value/True

value

1 E0 93 186 0.5 E1 105 209 0.5

2 E0 279 186 1.5 E1 313 209 1.5

3 E0 93 186 0.5 E1 313 209 1.5

4 E0 279 186 1.5 E1 105 209 0.5

2, 1.99% in group 6 of case 3 and 4.80% in group 1 of case 3, respectively; meanwhile, the maximum errors of the identified parameter E1 are 1.95% in group 2 of case 1, 4.34% in group 2 of case 4 and 7.19% in group 5 of case 4, respectively.

5.2.2 Exponential material variation

Tables 12–15 list the identified results of the parameters for the four cases, respectively. As can be seen, the material parameter can be successfully identified. Under the condition of noisefree displacements, the maximum errors of the identified parameters E0 and E1 are 0.11% and 0.06% in case 4, respectively. For the noisecontaminated displacements of 1%, 2% and 5% levels, the maximum errors of the identified parameter E0 are 1.54% in group 4 of case 3, 2.33% in group 3 of case 3 and 6.31% in group 4 of case 1, respectively, and in the meantime the maximum errors


Table 8: Identified parameters of a rectangular FGM plane structure with linear material variation (case 1: initial value E0 = 93 GPa, E1 = 105 GPa).

Noise levels

Groups

E0 (GPa)

E1 (GPa)

Iterations

True value Identified Errors (%) True value Identified Errors (%)

Noise free 1 186.00 185.97 -0.02 209.00 209.08 0.04 71% noise 1 186.00 186.93 0.50 209.00 205.33 -1.76 11

2 186.00 185.85 -0.08 209.00 204.92 -1.95 6 3 186.00 186.40 0.22 209.00 210.80 0.86 6 4 186.00 185.63 -0.20 209.00 207.69 -0.63 5 5 186.00 185.35 -0.35 209.00 206.11 -1.38 4 6 186.00 183.71 -1.23 209.00 209.29 0.14 7

2% noise 1 186.00 184.99 -0.54 209.00 207.10 -0.91 4 2 186.00 183.78 -1.19 209.00 208.89 -0.05 5 3 186.00 186.22 0.12 209.00 205.55 -1.65 6 4 186.00 183.28 -1.46 209.00 208.21 -0.38 5 5 186.00 186.79 0.42 209.00 204.98 -1.92 5 6 186.00 185.34 -0.35 209.00 213.77 2.28 6

5% noise 1 186.00 178.72 -3.91 209.00 209.24 0.11 5 2 186.00 188.42 1.30 209.00 203.44 -2.66 16 3 186.00 185.45 -0.30 209.00 195.37 -6.52 11 4 186.00 181.87 -2.22 209.00 209.01 0.01 5 5 186.00 193.85 4.22 209.00 203.40 -2.68 7 6 186.00 183.32 -1.44 209.00 206.59 -1.15 20


Noise levels

Groups

E0 (Gpa)

E1 (GPa)

Iterations


Noise free 1 186.00 185.92 -0.04 209.00 208.90 -0.05 41% noise 1 186.00 185.63 -0.20 209.00 210.47 0.70 6

2 186.00 185.65 -0.19 209.00 208.97 -0.01 5 3 186.00 185.83 -0.09 209.00 208.91 -0.04 6 4 186.00 188.62 1.44 209.00 209.19 0.09 4 5 186.00 186.04 0.02 209.00 206.43 -1.23 4 6 186.00 186.22 0.12 209.00 208.42 -0.28 5

2% noise 1 186.00 184.11 -1.02 209.00 209.21 0.10 4 2 186.00 183.89 -1.13 209.00 200.97 -3.84 15 3 186.00 182.45 -1.91 209.00 209.70 0.33 5 4 186.00 185.56 -0.24 209.00 204.88 -1.97 18 5 186.00 182.72 -1.76 209.00 207.23 -0.85 4 6 186.00 183.80 -1.18 209.00 205.59 -1.63 4

5% noise 1 186.00 191.40 2.90 209.00 197.02 -5.73 5 2 186.00 180.19 -3.12 209.00 201.15 -3.76 21 3 186.00 181.73 -2.30 209.00 210.16 0.56 19 4 186.00 185.62 -0.20 209.00 204.74 -2.04 9 5 186.00 188.86 1.54 209.00 196.54 -5.96 8 6 186.00 188.61 1.40 209.00 218.14 4.37 17

of the identified parameter E1 are 2.16% in group 4 of case two, 5.00% in group 3 of case 2 and 8.16% in group 2 of case 2, respectively.

For both linear material variation and exponential material variation, the identified results illustrate that the proposed algorithm has high accuracy; furthermore,


the material parameter can be successfully identified in a few iteration steps under the condition of noisefree displacements. With the noisecontaminated displacements,

the maximum errors of the identified parameters due to propagation of measurement errors are within the acceptable limit.


Noise levels

Groups

E0 (GPa)

E1 (GPa)

Iterations


Noise free 1 186.00 186.00 0.00 209.00 209.00 0.00 41% noise 1 186.00 188.48 1.33 209.00 210.55 0.74 4

2 186.00 186.52 0.28 209.00 206.28 -1.30 4 3 186.00 184.31 -0.91 209.00 206.29 -1.30 5 4 186.00 186.72 0.39 209.00 209.74 0.35 4 5 186.00 186.14 0.08 209.00 210.79 0.86 4 6 186.00 186.73 0.39 209.00 207.13 -0.89 5

2% noise 1 186.00 187.74 0.94 209.00 214.29 2.53 11 2 186.00 186.23 0.12 209.00 210.00 0.48 5 3 186.00 187.63 0.88 209.00 207.22 -0.85 5 4 186.00 184.71 -0.69 209.00 212.27 1.56 8 5 186.00 187.55 0.83 209.00 213.48 2.14 4 6 186.00 182.30 -1.99 209.00 200.82 -3.91 17

5% noise 1 186.00 194.93 4.80 209.00 195.33 -6.54 17 2 186.00 181.26 -2.55 209.00 210.12 0.53 15 3 186.00 179.49 -3.50 209.00 218.46 4.53 17 4 186.00 186.34 0.18 209.00 215.87 3.29 20 5 186.00 183.26 -1.47 209.00 197.23 -5.63 17 6 186.00 187.12 0.60 209.00 213.85 2.32 8


Noise levels

Groups

E0 (GPa)

E1 (GPa)

Iterations


Noise free 1 186.00 185.88 -0.06 209.00 208.91 -0.04 41% noise 1 186.00 186.42 0.23 209.00 208.27 -0.35 4

2 186.00 184.16 -0.99 209.00 206.19 -1.34 4 3 186.00 187.20 0.65 209.00 207.99 -0.48 4 4 186.00 186.33 0.18 209.00 209.33 0.16 5 5 186.00 187.50 0.81 209.00 212.94 1.89 4 6 186.00 186.84 0.45 209.00 205.75 -1.56 11

2% noise 1 186.00 185.21 -0.42 209.00 201.63 -3.53 6 2 186.00 185.40 -0.32 209.00 199.92 -4.34 5 3 186.00 185.65 -0.19 209.00 206.75 -1.08 6 4 186.00 187.56 0.84 209.00 215.26 3.00 4 5 186.00 187.41 0.76 209.00 213.06 1.94 10 6 186.00 186.98 0.53 209.00 207.64 -0.65 17

5% noise 1 186.00 190.46 2.40 209.00 206.15 -1.36 5 2 186.00 177.52 -4.56 209.00 204.69 -2.06 4 3 186.00 187.64 0.88 209.00 207.19 -0.87 8 4 186.00 180.34 -3.04 209.00 208.15 -0.41 5 5 186.00 189.97 2.13 209.00 224.03 7.19 15 6 186.00 190.89 2.63 209.00 220.73 5.61 19


Table 12: Identified parameters of a rectangular FGM plane structure with exponential material variation (case 1: initial value E0 = 93 GPa, E1 = 105 GPa).

Noise levels

Groups

E0 (GPa)

E1 (GPa)

Iterations


Noise free 1 186.00 186.00 0.00 209.00 209.09 0.04 51% noise 1 186.00 187.70 0.91 209.00 206.24 -1.32 9

2 186.00 184.27 -0.93 209.00 212.27 1.56 4 3 186.00 184.71 -0.69 209.00 208.51 -0.23 10 4 186.00 186.29 0.16 209.00 205.76 -1.55 18 5 186.00 186.49 0.26 209.00 206.45 -1.22 6 6 186.00 184.68 -0.71 209.00 206.56 -1.17 6

2% noise 1 186.00 186.14 0.08 209.00 212.08 1.47 5 2 186.00 187.37 0.74 209.00 207.16 -0.88 6 3 186.00 185.97 -0.02 209.00 209.30 0.14 4 4 186.00 182.91 -1.66 209.00 205.11 -1.86 11 5 186.00 182.65 -1.80 209.00 213.65 2.22 5 6 186.00 186.20 0.11 209.00 205.62 -1.62 5

5% noise 1 186.00 180.42 -3.00 209.00 195.44 -6.49 13 2 186.00 191.67 3.05 209.00 196.02 -6.21 21 3 186.00 192.00 3.23 209.00 222.24 6.33 13 4 186.00 197.74 6.31 209.00 223.58 6.98 17 5 186.00 180.35 -3.04 209.00 198.38 -5.08 12 6 186.00 182.10 -2.10 209.00 214.46 2.61 21


Noise levels

Groups

E0 (GPa)

E1 (GPa)

Iterations


Noise free 1 186.00 186.01 0.01 209.00 209.08 0.04 61% noise 1 186.00 184.04 -1.05 209.00 209.43 0.21 5

2 186.00 186.48 0.26 209.00 207.36 -0.78 11 3 186.00 185.82 -0.10 209.00 209.93 0.44 5 4 186.00 186.96 0.52 209.00 204.49 -2.16 5 5 186.00 186.57 0.31 209.00 206.30 -1.29 6 6 186.00 185.05 -0.51 209.00 204.79 -2.01 16

2% noise 1 186.00 189.05 1.64 209.00 206.92 -1.00 4 2 186.00 187.74 0.94 209.00 205.01 -1.91 8 3 186.00 184.61 -0.75 209.00 198.56 -5.00 5 4 186.00 188.44 1.31 209.00 219.31 4.93 15 5 186.00 185.90 -0.05 209.00 205.20 -1.82 18 6 186.00 185.60 -0.22 209.00 201.31 -3.68 5

5% noise 1 186.00 186.76 0.41 209.00 216.75 3.71 4 2 186.00 189.64 1.96 209.00 226.05 8.16 7 3 186.00 187.56 0.84 209.00 192.79 -7.76 17 4 186.00 186.88 0.47 209.00 224.66 7.49 19 5 186.00 174.69 -6.08 209.00 216.79 3.73 5 6 186.00 188.50 1.34 209.00 211.71 1.30 8

6 ConclusionsBased on the IGFEA, an algorithm is proposed to identify the material parameters of an FGM plane structure whose

elastic constants vary continuously according to linear or exponential form. The validity of this algorithm is illustrated by some numerical experiments. The numerical results demonstrate that the proposed algorithm for


parameter identification of the FGM plane structure has high accuracy and stable convergence.

The unavoidable noise in measurement is modeled by Gaussian noise. With the noisecontaminated

displacements, the numerical results remain stable, and the maximum errors of the identified parameter are within the acceptable limit. Those numerical experiments reveal that the proposed algorithm is robust to the effect


Noise levels

Groups

E0 (GPa)

E1 (GPa)

Iterations


Noise free 1 186.00 185.89 -0.06 209.00 209.07 0.03 41% noise 1 186.00 186.24 0.13 209.00 207.97 -0.49 4

2 186.00 188.33 1.25 209.00 209.84 0.40 4 3 186.00 184.87 -0.61 209.00 208.39 -0.29 5 4 186.00 188.87 1.54 209.00 210.34 0.64 4 5 186.00 186.64 0.34 209.00 209.99 0.47 7 6 186.00 185.91 -0.05 209.00 213.23 2.02 4

2% noise 1 186.00 185.63 -0.20 209.00 211.12 1.01 4 2 186.00 188.25 1.21 209.00 208.96 -0.02 4 3 186.00 181.67 -2.33 209.00 209.91 0.44 4 4 186.00 187.08 0.58 209.00 210.44 0.69 5 5 186.00 187.79 0.96 209.00 206.68 -1.11 28 6 186.00 186.68 0.37 209.00 205.72 -1.57 4

5% noise 1 186.00 177.88 -4.37 209.00 224.51 7.42 22 2 186.00 176.69 -5.01 209.00 219.96 5.24 21 3 186.00 175.73 -5.52 209.00 199.52 -4.54 20 4 186.00 186.07 0.04 209.00 197.33 -5.58 19 5 186.00 194.12 4.37 209.00 194.46 -6.96 20 6 186.00 187.66 0.89 209.00 214.45 2.61 15


Noise levels

Groups

E0 (GPa)

E1 (GPa)

Iterations


Noise free 1 186.00 185.80 -0.11 209.00 209.13 0.06 41% noise 1 186.00 184.36 -0.88 209.00 206.72 -1.09 9

2 186.00 187.78 0.96 209.00 210.97 0.94 6 3 186.00 188.13 1.15 209.00 209.74 0.35 5 4 186.00 184.93 -0.58 209.00 211.32 1.11 5 5 186.00 185.14 -0.46 209.00 206.34 -1.27 8 6 186.00 184.84 -0.62 209.00 208.22 -0.37 5

2% noise 1 186.00 184.96 -0.56 209.00 217.06 3.86 8 2 186.00 186.23 0.12 209.00 211.56 1.22 5 3 186.00 185.25 -0.40 209.00 203.45 -2.66 10 4 186.00 188.24 1.20 209.00 212.37 1.61 5 5 186.00 185.53 -0.25 209.00 214.66 2.71 5 6 186.00 185.73 -0.15 209.00 200.97 -3.84 7

5% noise 1 186.00 185.49 -0.27 209.00 196.42 -6.02 11 2 186.00 181.06 -2.66 209.00 199.57 -4.51 7 3 186.00 195.16 4.92 209.00 214.58 2.67 6 4 186.00 181.18 -2.59 209.00 195.99 -6.22 16 5 186.00 193.72 4.15 209.00 211.42 1.16 5 6 186.00 187.45 0.78 209.00 216.05 3.37 5


of measured displacement noise. The unavoidable measurement noise has great effect on the identified parameters and, therefore, some schemes should be taken to minimize the effect of measurement errors. In addition to improving the quality of measurements, the optimal measurement placement deserves to be studied in order to resolve this problem of error effect.

Acknowledgments: This work has been financially supported by the National Natural Science Foundation of China (Grant Number: 11262002) and Key Project of Guangxi Science Technology Lab Center (LGZX201101). The support is gratefully acknowledged.

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Lixin Huang*, Ming Yang, Xiaojun Zhou, Qi Yao and Lin Wang ...

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