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

  • Lixin Huang EMAIL logo , Ming Yang , Xiaojun Zhou , Qi Yao and Lin Wang
Published/Copyright: April 17, 2015
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Science and Engineering of Composite Materials
From the journal Science and Engineering of Composite Materials Volume 23 Issue 6

Abstract

An identification algorithm based on an isoparametric graded finite element model is developed to identify the material parameters of the plane structure of functionally 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 Levenberg-Marquardt 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; Levenberg-Marquardt method; material parameter identification; sensitivity analysis

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 well-defined 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 solution-based method to estimate the Young’s modulus variations of graded materials by experimental measurements. Combining instrumented micro-indentation 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 so-called mixed numerical-experimental method, which integrates experimental techniques, numerical method and optimization techniques, was first developed by Kavanagh and Clough [7], [8], [9] for characterizing material parameter of composite structures. In order to improve the efficiency and accuracy of the proposed method, many efforts have still actively been devoted to developing the mixed numerical-experimental method for the material parameter identification of composite structures ever since [10], [11], [12], [13], [14], [15], [16], [17], [18], [19], [20], [21], [22], [23], [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 integral-based graded element model for analyzing two-dimensional functionally graded solids. In their model, the governing equations of the problem are analytically satisfied in the element, with the fundamental solutions approximating the intra-element 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 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 Levenberg-Marquardt 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 Levenberg-Marquardt method used in this model belongs to the gradient-based search technique. Thus, the Levenberg-Marquardt 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 methods

As shown in Figure 1, the FGMs are microscopically heterogeneous and are composed of homogeneous isotropic metal and ceramic components, such as titanium aluminide (γ-TiAl) and yttria-stabilized tetragonal zirconia polycrystal (Y-TZP) [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.,

Figure 1: Metal-ceramic FGMs.
Figure 1:

Metal-ceramic FGMs.

(1)E(y)=E0+γEy
(2)ν(y)=ν0+γνy

for linear material variation and

(3)E(y)=E0exp(βEy),
(4)ν(y)=ν0exp(βνy),

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 γν are independent nonhomogeneity parameters characterized by

(5)γE=E(h)-E(0)h=E1-E0h,
(6)γν=ν(h)-ν(0)h=ν1-ν0h.

βE and βν are independent nonhomogeneity parameters characterized by

(7)βE=1hln[E(h)E(0)]=1hlnE1E0,
(8)βν=1hln[ν(h)ν(0)]=1hlnν1ν0.

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 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

(9)u=∑i=1mNiuie,
(10)v=∑i=1mNivie,

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

Figure 2: Quadrilateral element of eight nodes.
Figure 2:

Quadrilateral element of eight nodes.

(11)Ni=14(1+ξξi)(1+ηηi)(ξξi+ηηi-1),     i=1, 2, 3, 4
(12)Ni=12(1-ξ2)(1+ηηi),     i=5, 7
(13)Ni=12(1+ξξi)(1-η2),     i=6, 8

where (ξ, η) denote local coordinates with their limit from -1 to +1, and (ξi, ηi) are the local coordinates of node i. The so-called 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

(14)x=∑i=1mNixie,
(15)y=∑i=1mNiyie.

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

(16)E=∑i=1mNiEie,
(17)ν=∑i=1mNiνie.

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 stress-strain relations of the FMG plane structure can be given as

(18){σxσyτxy}=[D(y)]{εxεyγxy},

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

(19)[D(y)]=E(y)1-ν2(y)[1ν(y)0ν(y)10001-ν(y)2].

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

(20)[K[D(y)]]{u[D(y)]}={f},

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 least-squares problem expressed as

(21)Minimize F(p)=12∑i=1lri2(p),  p∈Rn, l≥n,
(22)Subjectto [K(p)]{u(p)}={f},

where p is the vector of parameter to be identified, i.e., p=[p1p2]T=[E0E1]T;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

(23)ri(p)=ui(p)-ui∗,

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

(24)r(p)=[r1(p)r2(p)⋯rl(p)]T.

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

(25)r(p)=u(p)−u∗=[u1(p)−u1∗  u2(p)−u2∗  …  ul(p)-ul∗]T.

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 ui∗(i=1, 2, ⋯, 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

(26)J(p)=[∂ui(p)∂pj]l×2=[∂u1(p)∂E0∂u1(p)∂E1∂u2(p)∂E0∂u2(p)∂E1⋮⋮∂ul(p)∂E0∂ul(p)∂E1].

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

(27)[∂K(p)∂pj]{u(p)}+[K(p)]{∂u(p)∂pj}=0.

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

(28)J(p)=[∂ui(p)∂pj]l×2=-[K(p)]-1[∂K(p)∂pj]{u(p)}.

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

(29)[Ke(p)]16×16=∬[Be]16×3T[De(y)]3×3[Be]3×16T t dxdy,

where t is the element thickness, [Be] is called the strain-displacement 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.,

(30)[D(y)]=∑i=18NiEi(yi)1-ν2[1ν0ν100012(1-ν)].

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

(31)[∂Ke(p)∂pj]16×16=∬[Be]16×3T[∂De(y)∂pj]3×3[Be]3×16T t dxdy,

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

(32)[D(y)]=∑i=18Ni(E0+E1-E0hyi)1-ν2[1ν0ν100012(1-ν)].

Differentiating Eq. (32) with respect to the design variables yields

(33)[∂D(y)∂E0]=∑i=18Ni(1-1hyi)1-ν2[1ν0ν100012(1-ν)],
(34)[∂D(y)∂E1]=1h∑i=18Niyi1-ν2[1ν0ν100012(1-ν)].

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

(35)[D(y)]=∑i=18NiE0exp(yihlnE1E0)1-ν2[1ν0ν100012(1-ν)].

Differentiating Eq. (35) with respect to the design variables yields

(36)[∂D(y)∂E0]=∑i=18Ni(1-yih)exp(yihlnE1E0)1-ν2[1ν0ν100012(1-ν)],
(37)[∂D(y)∂E1]=∑i=18NiE0E1yihexp(yihlnE1E0)1-ν2[1ν0ν100012(1-ν)].

The stiffness matrix of structure can be assembled by

(38)[K(p)]=∑e=1numel[Ke(p)],

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

(39)[∂K(p)∂pj]=∑e=1numel[∂Ke(p)∂pj].

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 least-squares problems defined in Eq. (21), the Levenberg-Marquardt method is convenient and effective in the small residual case. A sequence of steps to adjust the parameters can be obtained by using the Levenberg-Marquardt method until convergence is achieved according to the specified criteria. The adjusted parameters at iteration k are calculated from the equations given by

(40)[J(p(k))TJ(p(k))+α(k)I]δ(k)=-J(p(k))Tr(p(k)),
(41)p(k+1)=p(k)+δ(k),

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

(42)‖J(p(k))Tr(p(k))‖<ε1,
(43)∑i=1l|pi(k+1)-pi(k)pi(k)|<ε2,

where ε1 and ε2 are the specified accuracy requirements.

5 Numerical results and discussion

According 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 γ-TiAl/Y-TZP functionally graded material [4], where loads, geometry dimensions, and boundary conditions are specified. The true material parameters are listed in Table 1.

Figure 3: Square FGM plane structure.
Figure 3:

Square FGM plane structure.

Table 1

The true values of the material parameter.

Material componentγ-TiAlY-TZP
Young’s modulus E (GPa)186209

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

  • Poisson’s ratio ν=0.3

  • Initial value of the Levenberg-Marquardt parameter α(0)=10-18

  • Accuracy requirements of convergence criterions ε1=10-6 and ε2=10-3

  • Upper and lower bounds of the two parameters to be identified

  • 47 GPa≤E0≤372 GPa, 52 GPa≤E1≤418 GPa

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 noise-free and noise-contaminated displacements are considered for the parameter identification. The noise-contaminated displacements are generated by adding the Gauss noise of different levels to the IGFEA-computed displacements with the true values of material parameters. A series of pseudo-random numbers, i.e., the Gauss noise, are produced from a Gauss distribution with mean μ=0 and standard deviation σ^. The standard deviation σ^ is defined as [14, 29]

(44)σ^=pe×1n∑i=1n(uim)2,

where uim denotes the IGFEA-computed 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 noise-contaminated displacements for each noise level and one experiment group of noise-free 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 eight-node 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. Table 2 lists the initial values of parameter E1 and two cases are investigated.

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

CasesParameterInitial values (GPa)True values (GPa)Initial value/True value
1E11052090.5
2E13132091.5

5.1.1 Linear material variation

The identified results of case 1 and case 2 are listed in Tables 3 and 4, respectively. With noise-free 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.

Table 3

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

Noise levelsGroupsE1 (GPa)Iterations
True valueIdentifiedErrors (%)
Noise free1209.00208.98-0.00834
1% noise1209.00209.440.214
2209.00207.21-0.864
3209.00205.10-1.874
4209.00207.97-0.494
5209.00212.671.764
6209.00205.82-1.524
2% noise1209.00213.061.944
2209.00210.650.794
3209.00208.10-0.434
4209.00199.76-4.424
5209.00209.130.064
6209.00211.761.324
5% noise1209.00210.760.844
2209.00199.80-4.404
3209.00214.242.514
4209.00216.433.564
5209.00216.113.404
6209.00211.711.304
Table 4

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

Noise levelsGroupsE1 (GPa)Iterations
True valueIdentifiedErrors (%)
Noise free1209.00208.99-0.00624
1% noise1209.00208.03-0.464
2209.00205.55-1.654
3209.00208.12-0.424
4209.00209.560.274
5209.00211.811.344
6209.00209.950.454
2% noise1209.00202.67-3.034
2209.00211.991.434
3209.00208.48-0.254
4209.00203.84-2.474
5209.00213.101.964
6209.00217.464.054
5% noise1209.00202.28-3.224
2209.00219.004.784
3209.00211.981.434
4209.00215.813.264
5209.00208.45-0.264
6209.00203.72-2.534

5.1.2 Exponential material variation

Tables 5 and 6 present the identified results of case 1 and case 2, respectively. For noise-free 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.

Table 5

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

Noise levelsGroupsE1 (GPa)Iterations
True valueIdentifiedErrors (%)
Noise free1209.00208.85-0.074
1% noise1209.00210.440.694
2209.00208.13-0.424
3209.00214.112.444
4209.00212.081.474
5209.00210.170.565
6209.00207.43-0.754
2% noise1209.00216.753.714
2209.00212.421.645
3209.00210.010.484
4209.00210.030.495
5209.00204.61-2.104
6209.00214.282.534
5% noise1209.00211.931.404
2209.00208.30-0.334
3209.00208.17-0.404
4209.00203.53-2.624
5209.00205.76-1.554
6209.00223.506.944
Table 6

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

Noise levelsGroupsE1 (GPa)Iterations
True valueIdentifiedErrors (%)
Noise free1209.00208.98-0.014
1% noise1209.00210.910.914
2209.00206.68-1.114
3209.00213.141.984
4209.00210.080.524
5209.00208.18-0.394
6209.00210.800.864
2% noise1209.00207.35-0.794
2209.00212.081.474
3209.00218.084.344
4209.00212.501.674
5209.00211.000.964
6209.00211.791.334
5% noise1209.00197.14-5.674
2209.00203.44-2.664
3209.00215.393.064
4209.00213.492.154
5209.00215.643.184
6209.00200.66-3.995

For both linear material variation and exponential material variation, the identified results shown in Tables 3–6 illustrate that the proposed algorithm has high accuracy and stable convergence. Furthermore, the parameter identification process can be successfully finished in a few iteration steps (no more than five steps).

As further assessment, the iterative identification processes with the maximum errors are chosen to investigate the performance of convergence and stability of the proposed model. Thus, the relative errors of the identified parameter E1 are given by

(45)Err=|VI-VTVT|×100%,

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.

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

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

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

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

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 eight-node 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.

Figure 7: Rectangular FGM plane structure.
Figure 7:

Rectangular FGM plane structure.

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

CasesParametersInitial values (GPa)True values (GPa)Initial value/True value
1E0931860.5
E11052090.5
2E02791861.5
E13132091.5
3E0931860.5
E13132091.5
4E02791861.5
E11052090.5

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 noise-contaminated displacements of 1%, 2% and 5% levels, the maximum errors of the identified parameter E0 are 1.44% in group 4 of case 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.

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 levelsGroupsE0 (GPa)E1 (GPa)Iterations
True valueIdentifiedErrors (%)True valueIdentifiedErrors (%)
Noise free1186.00185.97-0.02209.00209.080.047
1% noise1186.00186.930.50209.00205.33-1.7611
2186.00185.85-0.08209.00204.92-1.956
3186.00186.400.22209.00210.800.866
4186.00185.63-0.20209.00207.69-0.635
5186.00185.35-0.35209.00206.11-1.384
6186.00183.71-1.23209.00209.290.147
2% noise1186.00184.99-0.54209.00207.10-0.914
2186.00183.78-1.19209.00208.89-0.055
3186.00186.220.12209.00205.55-1.656
4186.00183.28-1.46209.00208.21-0.385
5186.00186.790.42209.00204.98-1.925
6186.00185.34-0.35209.00213.772.286
5% noise1186.00178.72-3.91209.00209.240.115
2186.00188.421.30209.00203.44-2.6616
3186.00185.45-0.30209.00195.37-6.5211
4186.00181.87-2.22209.00209.010.015
5186.00193.854.22209.00203.40-2.687
6186.00183.32-1.44209.00206.59-1.1520
Table 9

Identified parameters of a rectangular FGM plane structure with linear material variation (case 2: initial value E0=279 GPa, E1=313 GPa).

Noise levelsGroupsE0 (Gpa)E1 (GPa)Iterations
True valueIdentifiedErrors (%)True valueIdentifiedErrors (%)
Noise free1186.00185.92-0.04209.00208.90-0.054
1% noise1186.00185.63-0.20209.00210.470.706
2186.00185.65-0.19209.00208.97-0.015
3186.00185.83-0.09209.00208.91-0.046
4186.00188.621.44209.00209.190.094
5186.00186.040.02209.00206.43-1.234
6186.00186.220.12209.00208.42-0.285
2% noise1186.00184.11-1.02209.00209.210.104
2186.00183.89-1.13209.00200.97-3.8415
3186.00182.45-1.91209.00209.700.335
4186.00185.56-0.24209.00204.88-1.9718
5186.00182.72-1.76209.00207.23-0.854
6186.00183.80-1.18209.00205.59-1.634
5% noise1186.00191.402.90209.00197.02-5.735
2186.00180.19-3.12209.00201.15-3.7621
3186.00181.73-2.30209.00210.160.5619
4186.00185.62-0.20209.00204.74-2.049
5186.00188.861.54209.00196.54-5.968
6186.00188.611.40209.00218.144.3717
Table 10

Identified parameters of a rectangular FGM plane structure with linear material variation (case 3: initial value E0=93 GPa, E1=313 GPa).

Noise levelsGroupsE0 (GPa)E1 (GPa)Iterations
True valueIdentifiedErrors (%)True valueIdentifiedErrors (%)
Noise free1186.00186.000.00209.00209.000.004
1% noise1186.00188.481.33209.00210.550.744
2186.00186.520.28209.00206.28-1.304
3186.00184.31-0.91209.00206.29-1.305
4186.00186.720.39209.00209.740.354
5186.00186.140.08209.00210.790.864
6186.00186.730.39209.00207.13-0.895
2% noise1186.00187.740.94209.00214.292.5311
2186.00186.230.12209.00210.000.485
3186.00187.630.88209.00207.22-0.855
4186.00184.71-0.69209.00212.271.568
5186.00187.550.83209.00213.482.144
6186.00182.30-1.99209.00200.82-3.9117
5% noise1186.00194.934.80209.00195.33-6.5417
2186.00181.26-2.55209.00210.120.5315
3186.00179.49-3.50209.00218.464.5317
4186.00186.340.18209.00215.873.2920
5186.00183.26-1.47209.00197.23-5.6317
6186.00187.120.60209.00213.852.328
Table 11

Identified parameters of a rectangular FGM plane structure with linear material variation (case 4: initial value E0=279 GPa, E1=105 GPa).

Noise levelsGroupsE0 (GPa)E1 (GPa)Iterations
True valueIdentifiedErrors (%)True valueIdentifiedErrors (%)
Noise free1186.00185.88-0.06209.00208.91-0.044
1% noise1186.00186.420.23209.00208.27-0.354
2186.00184.16-0.99209.00206.19-1.344
3186.00187.200.65209.00207.99-0.484
4186.00186.330.18209.00209.330.165
5186.00187.500.81209.00212.941.894
6186.00186.840.45209.00205.75-1.5611
2% noise1186.00185.21-0.42209.00201.63-3.536
2186.00185.40-0.32209.00199.92-4.345
3186.00185.65-0.19209.00206.75-1.086
4186.00187.560.84209.00215.263.004
5186.00187.410.76209.00213.061.9410
6186.00186.980.53209.00207.64-0.6517
5% noise1186.00190.462.40209.00206.15-1.365
2186.00177.52-4.56209.00204.69-2.064
3186.00187.640.88209.00207.19-0.878
4186.00180.34-3.04209.00208.15-0.415
5186.00189.972.13209.00224.037.1915
6186.00190.892.63209.00220.735.6119

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 noise-free displacements, the maximum errors of the identified parameters E0 and E1 are -0.11% and 0.06% in case 4, respectively. For the noise-contaminated 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 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.

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 levelsGroupsE0 (GPa)E1 (GPa)Iterations
True valueIdentifiedErrors (%)True valueIdentifiedErrors (%)
Noise free1186.00186.000.00209.00209.090.045
1% noise1186.00187.700.91209.00206.24-1.329
2186.00184.27-0.93209.00212.271.564
3186.00184.71-0.69209.00208.51-0.2310
4186.00186.290.16209.00205.76-1.5518
5186.00186.490.26209.00206.45-1.226
6186.00184.68-0.71209.00206.56-1.176
2% noise1186.00186.140.08209.00212.081.475
2186.00187.370.74209.00207.16-0.886
3186.00185.97-0.02209.00209.300.144
4186.00182.91-1.66209.00205.11-1.8611
5186.00182.65-1.80209.00213.652.225
6186.00186.200.11209.00205.62-1.625
5% noise1186.00180.42-3.00209.00195.44-6.4913
2186.00191.673.05209.00196.02-6.2121
3186.00192.003.23209.00222.246.3313
4186.00197.746.31209.00223.586.9817
5186.00180.35-3.04209.00198.38-5.0812
6186.00182.10-2.10209.00214.462.6121
Table 13

Identified parameters of a rectangular FGM plane structure with exponential material variation (case 2: initial value E0=279 GPa, E1=313 GPa).

Noise levelsGroupsE0 (GPa)E1 (GPa)Iterations
True valueIdentifiedErrors (%)True valueIdentifiedErrors (%)
Noise free1186.00186.010.01209.00209.080.046
1% noise1186.00184.04-1.05209.00209.430.215
2186.00186.480.26209.00207.36-0.7811
3186.00185.82-0.10209.00209.930.445
4186.00186.960.52209.00204.49-2.165
5186.00186.570.31209.00206.30-1.296
6186.00185.05-0.51209.00204.79-2.0116
2% noise1186.00189.051.64209.00206.92-1.004
2186.00187.740.94209.00205.01-1.918
3186.00184.61-0.75209.00198.56-5.005
4186.00188.441.31209.00219.314.9315
5186.00185.90-0.05209.00205.20-1.8218
6186.00185.60-0.22209.00201.31-3.685
5% noise1186.00186.760.41209.00216.753.714
2186.00189.641.96209.00226.058.167
3186.00187.560.84209.00192.79-7.7617
4186.00186.880.47209.00224.667.4919
5186.00174.69-6.08209.00216.793.735
6186.00188.501.34209.00211.711.308
Table 14

Identified parameters of a rectangular FGM plane structure with exponential material variation (case 3: initial value E0=93 GPa, E1=313 GPa).

Noise levelsGroupsE0 (GPa)E1 (GPa)Iterations
True valueIdentifiedErrors (%)True valueIdentifiedErrors (%)
Noise free1186.00185.89-0.06209.00209.070.034
1% noise1186.00186.240.13209.00207.97-0.494
2186.00188.331.25209.00209.840.404
3186.00184.87-0.61209.00208.39-0.295
4186.00188.871.54209.00210.340.644
5186.00186.640.34209.00209.990.477
6186.00185.91-0.05209.00213.232.024
2% noise1186.00185.63-0.20209.00211.121.014
2186.00188.251.21209.00208.96-0.024
3186.00181.67-2.33209.00209.910.444
4186.00187.080.58209.00210.440.695
5186.00187.790.96209.00206.68-1.1128
6186.00186.680.37209.00205.72-1.574
5% noise1186.00177.88-4.37209.00224.517.4222
2186.00176.69-5.01209.00219.965.2421
3186.00175.73-5.52209.00199.52-4.5420
4186.00186.070.04209.00197.33-5.5819
5186.00194.124.37209.00194.46-6.9620
6186.00187.660.89209.00214.452.6115
Table 15

Identified parameters of a rectangular FGM plane structure with exponential material variation (case 4: initial value E0=279 GPa, E1=105 GPa).

Noise levelsGroupsE0 (GPa)E1 (GPa)Iterations
True valueIdentifiedErrors (%)True valueIdentifiedErrors (%)
Noise free1186.00185.80-0.11209.00209.130.064
1% noise1186.00184.36-0.88209.00206.72-1.099
2186.00187.780.96209.00210.970.946
3186.00188.131.15209.00209.740.355
4186.00184.93-0.58209.00211.321.115
5186.00185.14-0.46209.00206.34-1.278
6186.00184.84-0.62209.00208.22-0.375
2% noise1186.00184.96-0.56209.00217.063.868
2186.00186.230.12209.00211.561.225
3186.00185.25-0.40209.00203.45-2.6610
4186.00188.241.20209.00212.371.615
5186.00185.53-0.25209.00214.662.715
6186.00185.73-0.15209.00200.97-3.847
5% noise1186.00185.49-0.27209.00196.42-6.0211
2186.00181.06-2.66209.00199.57-4.517
3186.00195.164.92209.00214.582.676
4186.00181.18-2.59209.00195.99-6.2216
5186.00193.724.15209.00211.421.165
6186.00187.450.78209.00216.053.375

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 noise-free displacements. With the noise-contaminated displacements, the maximum errors of the identified parameters due to propagation of measurement errors are within the acceptable limit.

6 Conclusions

Based 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 noise-contaminated 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 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.

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:

Funding source: National Natural Science Foundation of China

Award Identifier / Grant number: 11262002

Funding statement: 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.

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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Received: 2014-8-29
Accepted: 2015-1-1
Published Online: 2015-4-17
Published in Print: 2016-11-1

©2016 Walter de Gruyter GmbH, Berlin/Boston

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https://doi.org/10.1515/secm-2014-0289
Keywords for this article
functionally graded materials; isoparametric graded finite element; Levenberg-Marquardt method; material parameter identification; sensitivity analysis
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