The complex variable meshless local Petrov–Galerkin method for elasticity problems of functionally graded materials

Applied Mathematics and Computation 268 (2015) 1140–1151

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Applied Mathematics and Computation journal homepage: www.elsevier.com/locate/amc

The complex variable meshless local Petrov–Galerkin method for elasticity problems of functionally graded materials Dandan Wei, Weiwei Zhang, Linghui Wang, Baodong Dai∗ Department of Engineering Mechanics, Taiyuan University of Science & Technology, Taiyuan 030024, PR China

a r t i c l e

i n f o

Keywords: Meshless method Complex variable moving least-square method Complex variable meshless local Petrov–Galerkin method Functionally graded materials Elasticity

a b s t r a c t This paper proposed the complex variable meshless local Petrov–Galerkin (CVMLPG) method for the static analysis of functionally graded materials (FGMs). In the presented method, the complex variable moving least-square (CVMLS) approximation, which is established based on the moving least-square (MLS) approximation by introducing the complex variable theory, is adopted for construction of the field approximation function. Compared with the conventional MLS method, the number of the unknown coefficients in the trial function of the CVMLS method is less than that of the MLS approximation, thus higher efficiency and accuracy can be achieved under the same node distributions. One advantage of the CVMLPG method for the FGMs is that the variations of the functionally graded material properties are simulated by using material parameters at Gauss points, so it totally avoids the issue of the assumption of homogeneous in each element in the finite element method (FEM) for the FGMs. Some of selected benchmark examples are considered to confirm the validity and accuracy of the proposed method. © 2015 Elsevier Inc. All rights reserved.

1. Introduction Functionally graded materials (FGMs) are heterogeneous composite materials with gradient compositional variation of the constituents from one surface of the material to the other, which results in continuously varying material properties [1]. The materials are intentionally designed so as to improve strength, toughness, high temperature withstanding ability, etc. Since the concept of FGMs was introduced by Japanese scientists in 1984, the development and analyses of FGM have attracted much interest. However, because of the intrinsic complexity of the corresponding governing equation, analytical solutions to these boundary value and initial value problems are only in relatively few cases. Therefore, numerical methods with different discretization schemes, such as finite element method (FEM), are widely used in these analyses and have achieved great progress. Nevertheless, mesh generation in these mesh-based numerical methods is very time-consuming and burdensome and limit the application of these methods to some particular situations. In the past two decades, the development and application of meshless method have attracted considerable attention [2]. Some of the main reasons are that meshless method does not require a mesh to discretize the problem domain and the approximate solution is constructed entirely based on a set of scattered nodes. According to the published literature, the meshless methods can be classified into two general classes, namely the element free Galerkin (EFG) method [3] based on the global weak formulation and the meshless local Petrov–Galerkin (MLPG) method [4] based on the local weak formulation. Up to now, a wide range of meshless methods have been successfully used in most cases [5–18]. However, the computational cost of meshless methods is ∗

Corresponding author. Tel.: +8603516845416. E-mail address: [email protected] (B. Dai).

http://dx.doi.org/10.1016/j.amc.2015.07.020 0096-3003/© 2015 Elsevier Inc. All rights reserved.

D. Wei et al. / Applied Mathematics and Computation 268 (2015) 1140–1151

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generally higher than that of corresponding numerical methods. Consequently, any contribution made to the reduction of the computational cost for meshless methods can be regarded as an important progress. To overcome the aforementioned shortcoming of the higher computational cost of meshless methods, based on the complex variable moving least-square (CVMLS) approximation, the complex variable element free Galerkin method (CVEFG) has been developed by Liew et al. [19] and Cheng et al. [20]. Compared with the traditional MLS approximation, the advantages of the CVMLS approximation are that the trial function of a 2D problem is formed with 1D basis function, which leads to the fewer nodes used to form the meshless shape function. Consequently, the efficiency is improved. The CVEFG method in the published literature has been demonstrated to be quite successful in solving science and engineering problems [21–33]. Introducing the complex variable theory into the reproducing kernel particle method, Chen et al. [34] proposed the complex variable reproducing kernel particle method (CVRKPM) and take advantage of the CVRKPM in the analysis of transient heat conduction problems [35], potential problems [36], elastoplasticity problems [37]. Based on the CVMLS approximation and the MLPG method, Dai et al. proposed the CVMLPG method and made use of the proposed methods in the analysis of elasticity [38], potential problems [39], transient heat conduction problems [40] and dynamics [41]. In this paper, the CVMLPG method is extended to the plane elasticity problems of FGM beams under transverse mechanical loads. In this method, the CVMLS approximation is used for approximation of displacement field. Comparisons are made through some illustrative examples to show the validity of the present approach. 2. Complex variable moving least-square approximation The CVMLS approximation is the approximation of a vector function, but the MLS approximation is the approximation of a scalar function. The CVMLS approximation is described in the following [19]. Consider a domain of definition of point z or support domain z , which is located within the problem domain  and has a number of randomly located nodes zI (I = 1, 2, . . . , n). The approximation function uh (z) of u˜(z) in domain  is defined by

uh (z) = uh1 (z) + iuh2 (z) =

m 

p j (z)a j (z) = pT (z)a(z),

(z = x1 + ix2 ∈ ),

(1)

j=1

p(z) is the basis function vector of order m and a(z) is the corresponding coefficients vector (to be determined). The local approximation at point z is given by

uh (z, z¯) =

m 

p j (z¯)a j (z) = pT (z¯)a(z),

(2)

j=1

where z¯ is the node in the local support domain z of point z. The coefficient vector a(z) is determined by minimizing the difference between the local approximation and the function, and it is defined as

J=

n 

 2 n m   h 2  w(z − zI ) u (z, zI ) − u˜(zI ) = w(z − zI ) p j (z)a j (z) − u˜(zI )

I=1

I=1

j=1

= ( pa − u˜) W (z)( pa − u˜), T

(3)

where w(z − zI ) refers to a weight function with a domain of influence, zI (I = 1, 2, . . . , n) are the nodes in a domain of influence of point z for which the weight function w(z − zI ) > 0, n is the number of nodes zI , and

u˜(zI ) = u1 (zI ) + iu2 (zI ),

(I = 1, 2, . . . , n),

u˜ = (u˜(z1 ), u˜(z2 ), . . . , u˜(zn ))T = Q u, where

⎛

1 ⎜0 Q =⎜ . ⎝ .. 0

i 0 0 0 1 i .. .. .. . . . 0 0 0

0 0 ··· 0 0 ··· .. .. . . . . . 0 0 ···

(5)

⎞

0 0 0 0⎟ , .. .. ⎟ ⎠ . . 1 i n×2n

u = (u1 (z1 ), u2 (z1 ), u1 (z2 ), u2 (z2 ), . . . , u1 (zn ), u2 (zn ))T ,

⎛

p1 (z1 ) ⎜ p1 (z2 ) p=⎜ . ⎝ .. p1 (zn )

p2 (z1 ) p2 (z2 ) .. . p2 (zn )

··· ··· .. . ···

(4)

(6)

(7)

⎞

pm (z1 ) pm (z2 )⎟ .. ⎟ ⎠ , . pm (zn ) n×m

(8)

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D. Wei et al. / Applied Mathematics and Computation 268 (2015) 1140–1151

⎛

w(z − z1 ) 0 ⎜ W (z) = ⎜ .. ⎝ . 0

0 w(z − z2 ) .. . 0

··· ··· .. . ···

⎞

0 0 ⎟ ⎟ . .. ⎠ . w(z − zn ) n×n

(9)

In order to determine the coefficient vector a(z), we take J in (3) stationary with respect to a(z) as follows:

∂J = 0, ∂a

(10)

which yields

˜ A(z)a(z) = B(z)u,

(11)

A(z) = P T W (z)P,

(12)

B(z) = P T W (z).

(13)

where

Solving Eq. (11), we can obtain the coefficient vector a(z) as follows:

˜ a(z) = A−1 (z)B(z)u,

(14)

Once the a(z) is found and substituted into Eq. (2), we can obtain the final expression of the CVMLS local approximation as

uh (z) = (z)u˜ =

n 

φI (z)u˜(zI ) = uh1 (z) + iuh2 (z),

(15)

I=1

where (z) is called the shape function vector, and

(z) = (φ1 (z), φ2 (z), . . . , φn (z)) = pT (z)A−1 (z)B(z).

(16)

φI (z) = pT (z)A−1 (z)BI (z),

(17)

Thus

and the partial derivative of φI (z) is



φI,i (z) = pT (z)A−1 (z)BI (z) ,i −1 T = pT,i (z)A−1 (z)BI (z) + pT (z)A−1 ,i (z)BI (z) + p (z)A (z)BI,i (z).

From Eq. (15) we have

uh1









(z) = Re (z)u˜ = Re

n 

φI (z)u˜(zI ) ,





(z) = Im (z)u˜ = Im

n 

(19)





I=1

uh2

(18)

φI (z)u˜(zI ) .

(20)

I=1

This is called the CVMLS method. 3. CVMLPG formulation for 2D elasticity of FGMs In this section, we will illustrate the CVMLPG method. The problem domain is denoted by  which is bounded by boundaries including essential boundary u and natural boundary t . The weighted residual method is used to create the discrete system of equations. Consider a 2D elasticity problem, the corresponding equations of equilibrium and boundary conditions are as follows:

σi j, j + bi = 0, (in , i, j = 1, 2),

(21)

σi j n j = t¯i , (on t ),

(22)

ui = u¯ i ,

(on u ),

(23)

where σi j and bi are the stress tensor and body force tensor, respectively, t¯i and u¯ i denote the prescribed traction and displacement, n j represents the outwards unit normal to  ( = u ∪ t ).

D. Wei et al. / Applied Mathematics and Computation 268 (2015) 1140–1151

1143

t

q u

qu

qi

qt

Fig. 1. Boundary of sub-domain.

Here we use the local weighted residual method to create the discrete system of equations. A local weak form of the partial differential Eq. (21) over integral sub-domain q bounded by q can be written as



q

wI

 σi j, j + bi d = 0,

(24)

where wI is the test function. Integrating the first term in the left-hand side of Eq. (24) by parts, we can obtain



q

wI σi j, j d =



q

wI n j σi j d  −



q

vI σi j d,

(25)

where vI is the partial derivative of the test function wI . Substituting Eq. (25) into Eq. (24) leads to the following local weak form



qi

wI n j σi j d  −



q

 vI σi j − wI bi d = 0.

(26)

The global boundary q of the integral sub-domain q consists of three parts,

q = qi ∪ qt ∪ qu ,

(27)

where qi is part of the internal boundary of q , qt and qu the natural boundary and essential boundary which intersect with q as shown in Fig. 1. So Eq. (26) can be rewritten as



qi

wI n j σi j d  +



qu

wI n j σi j d  +

 qt

wI n j σi j d  −



q

 vI σi j − wI bi d = 0.

(28)

Using the natural boundary condition defined by Eq. (22), we can obtain



q

vI σ d  −



qi

wI t i d  −



qu

wI t i d  =



qt

wI t¯i d +



q

wI bd.

(29)

In this study, we chose the Heaviside step function as the test function



wI =

1, x ∈ q , 0, x ∈ / q

(30)

therefore vI = 0. Then Eq. (29) can be simplified as



−

qi

t i d −



qu

t i d =



qt

t¯i d +



q

bd.

(31)

From the geometric equation of 2D elasticity problems, the strain tensor can be defined as



⎛  ⎜Re

n 

⎞

φ1 (z)u˜(zI )

⎟ ⎟ ⎟ ⎟ ⎟ ⎟, ⎟ ⎟ ⎟ ⎟ ⎠

⎜ ⎜ I=1 ,1 ⎜ uh1,1 (z) n  ⎜ ⎠ = ⎜Im φ1 (z)u˜(zI ) ε(z) = Luh (z) = ⎝u2,2 ((z) ⎜ I=1 ⎜ h u1,2 ((z) + u2,1 (z) ⎜   ,2   ⎜ n n   ⎝ Re φ1 (z)u˜(zI ) + Im φ1 (z)u˜(zI ) ⎛

⎞

I=1

,2

I=1

,1

(32)

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D. Wei et al. / Applied Mathematics and Computation 268 (2015) 1140–1151

x2

E1

E ( x1 )

E2

l

x1

w

(a)

(b)

Fig. 2. (a) Non-homogeneous plate under uniform tensile traction. (b) Nodal arrangement.

in which L is differential operator,

⎛

∂ ⎜ ∂ x1 ⎜ ⎜ L=⎜ 0 ⎜ ⎝ ∂ ∂ x2

⎞

0

⎟ ⎟ ∂ ⎟ ⎟, ∂ x2 ⎟ ∂ ⎠ ∂ x1

(33)

ε(z) can be rewritten as

ε(z) = L((z)u˜) = L((z)Q u) = B(z)u =

n 

BI uI ,

(34)

I=1

where BI is the strain matrix about the node, which is given as follows:

⎞     Im φI,1 (z) φI,1 (z)     ⎟ ⎜ BI (z) = ⎝ Im φI,2 (z) Re φI,2 (z) ⎠,         Re φI,2 (z) + Im φI,1 (z) Im φI,2 (z) + Re φI,1 (z) ⎛

Re

B(z) = (B1 (z), B2 (z), . . . , Bn (z)).

(35)

(36)

From constitutive equation of 2D elasticity problems, the stress tensor can be written as

σ(z) = D · ε(z) = D · B(z) · u =

n  I=1

DBI uI ,

(37)

D. Wei et al. / Applied Mathematics and Computation 268 (2015) 1140–1151

1.4

CVMLPG MLPG FEM

1.2 1.0

( x1 )

1145

E2/E1=5

2

0.8 0.6 E2/E1=0.1

0.4 0.2

0.0

E2/E1=10

0.2

0.4

x1

0.6

0.8

1.0

Fig. 3. The comparison of the normal stresses along x2 = 1.

q

o

h 2

x1

l

x2

(a)

x1

0.1m 1m

x2

(b)

Fig. 4. (a) A FG beam loaded by a linear distributed force (b) Nodal arrangement.

where D is the material matrix for the plane stress problem, which is defined as follows:

⎡

D(z) =

μ(z)

1

E (z) ⎢μ(z) ⎣ 1 − μ2 (z) 0

0

1

0

0

1 − μ(z) 2

⎤

⎥ ⎦,

(38)

where E (z) and μ(z) are modulus of elasticity and Poisson’s ratio, respectively, and it is assumed that they are functions of the coordinate z. In the case of particular FGM beams, material matrix D is given as follows [42]:

⎡

S11

S12

S16

⎤−1

D = ⎣S21

S22

S26 ⎦

S61

S62

S66

,

(39)

Si j (i, j = 1, 2, 6) are the flexibility coefficients associated with the materials and certain functions of the coordinate z for the FGM beams. So the surface force tensor at arbitrary point z can be written as

 

t t= 1 t2 where

 n=



n1 = 0

n1

0

n2

0

n2

n1

0 n2

n2 n1

 σ11  σ22 = nDBu, τ12

(40)

 .

(41)

D. Wei et al. / Applied Mathematics and Computation 268 (2015) 1140–1151

CVMLPG MLPG Exact solution

0.10

vh

0.08 0.06 0.04 0.02 0.00 0.0

0.2

0.4

x1 l

0.6

0.8

1.0

Fig. 5. The comparison of the vertical displacements along the x1 -axis.

15

CVMLPG MLPG Exact solution

10 5

q

0

1

1146

-5 -10 -15 -0.6

-0.4

-0.2

0.0

x2 h

0.2

0.4

0.6

Fig. 6. The comparison of the normal stresses σ1 along x1 = 0.5 m.

0.0

CVMLPG MLPG Exact solution

-0.4 -0.8 -1.2 -1.6 -2.0 -0.6

-0.4

-0.2

0.0

x2 h

0.2

0.4

Fig. 7. The comparison of the shear stresses τ12 along x1 = 0.5 m.

0.6

D. Wei et al. / Applied Mathematics and Computation 268 (2015) 1140–1151

1147

x1 h

l

x2

(a)

x1

0.2m 1m

x2

(b) Fig. 8. (a) FG beam subjected to transverse sinusoidal loading (b) Nodal arrangement.

0.010 1 0.5

0.008

0

0.5

1

vh

0.006 0.004 0.002 0.000 0.0

(a)

0.2

0.4

x1 l

0.6

0.8

0.010

1.0

1

0.5 0

0.008

0.5

1

vh

0.006 0.004 0.002 0.000

(b)

0.0

0.2

0.4

x1 l

0.6

0.8

1.0

Fig. 9. (a) The vertical displacements at x2 = h/2 for different λ (the present method) (b) The vertical displacements at x2 = h/2 for different λ (exact solution) [44].

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D. Wei et al. / Applied Mathematics and Computation 268 (2015) 1140–1151

15

1 0.5

10

0

0.5

1

1

q

5 0 -5 -10 -15 0.0

0.2

0.4

(a)

x2 h

0.6

0.8

1.0

15

1 0.5

0

10

0.5

1

1

q

5 0 -5 -10 -15 0.0

0.2

0.4

(b)

x2 h

0.6

0.8

1.0

Fig. 10. (a) The normal stresses σ1 at x1 = 1/2 m for different λ (the present method) (b) The normal stresses σ1 at x1 = 1/2 m for different λ (exact solution) [44].

Combining Eq. (40) with Eq. (31), we can obtain that



−

qi

nDBud −



qu

nDBud =



qt

¯ + td

 q

bd.

(42)

The matrix form of Eq. (42) can be rewritten as

(kI )2×2n (u)2n×1 = ( f I )2×1 , (I = 1, 2, . . . , n).

(43)

This equation is the final equation system of the CVMLPG method for FGMs, where kI is the stiffness matrix and f I the load vector.



kI = −  fI =

qi

qt

nDBd −

¯ + td

 q



qu

bd.

nDBd ,

(44) (45)

4. Numerical examples In this section, three examples are proposed to demonstrate the validity and accuracy of the CVMLPG method for 2D elasticity of FGMs. Numerical results obtained from the present method are compared with the conventional MLPG solutions, the analytical solutions and FEM solutions. As is depicted in Fig. 2(a), the first example is a plate subjected to uniform tensile traction at the top and bottom end simultaneously. The Young’s modulus is assumed to be

E = E1 exp

x

1

w



ln

E2 , E1

(46)

D. Wei et al. / Applied Mathematics and Computation 268 (2015) 1140–1151

0.0

1149

1 0.5

0

-0.5

0.5

1

12

q

-1.0 -1.5 -2.0 -2.5 0.0

(a)

0.2

0.4

0.0

q

0.8

1.0

1 0.5

0

-0.5

12

0.6

x2 h

0.5

1

-1.0 -1.5 -2.0 -2.5 0.0

(b)

0.2

0.4

x2 h

0.6

0.8

1.0

Fig. 11. (a) The shearing stresses τ12 at x1 = 1/2 m for different λ (the present method) (b) The shearing stresses τ12 at x1 = 1/2 m for different λ (exact solution) [44].

where E1 = E (0), E2 = E (1). In the numerical computation, the parameters of the plate, which is shown in the Fig. 2(a), are taken as σ = 1, l = 4, w = 1, E1 = 1, and E2 /E1 = 0.1, 5, 10. The Poisson’s ratio is held constant with μ = 0.3. Due to the symmetry of geometry and load, only one half of the plate is considered and discretized by 18 × 8 nodes shown in Fig. 2(b). The comparison of the normal stresses σ2 along x2 = 1 obtained by using different methods is shown in Fig. 3. As it is observed from the comparison, the CVMLPG results agree well with the FEM solutions computed from the software ANSYS and have higher precision than the MLPG solutions. This example confirms that the developed method in this paper can give a good numerical solution indeed. As the second example [43], an anisotropic cantilevered beam with length l = 1 m, height h = 0.1 m, subjected to linear distributed loading at the top of the beam, is analyzed, as shown in Fig. 4(a). We consider a plane stress problem. si j = s0i j are

material parameters, which are listed in Table 1. Exceptionally, s11 is exponential function of x2 , that is s11 = s011 eλ(x2 /h+1/2) , here s011 is flexibility coefficient at x2 = −h/2 and λ is the gradient index. The maximum loading density is q0 = 1 × 106 N/m. In the computation, the problem domain is discretized by 22 × 8 nodes regularly (see Fig. 4(b)), and a plane stress condition is assumed. In order to verify the influence of different graded indexes λ on the distribution of stresses, several λ are considered (i.e. λ = 1, λ = 0 and λ = −1). Some of the results obtained for the vertical displacements along the x1 -axis and the stresses at x1 = 0.5 m are presented in Figs. 5–7. Once again, an excellent agreement is achieved with the exact solutions. It can also be seen that, for examples 1 and 2, the results obtained from CVMLPG method are better than those from MLPG method under the same node distribution. As the last example [44], the CVMLPG method is used to study the static behavior of the cantilevered beam under the transverse sinusoidal loading (i.e. q = q0 sin (π x1 /l )), q0 = 1 × 106 N/m2 , as shown in Fig. 8(a). The height h and length l of the beam are 0.2 m × 1 m. The material parameters obey exponential-law distribution of the volume fractions of the constituent, given by si j = s0i j eλx2 /h . Table 2 shows the coefficients s0i j , and λ is taken to be −1, −0.5, 0, 0.5, 1. As shown in Fig. 8(b), a regular distribution of 22 × 8 nodes is taken in the domain.

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D. Wei et al. / Applied Mathematics and Computation 268 (2015) 1140–1151 Table 1 Material parameters (unit: 10−10 m2 /N). s011

s012

s016

s022

s026

s066

0.1500

−0.0259

0

0.1032

0

0.1464

Table 2 Material parameters (unit: 10−11 m2 /N). s011

s012

s016

s022

s026

s066

1.029

−0.030

−0.246

0.890

0.072

2.124

In the Figs. 9(a)–11(b), the obtained dimensionless displacements and stresses components from the CVMLPG have been compared with the results from the analytical solutions. It is clear that the results of the CVMLPG in the vertical displacements, normal stresses and shearing stresses are in very good agreement with the ones of the analytical solutions. Through the comparisons between the results of the CVMPLG and that of other method, it is verified not only the feasibility but also the high efficiency of the current work. 5. Conclusions In this study, based on the MLPG method, by introducing the CVMLS approximation into the MLPG method, the CVMLPG method is proposed to the analysis of the FG elastic beams under transverse mechanical loads. In the construction of the shape functions, the trial function of a 2D problem is formed with a 1D basis function, thus the number of the unknown coefficients in the CVMLS approximation is less than that of the moving least-square approximation, and this substantially improves the computational efficiency for constructing the shape functions. In the numerical computation, different material parameters are considered. The comparisons have been made between results obtained by the CVMLPG method and that by other methods. These comparisons illustrate that the accuracy as well as the capability of the CVMLPG method has been improved compared with the conventional MLPG method due to the introduction of the CVMLS. Therefore, the present method has a good potential and can be extended to the investigation of FGM plates and shells. Acknowledgments The work in this project was fully supported by the Young Scientists Fund of the National Natural Science Foundation of China (Grant No. 11102125). References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21] [22]

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