- Email: [email protected]
Analysis of thermal and mechanical response in functionally graded cylinder using cell-based smoothed radial point interpolation method
Accepted Manuscript Analysis of thermal and mechanical response in functionally graded cylinder using cell-based smoothed radial point interpolation method
S.Z. Feng, A.M. Li
PII: DOI: Reference:
S1270-9638(16)30812-4 http://dx.doi.org/10.1016/j.ast.2017.02.009 AESCTE 3918
To appear in:
Aerospace Science and Technology
Received date: Revised date: Accepted date:
8 October 2016 19 January 2017 14 February 2017
Please cite this article in press as: S.Z. Feng, A.M. Li, Analysis of thermal and mechanical response in functionally graded cylinder using cell-based smoothed radial point interpolation method, Aerosp. Sci. Technol. (2017), http://dx.doi.org/10.1016/j.ast.2017.02.009
This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.
Analysis of thermal and mechanical response in functionally graded cylinder using cell-based smoothed radial point interpolation method S. Z. Feng∗, A. M. Li School of Mechatronic Engineering, China University of Mining and Technology, Xuzhou, 221116, P. R. China
Abstract This study aims to improve the accuracy and efficiency when dealing with thermal and mechanical response in functionally graded cylinder, which is very important in modern aerospace industry. The cell-based smoothed radial point interpolation method (CS-RPIM) is formulated for such analysis. In CS-RPIM, triangular meshes are utilized to discretize problem domains, which can be easily generated. Each triangular element is then partitioned into several smoothing cells. Field functions are constructed by RPIM shape functions and system equations are obtained based on these smoothing cells. Finally, the performances of CS-RPIM are fully investigated through several numerical examples.
Keywords: FGM; RPIM; Meshfree method; Thermal; Gradient smoothing technique
Aerospace industry
1. Introduction The functionally graded material (FGM) is a new kind of composite material, which has been successfully utilized in modern aerospace industry. In general, FGM is made up of metals and ceramics, in which the material volume fractions change continuously along desired directions [1, 2]. The metal component is utilized to prevent fracture caused by thermo-mechanical loads and the ceramic constituent can offer heat-resistant property
* Corresponding author. Tel: +86 516 83590718; fax: +86 516 8825016 E-mail address: [email protected] (S. Z. Feng).
owing to low thermo-conductivity. Therefore, engineering structures made by FGM can work under thermal and mechanical loads safely, such as FGM cylinders, in which the volume fraction varies continuously in the radius direction according to a power law. The thermal and mechanical analysis of FGM structures is an important issue in practical engineering [3, 4]. So many studies have been done to solve this problem in the past few years. For example, the thermo-elasticity analysis of functionally graded thick cylinder was studied by Hosseini [5]. Asgari utilized the graded finite elements to investigate the thermo-mechanical response of functionally graded cylinder [6]. Daneshmehr studied the dynamical thermo-elasticity responses of functionally graded cylinder, in which the time-dependent boundary conditions are considered [7]. It can be clearly seen that the analytical solution for such problems are usually very difficult to obtain and hence, numerical methods [8-12] are employed to deal with such analysis, such as the finite element method (FEM) and meshfree methods [13-15], which have become dominative tools in practical engineering. In order to further improve the efficiency of these methods, the gradient smoothing technique was proposed and applied to these methods [16, 17]. For example, Cui formulated an edge-based smoothed finite element method to deal with mechanical analysis of engineering structures [18, 19], which can provide much more accurate results than standard FEM. Wu utilized a node-based smoothed point interpolation method for thermal and mechanical problems [20]. Based on these studies, a cell-based smoothed radial point interpolation method (CS-RPIM) was then formulated to deal with static and free vibration analysis of structures [21-25] and numerical results with high quality are obtained. The main purpose of this study is to improve the accuracy, stability and convergence rate, when investigating thermal and mechanical responses of FGM cylinder. The validity of
2
this method is fully tested through several numerical examples and numerical results are compared with those obtained by existing fine numerical methods.
2.
Functionally graded cylinder
Functionally graded cylinders are made by metals and ceramics. Usually the outer layer is metal-rich and the inner layer is ceramic-rich. The material properties of FGM cylinder can be given as
Pf = Pm + ( Pc − Pm )Vc
Vc = (1 −
(1)
r − ri n ) ( n ≥ 0) , Vm = 1 − Vc ro − ri
(2)
where n represents the volume fraction exponent. Pf , Pc and Pm represent the material properties of ceramic and metal, respectively. ro is the outer radius, ri is the inner radius, c and m denote the ceramic and the metal, respectively. From Eq. (1) and Eq. (2), it is clearly seen that for for n = ∞ , Pf = Pm and n = 0 , Pf = Pc . These are two particular cases, which correspond to pure metal and ceramic, respectively. For different value of n, Fig. 1 further shows the variation of the volume fraction function versus non-dimensional radius.
3. Heat transfer analysis 3.1 Basic system equations When dealing with the transient heat transfer analysis of FGM cylinders, the governing differential equation can be given as
ȡcr
∂T ∂ ∂T ∂ ∂T = ( kr ) + ( kr ) ∂t ∂r ∂r ∂z ∂z
(3)
where, r represents the radius, k denotes the thermo-conductivity, ρ is the density and c is
3
the specific heat. Different thermal boundary conditions investigated in this study are directly listed as Initial condition : T = T0
∂T |ī = q0 ∂n0
(5)
∂T |ī = hc (T − T∞ ) ∂n0
(6)
Neumann boundary : − k
Robin boundary : − k
(4)
Adiabatic boundary : − k
∂T |ī = 0 ∂n0
(7)
where, T0 represents the initial temperature, q0 is the heat flux, T∞ is the environment temperature, hc denotes the convection coefficient and n0 is the unit outward normal.
3.2 Formulation of RPIM In this work, the temperature field function is approximated by the RPIM shape function, which is constructed using the local nodes and radial basis function (RBF). The polynomial basis function is also used to construct the RPIM shape function. For a problem domain including a set of arbitrarily scattered points xi ( i=1, 2, ..., n ), the temperature field function T can be approximated as n
m
i =1
j =1
T = ¦ Ri (x)ai + ¦ p j (x)b j = R T (x)a + P T (x)b
(8)
where Ri(x) denotes the radial basis function, ai is the unknown coefficient for functions Ri(x), bj represents the coefficient for polynomial basis Pj(x), and m is determined according to the polynomial basis. The multi-quadrics RBF (MQ-RBF) that used in RPIM can be given as
4
[
Ri ( x ) = ( x − xi ) 2 + ( y − yi ) 2 + (α c d c ) 2
]
q
(9)
where, dc denotes the equivalent length of triangular elements [21], q and Įc represent the real and arbitrary shape parameters. The polynomial basis function is directly given here PT (x) = [1, x, y,...]
(10)
By enforcing the field function T to be satisfied at the n nodes within the local support domain, the coefficients in Eq. (8) can then be computed. This procedure will bring about a set of linear equations, which is given as
Ts = R q a + Pmb
(11)
where Ts is the vector of function values and it is given as
Ts = {T1 , T2 ,..., Tn }T
(12)
Rq represents the moment matrix of RBFs,
ª R1 (x1 ) R2 (x1 ) « R (x ) R (x ) 2 2 Rq = « 1 2 « ... ... « ¬ R1 (x n ) R2 (x n )
... Rn (x1 ) º ... Rn (x 2 )»» ... ... » » ... Rn (x n )¼ n×n
(13)
Matrix Pm is defined as ª p1 (x1 ) « p (x ) Pm = « 1 2 « ... « ¬ p1 (x n )
p2 (x1 ) ... p2 (x 2 ) ... ...
...
p2 (x n ) ...
pm (x1 ) º pm (x 2 ) »» ... » » pm (x n )¼
(14)
The following constraint condition is utilized for the formulation [26]. n
¦ p (x j
k
)ai = 0 , j = 1,2,..., m
(15)
i =1
Combining Eq. (11) and Eq. (15) leads to
5
~ T ½ ªR Ts = ® s ¾ = « Tq ¯ 0 ¿ ¬ Pm
Pm º a ½ a ½ ® ¾ = G® ¾ » 0 ¼ ¯b ¿ ¯b ¿
(16)
By solving Eq. (16), the following equation can be obtained T a ½ −1 U s ½ a ½ −1 s ½ ® ¾=G ® ¾ ® ¾=G ® ¾ ¯0¿ ¯b ¿ ¯ 0 ¿ ¯b ¿
(17)
Therefore, the temperature field function T can be given as
T ½ PmT ]G −1 ® s ¾ = ijTs ¯0¿
T = [R Tq
(18)
where, ij(x) denotes the RPIM shape functions, which has the following form ij = [ϕ1 ϕ2 ... ϕn ]
(19)
where n
m
i =1
j =1
ϕk (x) = ¦ Ri (x)G(i ,k ) + ¦ p j (x)G( n + j ,k )
(20)
where G(i ,k ) represents the constituent of matrix G −1 . Finally, the temperature field function T is approximated as n
T = ¦ ϕ i Ti
(21)
i =1
3.3 CS-RPIM for heat transfer analysis The problem domain is first discretized by triangular elements, which is the same as standard FEM. The triangular element is called as “parent” cell for convenience in the discussion, which will be further divided into non-overlapping smoothing cells Ω k = * SC C =1 Ω k ( C ) , as shown in Fig. 2. The edge of the smoothing cell is called as
“segment”. The boundary īk(C) of the smoothing cell is made up of the segments of smoothing cell. It is assumed that the temperature gradient of the Cth smoothing cell 6
Ωk(C) is constant. By using the generalized gradient smoothing technique [27], it can be
given as İ kT( C ) ≡
1 n ⋅ TdΓ Ak ( C ) ³Γk ( C )
(22)
where T is the temperature field function of smoothing cell ȍk(C), Ak(C) is the area of the Cth smoothing cell and n is the outward normal. It has been proved that the SC=3 scheme can give accurate and stable results in the static and free vibration analysis [21] and hence, triangular elements with 3 smoothing cells are utilized, as shown in Fig. 3. Substituting Eq. (21) into Eq. (22), the smoothed temperature gradient can be obtained İ kT( C ) ≡
1 Ak ( C )
NP
NP
§¨ n ⋅ ϕ i dΓ ·¸Ti = ¦ g ki ( C ) Ti ¦ ³ Γk ( C ) © ¹ i =1 i =1
(23)
where NP denotes the number of the supporting nodes, Ti is the nodal temperature and
gki ( C ) is the smoothed temperature gradient interpolation matrix, which is given as
ªb i º g ki ( C ) = « kxi ( C ) » ¬bky ( C ) ¼
(24)
in which i kx ( C )
b
=
1 Ak ( C )
N sgem
¦ I =1
NG ª º 1 i n l « Ix I ¦ ϕ i ( x IJ ) » , bky (C ) = Ak ( C ) J =1 ¬ ¼
Nsgem
¦ I =1
NG ª º n l « Iy I ¦ ϕi ( x IJ ) » J =1 ¬ ¼
(25)
where lI denote the length of Ith segment, Nsgem represents the number of the segment, NG is the total number of Gauss points, nIx and nIy denote the components of the outward unit and xIJ is the coordinate vector of Jth gauss point. Smoothed Galerkin weak is employed to handle Eq. (3). A set of discretized system equations are then obtained
7
∂T ~ K T {T} + N T { } = {Q c } + {Q q } ∂t
(26)
where, ªk 0 º ~ K T = ³ 2ʌrg T « » gdȍi ȍi ¬0 k ¼
(27)
NT = ³ 2ʌrij T ijȡcdȍi
(28)
{Q c } = ³ 2ʌrij T hc (T∞ − T )dī
(29)
{Q q } = ³ 2ʌrij T q0 dī
(30)
ȍi
ī
ī
~ Where K T denotes the smoothed conduction matrix, N T represents the thermal capacitance matrix. {Q c } and {Q q } are the thermal load vectors. The thermal material parameters k, c, and ρ can be determined by Eq. (1) and Eq. (2). For the time approximation, the unconditionally stable implicit backward difference algorithm is employed. Then, {
{
∂T(t ) } can be expressed as ∂t
∂T(t ) {T(t )}- {T(t − Δt )} }= + ȅ(ǻt) ∂t Δt
(31)
Substituting Eq. (31) into Eq. (26) yields
N N ~ ( K T + T ){T(t )} = {Qc } + {Q q } + ( T ){T(t − Δt )} Δt Δt
(32)
When initial condition is given, the temperature field function at any time t can be solved by Eq. (32).
4.
CS-RPIM for mechanical analysis
For the mechanical analysis, the governing system equations are expressed as
8
Equilibrium equation: σ ij , j + bi = 0 ( ȍ )
(33)
Displacement boundary: ui = ui ( ī u )
(34)
Stress boundary: σ ij ni = f i ( ī f )
(35)
Stress-strain relations: σ ij = δ ij λε kk + 2 με ij − δ ij (3λ + 2 μ )αΔT ( δ ij = 1, i = j; δ ij = (36) 0, iĮ j)
Strain-displacement relations: ε ij = Lamé’s constants: λ =
μ=
1 (ui , j + u j ,i ) 2
Eυ (1 + υ )(1 − 2υ )
(37)
(38)
E 2(1 + υ )
(39)
where ui denotes the displacement, ui and f i are the prescribed displacement and force respectively. α represents the thermal expansion coefficient, ΔT is the change of temperature, λ and μ are Lamé’s constants. The total strain ε ij can be decomposed into two parts: the elastic strain ε ijE and the thermal strain ε ijT . İ ij = İ ijE + İ Tij
(40)
For the mechanical analysis, the deducing process of CS-RPIM is similar to the previous thermal analysis, which has been discussed in detail. Therefore, we just give a brief formulation for the mechanical analysis. The displacement field function u is also approximated by RPIM shape functions. n
m
i =1
j =1
U = ¦ Ri ( x )a i + ¦ p j ( x )b j = R T ( x )a + P T ( x )b
(41)
The generalized gradient smoothing technique is also utilized for the mechanical analysis 9
and then the strain field of Cth smoothing cell Ωk(C) can be calculated by
1
İk ( C ) =
Ak ( C )
NP
NP
i § · ¨ ³Γ n ⋅ ij i ( x ) dΓ ¸ u i = ¦ B k ( C ) u i ¦ k ( C ) ¹ i =1 © i =1
(42)
where İk ( C ) denotes the smoothed strain, NP represents the number of the supporting nodes of point x, ui is the nodal displacement, and B ik (C ) denotes the smoothed strain matrix
B ik ( C )
ªbkxi ( C ) « =« 0 «bkyi ( C ) ¬
0 º » bkyi ( C ) » bkxi ( C ) »¼
(43)
By using the smoothed strain, the smoothed Galerkin weak form can be further obtained
³ {δİ} {ı}dȍ = ³ {δu} {b}dȍ + ³ {δu} {f }dī T
T
ȍ
ȍ
T
ī
(44)
where ı denotes the smoothed stress that derived by the smoothed strain. b and f are the body force and boundary traction, respectively. By substituting Eq. (41), (42) and (43) into Eq. (44), the following sets of discretized equations can be obtained ~ K{U} = R f + R T
K ij =
(45)
T § SC · i ¨ ( ) 2 π r B D( B ki ( C ) ) Ak ( C ) ¸ ¦ ¦ ( ) k C ¨ ¸ k =1 © c =1 ¹
N cell
(46)
R f = ³ 2πrĭ T bdȍ + ³ 2πrĭT fdī ȍ
RT =
Ncell
¦³ k =1
(47)
ī
ȍk
2πr ( B k ) T Dİ kT dȍk
(48)
where Ncell denotes the number of the cells, K ij represents the smoothed stiffness matrix,
B k is the smoothed strain-displacement matrix of the Cth smoothing cell, D is the elasticity matrix, R f and R T are the load matrixes. The material parameters that
10
needed for the mechanical analysis can be determined by Eq. (1) and Eq. (2).
5. Results and discussion The accuracy, stability and convergence rate of CS-RPIM are fully investigated through several numerical examples. For thermal and mechanical analysis of structures, it has been demonstrated that the edge-based smoothed finite element method (ES-FEM) [28] is more efficient than FEM. Therefore, in order to verify the validity of CS-RPIM, ES-FEM and RPIM codes are also developed in FORTRAN for comparison purpose. The analytical solutions for such problems are usually not available and hence, reference solution is obtained using the finite element commercial software ABAQUS. In addition, thermal equivalent energy and elastic strain energy are also defined to study the convergence rate ªk x UT = ³ g T « ȍ ¬0
0º gdȍ k y »¼
(49)
1 E T (İ ) D(İ E )dȍ ȍ 2
(50)
UE = ³
In this study, RPIM shape functions are constructed using the MQ-RBF augmented with the linear polynomial basis (m=3). The shape parameters q is taken as 1.03, Įc is taken as 0.35. A more detailed discussion for such choices can be found in [29].
5.1 Example 1 In this section, a ZrO2/Ti-6Al-4V FGM cylinder is investigated. Thermal and mechanical boundary conditions are considered, as illustrated in Fig. 4. In the computation, the related parameters are taken as ri=0.1m, ro=0.2m, p=1.6h108N/m2. Both the upper and lower planes are adiabatic. For the Neumann boundary on the inner plane, q0=3000W/m2 and for the Robin boundary on the outer plane, T∞ =100ć and hc=200W/m2·ć. The 11
material parameters are provided in Table 1. The CS-RPIM, RPIM and ES-FEM solutions are obtained using the same triangular meshes with 231 nodes. For comparison purposes, the reference solution is obtained using ABAQUS quadrilateral elements with a large number of nodes (12506). In this numerical example, irregular meshes are utilized to investigate the sensitivity of present method to mesh distortion, which can be generated by the following equations xir = x + Δx ⋅ rc ⋅ α ir
(51)
yir = y + Δy ⋅ rc ⋅ α ir
where Δx and Δy denote the initial regular nodal spacing in x and y direction, respectively. rc is a random number between -1.0 and 1.0, Įir is a prescribed irregularity factor chosen
between 0.0 and 0.5. For example, irregular mesh with Įir=0.25 is plotted in Fig. 5. For n=0.2 and n=5, the temperature distributions along central line AB are plotted in Fig. 6 and Fig. 7, which is obtained by CS-RPIM, ES-FEM, RPIM and ABAQUS, respectively. It can be clearly seen that the results of CS-RPIM agree well with reference solutions. Fig. 8 shows the displacement of point A using irregularly distributed nodes for n=0.2, when the system has already achieved the stable state. The variation of
displacement of point B using irregular meshes for n=5 is plotted in Fig. 9. Based on these results, it can be proved that the CS-RPIM is insensitive to the mesh distortion even when the meshes are seriously distorted. Furthermore, Fig. 10 and Fig. 11 compare the thermal stress distributions along the central line AB for n=0.2 and n=5, respectively, which again verify that CS-RPIM can give more accurate results than ES-FEM and RPIM when using the same triangular meshes. It should be noted that the improvement of accuracy is mainly caused by the smoothing operation.
12
5.2 Example 2 A TiC/Ni FGM cylinder with mixed boundary conditions is studied here, as illustrated in Fig. 12. Some computational parameters are given as ri=0.2m, ro=0.5m. For the Robin boundary on the inner plane, T∞ =300ć and hc=600W/m2·ć and on the outer plane, T∞ =30ć and hc=100W/m2·ć. The inner pressure is taken as p=9.6h108N/m2. The
material parameters are listed in Table 2. The problem domain is discretized by triangular meshes using 273 nodes and reference solution is also obtained using ABAQUS quadrilateral elements with a number of nodes (13639). The comparisons of temperature distribution along the central line AB for different value of n are shown in Fig. 13 and Fig. 14, when the system has already achieved the stable state. From these results, it is clearly observed that the results obtained by CS-RPIM agree better with reference solutions than those obtained by ES-FEM and RPIM. For different value of n, the convergences of displacements versus the number of degrees of freedom are further plotted in Fig. 15 and Fig. 16, repectively. In addition, the convergence of thermal equivalent energy for n=0.2 and the convergence of elastic strain energy for n=5 are plotted in Fig. 17 and Fig. 18, respectively. These results circumstantiate that the convergence rate of CS-RPIM model is much higher than ES-FEM and RPIM models, when dealing with the thermo-mechanical analysis of FGM cylinders.
6. Conclusions The cell-based smoothed radial point interpolation method is formulated to deal with the thermal and mechanical analysis of FGM cylinders in this study. The accuracy, stability and convergence rate of CS-RPIM are fully compared with some existing methods, in
13
which different material models and boundary conditions are considered. Based on these studies, we can come to the conclusion that the CS-RPIM model is very accurate and stable for triangular meshes, even when the mesh quality is extremely poor. In addition, the convergence rate of CS-RPIM is proved to be much higher, when dealing with the thermal and mechanical analysis of FGM cylinders.
Acknowledgements The Fundamental Research Funds for the Central Universities and Priority Academic Program Development of Jiangsu Higher Education Institutions are gratefully acknowledged.
References [1] J. Ranjbar, A. Alibeigloo, Response of functionally graded spherical shell to thermo-mechanical shock, Aerospace Science and Technology 51 (2016) 61-69. [2] A.M. Zenkour, Mohammed Sobhy, Dynamic bending response of thermoelastic elastic functionally graded plates resting on elastic foundations, Aerospace Science and Technology 29 (2013) 7-17. [3] M. Lezgy-Nazargah, Fully coupled thermo-mechanical analysis of bi-directional FGM beams using NURBS isogeometric finite element approach, Aerospace Science and Technology 45 (2015) 154-164. [4] T. Singh and V.K. Gupta, Effect of anisotropy on steady state creep in functionally graded cylinder, Composite Structures 93 (2011) 747-58. [5] S.M. Hosseini, J. Sladek and V. Sladek, Meshless local Petrov-Galerkin method for coupled thermoelasticity analysis of a functionally graded thick hollow cylinder, Engineering Analysis with Boundary Elements 35 (2011) 827-835. [6] M. Asgari and M. Akhlaghi, Thermo-Mechanical Analysis of 2D-FGM Thick Hollow Cylinder using Graded Finite Elements, Advances in Structural Engineering 14 (2011) 1059-1073. [7] A. Daneshmehr and S.M. Bashusqeh, On the dynamical thermoelasticity problem for a hollow funtionally graded cylinder, Mechanics of Advanced Materials and Structures 23 (2016), 195-200. [8] G.Z. Xie, J.M. Zhang and Y.Q. Dong, An improved exponential transformation for nearly singular boundary element integrals in elasticity problems, International Journal of Solids and Structures 51 (2014) 1322-1329. [9] G.Z. Xie, J.M. Zhang and C. Huang, A direct traction boundary integral equation method for three-dimension crack problems in infinite and finite domains, Computational Mechanics 53 (2014) 575-586. [10] Y. Gu, H.W. Gao, W. Chen, C.J. Liu, C.Z. Zhang and X.Q. He, Fast-multipole accelerated singular boundary method for large-scale three-dimensional potential problems, International Journal of Heat and Mass Transfer 90 (2015) 291-301. [11] L.Y. Yao, J.W. Zhou, Z. Zhou, L. Li. Dispersion error reduction for acoustic problems using the finite element-least square point interpolation method, Journal of Vibration and Acoustics-Transactions of the ASME 137 (2015). [12] X.Y. Cui and G.Y. Li, Smoothed Galerkin methods using cell-wise strain smoothing technique, Engineering Analysis with Boundary Elements 36 (2012) 825-835.
14
[13] S.Z. Feng, X.Y. Cui, A.M. Li and G.Z. Xie, A face-based smoothed point interpolation method (FS-PIM) for analysis of nonlinear heat conduction in multi-material bodies, International Journal of Thermal Sciences 100 (2016) 430-437. [14] S.C. Wu, G.R. Liu, H.O. Zhang, X. Xu and Z.R. Li, A node-based smoothed point interpolation method (NS-PIM) for three-dimensional heat transfer problems, International Journal of Thermal Sciences 48 (2009) 1367-1376. [15] X.Y. Cui, G.R. Liu and G.Y. Li, A smoothed Hermite radial point interpolation method for thin plate analysis using gradient smoothing operation, Archive of Applied Mechanics 81 (2011) 1-18. [16] G.R. Liu, A G space theory and a weakened weak (W2) form for a unified formulation of compatible and incompatible methods: Part I theory, International Journal for Numerical Methods in Engineering 81 (2010) 1093-1126. [17] X.Y. Cui, G.R. Liu, G.Y. Li and G. Zheng, A rotation free formulation for static and free vibration analysis of thin beams using gradient smoothing technique, CMES-Computer Modeling in Engineering& Sciences 38 (2008) 217-229. [18] X.Y. Cui, G.R. Liu, G.Y. Li, G.Y Zhang and G.Y Sun, Analysis of elastic–plastic problems using edge-based smoothed finite element method, International Journal of Pressure Vessels and Piping 86 (2009) 711-718. [19] X.Y. Cui and G.Y. Li, Metal forming analysis using the edge-based smoothed finite element method, Finite Elements in Analysis and Design 63 (2013) 33-41. [20] S.C. Wu, G.R. Liu, H.O. Zhang and G.Y. Zhang, A node-based smoothed point interpolation method (NS-PIM) for thermoelastic problems with solution bounds, International Journal of Heat and Mass Transfer 52 (2009) 1464-1471. [21] X.Y. Cui, G.R. Liu and G.Y. Li, A cell-based smoothed radial point interpolation method (CS-RPIM) for static and free vibration of solids, Engineering Analysis with Boundary Elements 34 (2010) 144-157. [22] X.Y. Cui, G.R. Liu and G.Y. Li, Analysis of Mindlin-Reissner plates using cell-based smoothed radial point interpolation method, International Journal of Applied Mechanics 2 (2010) 653-680. [23] X.Y. Cui, S.Z. Feng and G.Y. Li, A cell-based smoothed radial point interpolation method (CS-RPIM) for heat transfer analysis, Engineering Analysis with Boundary Elements 40 (2014) 147-153. [24] X.Y. Cui, H. Feng, G.Y. Li and S.Z. Feng, A cell-based smoothed radial point interpolation method (CS-RPIM) for three-dimensional solids, Engineering Analysis with Boundary Elements 50 (2015) 474-485. [25] L.Y. Yao, D.J. Yu and J.W. Zhou, Numerical treatment of 2D acoustic problems with the cell-based smoothed radial point interpolation method, Applied Acoustics 73 (2012) 557-574. [26] M.A. Golberg, C.S. Chen and H. Bowman, Some recent results and proposals for the use of radial basis functions in the BEM, Engineering Analysis with Boundary Elements 23 (1999) 285-296. [27] G.R. Liu, A generalized gradient smoothing technique and the smoothed bilinear form for Galerkin formulation of a wide class of computational methods, International Journal of Computational Methods 5 (2008) 199-236. [28] S.Z. Feng, X.Y. Cui and G.Y. Li, Analysis of transient thermo-elastic problems using edge-based smoothed finite element method, International Journal of Thermal Sciences 65 (2013) 127-135. [29] J. Wang and G.R. Liu, On the optimal shape parameters of radial basis functions used for 2-D meshless methods, Computer Methods in Applied Mechanics and Engineering 191 (2002) 2611-2630.
15
Table 1 Material properties of ZrO2 and Ti-6Al-4V
Materials ZrO2 Ti-6Al-4V
E (GPa)
υ
k (W/m·ć)
ρ(kg/m3)
c (J/kg·ć)
α
210 115
0.3 0.289
1.78 7.955
5680 4420
420 612
9.4h10-6 7.89h10-6
Table 2 Material properties of TiC and Ni
Materials TiC Ni
E (GPa)
υ
k (W/m·ć)
ρ(kg/m3)
c (J/kg·ć)
α
460 199.5
0.336 0.312
20 89.5
4930 8900
560 440
7.3h10-6 17.5h10-6
Fig. 1. Variation of volume fraction versus non-dimensional radius
Field node parent cell k
"
Ω k ( SC )
Ω k (C )
Ω k (1)
"
cell edge
"
Γ k (C )
"
segment
Fig. 2. Illustration of constructing smoothing domains based on triangular meshes
Field node
Integration sampling point
SC=3 Fig. 3. A parent cell is divided into 3 smoothing cells
Fig. 4. Schematic of a FGM hollow cylinder with the Displacement, Neumann, Robin and force boundary conditions
Ir =
0.2
Fig. 5. Schematic of irregular mesh for Įir=0.2
Fig. 6. Comparison of Temperature distribution along the central line AB (AėB) for n=0.2
Fig. 7. Comparison of Temperature distribution along the central line AB (AėB) for n=5
Fig. 8. Displacement of point A for different methods using irregularly distributed field nodes for n=0.2
Fig. 9. Displacement of point B for different methods using irregularly distributed field nodes for n=5
Fig. 10. Comparison of thermal stress distribution along the central line AB (AėB) for n=0.2
Fig. 11. Comparison of thermal stress distribution along the central line AB (AėB) for n=5
z Robin boundary
Robin boundary Adiabatic
P A
B
ri TiC/Ni
ro
o
Adiabatic
r
Fig. 12. Geometry and the boundary conditions of a TiC/Ni FGM hollow cylinder
Fig. 13. Comparison of temperature distribution along the central line AB (AėB) for n=0.2
Fig. 14. Comparison of temperature distribution along the central line AB (AėB) for n=5
Fig. 15. Convergence of displacement versus the number of degrees of freedom of reference point A for n=0.2
Fig. 16. Convergence of displacement versus the number of degrees of freedom of reference point B for n=5
Fig. 17. Convergence of the thermal equivalent energy versus the number of degrees of freedom for n=0.2
Fig. 18. Convergence of the elastic strain energy versus the number of degrees of freedom for n=5
S.Z. Feng, A.M. Li
PII: DOI: Reference:
S1270-9638(16)30812-4 http://dx.doi.org/10.1016/j.ast.2017.02.009 AESCTE 3918
To appear in:
Aerospace Science and Technology
Received date: Revised date: Accepted date:
8 October 2016 19 January 2017 14 February 2017
Please cite this article in press as: S.Z. Feng, A.M. Li, Analysis of thermal and mechanical response in functionally graded cylinder using cell-based smoothed radial point interpolation method, Aerosp. Sci. Technol. (2017), http://dx.doi.org/10.1016/j.ast.2017.02.009
This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.
Analysis of thermal and mechanical response in functionally graded cylinder using cell-based smoothed radial point interpolation method S. Z. Feng∗, A. M. Li School of Mechatronic Engineering, China University of Mining and Technology, Xuzhou, 221116, P. R. China
Abstract This study aims to improve the accuracy and efficiency when dealing with thermal and mechanical response in functionally graded cylinder, which is very important in modern aerospace industry. The cell-based smoothed radial point interpolation method (CS-RPIM) is formulated for such analysis. In CS-RPIM, triangular meshes are utilized to discretize problem domains, which can be easily generated. Each triangular element is then partitioned into several smoothing cells. Field functions are constructed by RPIM shape functions and system equations are obtained based on these smoothing cells. Finally, the performances of CS-RPIM are fully investigated through several numerical examples.
Keywords: FGM; RPIM; Meshfree method; Thermal; Gradient smoothing technique
Aerospace industry
1. Introduction The functionally graded material (FGM) is a new kind of composite material, which has been successfully utilized in modern aerospace industry. In general, FGM is made up of metals and ceramics, in which the material volume fractions change continuously along desired directions [1, 2]. The metal component is utilized to prevent fracture caused by thermo-mechanical loads and the ceramic constituent can offer heat-resistant property
* Corresponding author. Tel: +86 516 83590718; fax: +86 516 8825016 E-mail address: [email protected] (S. Z. Feng).
owing to low thermo-conductivity. Therefore, engineering structures made by FGM can work under thermal and mechanical loads safely, such as FGM cylinders, in which the volume fraction varies continuously in the radius direction according to a power law. The thermal and mechanical analysis of FGM structures is an important issue in practical engineering [3, 4]. So many studies have been done to solve this problem in the past few years. For example, the thermo-elasticity analysis of functionally graded thick cylinder was studied by Hosseini [5]. Asgari utilized the graded finite elements to investigate the thermo-mechanical response of functionally graded cylinder [6]. Daneshmehr studied the dynamical thermo-elasticity responses of functionally graded cylinder, in which the time-dependent boundary conditions are considered [7]. It can be clearly seen that the analytical solution for such problems are usually very difficult to obtain and hence, numerical methods [8-12] are employed to deal with such analysis, such as the finite element method (FEM) and meshfree methods [13-15], which have become dominative tools in practical engineering. In order to further improve the efficiency of these methods, the gradient smoothing technique was proposed and applied to these methods [16, 17]. For example, Cui formulated an edge-based smoothed finite element method to deal with mechanical analysis of engineering structures [18, 19], which can provide much more accurate results than standard FEM. Wu utilized a node-based smoothed point interpolation method for thermal and mechanical problems [20]. Based on these studies, a cell-based smoothed radial point interpolation method (CS-RPIM) was then formulated to deal with static and free vibration analysis of structures [21-25] and numerical results with high quality are obtained. The main purpose of this study is to improve the accuracy, stability and convergence rate, when investigating thermal and mechanical responses of FGM cylinder. The validity of
2
this method is fully tested through several numerical examples and numerical results are compared with those obtained by existing fine numerical methods.
2.
Functionally graded cylinder
Functionally graded cylinders are made by metals and ceramics. Usually the outer layer is metal-rich and the inner layer is ceramic-rich. The material properties of FGM cylinder can be given as
Pf = Pm + ( Pc − Pm )Vc
Vc = (1 −
(1)
r − ri n ) ( n ≥ 0) , Vm = 1 − Vc ro − ri
(2)
where n represents the volume fraction exponent. Pf , Pc and Pm represent the material properties of ceramic and metal, respectively. ro is the outer radius, ri is the inner radius, c and m denote the ceramic and the metal, respectively. From Eq. (1) and Eq. (2), it is clearly seen that for for n = ∞ , Pf = Pm and n = 0 , Pf = Pc . These are two particular cases, which correspond to pure metal and ceramic, respectively. For different value of n, Fig. 1 further shows the variation of the volume fraction function versus non-dimensional radius.
3. Heat transfer analysis 3.1 Basic system equations When dealing with the transient heat transfer analysis of FGM cylinders, the governing differential equation can be given as
ȡcr
∂T ∂ ∂T ∂ ∂T = ( kr ) + ( kr ) ∂t ∂r ∂r ∂z ∂z
(3)
where, r represents the radius, k denotes the thermo-conductivity, ρ is the density and c is
3
the specific heat. Different thermal boundary conditions investigated in this study are directly listed as Initial condition : T = T0
∂T |ī = q0 ∂n0
(5)
∂T |ī = hc (T − T∞ ) ∂n0
(6)
Neumann boundary : − k
Robin boundary : − k
(4)
Adiabatic boundary : − k
∂T |ī = 0 ∂n0
(7)
where, T0 represents the initial temperature, q0 is the heat flux, T∞ is the environment temperature, hc denotes the convection coefficient and n0 is the unit outward normal.
3.2 Formulation of RPIM In this work, the temperature field function is approximated by the RPIM shape function, which is constructed using the local nodes and radial basis function (RBF). The polynomial basis function is also used to construct the RPIM shape function. For a problem domain including a set of arbitrarily scattered points xi ( i=1, 2, ..., n ), the temperature field function T can be approximated as n
m
i =1
j =1
T = ¦ Ri (x)ai + ¦ p j (x)b j = R T (x)a + P T (x)b
(8)
where Ri(x) denotes the radial basis function, ai is the unknown coefficient for functions Ri(x), bj represents the coefficient for polynomial basis Pj(x), and m is determined according to the polynomial basis. The multi-quadrics RBF (MQ-RBF) that used in RPIM can be given as
4
[
Ri ( x ) = ( x − xi ) 2 + ( y − yi ) 2 + (α c d c ) 2
]
q
(9)
where, dc denotes the equivalent length of triangular elements [21], q and Įc represent the real and arbitrary shape parameters. The polynomial basis function is directly given here PT (x) = [1, x, y,...]
(10)
By enforcing the field function T to be satisfied at the n nodes within the local support domain, the coefficients in Eq. (8) can then be computed. This procedure will bring about a set of linear equations, which is given as
Ts = R q a + Pmb
(11)
where Ts is the vector of function values and it is given as
Ts = {T1 , T2 ,..., Tn }T
(12)
Rq represents the moment matrix of RBFs,
ª R1 (x1 ) R2 (x1 ) « R (x ) R (x ) 2 2 Rq = « 1 2 « ... ... « ¬ R1 (x n ) R2 (x n )
... Rn (x1 ) º ... Rn (x 2 )»» ... ... » » ... Rn (x n )¼ n×n
(13)
Matrix Pm is defined as ª p1 (x1 ) « p (x ) Pm = « 1 2 « ... « ¬ p1 (x n )
p2 (x1 ) ... p2 (x 2 ) ... ...
...
p2 (x n ) ...
pm (x1 ) º pm (x 2 ) »» ... » » pm (x n )¼
(14)
The following constraint condition is utilized for the formulation [26]. n
¦ p (x j
k
)ai = 0 , j = 1,2,..., m
(15)
i =1
Combining Eq. (11) and Eq. (15) leads to
5
~ T ½ ªR Ts = ® s ¾ = « Tq ¯ 0 ¿ ¬ Pm
Pm º a ½ a ½ ® ¾ = G® ¾ » 0 ¼ ¯b ¿ ¯b ¿
(16)
By solving Eq. (16), the following equation can be obtained T a ½ −1 U s ½ a ½ −1 s ½ ® ¾=G ® ¾ ® ¾=G ® ¾ ¯0¿ ¯b ¿ ¯ 0 ¿ ¯b ¿
(17)
Therefore, the temperature field function T can be given as
T ½ PmT ]G −1 ® s ¾ = ijTs ¯0¿
T = [R Tq
(18)
where, ij(x) denotes the RPIM shape functions, which has the following form ij = [ϕ1 ϕ2 ... ϕn ]
(19)
where n
m
i =1
j =1
ϕk (x) = ¦ Ri (x)G(i ,k ) + ¦ p j (x)G( n + j ,k )
(20)
where G(i ,k ) represents the constituent of matrix G −1 . Finally, the temperature field function T is approximated as n
T = ¦ ϕ i Ti
(21)
i =1
3.3 CS-RPIM for heat transfer analysis The problem domain is first discretized by triangular elements, which is the same as standard FEM. The triangular element is called as “parent” cell for convenience in the discussion, which will be further divided into non-overlapping smoothing cells Ω k = * SC C =1 Ω k ( C ) , as shown in Fig. 2. The edge of the smoothing cell is called as
“segment”. The boundary īk(C) of the smoothing cell is made up of the segments of smoothing cell. It is assumed that the temperature gradient of the Cth smoothing cell 6
Ωk(C) is constant. By using the generalized gradient smoothing technique [27], it can be
given as İ kT( C ) ≡
1 n ⋅ TdΓ Ak ( C ) ³Γk ( C )
(22)
where T is the temperature field function of smoothing cell ȍk(C), Ak(C) is the area of the Cth smoothing cell and n is the outward normal. It has been proved that the SC=3 scheme can give accurate and stable results in the static and free vibration analysis [21] and hence, triangular elements with 3 smoothing cells are utilized, as shown in Fig. 3. Substituting Eq. (21) into Eq. (22), the smoothed temperature gradient can be obtained İ kT( C ) ≡
1 Ak ( C )
NP
NP
§¨ n ⋅ ϕ i dΓ ·¸Ti = ¦ g ki ( C ) Ti ¦ ³ Γk ( C ) © ¹ i =1 i =1
(23)
where NP denotes the number of the supporting nodes, Ti is the nodal temperature and
gki ( C ) is the smoothed temperature gradient interpolation matrix, which is given as
ªb i º g ki ( C ) = « kxi ( C ) » ¬bky ( C ) ¼
(24)
in which i kx ( C )
b
=
1 Ak ( C )
N sgem
¦ I =1
NG ª º 1 i n l « Ix I ¦ ϕ i ( x IJ ) » , bky (C ) = Ak ( C ) J =1 ¬ ¼
Nsgem
¦ I =1
NG ª º n l « Iy I ¦ ϕi ( x IJ ) » J =1 ¬ ¼
(25)
where lI denote the length of Ith segment, Nsgem represents the number of the segment, NG is the total number of Gauss points, nIx and nIy denote the components of the outward unit and xIJ is the coordinate vector of Jth gauss point. Smoothed Galerkin weak is employed to handle Eq. (3). A set of discretized system equations are then obtained
7
∂T ~ K T {T} + N T { } = {Q c } + {Q q } ∂t
(26)
where, ªk 0 º ~ K T = ³ 2ʌrg T « » gdȍi ȍi ¬0 k ¼
(27)
NT = ³ 2ʌrij T ijȡcdȍi
(28)
{Q c } = ³ 2ʌrij T hc (T∞ − T )dī
(29)
{Q q } = ³ 2ʌrij T q0 dī
(30)
ȍi
ī
ī
~ Where K T denotes the smoothed conduction matrix, N T represents the thermal capacitance matrix. {Q c } and {Q q } are the thermal load vectors. The thermal material parameters k, c, and ρ can be determined by Eq. (1) and Eq. (2). For the time approximation, the unconditionally stable implicit backward difference algorithm is employed. Then, {
{
∂T(t ) } can be expressed as ∂t
∂T(t ) {T(t )}- {T(t − Δt )} }= + ȅ(ǻt) ∂t Δt
(31)
Substituting Eq. (31) into Eq. (26) yields
N N ~ ( K T + T ){T(t )} = {Qc } + {Q q } + ( T ){T(t − Δt )} Δt Δt
(32)
When initial condition is given, the temperature field function at any time t can be solved by Eq. (32).
4.
CS-RPIM for mechanical analysis
For the mechanical analysis, the governing system equations are expressed as
8
Equilibrium equation: σ ij , j + bi = 0 ( ȍ )
(33)
Displacement boundary: ui = ui ( ī u )
(34)
Stress boundary: σ ij ni = f i ( ī f )
(35)
Stress-strain relations: σ ij = δ ij λε kk + 2 με ij − δ ij (3λ + 2 μ )αΔT ( δ ij = 1, i = j; δ ij = (36) 0, iĮ j)
Strain-displacement relations: ε ij = Lamé’s constants: λ =
μ=
1 (ui , j + u j ,i ) 2
Eυ (1 + υ )(1 − 2υ )
(37)
(38)
E 2(1 + υ )
(39)
where ui denotes the displacement, ui and f i are the prescribed displacement and force respectively. α represents the thermal expansion coefficient, ΔT is the change of temperature, λ and μ are Lamé’s constants. The total strain ε ij can be decomposed into two parts: the elastic strain ε ijE and the thermal strain ε ijT . İ ij = İ ijE + İ Tij
(40)
For the mechanical analysis, the deducing process of CS-RPIM is similar to the previous thermal analysis, which has been discussed in detail. Therefore, we just give a brief formulation for the mechanical analysis. The displacement field function u is also approximated by RPIM shape functions. n
m
i =1
j =1
U = ¦ Ri ( x )a i + ¦ p j ( x )b j = R T ( x )a + P T ( x )b
(41)
The generalized gradient smoothing technique is also utilized for the mechanical analysis 9
and then the strain field of Cth smoothing cell Ωk(C) can be calculated by
1
İk ( C ) =
Ak ( C )
NP
NP
i § · ¨ ³Γ n ⋅ ij i ( x ) dΓ ¸ u i = ¦ B k ( C ) u i ¦ k ( C ) ¹ i =1 © i =1
(42)
where İk ( C ) denotes the smoothed strain, NP represents the number of the supporting nodes of point x, ui is the nodal displacement, and B ik (C ) denotes the smoothed strain matrix
B ik ( C )
ªbkxi ( C ) « =« 0 «bkyi ( C ) ¬
0 º » bkyi ( C ) » bkxi ( C ) »¼
(43)
By using the smoothed strain, the smoothed Galerkin weak form can be further obtained
³ {δİ} {ı}dȍ = ³ {δu} {b}dȍ + ³ {δu} {f }dī T
T
ȍ
ȍ
T
ī
(44)
where ı denotes the smoothed stress that derived by the smoothed strain. b and f are the body force and boundary traction, respectively. By substituting Eq. (41), (42) and (43) into Eq. (44), the following sets of discretized equations can be obtained ~ K{U} = R f + R T
K ij =
(45)
T § SC · i ¨ ( ) 2 π r B D( B ki ( C ) ) Ak ( C ) ¸ ¦ ¦ ( ) k C ¨ ¸ k =1 © c =1 ¹
N cell
(46)
R f = ³ 2πrĭ T bdȍ + ³ 2πrĭT fdī ȍ
RT =
Ncell
¦³ k =1
(47)
ī
ȍk
2πr ( B k ) T Dİ kT dȍk
(48)
where Ncell denotes the number of the cells, K ij represents the smoothed stiffness matrix,
B k is the smoothed strain-displacement matrix of the Cth smoothing cell, D is the elasticity matrix, R f and R T are the load matrixes. The material parameters that
10
needed for the mechanical analysis can be determined by Eq. (1) and Eq. (2).
5. Results and discussion The accuracy, stability and convergence rate of CS-RPIM are fully investigated through several numerical examples. For thermal and mechanical analysis of structures, it has been demonstrated that the edge-based smoothed finite element method (ES-FEM) [28] is more efficient than FEM. Therefore, in order to verify the validity of CS-RPIM, ES-FEM and RPIM codes are also developed in FORTRAN for comparison purpose. The analytical solutions for such problems are usually not available and hence, reference solution is obtained using the finite element commercial software ABAQUS. In addition, thermal equivalent energy and elastic strain energy are also defined to study the convergence rate ªk x UT = ³ g T « ȍ ¬0
0º gdȍ k y »¼
(49)
1 E T (İ ) D(İ E )dȍ ȍ 2
(50)
UE = ³
In this study, RPIM shape functions are constructed using the MQ-RBF augmented with the linear polynomial basis (m=3). The shape parameters q is taken as 1.03, Įc is taken as 0.35. A more detailed discussion for such choices can be found in [29].
5.1 Example 1 In this section, a ZrO2/Ti-6Al-4V FGM cylinder is investigated. Thermal and mechanical boundary conditions are considered, as illustrated in Fig. 4. In the computation, the related parameters are taken as ri=0.1m, ro=0.2m, p=1.6h108N/m2. Both the upper and lower planes are adiabatic. For the Neumann boundary on the inner plane, q0=3000W/m2 and for the Robin boundary on the outer plane, T∞ =100ć and hc=200W/m2·ć. The 11
material parameters are provided in Table 1. The CS-RPIM, RPIM and ES-FEM solutions are obtained using the same triangular meshes with 231 nodes. For comparison purposes, the reference solution is obtained using ABAQUS quadrilateral elements with a large number of nodes (12506). In this numerical example, irregular meshes are utilized to investigate the sensitivity of present method to mesh distortion, which can be generated by the following equations xir = x + Δx ⋅ rc ⋅ α ir
(51)
yir = y + Δy ⋅ rc ⋅ α ir
where Δx and Δy denote the initial regular nodal spacing in x and y direction, respectively. rc is a random number between -1.0 and 1.0, Įir is a prescribed irregularity factor chosen
between 0.0 and 0.5. For example, irregular mesh with Įir=0.25 is plotted in Fig. 5. For n=0.2 and n=5, the temperature distributions along central line AB are plotted in Fig. 6 and Fig. 7, which is obtained by CS-RPIM, ES-FEM, RPIM and ABAQUS, respectively. It can be clearly seen that the results of CS-RPIM agree well with reference solutions. Fig. 8 shows the displacement of point A using irregularly distributed nodes for n=0.2, when the system has already achieved the stable state. The variation of
displacement of point B using irregular meshes for n=5 is plotted in Fig. 9. Based on these results, it can be proved that the CS-RPIM is insensitive to the mesh distortion even when the meshes are seriously distorted. Furthermore, Fig. 10 and Fig. 11 compare the thermal stress distributions along the central line AB for n=0.2 and n=5, respectively, which again verify that CS-RPIM can give more accurate results than ES-FEM and RPIM when using the same triangular meshes. It should be noted that the improvement of accuracy is mainly caused by the smoothing operation.
12
5.2 Example 2 A TiC/Ni FGM cylinder with mixed boundary conditions is studied here, as illustrated in Fig. 12. Some computational parameters are given as ri=0.2m, ro=0.5m. For the Robin boundary on the inner plane, T∞ =300ć and hc=600W/m2·ć and on the outer plane, T∞ =30ć and hc=100W/m2·ć. The inner pressure is taken as p=9.6h108N/m2. The
material parameters are listed in Table 2. The problem domain is discretized by triangular meshes using 273 nodes and reference solution is also obtained using ABAQUS quadrilateral elements with a number of nodes (13639). The comparisons of temperature distribution along the central line AB for different value of n are shown in Fig. 13 and Fig. 14, when the system has already achieved the stable state. From these results, it is clearly observed that the results obtained by CS-RPIM agree better with reference solutions than those obtained by ES-FEM and RPIM. For different value of n, the convergences of displacements versus the number of degrees of freedom are further plotted in Fig. 15 and Fig. 16, repectively. In addition, the convergence of thermal equivalent energy for n=0.2 and the convergence of elastic strain energy for n=5 are plotted in Fig. 17 and Fig. 18, respectively. These results circumstantiate that the convergence rate of CS-RPIM model is much higher than ES-FEM and RPIM models, when dealing with the thermo-mechanical analysis of FGM cylinders.
6. Conclusions The cell-based smoothed radial point interpolation method is formulated to deal with the thermal and mechanical analysis of FGM cylinders in this study. The accuracy, stability and convergence rate of CS-RPIM are fully compared with some existing methods, in
13
which different material models and boundary conditions are considered. Based on these studies, we can come to the conclusion that the CS-RPIM model is very accurate and stable for triangular meshes, even when the mesh quality is extremely poor. In addition, the convergence rate of CS-RPIM is proved to be much higher, when dealing with the thermal and mechanical analysis of FGM cylinders.
Acknowledgements The Fundamental Research Funds for the Central Universities and Priority Academic Program Development of Jiangsu Higher Education Institutions are gratefully acknowledged.
References [1] J. Ranjbar, A. Alibeigloo, Response of functionally graded spherical shell to thermo-mechanical shock, Aerospace Science and Technology 51 (2016) 61-69. [2] A.M. Zenkour, Mohammed Sobhy, Dynamic bending response of thermoelastic elastic functionally graded plates resting on elastic foundations, Aerospace Science and Technology 29 (2013) 7-17. [3] M. Lezgy-Nazargah, Fully coupled thermo-mechanical analysis of bi-directional FGM beams using NURBS isogeometric finite element approach, Aerospace Science and Technology 45 (2015) 154-164. [4] T. Singh and V.K. Gupta, Effect of anisotropy on steady state creep in functionally graded cylinder, Composite Structures 93 (2011) 747-58. [5] S.M. Hosseini, J. Sladek and V. Sladek, Meshless local Petrov-Galerkin method for coupled thermoelasticity analysis of a functionally graded thick hollow cylinder, Engineering Analysis with Boundary Elements 35 (2011) 827-835. [6] M. Asgari and M. Akhlaghi, Thermo-Mechanical Analysis of 2D-FGM Thick Hollow Cylinder using Graded Finite Elements, Advances in Structural Engineering 14 (2011) 1059-1073. [7] A. Daneshmehr and S.M. Bashusqeh, On the dynamical thermoelasticity problem for a hollow funtionally graded cylinder, Mechanics of Advanced Materials and Structures 23 (2016), 195-200. [8] G.Z. Xie, J.M. Zhang and Y.Q. Dong, An improved exponential transformation for nearly singular boundary element integrals in elasticity problems, International Journal of Solids and Structures 51 (2014) 1322-1329. [9] G.Z. Xie, J.M. Zhang and C. Huang, A direct traction boundary integral equation method for three-dimension crack problems in infinite and finite domains, Computational Mechanics 53 (2014) 575-586. [10] Y. Gu, H.W. Gao, W. Chen, C.J. Liu, C.Z. Zhang and X.Q. He, Fast-multipole accelerated singular boundary method for large-scale three-dimensional potential problems, International Journal of Heat and Mass Transfer 90 (2015) 291-301. [11] L.Y. Yao, J.W. Zhou, Z. Zhou, L. Li. Dispersion error reduction for acoustic problems using the finite element-least square point interpolation method, Journal of Vibration and Acoustics-Transactions of the ASME 137 (2015). [12] X.Y. Cui and G.Y. Li, Smoothed Galerkin methods using cell-wise strain smoothing technique, Engineering Analysis with Boundary Elements 36 (2012) 825-835.
14
[13] S.Z. Feng, X.Y. Cui, A.M. Li and G.Z. Xie, A face-based smoothed point interpolation method (FS-PIM) for analysis of nonlinear heat conduction in multi-material bodies, International Journal of Thermal Sciences 100 (2016) 430-437. [14] S.C. Wu, G.R. Liu, H.O. Zhang, X. Xu and Z.R. Li, A node-based smoothed point interpolation method (NS-PIM) for three-dimensional heat transfer problems, International Journal of Thermal Sciences 48 (2009) 1367-1376. [15] X.Y. Cui, G.R. Liu and G.Y. Li, A smoothed Hermite radial point interpolation method for thin plate analysis using gradient smoothing operation, Archive of Applied Mechanics 81 (2011) 1-18. [16] G.R. Liu, A G space theory and a weakened weak (W2) form for a unified formulation of compatible and incompatible methods: Part I theory, International Journal for Numerical Methods in Engineering 81 (2010) 1093-1126. [17] X.Y. Cui, G.R. Liu, G.Y. Li and G. Zheng, A rotation free formulation for static and free vibration analysis of thin beams using gradient smoothing technique, CMES-Computer Modeling in Engineering& Sciences 38 (2008) 217-229. [18] X.Y. Cui, G.R. Liu, G.Y. Li, G.Y Zhang and G.Y Sun, Analysis of elastic–plastic problems using edge-based smoothed finite element method, International Journal of Pressure Vessels and Piping 86 (2009) 711-718. [19] X.Y. Cui and G.Y. Li, Metal forming analysis using the edge-based smoothed finite element method, Finite Elements in Analysis and Design 63 (2013) 33-41. [20] S.C. Wu, G.R. Liu, H.O. Zhang and G.Y. Zhang, A node-based smoothed point interpolation method (NS-PIM) for thermoelastic problems with solution bounds, International Journal of Heat and Mass Transfer 52 (2009) 1464-1471. [21] X.Y. Cui, G.R. Liu and G.Y. Li, A cell-based smoothed radial point interpolation method (CS-RPIM) for static and free vibration of solids, Engineering Analysis with Boundary Elements 34 (2010) 144-157. [22] X.Y. Cui, G.R. Liu and G.Y. Li, Analysis of Mindlin-Reissner plates using cell-based smoothed radial point interpolation method, International Journal of Applied Mechanics 2 (2010) 653-680. [23] X.Y. Cui, S.Z. Feng and G.Y. Li, A cell-based smoothed radial point interpolation method (CS-RPIM) for heat transfer analysis, Engineering Analysis with Boundary Elements 40 (2014) 147-153. [24] X.Y. Cui, H. Feng, G.Y. Li and S.Z. Feng, A cell-based smoothed radial point interpolation method (CS-RPIM) for three-dimensional solids, Engineering Analysis with Boundary Elements 50 (2015) 474-485. [25] L.Y. Yao, D.J. Yu and J.W. Zhou, Numerical treatment of 2D acoustic problems with the cell-based smoothed radial point interpolation method, Applied Acoustics 73 (2012) 557-574. [26] M.A. Golberg, C.S. Chen and H. Bowman, Some recent results and proposals for the use of radial basis functions in the BEM, Engineering Analysis with Boundary Elements 23 (1999) 285-296. [27] G.R. Liu, A generalized gradient smoothing technique and the smoothed bilinear form for Galerkin formulation of a wide class of computational methods, International Journal of Computational Methods 5 (2008) 199-236. [28] S.Z. Feng, X.Y. Cui and G.Y. Li, Analysis of transient thermo-elastic problems using edge-based smoothed finite element method, International Journal of Thermal Sciences 65 (2013) 127-135. [29] J. Wang and G.R. Liu, On the optimal shape parameters of radial basis functions used for 2-D meshless methods, Computer Methods in Applied Mechanics and Engineering 191 (2002) 2611-2630.
15
Table 1 Material properties of ZrO2 and Ti-6Al-4V
Materials ZrO2 Ti-6Al-4V
E (GPa)
υ
k (W/m·ć)
ρ(kg/m3)
c (J/kg·ć)
α
210 115
0.3 0.289
1.78 7.955
5680 4420
420 612
9.4h10-6 7.89h10-6
Table 2 Material properties of TiC and Ni
Materials TiC Ni
E (GPa)
υ
k (W/m·ć)
ρ(kg/m3)
c (J/kg·ć)
α
460 199.5
0.336 0.312
20 89.5
4930 8900
560 440
7.3h10-6 17.5h10-6
Fig. 1. Variation of volume fraction versus non-dimensional radius
Field node parent cell k
"
Ω k ( SC )
Ω k (C )
Ω k (1)
"
cell edge
"
Γ k (C )
"
segment
Fig. 2. Illustration of constructing smoothing domains based on triangular meshes
Field node
Integration sampling point
SC=3 Fig. 3. A parent cell is divided into 3 smoothing cells
Fig. 4. Schematic of a FGM hollow cylinder with the Displacement, Neumann, Robin and force boundary conditions
Ir =
0.2
Fig. 5. Schematic of irregular mesh for Įir=0.2
Fig. 6. Comparison of Temperature distribution along the central line AB (AėB) for n=0.2
Fig. 7. Comparison of Temperature distribution along the central line AB (AėB) for n=5
Fig. 8. Displacement of point A for different methods using irregularly distributed field nodes for n=0.2
Fig. 9. Displacement of point B for different methods using irregularly distributed field nodes for n=5
Fig. 10. Comparison of thermal stress distribution along the central line AB (AėB) for n=0.2
Fig. 11. Comparison of thermal stress distribution along the central line AB (AėB) for n=5
z Robin boundary
Robin boundary Adiabatic
P A
B
ri TiC/Ni
ro
o
Adiabatic
r
Fig. 12. Geometry and the boundary conditions of a TiC/Ni FGM hollow cylinder
Fig. 13. Comparison of temperature distribution along the central line AB (AėB) for n=0.2
Fig. 14. Comparison of temperature distribution along the central line AB (AėB) for n=5
Fig. 15. Convergence of displacement versus the number of degrees of freedom of reference point A for n=0.2
Fig. 16. Convergence of displacement versus the number of degrees of freedom of reference point B for n=5
Fig. 17. Convergence of the thermal equivalent energy versus the number of degrees of freedom for n=0.2
Fig. 18. Convergence of the elastic strain energy versus the number of degrees of freedom for n=5