- Email: [email protected]
Stochastic transient analysis of thermal stresses in solids by explicit time-domain method
Theoretical & Applied Mechanics Letters 9 (2019) 293-296
Contents lists available at ScienceDirect
Theoretical & Applied Mechanics Letters journal homepage: www.elsevier.com/locate/taml
Letter
Stochastic transient analysis of thermal stresses in solids by explicit timedomain method Houzuo Guoa, Cheng Sua,b,*, Jianhua Xiana
a b
School of Civil Engineering and Transportation, South China University of Technology, Guangzhou 510640, China State Key Laboratory of Subtropical Building Science, South China University of Technology, Guangzhou 510640, China
H I G H L I G H T S
• A novel time-domain approach is proposed for non-stationary stochastic transient heat conduction and thermal response analysis. • The explicit expressions of temperatures and thermal displacements and stresses are established in terms of stochastic thermal excitations. • The dimension-reduced statistical moment analysis is conducted with high efficiency based on the explicit formulations. A R T I C L E I N F O
A B S T R A C T
Article history: Received 6 July 2019 Received in revised form 7 August 2019 Accepted 8 August 2019 Available online 15 August 2019
Keywords: Stochastic Non-stationary Heat conduction Thermal stress Explicit time-domain method
Stochastic heat conduction and thermal stress analysis of structures has received considerable attention in recent years. The propagation of uncertain thermal environments will lead to stochastic variations in temperature fields and thermal stresses. Therefore, it is reasonable to consider the variability of thermal environments while conducting thermal analysis. However, for ambient thermal excitations, only stationary random processes have been investigated thus far. In this study, the highly efficient explicit time-domain method (ETDM) is proposed for the analysis of non-stationary stochastic transient heat conduction and thermal stress problems. The explicit time-domain expressions of thermal responses are first constructed for a thermoelastic body. Then the statistical moments of thermal displacements and stresses can be directly obtained based on the explicit expressions of thermal responses. A numerical example involving non-stationary stochastic internal heat generation rate is investigated. The accuracy and efficiency of the proposed method are validated by comparison with the Monte-Carlo simulation. ©2019 The Authors. Published by Elsevier Ltd on behalf of The Chinese Society of Theoretical and Applied Mechanics. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/).
Traditional thermal analysis generally assumes deterministic thermal environments. In fact, the environmental parameters, e.g., internal heat generation rate, heat flux, boundary temperature and ambient temperature, etc., are difficult to measure and predict accurately. Therefore, the variability of the thermal environments should be considered to reflect the uncertainty propagation involved in the thermal stress problems. Over the past few decades, researchers have solved the problems of statistical moments of temperatures and thermal stresses of several simple structures under random internal heat genera
* Corresponding author. E-mail address: [email protected] (C. Su).
tion rate, heat flux, boundary temperature or ambient temperature with analytical methods, in which the thermal environments are modeled in the form of random variables or stationary random processes [1–3]. However, for practical problems, the thermal excitations involved often exhibit non-stationary characteristics due to the time-varying statistical properties of thermal environments, and therefore should be modelled as non-stationary random processes. In this study, the explicit time-domain method (ETDM) [4, 5], originally proposed for non-stationary random vibration analysis of structures, is applied to the analysis of non-stationary stochastic heat conduction and thermal stress problems. The explicit time-domain expressions of temperatures, thermal dis-
http://dx.doi.org/10.1016/j.taml.2019.05.007 2095-0349/© 2019 The Authors. Published by Elsevier Ltd on behalf of The Chinese Society of Theoretical and Applied Mechanics. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/).
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H.Z. Guo et al. / Theoretical & Applied Mechanics Letters 9 (2019) 293-296
placements and thermal stresses are first established. Then, the statistical moments of temperatures and thermal responses can be directly obtained from the statistical properties of non-stationary thermal excitations. The governing equation of a two-dimensional transient heat conduction problem can be expressed as ρc
( ) ( ) ∂T ∂ ∂T ∂ ∂T = k + k + q(x, y, t ), (x, y) ∈ Ω, ∂t ∂x ∂x ∂y ∂y
T = f T (x, y, t ), (x, y) ∈ ∂ΩE ,
(2)
−k∂T /∂n = f N (x, y, t ) + h[T − Ta (x, y, t )], (x, y) ∈ ∂ΩN ,
(3)
where ∂ΩE and ∂ΩN are the essential boundary and the natural boundary, respectively; n is the outward normal vector of the boundary; f T (x, y, t ) and f N (x, y, t ) are the temperature function and the heat flux function along ∂ΩE and ∂ΩN, respectively; h is the convective heat transfer coefficient; Ta (x, y, t ) is the ambient temperature. The finite element form of Eqs. (1)–(3) can be derived as [6] (4)
where T and T˙ are the nodal temperature vector and the nodal temperature rate vector, respectively; C T and K T are the heat capacity matrix and the heat conductivity matrix, respectively; L 1 = [l 1 l 2 · · · l m ], in which m is the number of the thermal excitations and l k (k = 1, 2, · · · , m) is the pattern vector of the k-th thermal excitation; P (t ) = [P 1 (t ) P 2 (t ) · · · P m (t )]T, in which T denotes the matrix transposition and P k (t ) (k = 1, 2, · · · , m) are the thermal excitations, including internal heat generation rate, heat flux, boundary temperature and ambient temperature, which can be modeled as non-stationary random processes. The transient temperature vector T (t ) of a plate will cause transient thermal displacements and stresses, and the equation of motion of the plate can now be expressed as M U¨ +C U˙ + K U = L 2 (T − Tref ),
(5)
where U , U˙ and U¨ are the nodal displacement, velocity and acceleration vector of the plate, respectively; M,C and K are the mass, damping and stiffness matrix of the plate, respectively; Tref is the reference temperature vector; and L 2 (T − Tref ) is the thermal load vector, in which L 2 is obtained by assembling all the element matrices L e2: Ï L e2 = αt¯
B T D N dxdy, ΩE
(7)
Then, Eqs. (4) and (5) can be combined into the following state equation: V˙ (t ) = HV (t ) + W F (t ),
(8)
where (1)
where Ω denotes the domain under consideration; T = T (x, y, t ) is the temperature field at time t ; ρ , c , and k are the mass density, the specific heat and the thermal conductivity, respectively; and q(x, y, t ) is the internal heat generation rate. The boundary conditions are given as follows:
C T T˙ + K T T = L 1 P (t ),
V = [U T U˙ T T T ]T ,
(6)
where B , D and N are the displacement-strain matrix, the stressstrain matrix and the temperature shape function matrix of the element, respectively; α is the linear thermal expansion coefficient; t¯ is the thickness of the plate. Introduce a state vector defined as
0 I 0 H = −M −1 K −M −1C M −1 L 2 , 0 0 −C T−1 K T 0 0 W = −M −1 L 2 0 , −1 0 CT L1 [ ] Tref F (t ) = . P (t )
(9)
For Eq. (8) , the initial condition can be assumed as V (0) = [U (0)T U˙ (0)T T (0)T ]T = [0 0 T0T ]T , in which T0 is the initial
temperature vector. Solving Eq. (8) for the state vector V , one can get the temperatures and thermal displacements of the plate. Using the constitutive relationship, one can further obtain the thermal stresses of the plate. To reflect the physical evolution process of the thermoelastic problem governed by Eq. (8) , the explicit time-domain expression of the state vector V (t ) is first derived. For linear problems, V (t ) can be expressed as a linear combination of F (t ) at different time instants as follows: Vi = Vi ,0 + A i ,0 F 0 + A i ,1 F 1 + · · · + A i ,i F i ,
i = 1, 2, · · · , n,
(10)
where Vi = V (t i ) and t i = i ∆t (i = 1, 2, · · · , n), in which ∆t and n are the time step and the number of time steps, respectively; Vi ,0 represents the contribution to Vi caused by the initial vector V0 ; F j = F (t j ) ( j = 0, 1, · · · , i ); and A i , j ( j = 0, 1, · · · , i ) are the coefficient matrices associated with the thermal and structural parameters, representing the influence of F j ( j = 0, 1, · · · , i ) on Vi . The coefficient matrices A i , j can be derived in closed forms from Eq. (8) as [4, 5] A 1,0 = Q 1 , A i ,0 = G A i −1,0 , 2 ≤ i ≤ n, A 1,1 = Q 2 , A 2,1 = GQ 2 +Q 1 , A i ,1 = G A i −1,1 , A i , j = A i −1, j −1 , 2 ≤ j ≤ i ≤ n,
3 ≤ i ≤ n,
(11)
where G , Q 1 and Q 2 can be obtained depending on different integration schemes. T P T (t )]T, as shown in Eq. (9). Without loss Note that F (t ) = [Tref of generality, assuming P 0 = P (0) = 0 , Eq. (10) can be further written as Vi = V¯i ,0 + A¯ i ,1 P 1 + A¯ i ,2 P 2 + · · · + A¯ i ,i P i ,
i = 1, 2, · · · , n,
(12)
where the coefficient matrices A¯ i , j can be extracted from the coefficient matrices A i , j ( j = 1, 2, · · · , i ); the influence of Tref at different time instants is included in V¯i ,0 . For better understanding, the coefficient matrices shown in
H.Z. Guo et al. / Theoretical & Applied Mechanics Letters 9 (2019) 293-296
Eq. (12) are also presented in Table 1. It can be seen from Table 1 that only the coefficient matrices A¯ i ,1 (i = 1, 2, · · · , n) need to be calculated and stored. Furthermore, the k -th columns in the coefficient matrices A¯ i ,1 represent the state vector of the plate at time t i caused by the unit impulse excitation p k (t ) corresponding to the k-th thermal excitation P k (t ) (k = 1, 2, · · · , m), as shown in Fig. 1. Therefore, the coefficient matrices A¯ i ,1 (i = 1, 2, · · · , n) can also be obtained by m transient analyses directly using commercial finite element software. Thus far, the physical evolution mechanism of the thermoelastic problem has been completely reflected by Eq. (12). On this basis, the statistical evolution mechanism can be further considered by moment operation rule. Note that, for structural design, not all the responses are required, and only a certain number of critical responses need to be focused on. Assume r i is one of the temperatures or thermal displacements in Vi . Then, the explicit expression for r i can be directly obtained from Eq. (12) as follows: r i = r i ,0 + a i ,1 P 1 + a i ,2 P 2 + · · · + a i ,i P i ,
i = 1, 2, · · · , n,
(13)
where r i ,0 and ai , j ( j = 1, 2, · · · , i ) are the corresponding rows extracted from V¯i ,0 and A¯ i , j ( j = 1, 2, · · · , i ), respectively. When r i denotes a thermal stress, the constitutive law needs to be considered for obtaining r i ,0 and ai , j ( j = 1, 2, · · · , i ) from the explicit expressions of the associated displacements. From Eq. (13), the mean and the variance of the response r i can be obtained as }
µr i = E (r i ) = r i ,0 + a i E (P [i ] ),
i = 1, 2, · · · , n,
σ2r i = cov(r i , r i ) = a i cov(P [i ] , P [i ] )(a i )T ,
(14)
where ai = [ai ,1 ai ,2 · · · ai ,i ], P [i ] = [P 1T P 2T · · · P iT ]T, and the mean vector and the covariance matrix of P [i ] can be expressed as E (P [i ] ) = [µ (t 1 ) µ (t 2 ) · · · µ (t i )] T P
T P
T P
T
295
cov(P [i ] , P [i ] ) =
R P P (t 1 , t 1 )−µP (t 1 )µTP (t 1 ) · · · R P P (t 1 , t i )−µP (t 1 )µTP (t i )
.. .
..
.. .
.
,
(16)
R P P (t i , t 1 )−µP (t i )µTP (t 1 ) · · · R P P (t i , t i )−µP (t i )µTP (t i )
respectively, in which µP (t ) and R P P (t , τ) are the mean function vector and the cross-correlation function matrix of P (t ), respectively. It can be seen from Eq. (14) that dimension-reduced statistical moment analysis can be easily conducted owing to the explicit formulation shown in Eq. (12), which will lead to significant reduction in computational cost for stochastic thermal analysis. Further, in the above formulation, the correlation functions rather than the power spectral density functions of the thermal excitations are required. Therefore, Eq. (14) can be applied to non-stationary stochastic analysis without any difficulties. Consider a square plate with thickness t¯ = 0.01 m , as shown in Fig. 2. The material parameters of the plate are listed in Table 2. The bottom edge of the plate is fixed, and all the plate edges are adiabatic. The plate is divided into 10 × 10 plate elements, as shown in Fig. 2. The initial temperature and the reference temperature are taken as 10 ◦ C for the whole domain. The plate is subjected to a uniform distributed internal heat generation rate modeled as a zero-mean non-stationary Gaussian process: Q(t ) = g (t )q(t ),
(17)
where g (t ) = 4(e−0.1t − e−0.2t ) is a modulation function; q(t ) is a stationary white noise process with spectral density being S 0 = 1.0 × 108 J2 /(m6 · s). y
1m
(15)
1m
and Table 1 Coefficient matrices for state vector
Time
P1
t1
A¯ 1,1
t2
A¯ 2,1
A¯ 1,1
t3
A¯ 3,1
A¯ 2,1
A¯ 1,1
.. .
.. .
.. .
.. .
..
tn
A¯ n,1
A¯ n−1,1
A¯ n−2,1
···
P2
…
P3
x
Fig. 2. A square plate
t2
t3
…
tn−1
Table 2 Material parameters
.
1 t1
B
pk(t)
0
Pn
tn
Time
Fig. 1. Unit impulse excitation applied at t 1
A¯ 1,1
Parameter
Value
Thermal conductivity (k)
1 J/(m·s·°C)
Specific heat (c)
1 J/(kg·°C)
Mass density (ρ)
1.0×103 kg/m3
Modulus of elasticity (E)
5.0×1010 N/m2
Poisson ratio (υ)
0.3
Thermal expansion coefficient (α)
2×10−5/°C
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H.Z. Guo et al. / Theoretical & Applied Mechanics Letters 9 (2019) 293-296 25
Standard deviation of thermal stress (MPa)
Standard deviation of temperature (°C)
MCS (with 500 samples) ETDM
20 15 10 5 0
0
5
10
15
Time/s
1.5 MCS (with 500 samples) ETDM 1.0
0.5
0
0
5
10
15
Time (s)
Fig. 5. Standard deviation of thermal normal stress in Y direction at node B
Fig. 3. Standard deviation of temperature at node B Standard deviation of thermal displacement (mm)
Table 3 Comparison of the time elapsed by different methods
0.30 MCS (with 500 samples) ETDM
0.25
Method
Elapsed time
MCS (with 500 samples)
95246 s (26.4 h)
ETDM
398 s
0.20
displacements and stresses are first established in terms of stochastic thermal excitations, and the dimension-reduced statistical moment analysis is then conducted based on the explicit formulations. A numerical example has been investigated to show the high accuracy and efficiency of the proposed method.
0.15 0.10 0.05 0
ACKNOWLEDGEMENTS 0
5
10
15
Time (s)
Fig. 4. Standard deviation of thermal displacement in Y direction at node B
The statistical moments of temperatures and thermal stresses of the plate are calculated by the proposed ETDM. For the purpose of comparison, the Monte-Carlo simulation (MCS) with 500 samples is conducted on ANSYS software platform. The duration of the time-history analysis is set to be 15 s with the time step being ∆t = 0.02 s . The standard deviations of the temperature and the thermal displacement and stress at node B are presented in Figs. 3–5, respectively. It can be observed from the figures that the results obtained with ETDM and MCS are in good agreement, indicating the high accuracy of the proposed method. For comparison of the calculation efficiency, the time elapsed by the two methods is presented in Table 3, from which it can be seen that the computational time of ETDM is 398 s, showing the high efficiency of the present approach. A highly efficient ETDM has been proposed for non-stationary stochastic transient heat conduction and thermal response analysis. The explicit expressions of temperatures and thermal
The research is funded by the National Natural Science Foundation of China (51678252) and the Guangzhou Science and Technology Project (201804020069)
References [1] R. Chiba, Y. Sugano, Stochastic thermoelastic problem of a func-
[2]
[3]
[4]
[5]
[6]
tionally graded plate under random temperature load, Archive of Applied Mechanics 77 (2007) 215–227. R. Chiba, Y. Sugano, Thermal stresses in bodies with random initial temperature distribution, Mathematics and Mechanics of Solids 15 (2010) 258–276. R. Chiba, Stochastic analysis of heat conduction and thermal stresses in solids. In: Heat Transfer Phenomena and Applications, edited by S.N. Kazi, pp. 243-266, Intech Open, Rijeka (2012). C. Su, R. Xu, Random vibration analysis of structures by a timedomain explicit formulation method, Structural Engineering & Mechanics 52 (214) 239–260. C. Su, H. Huang, H. Ma, Fast equivalent linearization method for nonlinear structures under non-stationary random excitations, Journal of Engineering Mechanics 142 (2016) 04016049. J.N. Reddy, D.K. Gartling, The Finite Element Method in Heat Transfer and Fluid Dynamics, CRC Press, Boca Raton (2000) .
Contents lists available at ScienceDirect
Theoretical & Applied Mechanics Letters journal homepage: www.elsevier.com/locate/taml
Letter
Stochastic transient analysis of thermal stresses in solids by explicit timedomain method Houzuo Guoa, Cheng Sua,b,*, Jianhua Xiana
a b
School of Civil Engineering and Transportation, South China University of Technology, Guangzhou 510640, China State Key Laboratory of Subtropical Building Science, South China University of Technology, Guangzhou 510640, China
H I G H L I G H T S
• A novel time-domain approach is proposed for non-stationary stochastic transient heat conduction and thermal response analysis. • The explicit expressions of temperatures and thermal displacements and stresses are established in terms of stochastic thermal excitations. • The dimension-reduced statistical moment analysis is conducted with high efficiency based on the explicit formulations. A R T I C L E I N F O
A B S T R A C T
Article history: Received 6 July 2019 Received in revised form 7 August 2019 Accepted 8 August 2019 Available online 15 August 2019
Keywords: Stochastic Non-stationary Heat conduction Thermal stress Explicit time-domain method
Stochastic heat conduction and thermal stress analysis of structures has received considerable attention in recent years. The propagation of uncertain thermal environments will lead to stochastic variations in temperature fields and thermal stresses. Therefore, it is reasonable to consider the variability of thermal environments while conducting thermal analysis. However, for ambient thermal excitations, only stationary random processes have been investigated thus far. In this study, the highly efficient explicit time-domain method (ETDM) is proposed for the analysis of non-stationary stochastic transient heat conduction and thermal stress problems. The explicit time-domain expressions of thermal responses are first constructed for a thermoelastic body. Then the statistical moments of thermal displacements and stresses can be directly obtained based on the explicit expressions of thermal responses. A numerical example involving non-stationary stochastic internal heat generation rate is investigated. The accuracy and efficiency of the proposed method are validated by comparison with the Monte-Carlo simulation. ©2019 The Authors. Published by Elsevier Ltd on behalf of The Chinese Society of Theoretical and Applied Mechanics. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/).
Traditional thermal analysis generally assumes deterministic thermal environments. In fact, the environmental parameters, e.g., internal heat generation rate, heat flux, boundary temperature and ambient temperature, etc., are difficult to measure and predict accurately. Therefore, the variability of the thermal environments should be considered to reflect the uncertainty propagation involved in the thermal stress problems. Over the past few decades, researchers have solved the problems of statistical moments of temperatures and thermal stresses of several simple structures under random internal heat genera
* Corresponding author. E-mail address: [email protected] (C. Su).
tion rate, heat flux, boundary temperature or ambient temperature with analytical methods, in which the thermal environments are modeled in the form of random variables or stationary random processes [1–3]. However, for practical problems, the thermal excitations involved often exhibit non-stationary characteristics due to the time-varying statistical properties of thermal environments, and therefore should be modelled as non-stationary random processes. In this study, the explicit time-domain method (ETDM) [4, 5], originally proposed for non-stationary random vibration analysis of structures, is applied to the analysis of non-stationary stochastic heat conduction and thermal stress problems. The explicit time-domain expressions of temperatures, thermal dis-
http://dx.doi.org/10.1016/j.taml.2019.05.007 2095-0349/© 2019 The Authors. Published by Elsevier Ltd on behalf of The Chinese Society of Theoretical and Applied Mechanics. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/).
294
H.Z. Guo et al. / Theoretical & Applied Mechanics Letters 9 (2019) 293-296
placements and thermal stresses are first established. Then, the statistical moments of temperatures and thermal responses can be directly obtained from the statistical properties of non-stationary thermal excitations. The governing equation of a two-dimensional transient heat conduction problem can be expressed as ρc
( ) ( ) ∂T ∂ ∂T ∂ ∂T = k + k + q(x, y, t ), (x, y) ∈ Ω, ∂t ∂x ∂x ∂y ∂y
T = f T (x, y, t ), (x, y) ∈ ∂ΩE ,
(2)
−k∂T /∂n = f N (x, y, t ) + h[T − Ta (x, y, t )], (x, y) ∈ ∂ΩN ,
(3)
where ∂ΩE and ∂ΩN are the essential boundary and the natural boundary, respectively; n is the outward normal vector of the boundary; f T (x, y, t ) and f N (x, y, t ) are the temperature function and the heat flux function along ∂ΩE and ∂ΩN, respectively; h is the convective heat transfer coefficient; Ta (x, y, t ) is the ambient temperature. The finite element form of Eqs. (1)–(3) can be derived as [6] (4)
where T and T˙ are the nodal temperature vector and the nodal temperature rate vector, respectively; C T and K T are the heat capacity matrix and the heat conductivity matrix, respectively; L 1 = [l 1 l 2 · · · l m ], in which m is the number of the thermal excitations and l k (k = 1, 2, · · · , m) is the pattern vector of the k-th thermal excitation; P (t ) = [P 1 (t ) P 2 (t ) · · · P m (t )]T, in which T denotes the matrix transposition and P k (t ) (k = 1, 2, · · · , m) are the thermal excitations, including internal heat generation rate, heat flux, boundary temperature and ambient temperature, which can be modeled as non-stationary random processes. The transient temperature vector T (t ) of a plate will cause transient thermal displacements and stresses, and the equation of motion of the plate can now be expressed as M U¨ +C U˙ + K U = L 2 (T − Tref ),
(5)
where U , U˙ and U¨ are the nodal displacement, velocity and acceleration vector of the plate, respectively; M,C and K are the mass, damping and stiffness matrix of the plate, respectively; Tref is the reference temperature vector; and L 2 (T − Tref ) is the thermal load vector, in which L 2 is obtained by assembling all the element matrices L e2: Ï L e2 = αt¯
B T D N dxdy, ΩE
(7)
Then, Eqs. (4) and (5) can be combined into the following state equation: V˙ (t ) = HV (t ) + W F (t ),
(8)
where (1)
where Ω denotes the domain under consideration; T = T (x, y, t ) is the temperature field at time t ; ρ , c , and k are the mass density, the specific heat and the thermal conductivity, respectively; and q(x, y, t ) is the internal heat generation rate. The boundary conditions are given as follows:
C T T˙ + K T T = L 1 P (t ),
V = [U T U˙ T T T ]T ,
(6)
where B , D and N are the displacement-strain matrix, the stressstrain matrix and the temperature shape function matrix of the element, respectively; α is the linear thermal expansion coefficient; t¯ is the thickness of the plate. Introduce a state vector defined as
0 I 0 H = −M −1 K −M −1C M −1 L 2 , 0 0 −C T−1 K T 0 0 W = −M −1 L 2 0 , −1 0 CT L1 [ ] Tref F (t ) = . P (t )
(9)
For Eq. (8) , the initial condition can be assumed as V (0) = [U (0)T U˙ (0)T T (0)T ]T = [0 0 T0T ]T , in which T0 is the initial
temperature vector. Solving Eq. (8) for the state vector V , one can get the temperatures and thermal displacements of the plate. Using the constitutive relationship, one can further obtain the thermal stresses of the plate. To reflect the physical evolution process of the thermoelastic problem governed by Eq. (8) , the explicit time-domain expression of the state vector V (t ) is first derived. For linear problems, V (t ) can be expressed as a linear combination of F (t ) at different time instants as follows: Vi = Vi ,0 + A i ,0 F 0 + A i ,1 F 1 + · · · + A i ,i F i ,
i = 1, 2, · · · , n,
(10)
where Vi = V (t i ) and t i = i ∆t (i = 1, 2, · · · , n), in which ∆t and n are the time step and the number of time steps, respectively; Vi ,0 represents the contribution to Vi caused by the initial vector V0 ; F j = F (t j ) ( j = 0, 1, · · · , i ); and A i , j ( j = 0, 1, · · · , i ) are the coefficient matrices associated with the thermal and structural parameters, representing the influence of F j ( j = 0, 1, · · · , i ) on Vi . The coefficient matrices A i , j can be derived in closed forms from Eq. (8) as [4, 5] A 1,0 = Q 1 , A i ,0 = G A i −1,0 , 2 ≤ i ≤ n, A 1,1 = Q 2 , A 2,1 = GQ 2 +Q 1 , A i ,1 = G A i −1,1 , A i , j = A i −1, j −1 , 2 ≤ j ≤ i ≤ n,
3 ≤ i ≤ n,
(11)
where G , Q 1 and Q 2 can be obtained depending on different integration schemes. T P T (t )]T, as shown in Eq. (9). Without loss Note that F (t ) = [Tref of generality, assuming P 0 = P (0) = 0 , Eq. (10) can be further written as Vi = V¯i ,0 + A¯ i ,1 P 1 + A¯ i ,2 P 2 + · · · + A¯ i ,i P i ,
i = 1, 2, · · · , n,
(12)
where the coefficient matrices A¯ i , j can be extracted from the coefficient matrices A i , j ( j = 1, 2, · · · , i ); the influence of Tref at different time instants is included in V¯i ,0 . For better understanding, the coefficient matrices shown in
H.Z. Guo et al. / Theoretical & Applied Mechanics Letters 9 (2019) 293-296
Eq. (12) are also presented in Table 1. It can be seen from Table 1 that only the coefficient matrices A¯ i ,1 (i = 1, 2, · · · , n) need to be calculated and stored. Furthermore, the k -th columns in the coefficient matrices A¯ i ,1 represent the state vector of the plate at time t i caused by the unit impulse excitation p k (t ) corresponding to the k-th thermal excitation P k (t ) (k = 1, 2, · · · , m), as shown in Fig. 1. Therefore, the coefficient matrices A¯ i ,1 (i = 1, 2, · · · , n) can also be obtained by m transient analyses directly using commercial finite element software. Thus far, the physical evolution mechanism of the thermoelastic problem has been completely reflected by Eq. (12). On this basis, the statistical evolution mechanism can be further considered by moment operation rule. Note that, for structural design, not all the responses are required, and only a certain number of critical responses need to be focused on. Assume r i is one of the temperatures or thermal displacements in Vi . Then, the explicit expression for r i can be directly obtained from Eq. (12) as follows: r i = r i ,0 + a i ,1 P 1 + a i ,2 P 2 + · · · + a i ,i P i ,
i = 1, 2, · · · , n,
(13)
where r i ,0 and ai , j ( j = 1, 2, · · · , i ) are the corresponding rows extracted from V¯i ,0 and A¯ i , j ( j = 1, 2, · · · , i ), respectively. When r i denotes a thermal stress, the constitutive law needs to be considered for obtaining r i ,0 and ai , j ( j = 1, 2, · · · , i ) from the explicit expressions of the associated displacements. From Eq. (13), the mean and the variance of the response r i can be obtained as }
µr i = E (r i ) = r i ,0 + a i E (P [i ] ),
i = 1, 2, · · · , n,
σ2r i = cov(r i , r i ) = a i cov(P [i ] , P [i ] )(a i )T ,
(14)
where ai = [ai ,1 ai ,2 · · · ai ,i ], P [i ] = [P 1T P 2T · · · P iT ]T, and the mean vector and the covariance matrix of P [i ] can be expressed as E (P [i ] ) = [µ (t 1 ) µ (t 2 ) · · · µ (t i )] T P
T P
T P
T
295
cov(P [i ] , P [i ] ) =
R P P (t 1 , t 1 )−µP (t 1 )µTP (t 1 ) · · · R P P (t 1 , t i )−µP (t 1 )µTP (t i )
.. .
..
.. .
.
,
(16)
R P P (t i , t 1 )−µP (t i )µTP (t 1 ) · · · R P P (t i , t i )−µP (t i )µTP (t i )
respectively, in which µP (t ) and R P P (t , τ) are the mean function vector and the cross-correlation function matrix of P (t ), respectively. It can be seen from Eq. (14) that dimension-reduced statistical moment analysis can be easily conducted owing to the explicit formulation shown in Eq. (12), which will lead to significant reduction in computational cost for stochastic thermal analysis. Further, in the above formulation, the correlation functions rather than the power spectral density functions of the thermal excitations are required. Therefore, Eq. (14) can be applied to non-stationary stochastic analysis without any difficulties. Consider a square plate with thickness t¯ = 0.01 m , as shown in Fig. 2. The material parameters of the plate are listed in Table 2. The bottom edge of the plate is fixed, and all the plate edges are adiabatic. The plate is divided into 10 × 10 plate elements, as shown in Fig. 2. The initial temperature and the reference temperature are taken as 10 ◦ C for the whole domain. The plate is subjected to a uniform distributed internal heat generation rate modeled as a zero-mean non-stationary Gaussian process: Q(t ) = g (t )q(t ),
(17)
where g (t ) = 4(e−0.1t − e−0.2t ) is a modulation function; q(t ) is a stationary white noise process with spectral density being S 0 = 1.0 × 108 J2 /(m6 · s). y
1m
(15)
1m
and Table 1 Coefficient matrices for state vector
Time
P1
t1
A¯ 1,1
t2
A¯ 2,1
A¯ 1,1
t3
A¯ 3,1
A¯ 2,1
A¯ 1,1
.. .
.. .
.. .
.. .
..
tn
A¯ n,1
A¯ n−1,1
A¯ n−2,1
···
P2
…
P3
x
Fig. 2. A square plate
t2
t3
…
tn−1
Table 2 Material parameters
.
1 t1
B
pk(t)
0
Pn
tn
Time
Fig. 1. Unit impulse excitation applied at t 1
A¯ 1,1
Parameter
Value
Thermal conductivity (k)
1 J/(m·s·°C)
Specific heat (c)
1 J/(kg·°C)
Mass density (ρ)
1.0×103 kg/m3
Modulus of elasticity (E)
5.0×1010 N/m2
Poisson ratio (υ)
0.3
Thermal expansion coefficient (α)
2×10−5/°C
296
H.Z. Guo et al. / Theoretical & Applied Mechanics Letters 9 (2019) 293-296 25
Standard deviation of thermal stress (MPa)
Standard deviation of temperature (°C)
MCS (with 500 samples) ETDM
20 15 10 5 0
0
5
10
15
Time/s
1.5 MCS (with 500 samples) ETDM 1.0
0.5
0
0
5
10
15
Time (s)
Fig. 5. Standard deviation of thermal normal stress in Y direction at node B
Fig. 3. Standard deviation of temperature at node B Standard deviation of thermal displacement (mm)
Table 3 Comparison of the time elapsed by different methods
0.30 MCS (with 500 samples) ETDM
0.25
Method
Elapsed time
MCS (with 500 samples)
95246 s (26.4 h)
ETDM
398 s
0.20
displacements and stresses are first established in terms of stochastic thermal excitations, and the dimension-reduced statistical moment analysis is then conducted based on the explicit formulations. A numerical example has been investigated to show the high accuracy and efficiency of the proposed method.
0.15 0.10 0.05 0
ACKNOWLEDGEMENTS 0
5
10
15
Time (s)
Fig. 4. Standard deviation of thermal displacement in Y direction at node B
The statistical moments of temperatures and thermal stresses of the plate are calculated by the proposed ETDM. For the purpose of comparison, the Monte-Carlo simulation (MCS) with 500 samples is conducted on ANSYS software platform. The duration of the time-history analysis is set to be 15 s with the time step being ∆t = 0.02 s . The standard deviations of the temperature and the thermal displacement and stress at node B are presented in Figs. 3–5, respectively. It can be observed from the figures that the results obtained with ETDM and MCS are in good agreement, indicating the high accuracy of the proposed method. For comparison of the calculation efficiency, the time elapsed by the two methods is presented in Table 3, from which it can be seen that the computational time of ETDM is 398 s, showing the high efficiency of the present approach. A highly efficient ETDM has been proposed for non-stationary stochastic transient heat conduction and thermal response analysis. The explicit expressions of temperatures and thermal
The research is funded by the National Natural Science Foundation of China (51678252) and the Guangzhou Science and Technology Project (201804020069)
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