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The Spline Method for the Solution of the Transient Heat Conduction Problem with Nonlinear Initial and Boundary Conditions for a Plate
Available online at www.sciencedirect.com
ScienceDirect Procedia Engineering 150 (2016) 1419 – 1426
International Conference on Industrial Engineering, ICIE 2016
The Spline Method for the Solution of the Transient Heat Conduction Problem with Nonlinear Initial and Boundary Conditions for a Plate V.M. Kudoyarovaa, V.P. Pavlova,* a
Ufa State Aviation Technical University, st. K. Marcsa,12, Ufa, 450000, Russian Federation
Abstract There are many cases in the engineering practice when it is necessary to know the transient temperature field in a flat plate. At that, temperature distribution is studied experimentally or by calculation. However, the obtained calculation results do not always provide the required accuracy. This paper obtains the spline-method grade 3 defect 1, which allows simulating the process of the thermal conductivity for the infinitely extended plate at nonlinear initial and boundary conditions of the first kind on the plate sides. For accurate evaluation of the solution of the heat conduction equation with the help of the spline-method, the standard problem is created for a one-dimensional infinite flat plane which has the exact analytical solution. This paper shows the use of spline method for the solution of transient heat conduction problem which gives an error of no more than 110-5 and thus satisfies obtaining of the sufficient accuracy to solve many practical problems. © 2016The TheAuthors. Authors. Published Elsevier © 2016 Published by by Elsevier Ltd.Ltd. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/). Peer-review under responsibility of the organizing committee of ICIE 2016. Peer-review under responsibility of the organizing committee of ICIE 2016 Keywords: Modeling; splines; thermal conductivity; differential equations of parabolic type.
1. Introduction Currently the spline-methods are widely applied in many fields of science with using methods of computer modeling. In turn, the numerical experiment is often used in the study and modeling of physical processes in dealing with a wide range of technical challenges, since it allows obtaining results of the study, even in a case when the real experiment is difficult to conduct. The main criteria of the effectiveness of a computer modeling is accuracy of the
* Corresponding author. Tel.: +7-347-273-0844; fax: +7-347-273-0844. E-mail address: [email protected].
1877-7058 © 2016 The Authors. Published by Elsevier Ltd. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/). Peer-review under responsibility of the organizing committee of ICIE 2016
doi:10.1016/j.proeng.2016.07.204
1420
V.M. Kudoyarova and V.P. Pavlov / Procedia Engineering 150 (2016) 1419 – 1426
solution, and the computational complexity of used numerical method. At that pursuing an objective of increasing the number of significant digits in the results of calculations using this method. The accuracy of the numerical simulation is characterized by a measure of the deviation of the numerical results obtained from the known exact analytical solutions [1]. Heat conduction problem is a system of differential equations, so one of the key purposes of numerical analysis is to develop methods to obtain reliable results close to accurate analytical values. 2. Problem Definition To evaluate the capacity of structures, working in conditions of heat, it is necessary to know the temperature field in it. The temperature distribution can be studied experimentally, or to be determined by solving the heat equation. The experimental method is very laborious and time-consuming, and besides there is not always able to realize it. Therefore, in most cases, the temperature field determined by calculation [2]–[6]. At that, it is important to choose the appropriate mathematical apparatus: analytical methods, finite-difference method, finite-element method [7]– [14]. In real thermal processes, the temperature on the solid surface is often changed under the influence of various factors such as the presence of the heat source in the body, or at the convective heat transfer when the surface temperature is dependent on flow regime at variable heat transfer intensity on its surface, etc. In this regard, this paper solves the boundary value problem of unsteady heat conduction with specified non-linear initial and boundary conditions of the first kind. In a case of absence of internal heat sources, the spline-method applied to solve the differential equation of heat conduction, and is given by [4]:
Uc
wT wt
w § wT ¨O wx © wx
· ¸, ¹
(1)
where t – time; x – position of a point; T T ( x, t ) – the function describing the temperature change by time t and space coordinate x ; U – material density; c – specific heat; O – thermal conductivity of the material along X - axis. In case when material properties are not dependent on temperature and time:
U
const, c
const, O
const
(2)
the equation (1) takes the following form:
Uc
wT wt
O
w 2T . wx 2
(3)
by replacing a2
O , Uc
(4)
the equation (3) is reduced to the form wT w 2T a2 2 wt wx
0.
(5)
This paper estimates the solution accuracy of the equation (5) of using splines of 3rd degree of defect 1 by comparing the numerical solution with the exact analytical solution of the test problem.
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3. Main Statements of Spline-method Fundamentals of splines method applied to the problems of the deformation of solids are stated in [15], [16]. One of the main features of the spline of 3rd degree defect 1 is the continuity of second derivative of splinefunction W3,1 ( x ) at all points of the domain that is highly desirable for solving thermal conduction differential equation (5) where the second derivative w 2T / wx 2 is present. Therein lies the advantage of splines method in comparison to the finite difference method and finite element method, which do not provide the continuity of the second derivatives. When constructing spline of 3rd degree defect 1 [15] the mesh 'x with N x -nodes is forming in interval [a, b].
'x : a
x1 x2 ... xNx
(6)
b
On the mesh base, the spline function of 3rd degree defect 1 W3,1 ( x ) is constructed with N s
N x 2 degrees of
freedom. In the limits of each interval [ xi , xi 1 ], i 1,..., N x 1 the spline-function W3,1 ( x ) is polynomial of 3rd degree. 3 (i ) D °W3,1 ( x) ¦ aD ( x xi ) , 0 D ® ° x [ xi , xi 1 ], i 1,..., N x 1. ¯
(7)
According to [15] the parameters determining a spline are accumulated in column vector Q , which consists of N s N x 2 spline-parameters: Q
(qk , k
1, 2,..., N x 2)T ,
(8)
where dW3,1 ( x1 ) , °q1 W3,1 ( x1 ), q2 dx ° ® d 2W3,1 ( xi ) °°®q , i 1,..., N x . °° i 2 dx 2 ¯¯
(9)
The number of components of Q -vector equals to number of degrees of spline freedom. For solutions of heat conduction equation (5) required the nodal values of the spline-function W3,1 ( x ) and its second derivatives in the nodes of the selected mesh ' : fi
W3,1 ( xi ),
f i II
d 2W3,1 ( xi ) dx 2
, i 1,..., N x .
(10)
Then form them as column vectors:
°V f ( fi , i 1,..., N x )T , ® I T °¯Vd 2 f ( fi , i 1,..., N x ) .
(11)
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The vectors of nodal values of spline-function W3,1 ( x ) and its second derivatives are determined by matrix representations with accordance to [15]:
°V f M f Q, ® °¯Vd 2 f M d 2 f Q,
(12)
where M f , M d 2 f - rectangular matrixes N u ( N 2) , which formed on a base of coordinates xn , n 1,..., N x by methodology, given in [15]. 4. Test Problem of Heat Conductivity
Accuracy evaluation of the proposed spline-method is performed for solving of the test problem, which has the exact solution. At that, the flat plate with infinite length and thickness H (with coordinates x 0 and x H at edges) is considered. Private exact solution of the equation (5) is sought in the form of:
T
e P
2 a2t
( A cos P x B sin P x)
(13)
where P , A, B – some constant values:
P
const,
A
const, B
const
(14)
Let us derive the derivative to check the suitability of the expression (13) as an exact solution of the differential equation (5): wT 2 2 P 2 a 2t ( A cos P x B sin P x), ° wt P a e ° 2 2 ° wT e P a t ( P A sin P x P B cos P x), ® w x ° ° w 2T P 2 a 2t ( P 2 A cos P x P 2 B sin P x). ° 2 e ¯ wx
(15)
Substituting (15) into (5) we obtain the identity, which testifies to the validity of the adoption of the expression (13) as a particular solution of the differential equation (5). To specify the problem assume that the plate is made of polypropylene with the following thermal properties [17]: density U 0,9 103 kg/m3 , heat conductivity O 0,128 W/(m K) , specific heat c 1,93 103 J/(kg K) . On the base of (4) a 2 is calculated for a certain quantity of material given characteristics: a2
0,128 0,9 103 1,93 103
7,37 108 ɦ 2 /s.
(16)
Coefficient ʅis determined by following equation
P
50S .
For the chosen coefficients at x0 based on (13):
(17) 0 and xN
H
0, 01 m the nonlinear boundary conditions are given as
V.M. Kudoyarova and V.P. Pavlov / Procedia Engineering 150 (2016) 1419 – 1426
T ° 0 °T ® N ° ° ¯
e P e
2 a2t
( A cos P x0 B sin P x0 )
P 2 a 2t
Ae P
2 a2t
,
( A cos P xN B sin P xN )
e P
2 a 2t
( A cos
S 2
S
B sin ) 2
(18) Be P
2 a 2t
,
Then, the equation of the initial conditions for the initial time t T (0)
1423
0 is determined from (13):
A cos 50S x B sin 50S x .
(19)
After setting A 500 , B 100 and the final time of calculations tk 1000 c , this paper constructs the charts (fig. 1, a) of the exact functions of temperature change inside the wall thickness for calculated points of time t 0, 200, 400, 600, 800, 1000 c . The chart of temperature changes at the initial time determines the initial conditions of the problem being solved. In fig.1, b is a graph of temperature change on the wall boundaries that define the boundary conditions of the problem.
Fig. 1. (a) temperature distribution inside the wall thickness; (b) temperature change on the wall surfaces over time.
5. Discrete Analog of Heat Conduction Equation for Spline-method
Traditionally, for the solution of unsteady heat conduction problem the calculation method "step-by-step" is applied, which is formed by a rectangular matrix of calculated temperatures T in size N x u M t : T
(Tn( m ) , n 1,..., N x , m 1,..., M t ) ,
(20)
where N x – number of nodes along X -axis; M t – number of nodes along time axis t . In the first step, the temperatures Tn(1) , n 1,..., N x are determined by equation of initial conditions (19). In the second and following steps ( m 2,..., M t ) the discrete analog is built for differential equation (5) for determined temperatures Tn( m 1) , n 1,..., N x on the previous step by this time. The first derivatives wT / wt at the nodes xn are replaced by their difference analogues for the formation of discrete analog of the heat conduction equation (5): wT ( xn ) Tn( m ) Tn( m 1) # , n 1,..., N x . wt 't
(21)
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(m) Forming the vectors of unknown up to now temperatures Tn , n 1,..., N x and of determined temperatures Tn( m 1) , n 1,..., N x on previous m - 1 step:
°VT (Tn( m ) , n 1,..., N x ), ® ( m 1) (Tn( m 1) , n 1,..., N x ). °¯VT
(22)
On a base of (12) the vector VT is determined by matrix equation: M fQ.
VT
(23)
The vector Vw 2T is formed of the second derivatives of temperature function T nodes ( xn , n 1,..., N x ) for time tm : Vw 2T
(
w 2 ( xn , tm ) , n 1,..., N x ). wx 2
T ( x, t ) for x-coordinate in xn
(24)
On a base of (12) the vector Vw 2T is determined by matrix equation: Vw 2T
Md2 f Q.
(25)
Taking into account (21), (22), (23), (24) and (25) the difference analog of the heat conduction equation (5) takes the following form: 1
't
M f Q a2 M d 2 f Q
1
't
VT( m 1) .
(26)
The system (26) of N x -equations with N x 2 unknowns. Supplement it with two equations that take into account the boundary conditions: Nx f °° ¦ M 1, k qk k 1 ® Nx °¦ M f q °¯ k 1 2, k k
T1( m ) , (27) (m) Nx
T
.
In that way the solution of set of N x 2 equations (26) and (27) with N x 2 unknowns gives the temperature values for the m - time layer. Then the similar calculations are performed for the next time tm . 6. Results and Discussion
As a result of numerical calculations by using the proposed spline-method the temperatures Tn( m ) in nodes xn , n 1,..., N x of mesh 'x for all rated time moments tm , m 1,..., M t are determined and reduced to a matrix structure T which is given in (20). In these nodes xn for the same time moments tm the exact values of temperature T ( m ) are calculated by the formula (20), which are accumulated in a matrix T : n
T
V.M. Kudoyarova and V.P. Pavlov / Procedia Engineering 150 (2016) 1419 – 1426
TT
(Tn( m ) , n 1,..., N x , m 1,..., M t ).
1425
(28)
To quantify precisely of implemented spline-method the method consisting of a series of steps is performed. Step 1. The maximum value of the absolute values of all the components of the matrix that contains the exact temperature values TT is determined: TTM
max
m 1,..., M t ; n 1,..., N x
| Tn( m ) | .
(29)
Step 2. Determine the matrix ǻ of absolute differences between the exact Tn( m ) and calculated Tn( m ) values of temperatures at all nodes of the mesh 'x and for all times:
' = TT T
(30)
where the matrix ǻ have the structure: ǻ
('n( m ) | Tn( m ) Tn( m ) |, n 1,..., N x , m 1,..., M t ).
(31)
Step 3. Determine the logarithm of the maximum value of all values of the relative error: lg |
§ TT T | log10 ¨ M TT ©m
max
1,..., M t ; n 1,..., N x
|
'n( m ) ·
|¸. TTM ¹
(32)
TT T | calculated for different combinations of N x and M t , the results of calculations TTM presented as points in fig. 2.
The values of lg |
7. Conclusion
This paper proposes the spline-method to determine the temperature fields. It is shown that this method yields the best order of convergence for the number of nodes M t along the time axis, that is almost equals to one, regardless of the number N x nodes along the X axis. Therefore, if a setting goal is to increase the order of convergence by the mesh step along the time axis, it is necessary to find for the temperature derivative with respect to time wT / wt the difference analogue better than formula (21) used by us. In addition, the resulting numerical solution can be refined by applying the Romberg's method [18], [19] with re-extrapolation for a given number of time steps for different spatial mesh [20]. Analysis of the calculation results shown in fig. 2, leads to the following conclusions: 1. Regardless of the number of nodes N x along the X -axis, the best order of convergence by the number of nodes M t along the time axis is almost equal to one. Thus, if we set the target of increasing convergence of the mesh step along the time axis, then it is necessary to find higher-end of the difference analogue for the temperature derivative with respect to time wT / wt than formula (21) used by us. 2. Fig. 2 shows, that using the spline-function 3rd degree defect 1, even at number of nodes N x 11 allows achieving the accuracy of calculations with a relative error not exceeding than 110-3, and at N x 501 the calculations with accuracy no more than 110-5, that is quite enough to solve many practical problems. Thus, the high efficiency of using this spline-method for solving the considered specific problem allows predicting its high efficiency in solving other problems of heat conduction.
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Fig. 2. Change of the logarithm of the relative error for the temperature, depending on the number of nodes
N x and M t .
Acknowledgement
The authors acknowledge receiving support base part of state-funded research program of The Ministry of Education and Science of the Russian Federation for the years 2014-2016. References [1] V.P. Zhitnikov, N.M. Sheryhalina, Evaluation of the reliability of the numerical results obtained by several methods of problem solution, ZhVT. 6 (1999) 7787. [2] V.P. Isachenko, Heat transfer, Jenergija, Moscow, 1979. [3] A.V. Lykov, Theory of heat conduction, Vysshaja shkola, Moscow, 1967. [4] B.N. Judaev, Heat transfer, Vyssh. shkola, Moscow, 1973. [5] N.M. Tsirelman, The theory and applied heat and mass transfer problems, Mashinostroenie, Moscow, 2011. [6] F.F. Cvetkov, B.A. Grigoriev, Heat and mass transfer, Izdatelstvo MJeI, Moscow, 2005. [7] N.S. Bahvalov, N.P. Zhidkov, G.M. Kobelkov, Numerical methods, Laboratorija znanij, Moscow, 2015. [8] G. Hall, J.M. Watt, Modern Numerical Methods for Ordinary Differential Equations, Clarendon Press, Oxford, 1976. [9] A.A. Amosov, Ju.A. Dubinskij, N.V. Kopchenova, Computational Methods for Engineers, Vysshaja shkola, Moscow, 1994. [10] A.A. Samarskij, Introduction to Numerical Methods, Lan, St. Petersburg, 2005. [11] B.P. Demidovich, I.A. Maron, Je.Z. Shuvalova, Numerical methods of analysis, Approximation of functions, differential and integral equations, Lan, St. Petersburg, 2010. [12] I.S. Berezin, N.P. Zhidkov, Calculation methods, vol. 1, GIFML, Moscow, 1962. [13] V.S. Rjabenkij, A.F. Filippov, About stability of difference equations, Gostehizdat, Moscow, 1956. [14] E.A. Volkov, Numerical methods, Nauka, Moscow, 1987. [15] V.P. Pavlov, Spline-methods and other numerical methods for solving one-dimensional problems of deformable solids mechanics, USATU, Ufa, 2003. [16] V.P. Pavlov, A.A. Abdrahmanova, R.P. Abdrahmanova, The problem of calculating rods by one-dimensional spline fifth degree of the defect two, Matematicheskie zametki JaGU. 1 (2013) 5059. [17] A.I. Semenov, K.K. Poljakova, Foreign industrial polymeric materials and components, Collegiate Dictionary Directory, Izdatelstvo Akademii Nauk SSSR, Moscow, 1963. [18] I.I. Blehman, A.D. Myshkis, Ja.G. Panovko, Mechanics and Applied Mathematics, The subject of logic and especially the approaches, Izdatelstvo URSS, Moscow, 2005. [19] N.N. Kalitkin, Numerical methods, Nauka, Moscow, 1978. [20] V. Kudoyarova, V. Pavlov, Refining of numerical solution for nonlinear transient heat conduction in a plate made of polymer composite material, International Journal of Applied Engineering Research. 18 (2015) 3946639470.
ScienceDirect Procedia Engineering 150 (2016) 1419 – 1426
International Conference on Industrial Engineering, ICIE 2016
The Spline Method for the Solution of the Transient Heat Conduction Problem with Nonlinear Initial and Boundary Conditions for a Plate V.M. Kudoyarovaa, V.P. Pavlova,* a
Ufa State Aviation Technical University, st. K. Marcsa,12, Ufa, 450000, Russian Federation
Abstract There are many cases in the engineering practice when it is necessary to know the transient temperature field in a flat plate. At that, temperature distribution is studied experimentally or by calculation. However, the obtained calculation results do not always provide the required accuracy. This paper obtains the spline-method grade 3 defect 1, which allows simulating the process of the thermal conductivity for the infinitely extended plate at nonlinear initial and boundary conditions of the first kind on the plate sides. For accurate evaluation of the solution of the heat conduction equation with the help of the spline-method, the standard problem is created for a one-dimensional infinite flat plane which has the exact analytical solution. This paper shows the use of spline method for the solution of transient heat conduction problem which gives an error of no more than 110-5 and thus satisfies obtaining of the sufficient accuracy to solve many practical problems. © 2016The TheAuthors. Authors. Published Elsevier © 2016 Published by by Elsevier Ltd.Ltd. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/). Peer-review under responsibility of the organizing committee of ICIE 2016. Peer-review under responsibility of the organizing committee of ICIE 2016 Keywords: Modeling; splines; thermal conductivity; differential equations of parabolic type.
1. Introduction Currently the spline-methods are widely applied in many fields of science with using methods of computer modeling. In turn, the numerical experiment is often used in the study and modeling of physical processes in dealing with a wide range of technical challenges, since it allows obtaining results of the study, even in a case when the real experiment is difficult to conduct. The main criteria of the effectiveness of a computer modeling is accuracy of the
* Corresponding author. Tel.: +7-347-273-0844; fax: +7-347-273-0844. E-mail address: [email protected].
1877-7058 © 2016 The Authors. Published by Elsevier Ltd. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/). Peer-review under responsibility of the organizing committee of ICIE 2016
doi:10.1016/j.proeng.2016.07.204
1420
V.M. Kudoyarova and V.P. Pavlov / Procedia Engineering 150 (2016) 1419 – 1426
solution, and the computational complexity of used numerical method. At that pursuing an objective of increasing the number of significant digits in the results of calculations using this method. The accuracy of the numerical simulation is characterized by a measure of the deviation of the numerical results obtained from the known exact analytical solutions [1]. Heat conduction problem is a system of differential equations, so one of the key purposes of numerical analysis is to develop methods to obtain reliable results close to accurate analytical values. 2. Problem Definition To evaluate the capacity of structures, working in conditions of heat, it is necessary to know the temperature field in it. The temperature distribution can be studied experimentally, or to be determined by solving the heat equation. The experimental method is very laborious and time-consuming, and besides there is not always able to realize it. Therefore, in most cases, the temperature field determined by calculation [2]–[6]. At that, it is important to choose the appropriate mathematical apparatus: analytical methods, finite-difference method, finite-element method [7]– [14]. In real thermal processes, the temperature on the solid surface is often changed under the influence of various factors such as the presence of the heat source in the body, or at the convective heat transfer when the surface temperature is dependent on flow regime at variable heat transfer intensity on its surface, etc. In this regard, this paper solves the boundary value problem of unsteady heat conduction with specified non-linear initial and boundary conditions of the first kind. In a case of absence of internal heat sources, the spline-method applied to solve the differential equation of heat conduction, and is given by [4]:
Uc
wT wt
w § wT ¨O wx © wx
· ¸, ¹
(1)
where t – time; x – position of a point; T T ( x, t ) – the function describing the temperature change by time t and space coordinate x ; U – material density; c – specific heat; O – thermal conductivity of the material along X - axis. In case when material properties are not dependent on temperature and time:
U
const, c
const, O
const
(2)
the equation (1) takes the following form:
Uc
wT wt
O
w 2T . wx 2
(3)
by replacing a2
O , Uc
(4)
the equation (3) is reduced to the form wT w 2T a2 2 wt wx
0.
(5)
This paper estimates the solution accuracy of the equation (5) of using splines of 3rd degree of defect 1 by comparing the numerical solution with the exact analytical solution of the test problem.
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3. Main Statements of Spline-method Fundamentals of splines method applied to the problems of the deformation of solids are stated in [15], [16]. One of the main features of the spline of 3rd degree defect 1 is the continuity of second derivative of splinefunction W3,1 ( x ) at all points of the domain that is highly desirable for solving thermal conduction differential equation (5) where the second derivative w 2T / wx 2 is present. Therein lies the advantage of splines method in comparison to the finite difference method and finite element method, which do not provide the continuity of the second derivatives. When constructing spline of 3rd degree defect 1 [15] the mesh 'x with N x -nodes is forming in interval [a, b].
'x : a
x1 x2 ... xNx
(6)
b
On the mesh base, the spline function of 3rd degree defect 1 W3,1 ( x ) is constructed with N s
N x 2 degrees of
freedom. In the limits of each interval [ xi , xi 1 ], i 1,..., N x 1 the spline-function W3,1 ( x ) is polynomial of 3rd degree. 3 (i ) D °W3,1 ( x) ¦ aD ( x xi ) , 0 D ® ° x [ xi , xi 1 ], i 1,..., N x 1. ¯
(7)
According to [15] the parameters determining a spline are accumulated in column vector Q , which consists of N s N x 2 spline-parameters: Q
(qk , k
1, 2,..., N x 2)T ,
(8)
where dW3,1 ( x1 ) , °q1 W3,1 ( x1 ), q2 dx ° ® d 2W3,1 ( xi ) °°®q , i 1,..., N x . °° i 2 dx 2 ¯¯
(9)
The number of components of Q -vector equals to number of degrees of spline freedom. For solutions of heat conduction equation (5) required the nodal values of the spline-function W3,1 ( x ) and its second derivatives in the nodes of the selected mesh ' : fi
W3,1 ( xi ),
f i II
d 2W3,1 ( xi ) dx 2
, i 1,..., N x .
(10)
Then form them as column vectors:
°V f ( fi , i 1,..., N x )T , ® I T °¯Vd 2 f ( fi , i 1,..., N x ) .
(11)
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The vectors of nodal values of spline-function W3,1 ( x ) and its second derivatives are determined by matrix representations with accordance to [15]:
°V f M f Q, ® °¯Vd 2 f M d 2 f Q,
(12)
where M f , M d 2 f - rectangular matrixes N u ( N 2) , which formed on a base of coordinates xn , n 1,..., N x by methodology, given in [15]. 4. Test Problem of Heat Conductivity
Accuracy evaluation of the proposed spline-method is performed for solving of the test problem, which has the exact solution. At that, the flat plate with infinite length and thickness H (with coordinates x 0 and x H at edges) is considered. Private exact solution of the equation (5) is sought in the form of:
T
e P
2 a2t
( A cos P x B sin P x)
(13)
where P , A, B – some constant values:
P
const,
A
const, B
const
(14)
Let us derive the derivative to check the suitability of the expression (13) as an exact solution of the differential equation (5): wT 2 2 P 2 a 2t ( A cos P x B sin P x), ° wt P a e ° 2 2 ° wT e P a t ( P A sin P x P B cos P x), ® w x ° ° w 2T P 2 a 2t ( P 2 A cos P x P 2 B sin P x). ° 2 e ¯ wx
(15)
Substituting (15) into (5) we obtain the identity, which testifies to the validity of the adoption of the expression (13) as a particular solution of the differential equation (5). To specify the problem assume that the plate is made of polypropylene with the following thermal properties [17]: density U 0,9 103 kg/m3 , heat conductivity O 0,128 W/(m K) , specific heat c 1,93 103 J/(kg K) . On the base of (4) a 2 is calculated for a certain quantity of material given characteristics: a2
0,128 0,9 103 1,93 103
7,37 108 ɦ 2 /s.
(16)
Coefficient ʅis determined by following equation
P
50S .
For the chosen coefficients at x0 based on (13):
(17) 0 and xN
H
0, 01 m the nonlinear boundary conditions are given as
V.M. Kudoyarova and V.P. Pavlov / Procedia Engineering 150 (2016) 1419 – 1426
T ° 0 °T ® N ° ° ¯
e P e
2 a2t
( A cos P x0 B sin P x0 )
P 2 a 2t
Ae P
2 a2t
,
( A cos P xN B sin P xN )
e P
2 a 2t
( A cos
S 2
S
B sin ) 2
(18) Be P
2 a 2t
,
Then, the equation of the initial conditions for the initial time t T (0)
1423
0 is determined from (13):
A cos 50S x B sin 50S x .
(19)
After setting A 500 , B 100 and the final time of calculations tk 1000 c , this paper constructs the charts (fig. 1, a) of the exact functions of temperature change inside the wall thickness for calculated points of time t 0, 200, 400, 600, 800, 1000 c . The chart of temperature changes at the initial time determines the initial conditions of the problem being solved. In fig.1, b is a graph of temperature change on the wall boundaries that define the boundary conditions of the problem.
Fig. 1. (a) temperature distribution inside the wall thickness; (b) temperature change on the wall surfaces over time.
5. Discrete Analog of Heat Conduction Equation for Spline-method
Traditionally, for the solution of unsteady heat conduction problem the calculation method "step-by-step" is applied, which is formed by a rectangular matrix of calculated temperatures T in size N x u M t : T
(Tn( m ) , n 1,..., N x , m 1,..., M t ) ,
(20)
where N x – number of nodes along X -axis; M t – number of nodes along time axis t . In the first step, the temperatures Tn(1) , n 1,..., N x are determined by equation of initial conditions (19). In the second and following steps ( m 2,..., M t ) the discrete analog is built for differential equation (5) for determined temperatures Tn( m 1) , n 1,..., N x on the previous step by this time. The first derivatives wT / wt at the nodes xn are replaced by their difference analogues for the formation of discrete analog of the heat conduction equation (5): wT ( xn ) Tn( m ) Tn( m 1) # , n 1,..., N x . wt 't
(21)
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(m) Forming the vectors of unknown up to now temperatures Tn , n 1,..., N x and of determined temperatures Tn( m 1) , n 1,..., N x on previous m - 1 step:
°VT (Tn( m ) , n 1,..., N x ), ® ( m 1) (Tn( m 1) , n 1,..., N x ). °¯VT
(22)
On a base of (12) the vector VT is determined by matrix equation: M fQ.
VT
(23)
The vector Vw 2T is formed of the second derivatives of temperature function T nodes ( xn , n 1,..., N x ) for time tm : Vw 2T
(
w 2 ( xn , tm ) , n 1,..., N x ). wx 2
T ( x, t ) for x-coordinate in xn
(24)
On a base of (12) the vector Vw 2T is determined by matrix equation: Vw 2T
Md2 f Q.
(25)
Taking into account (21), (22), (23), (24) and (25) the difference analog of the heat conduction equation (5) takes the following form: 1
't
M f Q a2 M d 2 f Q
1
't
VT( m 1) .
(26)
The system (26) of N x -equations with N x 2 unknowns. Supplement it with two equations that take into account the boundary conditions: Nx f °° ¦ M 1, k qk k 1 ® Nx °¦ M f q °¯ k 1 2, k k
T1( m ) , (27) (m) Nx
T
.
In that way the solution of set of N x 2 equations (26) and (27) with N x 2 unknowns gives the temperature values for the m - time layer. Then the similar calculations are performed for the next time tm . 6. Results and Discussion
As a result of numerical calculations by using the proposed spline-method the temperatures Tn( m ) in nodes xn , n 1,..., N x of mesh 'x for all rated time moments tm , m 1,..., M t are determined and reduced to a matrix structure T which is given in (20). In these nodes xn for the same time moments tm the exact values of temperature T ( m ) are calculated by the formula (20), which are accumulated in a matrix T : n
T
V.M. Kudoyarova and V.P. Pavlov / Procedia Engineering 150 (2016) 1419 – 1426
TT
(Tn( m ) , n 1,..., N x , m 1,..., M t ).
1425
(28)
To quantify precisely of implemented spline-method the method consisting of a series of steps is performed. Step 1. The maximum value of the absolute values of all the components of the matrix that contains the exact temperature values TT is determined: TTM
max
m 1,..., M t ; n 1,..., N x
| Tn( m ) | .
(29)
Step 2. Determine the matrix ǻ of absolute differences between the exact Tn( m ) and calculated Tn( m ) values of temperatures at all nodes of the mesh 'x and for all times:
' = TT T
(30)
where the matrix ǻ have the structure: ǻ
('n( m ) | Tn( m ) Tn( m ) |, n 1,..., N x , m 1,..., M t ).
(31)
Step 3. Determine the logarithm of the maximum value of all values of the relative error: lg |
§ TT T | log10 ¨ M TT ©m
max
1,..., M t ; n 1,..., N x
|
'n( m ) ·
|¸. TTM ¹
(32)
TT T | calculated for different combinations of N x and M t , the results of calculations TTM presented as points in fig. 2.
The values of lg |
7. Conclusion
This paper proposes the spline-method to determine the temperature fields. It is shown that this method yields the best order of convergence for the number of nodes M t along the time axis, that is almost equals to one, regardless of the number N x nodes along the X axis. Therefore, if a setting goal is to increase the order of convergence by the mesh step along the time axis, it is necessary to find for the temperature derivative with respect to time wT / wt the difference analogue better than formula (21) used by us. In addition, the resulting numerical solution can be refined by applying the Romberg's method [18], [19] with re-extrapolation for a given number of time steps for different spatial mesh [20]. Analysis of the calculation results shown in fig. 2, leads to the following conclusions: 1. Regardless of the number of nodes N x along the X -axis, the best order of convergence by the number of nodes M t along the time axis is almost equal to one. Thus, if we set the target of increasing convergence of the mesh step along the time axis, then it is necessary to find higher-end of the difference analogue for the temperature derivative with respect to time wT / wt than formula (21) used by us. 2. Fig. 2 shows, that using the spline-function 3rd degree defect 1, even at number of nodes N x 11 allows achieving the accuracy of calculations with a relative error not exceeding than 110-3, and at N x 501 the calculations with accuracy no more than 110-5, that is quite enough to solve many practical problems. Thus, the high efficiency of using this spline-method for solving the considered specific problem allows predicting its high efficiency in solving other problems of heat conduction.
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Fig. 2. Change of the logarithm of the relative error for the temperature, depending on the number of nodes
N x and M t .
Acknowledgement
The authors acknowledge receiving support base part of state-funded research program of The Ministry of Education and Science of the Russian Federation for the years 2014-2016. References [1] V.P. Zhitnikov, N.M. Sheryhalina, Evaluation of the reliability of the numerical results obtained by several methods of problem solution, ZhVT. 6 (1999) 7787. [2] V.P. Isachenko, Heat transfer, Jenergija, Moscow, 1979. [3] A.V. Lykov, Theory of heat conduction, Vysshaja shkola, Moscow, 1967. [4] B.N. Judaev, Heat transfer, Vyssh. shkola, Moscow, 1973. [5] N.M. Tsirelman, The theory and applied heat and mass transfer problems, Mashinostroenie, Moscow, 2011. [6] F.F. Cvetkov, B.A. Grigoriev, Heat and mass transfer, Izdatelstvo MJeI, Moscow, 2005. [7] N.S. Bahvalov, N.P. Zhidkov, G.M. Kobelkov, Numerical methods, Laboratorija znanij, Moscow, 2015. [8] G. Hall, J.M. Watt, Modern Numerical Methods for Ordinary Differential Equations, Clarendon Press, Oxford, 1976. [9] A.A. Amosov, Ju.A. Dubinskij, N.V. Kopchenova, Computational Methods for Engineers, Vysshaja shkola, Moscow, 1994. [10] A.A. Samarskij, Introduction to Numerical Methods, Lan, St. Petersburg, 2005. [11] B.P. Demidovich, I.A. Maron, Je.Z. Shuvalova, Numerical methods of analysis, Approximation of functions, differential and integral equations, Lan, St. Petersburg, 2010. [12] I.S. Berezin, N.P. Zhidkov, Calculation methods, vol. 1, GIFML, Moscow, 1962. [13] V.S. Rjabenkij, A.F. Filippov, About stability of difference equations, Gostehizdat, Moscow, 1956. [14] E.A. Volkov, Numerical methods, Nauka, Moscow, 1987. [15] V.P. Pavlov, Spline-methods and other numerical methods for solving one-dimensional problems of deformable solids mechanics, USATU, Ufa, 2003. [16] V.P. Pavlov, A.A. Abdrahmanova, R.P. Abdrahmanova, The problem of calculating rods by one-dimensional spline fifth degree of the defect two, Matematicheskie zametki JaGU. 1 (2013) 5059. [17] A.I. Semenov, K.K. Poljakova, Foreign industrial polymeric materials and components, Collegiate Dictionary Directory, Izdatelstvo Akademii Nauk SSSR, Moscow, 1963. [18] I.I. Blehman, A.D. Myshkis, Ja.G. Panovko, Mechanics and Applied Mathematics, The subject of logic and especially the approaches, Izdatelstvo URSS, Moscow, 2005. [19] N.N. Kalitkin, Numerical methods, Nauka, Moscow, 1978. [20] V. Kudoyarova, V. Pavlov, Refining of numerical solution for nonlinear transient heat conduction in a plate made of polymer composite material, International Journal of Applied Engineering Research. 18 (2015) 3946639470.