A METHOD FOR TRANSIENT HEAT CONDUCTION · A METHOD FOR TRANSIENT HEAT CONDUCTION Jian Song*, Lili Yan and Yingying Yin School of Mathematics and Statistics, Shandong University of

IJRRAS 42 (1) ● Jan 2020 www.arpapress.com/Volumes/Vol42Issue1/IJRRAS_42_1_04.pdf

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A METHOD FOR TRANSIENT HEAT CONDUCTION

Jian Song*, Lili Yan and Yingying Yin

School of Mathematics and Statistics, Shandong University of Technology, Zibo 255049, China

ABSTRACT

In this paper, we proposed a calculation method which combining radial basis function method and Houbolt method

to solve the transient heat conduction problem. Houbolt method is used for discrete differences in time domain. In

order to reduce the influence of shape parameter, this study proposed an improved radial basis function, and the

improved radial basis function is used to compute the value of the problem. This method is accurate and stable, and

it can solve the transient heat conduction problem excellently. Finally, three numerical examples are applied to

illustrate the effectiveness of the proposed method.

Key words: transient heat conduction, radial basis function, Houbolt method.

1. INTRODUCTION

It is difficult to solve the transient heat conduction because of its inherent complexity. But with the development of

science and technology, there are more and more solutions to solve this problem, for example, the finite element

method (FEM)[1], boundary element method (BEM)[2-4], the finite difference method (FDM), and meshfree

method[9-15] have been well displayed in recent years and have been frequently used in heat conduction problems.

All these methods display different advantages and disadvantages. BEM has a distinct advantage is its property in

reducing the dimensionality of heat conduction problem by one when compare with FDM and FEM. However, the

obvious advantage is gradually disappearing when this method is used in solving transient and non-homogeneous

problems. Meshfree method is used in interpolating the field function by the fitting function, and it ensures the high-

order continuity of the basic field variables either in the entire computation domain or on the computation boundary.

In recent years, mesh-free multiquadric radial basis functions (MQ-RBFs) method which developed by Kansa [5]

has attracted the attention of the researchers since it is easy to calculation and has higher precision. However, it has a

disadvantage that can not be ignored: it is seriously affected by shape parameters.

In this paper, we propose a combine method based on RBFs[6] and Houbolt method, which overcome the sensitivity

of parameter and get excellent precision.

Section two introduces the governing equation of the transient heat conduction. Section three presentation the

combine method and the improved RBF. In order to prove its performance, section four test three numerical

examples. And in section five, we summarize the advantages and disadvantages of the method in this paper.

2. GOVERNING EQUATION

Assuming that the computed domain of the transient heat conduction is which bounded by 21 =, where

21, are the temperature boundary, the heat flux boundary respectively. The governing equation of the problem is

( ) ( ) ( ) ( )

t

tucfukuktfuk

=++=+

),(, 2 x

xxx , in (1)

where2 is the Laplace operator,

ft,are time and heat source, respectively,

, k , c denotes mass density,

thermal conductivity, and specific heat respectively, and may vary with ( )21, xx=x

.

The boundary condition can be written as follows:

),(),( tutu xx = on 1 (2a)

),(),( tqtq xx = on 2 (2b)

and initial condition

)(),( 00 xx utu = in (3)

where u ,q

are specified value of temperature on 1 and the given heat flux on 2 , respectively. And

n

ukukq

=−= n

, ( )21,nn=n

is the outward unit normal to the boundary.

IJRRAS 42 (1) ● Jan 2020 Song et al. ● A Method for Transient Heat Conduction

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3. SOLVING THE TRANSIENT HEAD CONDUCTION WITH A COMBINE METHOD

3.1 the combine method

In order to solve the time-dependent terms, we use the following method which is in Ref [7] to discretize the time

domain: 1

1 1 21(11 18 9 2 )

6

n

n n n nuu u u u

t t

+

+ − − − + −

, (4)

and when 2n ,

1

11( )

n

n nuu u

t t

+

+ −

, where t is the time step, 1

1( , )n

nu u t+

+= x, 1n nt t t n t t+ = + = +

By using Eq(4) into Eq(1), we can obtain the following equation:

( ) ( )2 1 1 1 1 2 11 1( 18 9 2 )

6 6

n n n n n n nk u k u u c u u u ft t

+ + + − − + + − = − + − +

x x

( ) ( )2 1 1 1 11 1n n n n nk u k u u c u ft t

+ + + + + − = +

x x

and we can use Radial basis function method to solve nu .

We assumed that RRd →:

is a continunous mapping and 0)0(

, then a Radial basis function on dR is

)(r,where ir xx−=

. If N points NiRd

i ,1, =x are closed in

dR , then

+1 +1

1

( , ) ( )N

n n

i i

i

u u t r=

= =x

(5)

also can be called a radial basis function. Radial basis functions can be divided into two categories. The first

category is infinitely differentiable and seriously affected by shape parameters c .The second category is not

infinitely differentiable and is not affected by shape parameters c , but has poor accuracy. In this paper, we focus on

the first category.

Eq(5) also can be written as the form:

+1

1

( , ) ( , ) ( )N

n n

i i

i

f t Lu t r=

= =x x

(6)

where L is a linear operator,)()( ii r xx −=

, and i is a set of undetermined coefficients, and 1( , )nf t +x

is

a known function. If we choose N collocation points, then Eq(6) can be written as

+1

1

( , ) ( , ) ( )N

n n

j j i i j

i

f t Lu t r=

= =x x

therefore, we define

= )(rψy (7)

where T

n ],,,[ 21 =,

=

)()()(

)()()(

)()()(

)(

21

22221

11211

NNNN

N

N

rrr

rrr

rrr

r

ψ

,

ijjr xx −=.

By the Eq(7), the governing equation and boundary condition can be transformed into the following form: +1 +1( , ) ( , )n nLu t f t=x x , x

+1 +1( , ) ( , )n nGu t g t=x x , x

where, GL, are linear operator. The above equations produce a system of linear equations:

BA = where


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−

−=

))((

))((

ij

ij

xxG

xxL

A

,

ij

ij

xx

xx

,

,

+1

+1

( , )

( , )

n

n

f t

g t

=

xB

x

3.2 An improved RBF

Some well-known RBFs has been proposed already, for example, Multiquadratic RBF(MQ-RBF), Inverse

multiquadratic RBF(IMQ-RBF,), Gaussian RBF(GA-RBF), and Inverse quadratic RBF, but, usually, the RBFs

generated by them often have an fatal flaw: heavily influenced by shape parameters c . Then we use the improved

RBF to avoid the influence of shape parameters [7]:

=)(rj

2

5

1ln( ( ))

1

2 2

j

j

c r fcf r

r

+

+ +

where 2 2 2( ) ( )j jr x x y y= − + −

,

22 crf j +=.

In order to verify the effectiveness of the proposed method, we have tried the following three numerical experiments

4. NUMERICAL EXAMPLE

In this section, in order to make comparison with the exact solution, we set the root mean square( RMSE ) as

follows:

N

II

RMSE

n

i

numexa=

−

= 1

2)(

where exaI and numI

are obtained by exact and numerical solution, N is the number of center points.

Example 1. Consider the problem in Ref [6]:

0u

u ft

+ − =

,

the computational domain is[0,1] [0,1]

, the source term as follows:

1 2sin( )sin( )(2sin( ) cos( ))f x x t t= +

The analytical solution is given by 1 2sin( )sin( )sin( )u x x t=, thus we can obtain the boundary condition and initial

condition.

In this problem, the time step 0.5t s = , we evenly selected 32 boundary points and 36 interior points in the

calculation domain. In order to verify the accuracy of this method, we randomly selected three calculation points in

the calculation domain, Fig.1 have shown the absolute error curves with respect to time when the shape parameter

10c = . And we selected 225 uniformly distributed points over the region 1 2 1 2( , ) 0.1 , 0.9x x x x

at 10t s= , the

distribution of absolute errors are shown in Fig.2. And Fig.3 gives the distribution of RMSE in regard to shape

parameter c at 10t s= . Among them, the method proposed in this paper can get perfect and stable accuracy.


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Fig.1 The distribution of absolute error at different points in regard to time.

Fig.2 Absolute error surface in the domain 1 2 1 2( , ) 0.1 , 0.9x x x x

at 10t s=


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Fig.3 Distribution of RMSE with respect to shape parameter c at 10t s=

Example 2. In this example, we consider the problem in Ref[8], The form of governing equation are as follow:

2

2 1

1 2

cos sin ( , )u u u

u x x h tt x x

= − − +

x

,, 0t x

and the analytical solution is 1 2sin sin cosu x x t=. Dirichlet boundary condition is given, and the boundary of

computed domain (can be seen in Fig.4) can be written as

1 2 1 2= ( , ) = cos , = sin , 0 2x x x x

3 cos( / 7.0)sin(4 )=

5 sin(2 )

+ −

+

25 boundary center points and only one internal center points are assumed, and the time step 0.5t s = . Fig.5 gives

the absolute error surface obtained by the method in this paper when shape parameter 10c = , and Fig.6 gives the

distribution of RMSE in regard to shape parameter c at 10t s= . We can see the method in this paper is effective

and stable , it can solve the problem excellently.


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Fig.4 The computed domain

Fig.5 Absolute error surface in the domain 1 2 1 2( , ) -1 , 1x x x x

at 10t s=


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Example 3. In this example can be found in Ref [8], the form of governing equation are as follow:

2

2 2 2 1

1 2

sin cos ( , )u u u

u x x x x h tt x x

= − + +

x

, , 0t x

and the analytical solution is 2 1 1 2( sin cos )cosu x x x x t= +.Dirichlet boundary condition is given, and the

boundary of computed domain (can be seen in Fig.7) can be written as:

1 2 1 2= ( , ) = cos , = sin , 0 2x x x x

sin 2 cos 2=e sin (2 ) cos (2 )e +

30 boundary center points and 9 internal center points are assumed, and the time step 0.1t s = . The distribution of

the exact solution and the numerical solution in the computer domain are compared in Fig. 8 when shape

parameter 10c = and 2t s= , the corresponding relative errors are shown in Fig.9. and Fig.10 gives the distribution

of RMSE with respect to shape parameter c at 2t s= .We can see the results computed by the method of this paper

have a good agreement with the analytical solutions.


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Fig.7 The computed domain

(a) (b)

Fig.8 The distribution of (a) numerical solution (b) analytical solution


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Fig.9 The distribution of absolute error at 2t s=


5. Conclusion

In this paper, a calculation method combining radial basis function method and Houbolt method is proposed.

Houbolt method is used for discrete difference in time area, and then the improved radial basis function method is

used to calculate the function value. This method is simple, effective and practical. But at the same time, the

distribution of internal points is more sensitive, which needs to be improved.

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A METHOD FOR TRANSIENT HEAT CONDUCTION · A METHOD FOR TRANSIENT HEAT CONDUCTION Jian Song*, Lili Yan and Yingying Yin School of Mathematics and Statistics, Shandong University of