Application of the homotopy perturbation method for the solution of inverse heat conduction problem

International Communications in Heat and Mass Transfer 39 (2012) 30–35

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Application of the homotopy perturbation method for the solution of inverse heat conduction problem☆ Edyta Hetmaniok, Iwona Nowak, Damian Słota ⁎, Roman Wituła Institute of Mathematics, Silesian University of Technology, Kaszubska 23, 44-100 Gliwice, Poland

a r t i c l e

i n f o

Available online 23 September 2011 Keywords: Heat transfer Inverse problem Homotopy perturbation method

a b s t r a c t This paper deals with an application of the homotopy perturbation method for the solution of inverse heat conduction problem. This problem consists in the calculation of temperature distribution in the domain, as well as in the reconstruction of functions describing the temperature and heat flux on the boundary, when the temperature measurements in the domain are known. Examples illustrating discussed application and confirming utility of this method in such a type of problem was also presented. © 2011 Elsevier Ltd. All rights reserved.

1. Introduction Methods that allow to find solutions of any type of nonlinear physical and technical problems have been utilized in many application, recently. These methods include, among the others: the Adomian decomposition method [1–3], variational iteration method [4–7], and homotopy perturbation method [8–13]. In general, a mathematical formulation of those methods makes it possible to solve nonlinear operator equations. In this type of methods a sequence or functional series is constructed, whose limit is a function which is the solution of discussed problem (with appropriate assumptions). Usually, due to a quite fast convergence of appropriate sequences or series, determining of their few first components leads to a very good approximation of solution searched. The homotopy perturbation is an effective solution method for a broad class of problems. Applications of this method for solving nonlinear ordinary differential equations with boundary conditions or similar problems are presented in papers [14,15]. In the literature one can also find its applications in wave and diffusion equations [16–18], inverse problem of diffusion equation [19], Laplace equation [20] and hyperbolic partial differential equation [21]. Ganji and his colleagues in many works [22–26] have dealt with homotopy perturbation method used in solution of various problems connected with heat transfer processes. Slota [27,28] applied the method for determination of exact (or approximate) solution of one- or two-phase inverse Stefan problem. Another work [29], shows an utilization of the method mentioned for finding the temperature distribution in the cast-mould heterogeneous

☆ Communicated by W.J. Minkowycz ⁎ Corresponding author. E-mail address: [email protected] (D. Słota). 0735-1933/$ – see front matter © 2011 Elsevier Ltd. All rights reserved. doi:10.1016/j.icheatmasstransfer.2011.09.005

domain. Information about convergence of homotopy perturbation method can be found in papers [8,30]. Mathematical models of thermal processes are formulated as systems of ordinary or partial differential equations, as well as integral ones. In order to provide a uniqness of the solution, apart from the geometry of the domain, equations defining the temperature field, boundary conditions and initial condition for transient phenomena have to be specified. Research conducted on real objects very often forces to formulate and solve the heat conduction inverse problems in which lost or unsure elements of mathematical model are retrieved on the basis of measurement data. Inverse solution consists not only in estimation of unknown information but also in finding full temperature field. High interest in this kind of problems makes that inverse problems in heat conduction are widely discussed in the literature (see for example [31–39]). Most of works are devoted to steady or transient heat conduction models. Huang and Wang [40] estimate the unknown surface heat fluxes in the solid. Huang and Tsai [41] employed an inverse method to determine the time-dependent local heat transfer coefficients for a plate fin. Chen et al. [42] estimated the heat flux and temperature on the external surface in laminar pipe flow using an inverse algorithm. There were also papers that addressed three-dimensional Stefan problems. Nowak et al. solve the inverse boundary problem [43] as well as geometry one [44] for a continuous casting process of an aluminum alloy. Slota [45,46] evaluate the cooling strategy for the primary and secondary cooling zone in the continuous casting of pure metals, whereas in [47,48] the inverse design Stefan problem was considered, where heat transfer coefficient was designated on the boundary of the domain. Apart from the examples of numerical solutions of inverse problems connected with real engineering problem, in the literature one can find examples of effective analytical methods based on Laplace transformation technique [49,50] or homotopy perturbation method. The latter one has been used as a solutions procedure in inverse problem for diffusion equation [19] and parabolic partial differential equations with nonlinear conductivity term [51]. High efficiency of the

E. Hetmaniok et al. / International Communications in Heat and Mass Transfer 39 (2012) 30–35

−k

Nomenclature a D f H k L N p q t t⁎ u x

thermal diffusivity domain of the problem function homotopy operator thermal conductivity linear operator nonlinear operator homotopy parameter (real number) function describing the boundary condition time end of the process temperature space variable

  u xp ; t ¼ ψp ðt Þ;

LðuÞ þ N ðuÞ ¼ f ðzÞ;

ð8Þ

H ðv; pÞ≡

! 2 2 2 ∂ v ∂ u0 ∂ u0 1 ∂v : − þ p − a ∂t ∂x2 ∂x2 ∂x2

ð9Þ

The solution of equation H(v, p) = 0 is searched in the form: ∞

j

v ¼ ∑ p uj :

ð1Þ

ð10Þ

j¼0

If the above series is convergent, then substituting p = 1 the solution of considered equation is obtained: ∞

where a is the thermal diffusivity, u is temperature, and t and x refer to time and spatial location, respectively. On the boundary Γ0 the initial condition is defined: ð2Þ

On boundary Γ1 the Dirichlet boundary condition is assumed: ð3Þ

u ¼ lim v ¼ ∑ uj : p→1

ð11Þ

j¼0

Information about the convergence of the series (Eq. (10)) is contained in works [8,30]. In many cases this series is rapidly convergent, therefore the sum reduced to a few initial components provides a very good approximation of the desired solution. If we limit the sum to the first n + 1 components, we obtain so-called n-order approximate solution: n

uˆ n ¼ ∑ uj :

ð12Þ

j¼0

In the discussed inverse problem the temperature distribution u(x, t) in region D is determined as well as temperature θ(t) and heat flux q(t) on boundary Γ2, which define the Dirichlet and the Neumann boundary condition: uðb; t Þ ¼ θðt Þ;

ð7Þ

where p∈[0,1] is so-called homotopy parameter, vðz; pÞ : Ω×½0; 1→R, and u0 is the initial approximation of a solution of Eq. (7). Because H(v,0)=L(v)−L(u0), that for p=0 the solution of operator equation H (v,0)=0 is equivalent to solution of a trivial problem L(v)−L(u0)=0, where we get v=u0. For p=1 the solution of operator equation H (v,1)=0 is also equivalent but to solution of the initial equation. Thus, changing the parameter p between 0 and 1 means changing of equation between trivial L(v)−L(u0)=0 and given one (i.e. solution v from u0 to u). In our case, for heat transfer Eq. (1), we obtain:

where the initial and boundary conditions are given. In the region D we consider the heat conduction equation:


 t ∈ 0; t ;

z ∈ Ω:

H ðv; pÞ≡LðvÞ−Lðu0 Þ þ pðLðu0 Þ þ NðvÞ−f ðzÞÞ;

Γ0 ¼ fðx; 0Þ; x ∈ ½0; bg; Γ1 ¼ fð0; t Þ; t ∈ ½0; t  Þg;  Γ2 ¼ fðb; t Þ; t ∈ ½0; t Þg;


 t ∈ 0; t :

ð6Þ

where L and N are a linear and nonlinear operator, respectively. Function f is given while u is a searched function. A new operator H, called homotopy operator, is now defined as follows:

Let's start with a formulation of a mathematical model of the considered problem. Let D = {(x, t) ; x ∈ [0, b], t ∈ [0, t ∗)}. On the boundary of this domain three components are distributed:

uð0; t Þ ¼ ψðt Þ;


 t ∈ 0; t :

Using the homotopy perturbation method the solution of nonlinear operational equation can be found in the following form:

2. Problem formulation

x ∈ ½0; b:

ð5Þ

3. The homotopy perturbation method

homotopy perturbation method, in direct, as well as, inverse problems, gives a chance of solution of inverse thermal problem which is presented in the paper.

uðx; 0Þ ¼ φðxÞ;


 t ∈ 0; t :

The incomplete mathematical description is supplemented by temperature value at some fixed point x = xp, where xp ∈ (0, b):

Greek symbols Γi boundary of the domain δ absolute error Δ relative error θ function describing the boundary condition φ function describing the initial condition ψ function describing the boundary condition

∂u ∂2 uðx; t Þ ðx; t Þ ∈ D; ðx; t Þ ¼ a ∂t ∂x2

∂uðb; t Þ ¼ qðt Þ; ∂x

31

ð4Þ

In order to find the function uj, the relation (Eq. (10)) is put into the equation H(v, p) = 0 which finally leads to:

∞

∑p j¼0

j

∂2 uj ∂x

2

¼

∂2 u0 ∂2 u 1 ∞ j ∂uj−1 : −p 20 þ ∑ p 2 a j¼1 ∂t ∂x ∂x

ð13Þ

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E. Hetmaniok et al. / International Communications in Heat and Mass Transfer 39 (2012) 30–35

Comparison of the expressions with the same powers of the parameter p gives the following equations:

In next step, the functions uj(x, t), j ≥ 2 are determined recursively by solving the Eq. (15) with boundary condition (Eq. (17)). Finally we obtain:

∂2 u1 1 ∂u0 ∂2 u0 ¼ − 2 a ∂t ∂x2 ∂x

u2 ðx; t Þ ¼

ð14Þ

and:

  2 3 1 t tx x þ ; − − xþ 12 2 2 12 3 4 x x x − þ ; u3 ðx; t Þ ¼ 24 12 24

and ∂2 uj

1 ∂uj−1 ¼ a ∂t ∂x2

for j ≥ 2:

ð15Þ

The above partial differential equations must be supplemented by conditions ensuring a uniqness of the solution. For Eq. (14) we assume the following conditions: 8 < u0 ð0; t Þ þ u1 ð0; t Þ ¼ ψðt Þ;     : u0 xp ; t þ u1 xp ; t ¼ ψp ðt Þ;

∞

1 4 1 2 1 2 x þ tx þ t : 24 2 2

Knowing the exact temperature distribution in the whole area we can easily determine the functions describing the boundary conditions: 2 1 2 þ 2t þ t ; 3 2 4 qðt Þ ¼ − −2t: 3

θðt Þ ¼ ð17Þ

ð18Þ

Knowing the exact temperature distribution u(x, t) or its approximation uˆ n ðx; t Þ, which is defined by continuous functions, missing boundary conditions could be determined: θ(t) = u(b, t), ðb;t Þ or adequately θðt Þ ¼ uˆ n ðb; t Þ, qðt Þ ¼ −k ∂ˆun∂xðb;tÞ. −k ∂u∂x

Thus we find the exact temperature distribution in the whole region under consideration:

j¼0

ð16Þ

In this way, the solution of given problem was brought to the sequence of easy to solve partial differential equations. Looking for the solution of the formulated problem, we need to define yet an initial approximation u0, which we can assume as the function determining the initial condition: u0 ðx; t Þ ¼ φðxÞ:

j ≥ 4:

uðx; t Þ ¼ ∑ uj ðx; t Þ ¼

while for Eq. (15) conditions are in the form (j ≥ 2): 8 < uj ð0; t Þ ¼ 0;   : uj xp ; t ¼ 0:

uj ðx; t Þ ¼ 0;

qðt Þ ¼

4. Computational examples

4.2. Example 2 1 In the next example, we assume: b = 1.2, xp = 1, a ¼ 10 , k = 2, ⁎ t = 2 and: 1−x

φðxÞ ¼ e ; t=10þ1 ψðt Þ ¼ e ; ψp ðt Þ ¼ et=10 : The exact solution of the problem formulated above is defined by functions: uðx; t Þ ¼ e1−xþt=10 ; θðt Þ ¼ eðt−2Þ=10 ; qðt Þ ¼ 2eðt−2Þ=10 :

4.1. Example 1 The application of the method described will be illustrated in the example in which: b = 2, xp = 1, a = 1, k = 1, t * = 2 and: 1 4 x ; 24 1 2 ψðt Þ ¼ t ; 2 1 1 1 2 þ tþ t : ψp ðt Þ ¼ 24 2 2 φðxÞ ¼

As the initial approximation u0 the function that satisfies the initial condition is taken: u0 ðx; t Þ ¼

1 4 x : 24

Solving now Eq. (14) with boundary conditions (Eq. (16)) we find: u1 ðx; t Þ ¼

  2 4 t 1 t x þ x− : þ 24 2 2 24

As the initial approximation u0 we assume the function satisfying the initial condition: u0 ðx; t Þ ¼ e

1−x

:

Solving now Eq. (14) with boundary condition (Eq. (16)) we determine: 1−x

u1 ðx; t Þ ¼ −e

t=10

þe

ðe þ x−exÞ:

Again the functions uj(x, t), j ≥ 2 are determined recursively by solving the Eq. (15) with boundary conditions (Eq. (17)). For example if j = 2 and j = 3 we obtain:  1 t=10  3 2 e ð1−eÞx þ 3ex −ð1 þ 2eÞx ; 6  1 t=10  5 4 3 e 3ð1−eÞx þ 15ex −10ð1 þ 2eÞx þ ð7 þ 8eÞx : u3 ðx; t Þ ¼ 360

u2 ðx; t Þ ¼

In Table 1 errors of retrieving function describing the temperature distribution in the considered area are presented. The Table 2 contains errors of reconstructed missing boundary conditions i.e. functions θ(t) and q(t), for different number of components n (see the relationship (Eq. (12))). The results placed in the paper show, that

E. Hetmaniok et al. / International Communications in Heat and Mass Transfer 39 (2012) 30–35 Table 1 Errors of the reconstructed temperature distribution (Δ — absolute error, δ — relative error). n

Δu

δu

1 2 3 4 5 6 7 8

0.162 0.016 1.638 · 10− 3 1.660 · 10− 4 1.682 · 10− 5 1.704 · 10− 6 1.657 · 10− 7 5.085 · 10− 8

8.714% 0.871% 0.088% 8.948 · 10− 3% 9.067 · 10− 4% 9.186 · 10− 5% 8.931 · 10− 6% 2.741 · 10− 6%

a

33

6.8 × 10−7 6.6 × 10−7 6.4 × 10−7

Δθ

6.2 × 10−7 6. × 10−7 5.8 × 10−7 5.6 × 10−7 0.0

0.5

1.0

1.5

2.0

1.5

2.0

t Table 2 Errors of the reconstructed boundary condition (Δ — absolute error, δ — relative error).

b

2.8 × 10−11

Δθ

δθ

Δq

δq

2.7 × 10−11

1 2 3 4 5 6 7 8

0.180 0.014 1.449 · 10− 3 1.474 · 10− 4 1.495 · 10− 5 1.516 · 10− 6 1.536 · 10− 7 1.556 · 10− 8

19.834% 1.539% 0.160% 0.016% 1.647 · 10− 3% 1.669 · 10− 4 1.692 · 10− 5% 1.714 · 10− 6%

1.995 0.116 0.013 1.276 · 10− 3 1.293 · 10− 4 1.311 · 10− 5 1.328 · 10− 6 1.346 · 10− 7

109.871% 6.408% 0.692% 0.070% 7.123 · 10− 3% 7.219 · 10− 4% 7.314 · 10− 5% 7.411 · 10− 6%

2.6 × 10−11

Δθ

n

2.5 × 10−11 2.4 × 10−11 2.3 × 10−11 0.0

0.5

1.0

t errors decrease rapidly with increase of component number in the sum (Eq. (12)). The graph of errors of boundary condition reconstruction on Γ2 is shown also in Figs. 1 and 2. They present the approximations of orders 5 and 7. The error of the initial condition conformity for 5 and 7-order approximations is shown in Fig. 3. The boundary condition on Γ1 as well as the additional condition for x = xp given in the inverse problem are satisfied precisely. Additionally, same calculations were done for input data burdened with random error of size 0.1, 0.5, 1.0, 2.0 and 5.0%. Because of some

Fig. 2. Errors of reconstructed temperature on the boundary Γ2 (a — for n = 5, b — for n = 7).

requirements of the discussed method, disturbed measurement data were approximated by combination of Trefftz functions for heat conduction equation [37,52,53]. In the next step, just obtained continuous function was used in calculations as the function ψp. In Table 3 one can find the absolute and relative errors of reconstructed temperature distribution and heat flux for different level of

a a

0.000160

0.00002

0.000150

0.000015

Δϕ

Δq

0.000155

0.000145

0.00001

0.000140 5. × 10−6

0.000135 0.0

0.5

1.0

1.5

2.0

0

t

b

0.0

0.2

0.4

0.6

0.8

1.0

1.2

0.8

1.0

1.2

x 1.8 × 10−6

b

1.75 × 10−6 2. × 10−7

1.65 × 10−6

1.5 × 10−7

Δϕ

Δq

1.7 × 10−6

1.6 × 10−6 1.55 × 10−6

1. × 10−7 5. × 10−7

1.5 × 10−6 0.0

0.5

1.0

1.5

2.0

t

0 0.0

0.2

0.4

0.6

x Fig. 1. Errors of reconstructed heat flux on the boundary Γ2 (a — for n = 5, b — for n = 7).

Fig. 3. Error of initial condition conformity (a — for n = 5, b — for n = 7).

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E. Hetmaniok et al. / International Communications in Heat and Mass Transfer 39 (2012) 30–35

Table 3 Errors of the reconstructed boundary condition for the burdened input data (n = 6, Δ — absolute error, δ — relative error).

2.00 1.95

Δθ

δθ

Δq

δq

1.90

0.000179 0.001379 0.002161 0.003764 0.005490

0.019674% 0.151914% 0.238436% 0.414562% 0.604684%

0.001491 0.008687 0.008358 0.013445 0.025805

0.082094% 0.478418% 0.460337% 0.740506% 1.421229%

1.85

q(t)

0.1% 0.5% 1.0% 2.0% 5.0%

a

1.80 1.75 1.70 1.65

Using the homotopy perturbation method the function series is obtained which converges to the solution of discussed problem (with appropriate assumptions). In many cases it is possible to find analytically the sum of the series and in consequence to determine the exact solution of problem. If it is impossible to obtain exact solution by analytical way, one can use a few initial components of the series and build the approximated solution. Because of the fast convergence of the series considered, a few initial components provide a very small error of the reconstructed solution. In the paper an application of the homotopy perturbation method for the solution of inverse heat conduction problem is presented. Problem consists in the calculation of temperature distribution in

θ (t)

0.90 0.85

0.0

0.5

1.0

1.5

2.0

1.5

2.0

t 0.006 0.005 0.004

Δθ

1.0

1.5

2.0

1.5

2.0

t

b 0.020 0.015

0.005 0.0

0.5

1.0

t Fig. 5. Heat flux on boundary Γ2 (a) and error of its reconstruction (b) for input data burned by error 2% (solid line — exact values, doted line — reconstructed values, n = 6).

the domain, as well as in the reconstruction of functions describing the temperature and heat flux on the boundary, when the temperature measurements in the domain are known. The examples presented in the paper confirm utility of the homotopy perturbation method for solving problems discussed. References

1.00 0.95

b

0.5

0.010

5. Conclusion

a

0.0

Δq

input data disturbance and the sixth-order approximate solution. Next, in Figs. 4 and 5 the exact and reconstructed distribution of functions defining boundary conditions are drown as well as an absolute errors of reconstruction in case of input data burdened with error 2.0% and for the sixth-order approximate solution. As can be seen from the results obtained the discussed method is stable with respect to errors of input data. Each time when the input data were burdened with errors, the error of the boundary condition reconstruction did not exceed the error of the input data.

0.003 0.002 0.001 0.0

0.5

1.0

t Fig. 4. Temperature on boundary Γ2 (a) and error of its reconstruction (b) for input data burned by error 2% (solid line — exact values, doted line — reconstructed values, n = 6).

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