Heat Source Estimation with the Conjugate Gradient Method in Inverse Linear Diffusive Problems

Heat Source Estimation with the Conjugate Gradient Method in Inverse Linear Diffusive Problems

Jian Su

Nuclear Engineering Program, COPPE

Universidade Federal do Rio de Janeiro

Caixa Postal 68509

21945-970 Rio de Janeiro, RJ. Brazil

[e-mail]

Antônio J. Silva Neto

Instituto Politécnico

Universidade do Estado do Rio de Janeiro

Caixa Postal 97282

28601-970 Nova Friburgo, RJ. Brazil

[e-mail]

Nuclear Engineering Program, COPPE

Universidade Federal do Rio de Janeiro

Caixa Postal 68509

21945-970 Rio de Janeiro, RJ. Brazil

[e-mail]

In this work, we present the solution of a class of linear inverse heat conduction problems for the estimation of unknown heat source terms, with no prior information of the functional forms of timewise and spatial dependence of the source strength, using the conjugate gradient method with an adjoint problem. After describing the mathematical formulation of a general direct problem and the procedure for the solution of the inverse problem, we show applications to three transient heat transfer problems: a one-dimensional cylindrical problem; a two-dimensional cylindrical problem; and a one-dimensional problem with two plates.

Keywords: Inverse problem, heat conduction, conjugate gradient method, adjoint problem

Introduction

The inverse analysis of heat transfer problems has received a lot of attention recently with several relevant applications in engineering, with the design of thermal equipment, systems and instruments ranking high among them.

Inverse heat conduction problems are known to be ill-posed (Tikhonov and Arsenin, 1977; Beck et al., 1985), in contrast to the direct heat conduction problems, which are well-posed, that is, there exists an unique solution which is stable to small changes in the input data.

A variety of numerical and analytical techniques have been proposed for the solution of inverse problems. Alnajem and Özisik (1986) used a direct analytic approach, Murio (1993) developed the mollification method, Frankel and Keyhani (1997) proposed a method in which a global basis temperature representation is made, Dorai and Tortorelli (1997) applied unconstrained optimisation algorithms utilising analytically determined first and second order sensitivities, Al-Khalidy (1998) used a hyperbolic equation as a stabilisation method, Mikhalev and Reznik (1988) used a gradient algorithm with the search direction given by the Davidson-Fletcher-Powell method.

Alifanov suggested and developed the iterative regularisation method (Alifanov 1974; Alifanov et al., 1995). In this approach an optimisation problem is solved, with the conjugate gradient method for example, being the gradient equation determined by an adjoint problem (Jarny et al., 1991). This method has been applied with success in a variety of inverse heat transfer problems (Alifanov and Egorov, 1985; Huang and Özisik, 1992; Silva Neto and Özisik, 1994a; Huang et al., 1999; Colaço and Orlande,1999; Su and Silva Neto, 2001).

Most of the work in the solution of multidimensional inverse heat conduction problems are devoted to wall heat flux or surface temperature estimation, in Cartesian coordinates (Kerov, 1984; Alifanov and Nenarokomov, 1993; Woodbury and Thakur, 1993; Huang and Wu, 1994).

Even though the estimation of internal heat source strength and/or location has a pratical strong appeal in areas such as chemical and mechanical engineering, it has been observed that less effort has been dedicated to the solution of this problem in comparison with other inverse heat conduction problems. This is probably due to the complexity of the problem at hand. Borukhov and Kolesnikov (1988) developed an analytical solution for a certain class of such problems. Silva Neto and Özisik (1992, 1993a, 1993b, 1994b) have applied with success Alifonov's iterative regularisation method, here called conjugate gradient method with an adjoint equation (CGMAE), to the estimation of heat source intensities.

In the present work we extend a previous work by Silva Neto and Özisik (1994b), considering a class of inverse linear heat conduction problems for the estimation of unknown heat source terms, g(x, t), with no prior information on the functional forms of timewise and spatial dependence of the source strength using the CGMAE. In a recent paper, Silva Neto and Su (1999) have considered the one-dimensional case for fuel elements with cladding. Here we present a unified formalism for one and two dimensional cylindrical coordinates as well as a two slab composite medium in cartesian coordinates. To the best of our knowledge such problems and unified formalism have not been presented before.

In the next sections the mathematical formulations of the direct and inverse problems are presented, as well as the results for some test cases.

Nomenclature

c_p = specific heat

e = random number

g = heat source intensity

G = heat source intensity divided by the thermal conductivity

h = heat transfer coefficient

J = squared residue functional

k = thermal conductivity

L = linear operator

M = number of temperature sensors

n = iteration index

n = outward normal vector

P = direction of descent or linear

operator

R = Radius of a rod

r = radial coordinate

S = source term of the adjoint problem

t = time

T = temperature

x = spatial coordinate

x = spatial coordinate vector

Superscripts

' = gradient

* = adjoint operator

Subscripts

0 = initial condition

f = final time

1,2 = different boundaries

m = temperature sensor

f = fuel

c = cladding

g = gap

w = external cladding surface

ref = reference

¥ = surrounding fluid condition

Greek letters

a = thermal diffusivity

b = step size

g = conjugate coefficient

G = boundary

e = real valued parameter

l = adjoint function

r = density

s = standard derivation of measurement errors

w = weighting function

W = spatial domain

Ѳ = laplacian operator

Mathemical Formulation

We consider transient heat conduction in a constant thermal property material with a uniform initial temperature distribution, and adiabatic or convective boundary conditions. This linear diffusive problem is described by the following partial differential equation

defined on the domain W x [0, t_f], subject to appropriate boundary and initial conditions

where T is the temperature, G(x, t) is a generic source term proportional to the volumetric rate of heat generation, [0, t_f] is the interval of observation, h is the convective heat transfer coefficient with the ambient around the medium at temperature T_¥, G = G₁ + G₂ is the boundary of the spatial domain W and n is the outward normal vector. The linear differential operator is defined as

where a (= k /r c_p) is the thermal diffusivity of the material, with k being the thermal conductivity, r the density, and c_p the specific heat.

When the operator L, the geometry, the initial condition, the boundary condition, material properties k and a and the heat source term G are known, the temperature distribution, T(x, t), can be calculated. Problem (1) - (3) is then called the direct problem. On the other hand, when any of these information, or a combination of them, is unknown, but the field T(x, t) is known somewhere in the space-time domain an estimation of the unknown quantities may be attempted. This is known as the inverse problem.

We consider here the estimation of the heat source strength, g(x, t), from temperature measurements taken at the boundary and interior of the medium, using Alifanov's iterative regularisation method(CGMAE). For implementation of this method, the following steps are required: (i) the conjugate gradient method of minimisation; (ii) the sensitivity problem and (iii) the adjoint problem and the gradient equation, and (iv) the stopping criterion. A brief description of each of these steps will now be provided. Afterwards a solution algorithm will be presented, linking these basic steps.

The Conjugate Gradient Method of Minimisation

The inverse problem at hand, classified as a function estimation in inverse input data estimation, is solved as an infinite dimensional optimisation problem where we search in a space of functions for the solution estimate that minimises the functional

where T_mand Z_m are the calculated and measured temperatures, respectively, M is the total number of temperature sensors considered, x_m represents their spatial position, w(x_m) a weighting function, and [0, t_f] is the interval of time in which experimental data are taken.

We now consider the conjugate gradient iterative procedure to obtain the estimate for G(x, t)that minimises the functional J(x, t) (Polak, 1971),

where n is the iteration index, bⁿ is the step size in going from step n to step n+1, and P(x, t)ⁿ is the direction of descent defined as

where J'_G is the gradient of the functional and gⁿ is the conjugate coefficient.

There are several ways of computing the conjugate coefficient, being the most frequently used

The step size, bⁿ, is determined by minimising J[G(x, t)^n-+1], yielding (Silva Neto and Moura Neto, 1999).

For the implementation of the iterative procedure described here the sensitivity term (P(x, t)ⁿ) and the gradient J'_G(x, t) are required. The former is determined with the sensitivity problem and the latter with the adjoint problem. Silva Neto et. al. (1999) have demonstrated that the adjoint problem is adjoint to the sensitivity problem. Both problems will be presented next.

The Sensitivity Problem

By forcing a small perturbation on the heat source in the direct problem, i.e. G ® G +, a small perturbation on the temperature is expected, T ® T + . Subtracting the original problem from the perturbed one the following sensitivity problem is obtained

The Adjoint Problem and the Gradient Equation

The adjoint operator L* is defined in an appropriate space of functions, V, such that for a function l that belongs to this space,

where U is the space of functions that satisfy problem (9).

Using the standard L² inner product we get

After integrating by parts(what is meant when operator L is moved from the right entry to the left entry in the inner product given by Eq. (10)), and using the initial and boundary conditions of the sensitivity problem, we conclude that the adjoint operator is

and V is the space of functions that satisfy

Let us look now at the determination of the gradient J'_G of the functional given by Eq.(4), using a one parameter family G^Î

where x_m defines the location of the temperature sensors. Differentiating Eq.(14) with respect to Î and setting Î = 0, we obtain the directional derivative of the functional G in the direction

where(x,t) is the solution of the sensitivity problem.

Denote by P and P* the linear operators that give the solutions of the sensitivity and adjoint problems respectively. Therefore

and

where

From the definition of gradient,

Comparing Eqs.(17) and (19) we conclude that

where the adjoint problem is

with the boundary condition and final conditions given by Eqs.(13a) through (13c).

The Stopping Criterion

As the experimental data in real applications is always noisy, there is no reason to require that the residue between the calculated and measured temperatures become smaller than a number related to the experimental error (Engl et al., 1996, Silva Neto and Moura Neto, 1999). This approach known as discrepancy principle is now described following the work of Alifanov(1974). Let the standard deviation of the measurement errors, s , be the same for all sensors and measurements. Therefore,

Introducing Eq.(22) into Eq.(4), we calculate a lower acceptable threshold for the functional

The iterative procedure is interrupted when

The Solution Algorithm

We will now describe how to link the several steps described above:

1 - Choose an initial guess G(x,t)⁰, say, G (x,t)⁰ = constant. Set n = 0;

2 - Solve the direct problem, Eqs.(1)-(3), to obtain T(x,t);

3 - Solve the adjoint problem, Eqs.(21) and (13a-c), to obtain l(x,t)

4 - Compute the gradient, J'_G(x,t)ⁿ, with Eq.(20);

5 - Compute the conjugate coefficient, gⁿ, with Eq.(7);

6 - Compute the direction of descent, P(x,t)ⁿ with Eq.(6);

7 - Solve the sensitivity problem, Eqs.(9a-c), with the source term given by (x,t) = P(x,t)ⁿ to obtain;(x,t);

8 - Compute the step size, , with Eq.(8);

9 - Compute a new estimate, G(x,t)ⁿ⁺¹, with Eq.(5);

10 - Interrupt the iterative procedure if the stopping criterion, Eq. (24), is satisfied.

Otherwise,increase n by 1 and go back to Step 2.

Results and Discussion

In order to investigate the behavior of the conjugate gradient method with an adjoint equation presented in the previous section we have considered the estimation of heat sources with spatial and timewise dependence in three inverse heat conduction problems.

In all cases presented, the simulated transient temperature, Z_m,j, containing measurement errors are generated by adding random errors to a computed exact temperature, T_m,j

where s is the standard deviation of measurement errors, which is assumed to be the same for all measurements, and e_m,j is a normally distributed random error, M is the total number of sensors and N the total number of measurements taken with each sensor in the time interval [0,tf]. For normally distributed error, there is a 99% probability of the value of e_m,jlying in the range

One-Dimensional Rod

We consider a one-dimensional rod with a radius R, initially at a uniform temperature T₀, containing a volumetric heat source g(r, t) with r being the spatial coordinate. The efficiency of the conjugate gradient method for the estimation of an unknown volumetric heat source strength is examined for several test cases by using simulated transient temperature readings, Z_m(t).

In Figs. 1 and 2 we examine the effects of the number of sensors on the estimation of a heat source with a 1/8 cosine radial dependence and a linearly decreasing time dependence reaching zero at t=0.5t_f The solid lines in Figs. 1 and 2 represent the exact values for the unknown strength of the heat source.

In Fig. 1(a), as well as in Fig. 2(a), the estimates at t = 0.1t_f and 0.3t_f are shown, for three different values of the standard deviation of the measurement errors, s = 0.0, 0.02, and 0.05, corresponding, respectively, to measurement errors of 0%, 5% and 13% with respect to the largest value of measured temperatures.

In Fig. 1(b), as well as in Fig. 2(b), the estimates obtained for two different radial positions, r = 0.2R and r are shown, with the same values for the same standard deviations of measurement errors used before.

The results shown in Fig. 1 were obtained with 3 temperature sensors evenly distributed in the medium. A good agreement between exact and estimated values for the heat source strength is observed.

In Fig. 2 the results are shown for the same test case considered in Fig 1, but now using 6 sensors evenly distributed .

In this test case where the unknown heat source strength presents a smooth dependence along the radial coordinate, an excess of temperature sensors causes a degradation on the estimates. In the example presented here, better estimates are obtained with 6 temperature sensors than with 3 temperature sensors.

From a practical point of view, the number of temperature sensors should be kept at a minimum.

In cases where sharp variations of the heat source strength with the spatial coordinate is observed, Silva Neto and Ozisik (1994b) have shown that a temperature sensor should be located at each point where a discontinuity on the heat source strength spatial derivative takes place.

Two-Dimensional Problem in Cylindrical Coordinate System

In Fig. 3 we present the estimates for a heat source with a sinusoidal azimuthal variation. As in the examples previously presented, a linearly decreasing timewise dependence was considered. The solid lines in Fig. 3 represent the exact value for the unknown strength of the heat source.

As no a priori assumption is made on the functional form of the azimuthal and radial dependence, an extremely large number of temperature sensors (81) was required in order to obtain fairly reasonable estimates. As mentioned before, this would be of little practical interest. In such a case, one should try to approximate the heat source with known functions, such as polynomials or periodic functions, leaving only a finite number of coefficients to be recovered.

As a matter of fact strong azimuthal variations such as those presented in Fig. 3 are not expected in real applications.

As our main interest is to investigate the behaviour of the metodology presented in a wide range of applications, we have considered such severe test cases.

One-Dimensional Problem - Two Plates

Here we consider the estimation of intensity of heat sources imbeded in one plate (fuel) using temperature measurements taken at a second plate coupled to the first one (cladding) with no internal sources in it. In this case a slightly different formulation is required because two media are involved, i.e., heat conduction takes place through two plates coupled through a gap modelled as a thermal resistance.

The mathematical formulation of this transient heat conduction problem is given as

where k and a are the thermal conductivity and diffusivity, respectively, h_g is the heat transfer coefficient within the gap, h_w is the convection heat transfer coefficient from the cladding external surface to the fluid around it, and the subscripts f and c represents fuel and cladding, respectively.

In Figs. 4-6 we examine the effects of the number of sensors on the estimation of the heat source in the fuel g(x, t) using temperature measurements only in the cladding or in both fuel and cladding. The solid lines in Figs. 4-6 represent the exact value for the heat source.

The results are shown for the estimation of a heat source with a parabolic spatial dependence and a linearly decreasing temporal dependence, with g = 0 for t ³ 0.5 t_f. In Fig. 4 are shown the results obtained using just two temperature sensors in the cladding, one on each side of it, i.e. x = b and x = c. The standard deviations s = 0, 0.02 and 0.05 correspond to experimental errors of 0%, 5% and 13%, respectively.

If we keep increasing the number of temperature sensors in the cladding, we keep getting increasingly better estimates, even at early times such as t = 0.2 t_fNonetheless, a strong deviation is always observed at x = a.

In Fig. 5 results are presented for the same test case, using five temperature sensors at the boundaries and in the interior of the cladding. The experimental errors are the same as those considered before for the test cases with one plate or rod.

In all test cases considered, the temperature profiles are spatially concave down. In addition to that, the temperature information gets to the sensors located in the cladding only after undergoing a further decrease due to the gap thermal resistance. As the test functions are spatially concave up, it is not surprising that such difficulties come up mainly at the external surface of the fuel, i.e. x = a. To further clarify this matter, we stress that the estimates for the heat source intensity are directly related to the solution of the adjoint problem, that is very similar to the direct problem whcih provides the temperature profiles. Therefore, a spatially concave down temperature profile implicates on an adjoint function with the same behaviour, what may lead to the algorithm difficulty in recovering the right tendency mainly near the location x = a.

We have tried a few test cases with spatially concave down functions and the estimates improved throughout the spatial domain, but discrepancies could still be observed, and once again mainly on the vicinity of x = a. This fact demonstrates the effects of the gap thermal resistance. With increasing values of the heat transfer coefficient, h_g, the estimates get increasingly better. But one has to keep in mind that in all these attempts we are trying to estimate the spatial dependence of the heat source in the fuel with no temperature sensors located there. Another aspect that can be predicted just by looking at the adjoint problem boundary conditions at x = b and x = c, is that for ratios k_c / k_fsignificantly different from unity difficulties may be expected if only cladding temperature measurements are considered. Numerical experiments have confirmed this behaviour.

In Fig. 6 results are presented for the same test case considered before, using this time two sensors in the cladding, in x = b and x = c, and eleven sensors evenly distributed within the fuel. The standard deviations s = 0, 0.02 and 0.05 correspond to experimental errors of 0%, 5% and 13% respectively.

We observe here that the inclusion of temperature measurements obtained with temperature sensors located inside the fuel did not improve the estimates.

Two comments are in order. The first comment is that in real applications the cladding may be thin, it may not be possible to use more than two or three temperature sensors in the cladding (boundaries included). Difficulties of practical order may also come up if attempts are made to locate temperature sensors within the fuel. In short, all efforts (mathematical and experimental) should be directed towards the use of only two temperature sensors located somewhere in the cladding.

The other aspect to be considered in such situations is whether non-uniqueness becomes a problem. If two or more different heat sources in the fuel could yield the same temperature profile in the cladding, with temperature measurements taken only at the cladding, there would be no way to tell them apart.

Conclusions

The results presented here are part of an ongoing research project that aims a comprehensive analysis of heat source in diffusive processes through the solution of inverse heat transfer problems.

Aiming the goal of equipment design, as well as operation surveillance, we are examining the feasibility of considering only temperature sensors located at external boundaries. In such a case the non-uniqueness of the inverse problem becomes a real concern.

It is worthy to mention here that the weighting factor w(x) included in the squared residue functional was instrumental in obtaining good quality estimations. This fact deserves a better investigation and that will be done in a future work .

We must stress that in the solution of the inverse problems presented here we deal with more unknowns (the source intensity at each node of the computational mesh) than available experimental data (transient temperature measurements taken at a limited number of location). For the smoothly varying functional dependence with space we are able to get reasonable estimates for the heat source.This is not the case if the heat source presents strong variations. It seems though that such strong variations are not expected in most real applications. This subject deserves also futher investigation.

Acknowledgements

The authors wish to acknowledge Professors F.D.Moura Neto and Y. Jarny for helpful discussions, and students J.L.M.A.Gomes, C.R.Regis, and F.C.Rezende for assistance in computational work.

Article received: February, 2000. Technical Editor: Átila P. S. Freire

This paper has been presented at the 1st International Conference on Engineering Thermophysics, August, 1999, Beijing, China (Su and Silva Neto, 1999).

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Publication Dates

• Publication in this collection
  19 Aug 2002
• Date of issue
  2001

History

• Received
  Feb 2000