A two-dimensional inverse problem in imaging the thermal conductivity of a non-homogeneous medium

doi:10.1016/S0017-9310(00)00044-2

International Journal of Heat and Mass Transfer

Volume 43, Issue 22, 15 November 2000, Pages 4061-4071

https://doi.org/10.1016/S0017-9310(00)00044-2 Get rights and content

Abstract

A two-dimensional inverse heat conduction problem is solved successfully by the conjugate gradient method (CGM) of minimization in imaging the unknown thermal conductivity of a non-homogeneous material. This technique can readily be applied to medical optical tomography problem. It is assumed that no prior information is available on the functional form of the unknown thermal conductivity in the present study, thus, it is classified as the function estimation in inverse calculation. The accuracy of the inverse analysis is examined by using simulated exact and inexact measurements obtained on the medium surface. The advantages of applying the CGM in the present inverse analysis lie in that the initial guesses of the unknown thermal conductivity can be chosen arbitrarily and the rate of convergence is fast. Results show that an excellent estimation on the thermal conductivity can be obtained within a couple of minutes CPU time at Pentium II-350 MHz PC. Finally the exact and estimated images of the thermal conductivity will be presented.

Introduction

A two-dimensional inverse heat conduction problem (IHCP) is examined in the present study by the conjugate gradient method to estimate the thermal conductivity of a non-homogeneous medium. In addition, temperature readings using infrared scanners taken at some appropriate locations and time on the medium surface are also considered available.

Numerous engineering and mathematical researchers have considered problems equivalent to estimating the thermal conductivity. For instance, Huang and Ozisik [1], [2] used direct integration and Levenberg–Marquardt methods to estimate thermal conductivity and heat capacity simultaneously; Beck and Al-Araji [3] determined the constant thermal conductivity, heat capacity and contact conductance at one time; Terrola [4] used Davidon–Fletcher–Powell method to determine temperature-dependent thermal conductivity. All the above references belong to parameter estimations, i.e., the functional form for the unknown quantities should be assigned before the inverse calculations. However, when the thermal conductivity of a non-homogeneous or composite material is to be estimated, then from parameter estimation, it is difficult to achieve the goal, especially for multi-dimensional problems. Thus, function estimation with conjugate gradient method (CGM) [5] should be used in this inverse heat conduction problem to estimate the unknown thermal conductivity.

Recently, Huang and Yuan have developed an efficient function estimation algorithm based on the CGM in determining the thermal properties of the material in one-dimensional inverse problems. For example, Huang and Yuan [6] used the CGM in estimating temperature-dependent thermal conductivity. Huang and Yuan [7] used the same technique to estimate the temperature-dependent heat capacity per unit volume. Finally, Huang and Yuan [8] determined simultaneously the temperature-dependent thermal conductivity and heat capacity per unit volume.

The estimation of the thermal properties for the multi-dimensional problems is very limited in the literature. The purpose of the present study is to extend our previous algorithm to a two-dimensional IHCP to estimate the spatial and time-varying thermal conductivity in a non-homogeneous medium.

The CGM derives basis from the perturbation principle [5] and transforms the direct problem to the solution of two other related problems in the inverse analysis, namely, the sensitivity problem and the adjoint problem, which will be discussed in details in text.

Once this technique is established, it can also be applied to many applications in estimating the diffusion coefficients, such as the medical optical tomography problem [9], [10], since the governing equations for those fields are very similar or even identical (for example in Ref. [10]) to the present study.

Section snippets

Direct problem

To illustrate the methodology of the present two-dimensional IHCP in determining unknown spatial and time varying thermal conductivity $k x, y, t$ in a non-homogeneous medium, we consider the following transient heat conduction problem.

A square plane $Ω$ with side length L̄ is initially at temperature $T ̄ x ̄, y ̄, 0 = T ̄_{0}$ . For time $t ̄ >0$ , the boundary surfaces along $x ̄ =0$ and $y ̄ =0$ are subjected to a prescribed constant heat flux $q ̄_{1}$ and $q ̄_{3}$ , respectively, while along boundary surfaces $x ̄ = L ̄$ and $y ̄ = L ̄$ , a

Inverse problem

For the inverse problem, the thermal conductivity $k x, y, t$ is regarded as being unknown, but , , , , , quantities in are known. In addition, temperature readings using infrared scanner taken at some appropriate grid locations and time on the medium surface are also considered available.

We assumed that the temperatures obtained from infrared scanner at the grid point are used to identify $k x, y, t$ in the inverse calculations. Let the temperature reading taken at these grid points over the time

Conjugate gradient method for minimization

The following iterative process based on the CGM [5] is now used for the estimation of $k x, y, t$ by minimizing the above functional $J k x, y, t$ $k ̂^{n+1} x, y, t = k ̂^{n} x, y, t −β^{n} P^{n} x, y, t for n=0, 1, 2,…$ where βⁿ is the search step size in going from iteration n to iteration n+1, and $P^{n} x, y, t$ is the direction of descent (i.e., search direction) given by $P^{n} x, y, t =J′^{n} x, y, t +γ^{n} P^{n−1} x, y, t$ which is a conjugation of the gradient direction $J′^{n} x, y, t$ at iteration n and the direction of descent $P^{n−1} x, y, t$ at iteration n−1. The conjugate

Sensitivity problem and search step size

The sensitivity problem is obtained from the original direct problem defined by , , , , , in the following manner. It is assumed that when $k x, y, t$ undergoes a variation $Δ k x, y, t, T x, y, t$ is perturbed by $Δ T x, y, t$ . Then in the direct problem, replacing k by k+Δk and T by T+ΔT, subtracting from the resulting expressions the direct problem and neglecting the second-order terms, the following sensitivity problem for the sensitivity function ΔT is obtained: $∂ ∂x k x, y, t ∂ Δ T x, y, t ∂x + ∂ ∂y k x, y, t ∂ Δ T x, y, t ∂y + ∂ ∂x Δ k x, y,$

Adjoint problem and gradient equation

To obtain the adjoint problem, Eq. (1a) is multiplied by the Lagrange multiplier (or adjoint function) $λ x, y, t$ and the resulting expression is integrated over the correspondent time and space domains. Then the result is added to the right-hand side of Eq. (2) to yield the following expression for the functional $J k x, y, t$ : $J k x, y, t =∫^{t_{f}}_{t=0} ∑ i=1 I ∑ j=1 I T_{i,j} −Y_{i,j}^{2} d t+∫^{L}_{x=0} ∫^{L}_{y=0} ∫^{t_{f}}_{t=0} λ x, y, t ∂ ∂x k x, y, t ∂T x, y, t ∂x + ∂ ∂y k x, y, t ∂T x, y, t ∂y − ∂T x, y, t ∂t d t d x d y$ The variation ΔJ is obtained by perturbing k by Δk and T by ΔT in

Stopping criterion

If the problem contains no measurement errors, the traditional check condition is specified as $J⌊ k ̂^{n+1} x, y, t ⌋<ε$ where ε is a small-specified number. However, the observed temperature data may contain measurement errors. Therefore, we do not expect the functional equation (2) to be equal to zero at the final iteration step. Following the experience of the authors [5], [6], [7], [8], we use the discrepancy principle as the stopping criterion, i.e., we assume that the temperature residuals may be

Computational procedure

The computational procedure for the solution of this inverse problem may be summarized as follows:

Suppose $k ̂^{n} x, y, t$ is available at iteration n.

Step 1. Solve the direct problem given by , , , , , for $T x, y, t$ .
Step 2. Examine the stopping criterion given by Eq. (12) with ε given by Eq. (14). Continue if not satisfied.
Step 3. Solve the adjoint problem given by , , , , , for $λ x, y, t$ .
Step 4. Compute the gradient of the functional $J′ x, y, t$ from Eq. (11).
Step 5. Compute the conjugate coefficient γⁿ and

Results and discussions

To demonstrate the validity of the present CGM in predicting $k x, y, t$ for a two-dimensional non-homogeneous material from the knowledge of transient temperature recordings on the medium surface, we consider two specific examples where a drastic change of the thermal conductivity is considered.

The objective of this article is to show the accuracy of the present approach in estimating $k x, y, t$ with no prior information on the functional form of the unknown quantities, which is the so-called function

Conclusions

The CGM with adjoint equation was successfuly applied for the solution of the inverse problem to determine the non-homogeneous thermal conductivity. Several test cases involving different functional forms of non-homogeneous thermal conductivity and measurement errors were considered. The results show that the CGM does not require a priori information for the functional form of the unknown quantities and needs very short CPU time at Pentium II-350 MHz PC to perform the inverse calculations.

Acknowledgements

This work was supported in part through the National Science Council, Taiwan, ROC, Grant number NSC-89-2611-E-006-004.

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View full text

A two-dimensional inverse problem in imaging the thermal conductivity of a non-homogeneous medium

Abstract

Introduction

Section snippets

Direct problem

Inverse problem

Conjugate gradient method for minimization

Sensitivity problem and search step size

Adjoint problem and gradient equation

Stopping criterion

Computational procedure

Results and discussions

Conclusions

Acknowledgements

Int. J. of Heat and Fluid Flow

Int. J. Heat and Mass Transfer

A direct integration approach for simultaneously estimating temperature dependent thermal conductivity and heat capacity

Numerical Heat Transfer, Part A

Investigation of a new simple transient method of thermal property measurement

ASME J. Heat Transfer

A method to determine the thermal conductivity from measured temperature profiles

Int. J. Heat Mass Transfer

Solution of an inverse problem of heat conduction by iteration methods

J. of Engineering Physics

The function estimation in predicting temperature dependent thermal conductivity without internal measurements, AIAA

J. Thermophysics and Heat Transfer