Inverse estimation of surface heating condition in a three-dimensional object using conjugate gradient method

https://doi.org/10.1016/j.ijheatmasstransfer.2010.02.048Get rights and content

Abstract

Temperature and heat flux on inaccessible surfaces can be estimated by solving an inverse heat conduction problem (IHCP) based on the measured temperature and/or heat flux on accessible surfaces. In this study, the heat flux and temperature on the front (heated) surface of a three-dimensional (3D) object is recovered using the conjugate gradient method (CGM) with temperature and heat flux measured on back surface (opposite to the heated surface). The thermal properties of the 3D object are considered to be temperature-dependent. The simulated measurement data, i.e., the temperature and heat flux on the back surface, are obtained by numerically solving a direct problem where the front surface of the object is subjected to high intensity periodic laser heat flux with a Gaussian profile. The robustness of the formulated 3D IHCP algorithm is tested for two materials. The effects of the uncertainties in thermophysical properties on the inverse solutions are also examined. Efforts are made to reduce the total number of heat flux sensors on the back surface required to recover the front-surface heating condition.

Introduction

High-Energy Laser (HEL) weapons can remotely deliver high-power laser at the speed of light onto a military target. It is critical to know the transient of temperature in the target in order to accurately assess the resulting thermomechanical response. However, the conventional temperature sensors cannot be used to directly measure the surface temperature since the sensors can be easily destroyed or interfere with the laser beam. Similar problems can be found during reentry of a space vehicle into the atmosphere as well as in high-power laser manufacturing processes [1]. For these circumstances, however, the heated surface temperature can be determined indirectly by solving an inverse heat conduction problem (IHCP) [2], [3], [4] based on the transient temperature and/or heat flux measured on the back surface.

To formulate the IHCP, either temperature or heat flux at some locations should be measured to provide information for solving the ill-posed problem. Between them, temperature is often preferred because it can be measured with less uncertainty compared to the heat flux [5], [6], [7], [8]. Recent studies, however, have shown that using the measured heat flux as additional information in an IHCP can reduce the proneness to the inherent instability of the ill-posed problem [9], [10].

Although the IHCPs have been extensively studied for different applications in the past decades (e.g., [11], [12], [13], [14], [15], [16], [17], [18], [19]), little work has been done for the inverse numerical algorithm using heat flux measurement data in the objective functional. Furthermore, in HEL weapon applications, the laser energy may be delivered to the surface in a periodic way because of the target-spinning or atmosphere variations. Since the formulation of the IHCP is quite subjective, it is necessary to determine which formulation is more appropriate for applications with a periodic heat flux that may pose extra difficulties in the solution of the inverse problems.

Recently, the authors proposed a stable 1D IHCP formulation to reconstruct the front-surface heating condition with back-surface measurement data [20]. After an optimal investigation on the choice of the boundary condition and objective function variable, it was demonstrated that the most accurate solution can be obtained by choosing the front-surface heat flux as an unknown function, using the temperature measurement data as the boundary condition at the back surface, and employing the heat flux measurement data in the objective function. In Ref. [20] thermophysical properties were assumed to be constant. In reality, those properties could vary with temperature during a high-power laser interaction.

The 1D IHCP model can be applied to the situation that the flat-top laser beam diameter is much larger than the thickness of the target. For the case that the laser beam profile is Gaussian and/or the laser diameter is comparable to the thickness of the target, the temperature distribution in the target is 3D. The objective of this paper is to develop a 3D IHCP formulation that can accurately recover the front surface temperature based on measured temperature and heat flux on the back surface for a target subjected to a periodic heat flux on the front surface and with temperature-dependent thermophysical properties.

Section snippets

Model description

To illustrate the methodology of the inverse heat transfer algorithm employed in this study, a three-dimensional object shown in Fig. 1 is considered. Initially, the object is under a uniform temperature T0 and then is subjected to a high intensity, Gaussian laser beam qlaser on the front surface from t = 0+. The purpose of this study is to demonstrate the effectiveness and accuracy of the proposed IHCP formulation in reconstructing the observed heat flux q1(y,z,t) and temperature T1(y,z,t)

Computational procedure

The solution procedure of the IHCP above using the CGM is summarized as follows. Start with an initial guess q10(y,z,t) for q1(y, z, t), set k = 0, and then perform the steps below:

  • -

    Step 1. Solve the direct problem given by Eqs. (2), (3), (4), (5), (6), (7) for T(x, y, z, t) based on the value q1k(y,z,t).

  • -

    Step 2. Check the stopping criterion Eq. (45). Stop the iteration if satisfied; otherwise, continue the following solution procedure.

  • -

    Step 3. Solve the adjoint problem given by Eqs. (29), (30), (31),

Generation of simulated measurement data

Instead of conducting actual experiment, the measurement data of temperature and heat flux are generated numerically from solving the direct problem described by the governing Eq. (2) with initial condition and boundary conditions at side surfaces given by Eqs. (3), (6), (7) and the following boundary conditions on front and back surfaces-k(T)Tx=qlaser-h(T-T)-εσ(T4-T4)forx=0,t>0k(T)Tx=h(T-T)+εσ(T4-T4)forx=L,t>0where qlaser is the heat flux imposed on the front surface, which is assumed

Conclusions

A conjugate gradient method (CGM) algorithm is presented to recover the heat flux and temperature at the front (heated) surface of a 3D object with temperature-dependent thermophysical properties, based on the temperature and heat flux measurements at the back surface (opposite to the heated surface). The inverse problem is formulated in such a way that the front-surface heat flux is chosen as the unknown function to be recovered, and the front surface temperature is computed as a by-product of

Acknowledgement

The authors would like to thank the Test Resource Management Center (TRMC) Test and Evaluation/Science & Technology (T&E/S&T) Program for their support. This work is funded by the T&E/S&T Program through the US Army Program Executive Office for Simulation, Training and Instrumentation’s contract number W900KK-08-C-0002. The authors would also like to express their gratitude to Dr. James L. Griggs for his valuable discussions.

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