Inverse radiation problems

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It is usually stated that to solve any mathematical problem described by a differential equation, we must have the number of boundary conditions equal to the order of the PDE that is to be solved; in addition, if the problem is transient in nature, an initial condition must also be prescribed. Further, the number of unknowns being sought must be equal to the number of available equations. For integral equations, the boundary conditions are usually implicit in the limits of the integrals.

Determining the heat shield hot surface temperature Ts(t) and heat flux q"s(t) from measurement of remote temperature and heat flux (T(t) and Failed to parse (lexing error): q"(t) )
Figure 1: Determining the heat shield hot surface temperature Ts(t) and heat flux q"s(t) from measurement of remote temperature and heat flux (T(t) and q"(t))

Suppose, however, that these conditions are not met; for example, more than the required number of boundary conditions are prescribed on some surface or surfaces, and it is desired to find the missing boundary conditions on other surfaces. In some cases of this type, the number of equations may not match the number of unknowns. These problems fall under the class of inverse problems, because the requirements for standard solution are not met. Problems of this type may be very sensitive to the imposed values of the given boundary conditions, and solutions may vary wildly. This type of problem is referred to as ill-posed.

Inverse solutions in heat transfer may be required when experimental observations of temperature or heat flux are not available at the physical location where they are needed, or radiative property distributions in a participating medium must be obtained from remote measurements. For example on the re-entry heat shield of a spacecraft (Fig. 1), it may be impossible to measure temperature Ts(t) or heat flux q"s(t) on the ablating surface of the shield, but transient temperatures T(t) and heat fluxes q"(t) can be measured on the interior surface. Can these remote measurements be used to determine the unknown temperature and heat flux on the heat shield surface? Intuitively, it would seem so: It isn't so easy! The governing equations for inverse problems tend to be mathematically ill-posed, and predicting conditions on the remote boundary can result in multiple solutions, physically unrealistic solutions, or solutions that oscillate greatly in space and time. In this Section, we examine why these difficulties arise, and show one solution method for handling them.

Various methods may be applied for overcoming the ill-posed nature of the governing equations. Tikhonov (1963) and Phillips (1962) are often credited with developing the first systematic treatment for these types of inverse problems. For problems dominated by conduction, there are texts and monographs available that demonstrate many of these methods (Tikhonov, 1963; Alifanov, 1994; Alifanov et al. 1995; Beck et al., 1995; Özişik and Orlande, 2000).

For determining radiative properties from remote measurements, various inverse techniques have been employed. This type of problem is required in analyzing temperature and soot distributions in flames, accounting for atmospheric attenuation effects in remote sensing from satellites, and is also closely related to problems in x-ray tomography.

Aside from determining conditions at an inaccessible boundary or radiative properties, inverse problems arises in the design and control of thermal systems. In these problems, the designer specifies the desired output of the thermal system being designed; in most cases, this is a desired temperature and heat flux distribution over a product located on a design surface. The designer must then predict the necessary energy inputs to the thermal system that will produce the desired distributions over the design surface. For example, a semiconductor wafer of known properties is to be heated following a specified time-temperature curve, while holding the entire wafer at uniform temperature at every instant to avoid thermal stresses or deformation. Thus, the local temperature and heat flux necessary to heat at the prescribed rate are specified at every time and position on the wafer: Two boundary conditions on a single boundary! The unknown inputs may be the required temperature and energy input distributions to radiant heaters, or heater locations, or oven geometry.

For thermal systems dominated by radiative transfer, the problem is complicated because the thermal input at any location on the design surface may be affected by some or all radiant sources in the system, depending on the presence of blocking or shading. The mathematical form of the inverse solution is the set of algebraic or integral equations derived in Section 10.2.4 that must be solved simultaneously. For the inverse problem, the missing boundary conditions cause some of the integrals to be Fredholm integral equations of the first kind, which are notoriously ill-conditioned Wing (1991), Hansen (1998). In addition, depending on how the design surface and the heater surfaces are subdivided, the number of unknowns in the governing radiative exchange equations may be less than, equal to, or greater than the number of equations that describe the system. These factors imply that design and control of distributed radiative sources may be difficult, especially in problems where both a transient temperature and heat flux distribution are prescribed over the design surface, and where other heat transfer modes play a role. The traditional design strategy has been to use trial-and-error solutions with guidance by experience to attain a viable solution. However, this can lead to sub-optimal solutions and can be quite time consuming. Some problems are simply not amenable to trial-and-error approaches.

Reviews of inverse methods for design of radiative transfer systems are in França et al. (2002); Daun et al. (2003; 2006). These discuss various methods for solving inverse radiation problems, and cover the work of many contributors.


Alifanov, O.M., 1994, Inverse Heat Transfer Problems, Springer, Berlin.

Alifanov, O.M., Artyukhin, E.A., and Rumyantsev, S.V., 1995, Extreme Methods for Solving Ill-Posed Problems with Applications to Inverse Heat Transfer Problems, Begell House, New York, NY.

Beck, J.V., Blackwell, B., and St. Clair, Jr., C.R., 1995, Inverse Heat Conduction: Ill-Posed Problems, Wiley-Interscience, New York, NY.

Daun, K.J., Erturk, H., Gamba, M. Hosseini Sarvari, M. and Howell, J.R., 2003, “The Use of Inverse Methods for the Design and Control of Radiant Sources,” JSME International Journal, Series B, 46, no. 4, 470-478, 2003.

Faghri, A., Zhang, Y., and Howell, J. R., 2010, Advanced Heat and Mass Transfer, Global Digital Press, Columbia, MO.

França, F.H.R., Howell, J.R., Ezekoye, O.A., and Morales, J.C., 2002, “Inverse Design of Thermal Systems,” Chap. 1, Advances in Heat Transfer, J.P. Hartnett and T.F. Irvine, eds., Elsevier, Vol. 36, 1-110.

Hansen, P.C., 1998, Rank-Deficient and Discrete Ill-Posed Problems: Numerical Aspects of Linear Inversion, SIAM, Philadelphia, PA.

Özişik, M.N. and Orlande, H.R.B., 2000, Inverse Heat Transfer: Fundamentals and Applications, Taylor and Francis, New York, NY.

Phillips, D.L., 1962, “A Treatment for the Numerical Solution of Certain Integral Equations of the First Kind,” J. Assoc. Computing Machinery, Vol. 9, pp. 84-97.

Tikhonov, A.N., Solution of Incorrectly Formulated Problems and the Regularization Method, Soviet Math. Dokl., 4, 1035-1038, 1963,. [Engl. trans. Dokl. Akad. Nauk. SSSR, 151, 501-504].

Wing, G.W., 1991, A Primer on Integral Equations of the First Kind, SIAM, Philadelphia, PA.

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