Inverse radiation design problem
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Analytical techniques used for inverse design and control of radiative systems are similar to those used in inverse data analysis, property determination, and remote measurement, but there are significant differences in how the techniques are implemented. The latter types of problem should have a single solution; i.e., we don’t expect a material to have multiple properties that give the same measurements, nor multiple temperatures or heat fluxes at a remote boundary that give the same measurements at an accessible location. However, design problems may allow significantly wider tolerances in specification of acceptable results. In design, solving the inverse problem may produce multiple solutions that fall within the allowable tolerances but are very different in form.
Figure 1 shows the usual two-dimensional enclosure. Consider an annealing furnace, where a billet of metal is to be heated to a specified temperature. The heat capacity, volume and density of the billet then impose the required heat flux. As the allowable tolerances on q"1(x1) and T1(x1) are relaxed, multiple allowable solutions for the heater power distribution may be found. Multiple acceptable solutions are desirable, as they allow the designer to choose among the solutions based on considerations such as smoothness and ease of implementation.
It is possible in design problems to specify conditions on the design surface for which no acceptable physical solution on Surface 2 exists. The designer may specify design surface characteristics of q1"(x1) and T(x1) that cannot be obtained (at least within acceptable error limits around the desired distributions) by any distribution of heater settings. Just because the designer wants a particular outcome, there is no a priori guarantee that it can be obtained! This is in contrast to data analysis problems, where in most cases a feasible solution to the inverse problem is known to exist because some set of physical variables gave rise to the measured experimental data.
Finally, designing radiant systems involves inverse solution of integral equations rather than the differential equations that arise in applications involving other heat transfer modes. Since there are few radiation heat transfer problems where conduction and convection can be completely neglected, the governing energy relations are often highly nonlinear integral-differential equations.
Direct Inverse Solutions: A direct or explicit solution to inverse problems requires use of an inverse formulation. Inverse problems are inherently ill-posed, and the corresponding discretized set of equations is ill-conditioned (i.e., the matrix of coefficients in the discretized solution is singular or near-singular). Ordinary techniques for solving the discretized set of integral equations (e.g., Gauss-Seidel, Gauss elimination or LU decomposition) are likely to either identify non-physical solutions with high amplitude fluctuations and/or imaginary absolute temperatures, or may completely fail to find a solution. Before considering direct inverse solution techniques, the ill-conditioned behavior of a system of equations of the type arising in radiative systems can be diagnosed by carrying out a singular value decomposition (SVD) (Hansen, 1998). The SVD of an arbitrary MN matrix A is A = USVT, where U and V are orthogonal matrices and S is the diagonal matrix of singular values, and the elements Si,j of the S matrix are arranged so that S1,1 > S2,2 > . > SN,N >= 0. The inverse of A is then given by A − 1 = VS'UT, where Si,j = 1 / Si,j. If the condition number of this matrix (S1,1 / SN,N) is large, small singular values dominate the inverse matrix and the solution becomes unstable. If A is rank-deficient, some of the singular values equal zero and the conventional inversion process fails completely. Most contemporary mathematical packages (Matlab, Mathcad, etc.) contain commands for extracting U, V, and S and the singular values.
References
Faghri, A., Zhang, Y., and Howell, J. R., 2010, Advanced Heat and Mass Transfer, Global Digital Press, Columbia, MO.
Hansen, P.C., 1998, Rank-Deficient and Discrete Ill-Posed Problems: Numerical Aspects of Linear Inversion, SIAM, Philadelphia, PA.
Wing, G.W., 1991, A Primer on Integral Equations of the First Kind, SIAM, Philadelphia, PA.