Contemporary analysis and design of engineering systems is generally based on a mathematical model of the system. Because of this, the approach to solution is greatly influenced by (i) the background we have been taught in engineering mathematics and (ii) the methods of solution that have evolved for the differential and integral equations that describe the behavior of engineering systems.

We are generally taught throughout our exposure to mathematical modeling and analysis of engineering systems the following:

- One and only one boundary condition or initial condition must be prescribed for each order of derivative in each dimension of a differential equation describing a system
- The number of equations available must be equal to the number of unknowns being sought.

These constraints are not necessary for solution of equations unless we use only conventional methods. Imposing these constraints causes us to approach engineering design from a direction that is often less than optimal. Eliminating these constraints allows more accurate analysis and design of thermal systems, and also allows the designer more freedom in design, the possibility of increased creativity, and the discovery of unique designs that could not be found through conventional analysis.

Consider the following design problem. A utility boiler is to be constructed. To simplify piping and water wall design, it is desired to have uniform temperature and heat flux on all of the boiler water walls. Where should the burners be placed in the furnace to best achieve this result?

To design such a system, the engineer would probably specify the desired temperature of the constant-temperature surfaces and all of the surface and medium physical and transport properties. Then, experience would be used to generate an initial estimate of the burner characteristics and locations. Commercial codes or specific codes generated for solving this problem then predict the heat flux distribution on the furnace walls. In this conventional or forward approach, the geometry, properties, standard boundary conditions (one on each surface), and governing equations are all specified. All major CFD/heat transfer codes require input of this type, and specification of one boundary condition for each variable on each boundary element and one condition in each volume element. This automatically results in having an equal number of equations and unknowns so that standard solution methods for systems of equations can be employed.

If the predicted boundary heat flux distribution is unsatisfactory, then the engineer must try new burner locations, and rerun the analysis. This trial and error approach continues until a satisfactory solution is found, or until money and patience are exhausted. The best solution is chosen from among the design sets that have been specified and solved. It is possible that no acceptable solution exists for such a design problem, and the desired conditions must be modified.

A better approach might be to specify the distributions of *both* the desired boundary temperature and heat flux on the water walls, and then solve directly for the necessary heat source distribution in the furnace. The governing equations are the same as for conventional design, but the specified boundary conditions are not. This is the inverse design approach. Why do not we approach design in this way, since clearly it is better to specify the desired outcome and solve directly for the required design that will achieve it? First, because we have been taught that we must not specify two conditions on a boundary. In addition, we find that it is quite possible, depending on the prescribed volume and surface discretization, that setting up the inverse equations in this way can result in many more unknowns than equations or, in other situations, the number of equations may exceed the number of unknowns. These inequalities are another thing we have been warned against. Indeed, if we do set up an inverse solution and then apply standard matrix solution techniques to the resulting set of equations, we will find that solution will quite often be extremely difficult or impossible. The equation set is found to be ill conditioned because the matrix of coefficients of the equation set is near-singular or singular. In such a case, conventional matrix inversion techniques and iterative solvers will fail.

The advantages of finding an inverse design solution for such a problem are great, if it can be achieved. Inverse design avoids expensive conventional iterative solutions, provides an optimal solution rather than the best solution from among the solutions for a limited number of design sets generated by trial and error, and may generate solutions that are not intuitively obvious and would not result from the trial and error approach.

In some cases, inverse solutions are forced on the engineer. Experimental observations of temperature or heat flux may not be available at the physical location where they are needed, but are taken at remote locations. Radiative property distributions in a participating medium must often be obtained from remote measurements. These situations belong to the class of inverse problems. Solution is difficult because the governing equations 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 in space and time. Various methods may be applied for overcoming the ill-posed nature of the governing equations. 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; Morozov, 1995; Özişik and Orlande, 2000). Tikhonov (1963) and Phillips (1962) are often credited with developing the first systematic treatment for these types of inverse conduction problems.

For thermal systems dominated by radiative transfer, the problem is even more 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 same set of equations that are found for the forward problem when one boundary condition is set for each surface. However, when the inverse problem is formulated, some of the equations in the set take the form of Fredholm equations of the first kind, which are notoriously ill conditioned (Wing, 1991; Hansen, 1998). In addition, 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. Reviews of inverse methods for radiative transfer systems (França et al., 2002; Daun et al., 2003a,b; Daun and Howell, 2005) cover the work of many contributors.

Analytical inverse techniques used for radiative systems are similar to those used in inverse data analysis, property determination, and remote measurement. Data analysis problems should have a single solution; i.e., we do not 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. 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.

Consider the annealing furnace below, where a billet of metal is to be heated according to a specified temperature recipe. Once the temperature history is prescribed, a first law energy balance on the billet imposes a second boundary condition because the required heat flux is specified by the heat capacity, volume, density, and transient temperature trajectory of the billet (see Fig. 1). The designer may therefore prescribe both the required radiative heat flux *q*_{1}(*x*_{1},t) and temperature distribution *T*_{1}(*x*_{1},t) for design surface 1, and seek to find the energy input distributions required on the heater (upper) surface 2 that will provide the desired result. As the allowable tolerances on *q*_{1}(*x*_{1},t) and *T*_{1}(*x*_{1},t) are relaxed, multiple allowable solutions for the heater temperature distribution *T*_{2}(*x*_{
2},t) may result. For the designer, such multiple acceptable solutions are desirable, since they allow the designer to choose among the solutions based on considerations such as smoothness and ease of implementation.

**Figure 1. Enclosure with two boundary conditions specified on surface 1.**

It is possible in design problems to specify conditions on the design surface for which no acceptable physical solution for *T*_{2}(*x*_{2}) exists, that is, the designer may specify design surface characteristics of *q*_{1}(*x*_{1}) and *T*_{1}(*x*_{1}) 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 must have produced the observed experimental data.

A direct or explicit solution to inverse problems requires the use of an inverse formulation. Because inverse problems are inherently ill posed, they may not possess a mathematical or physically reasonable solution. The corresponding discretized set of equations may be ill conditioned, in that the matrix of coefficients defining the solution may be singular or near-singular. Ordinary techniques for solving the discretized set of integral equations (e.g., Gauss-Seidel, Gauss elimination, or lower-upper (LU) decomposition) are likely to result in solutions to the prescribed inverse problem that have negative absolute temperatures, heat sinks rather than sources, or other conditions that are physically impossible, cannot be obtained in practice, or are very undesirable in a practical design. Sometimes, these results come about because the designer has imposed unrealistically tight requirements on the comparison between the inverse solution and the imposed conditions. In that case, a physically real solution may be salvaged by allowing somewhat greater disagreement between the imposed conditions and the predicted results from the inverse solution. (After all, a 5% variation from the imposed conditions may be quite acceptable in a design because there may and easily be that much uncertainty in properties or other factors used in the system model.) However, there is no guarantee that a solution exists to a particular inverse design problem.

Radiant systems involve the inverse solution of integral equations rather than the differential equations that describe 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.

The question of uniqueness is often brought up with regard to inverse solutions of experiment-based problems. Is it not possible to find many solutions that satisfy the imposed conditions? The answer is often yes, and this bedevils the researcher who is using inverse methods to determine physical properties or boundary conditions at a remote location. How can the researcher know whether a found solution reflects the real physical value that is being sought by inverse solution?

The presence of multiple solutions is another reason that inverse design differs, at least philosophically, from inverse solutions applied to experiments. The designer using inverse methods is happy to have multiple solutions, so long as they satisfy the prescribed design set within some prescribed error bounds and are physically attainable. Multiple solutions allow the designer a choice, and the solution that is cheapest and easiest to implement can be chosen from among them. These solutions may have much different shapes and/or magnitudes from the solutions that provide only slightly better agreement with the desired input design conditions. The designer may pose a problem in which the conditions on the design surface are prescribed (given temperature and heat flux distributions, for example). If the designer also specifies the system geometry in such a way that the heaters to be designed have small influence on the design surface, then the inverse solution may appear to give poor results. A wide range of solutions for the heater characteristics may provide almost the same conditions on the design surface, because the influence of the heaters is small. This result is not a failure in inverse design; rather, it indicates that the imposed system characteristics were poorly thought out. The characteristics of an inverse solution provide guidance to the designer in such a case, but as with all design problems, some careful preliminary analysis can eliminate some less-than-useful effort. The designer or analyst still needs to bring experience and understanding to bear whether forward or inverse design is to be used.

If radiation is the sole heat transfer mechanism, and the medium if present is gray, the resulting system of equations is linear. In most practical situations, however, the resulting system of equations is nonlinear, as happens with combined heat transfer problems and with nongray media and surfaces. In these cases, the system of equations will be both nonlinear and ill conditioned. Both the first and fourth powers of the unknown temperatures arise in the numerical discretization of the system, resulting in a highly nonlinear system of equations. The same is true for the case where the thermal properties are dependent on the unknown temperatures. Modeling a nongray medium where the absorption coefficient is dependent on the wavelength of radiation is a special case because the local mean radiative properties then depend on the temperature.

The equation set describing the physics of the problem can be arranged in many ways. The portions of the equation set that are ill conditioned require the most care in solution, and can often be identified by careful thought on the part of the designer. The equations can then be set up in such a way that the inverse portion of the solution is required for only a subset of the entire equation set. After inverse solution of the subset, the remaining equations may be solved by more conventional schemes.

Most inverse solution methods were developed for systems of linear discrete equations, and the mathematical basis and proofs are based on the assumption of linearity. As noted, interesting engineering design problems involve mixed-mode heat transfer are highly nonlinear because of the fourth-power dependence of radiation coupled with first-power temperatures and their derivatives through the advection and conduction terms. There is little guidance from the mathematical literature on inverse solution of these problems.

As with forward solution of nonlinear problems, inverse solutions are also almost always iterative. Linearization of the equations describing a thermal system provides an obvious approach to solving inverse problems, since the linear inverse solvers can be applied iteratively. However, care must be used in this approach. Some linearization procedures may cause the form of the resulting linearized equations to introduce solution-dependent terms into the coefficient matrix **A**, i,e., **A** = **A**(**x**). This means, for example, that expensive inverse operations would have to be performed at each iteration step, making the inverse solution quite costly. Thus, wherever possible, it is desirable to arrange the linearized equation into the form

(1) |

by placing the **x**-dependent terms on the right-hand side of the equation, and leaving the coefficients in **A** as constants.

The highly nonlinear nature of mixed-mode heat transfer relations causes well-known convergence difficulties, even for forward solutions. Relaxation factors must often be introduced to prevent divergence of the solutions during iterative solution, and the same is true for inverse solutions.

Within these constraints, there is still wide latitude in solution approaches, as well as a number of potential pitfalls in arranging the solution algorithm. For either forward or inverse solutions, the system of equations must be arranged in such a way that the guessed terms are not the dominant information of the system. For instance, if radiation is the dominating heat transfer mechanism, then the conduction and convection terms should form the guessed part of the system, not the contrary. In matrix representation, the system of equations becomes

(2) |

where matrix **A** and vector **b** are dependent on the unknown vector **x**.

Remaining sections of this portion of *Thermopedia* describe solution methods for inverse problems, and applications of these methods to treatment of engineering inverse problems.

#### REFERENCES

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

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

Beck, J. V., Blackwell, B., and St. Clair, Jr., C. R., Inverse Heat Conduction: Ill-Posed Problems, Wiley, Hoboken, NJ, 1995.

Daun, K. J. and Howell, J. R., Inverse design methods for radiative transfer systems, *J. Quant. Spectrosc. Radiat. Transfer*, vol. **93**, no.1-3, pp. 43–60, 2005.

Daun, K. J., Erturk, H., and Howell, J. R., Inverse design methods for high-temperature systems, *Arabian J. Sci. and Tech*, vol. **27**, no. 2C, pp. 3–48, 2003a.

Daun, K. J., Erturk, H., Gamba, M. Hosseini Sarvari, M., and Howell, J. R., The use of inverse methods for the design and control of radiant sources, *JSME Int. J., Ser. B*, vol. **46**, no. 4, pp. 470–478, 2003b.

França, F. H. R., Howell, J. R., Ezekoye, O. A., and Morales, J. C., Inverse design of thermal systems, *Advances in Heat Transfer,* vol. **36**, J. P. Hartnett and T. F. Irvine (eds.), Elsevier, New York, pp. 1–110, 2002,

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

Morozov, V. A., Methods for Solving Incorrectly Posed Problems, Springer-Verlag, New York, 1984.

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

Phillips, D. L., A treatment for the numerical solution of certain integral equations of the first kind, *J. Assoc. Comput. Mach*., vol. **9**, pp. 84–97, 1962.

Tikhonov, A. N., Solution of Incorrectly Formulated Problems and the Regularization Method, *Sov. Math. Dokl.,* vol. **4**, 1035–1038, 1963; Engl. trans., *Dokl. Akad. Nauk. SSSR*, vol. **151**, pp. 501–504, 1963.

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