ABSTRACT
The analysis of diffusion problems in heterogeneous media involves formulations with spatially dependent thermophysical properties in different ways, such as large-scale variations in functionally graded materials (FGM), abrupt variations in layered composites, and random variations due to local concentration fluctuations in dispersed phase systems (Lin 1992, Divo and Kassab 1998, Fudym et al. 2002, Chen et al. 2004, Kumlutas and Tavman 2006, Fang et al. 2009). For instance, composite materials consisting of a dispersed reinforcement phase embedded in a bulk matrix phase have been providing engineers with increased opportunities for tailoring structures to meet a variety of property and performance requirements. As the composite material morphology in the realm of applications presents endless possibilities of design tailoring, manufacturing processes, and even selfstructuring, the characterization of their physical properties is to be made almost case to case (Progelhof et al. 1976, Tavman 1996, Tavman and Akinci 2000, Danes et al. 2003, Kumlutas et al. 2003, Weidenfeller et al. 2004, Serkan Tekce et al. 2007). The usefulness of such materials in heat transfer applications is nevertheless limited by
the precise knowledge of the corresponding thermophysical properties and boundary condition coefficients that are fed into the corresponding models, and quite often need to be determined by the appropriate inverse problem analysis (Flach and Ozisik 1989, Huang and Ozisik 1990, Lesnic et al. 1999, Divo et al. 2000, Huang and Chin 2000, Rodrigues et al. 2004, Huttunen et al. 2006, Huang and Huang 2007). Among the various available solution techniques of inverse problems (Beck and Arnold 1977, Alifanov 1994, Ozisik and Orlande 2000, Kaipio and Somersalo 2004, Zabaras 2006), a fairly common approach is related to the minimization of an objective function that usually involves the quadratic difference
some ations with the inclusion of regularization terms. Although very popular and useful in many situations, the minimization of the least squares norm is a non-Bayesian estimator. A Bayesian estimator is basically concerned with the analysis of the posterior probability density, which is the conditional probability of the parameters given the measurements, while the likelihood is the conditional probability of the measurements given the parameters (Kaipio and Somersalo 2004). This work illustrates the use of Bayesian inference, already discussed in detail in Chapter 12, in the estimation of spatially variable equation and boundary condition coefficients in diffusion problems, by employing the Markov chain Monte Carlo (MCMC) method (Migon and Gamerman 1999, Kaipio and Somersalo 2004, Gamerman and Lopes 2006, Fudym et al. 2008, Orlande et al. 2008, Paez 2010). The Metropolis-Hastings algorithm is applied for the sampling procedure (Metropolis et al. 1953, Hastings 1970), implemented in the Mathematica platform (Wolfram 2005). This sampling procedure used to recover the posterior distribution is in general the most expensive computational task in solving an inverse problem by Bayesian inference, since the direct problem is calculated for each state of the Markov chain. In the context of variable properties identification, the use of a fast, accurate, and robust
computational implementation of the direct solution is extremely important. The accurate representation of the heat conduction phenomena requires a detailed local solution of the temperature distribution, generally with the aid of discrete numerical solutions with sufficient mesh refinement and computational effort and=or semi-analytical approaches for specific or simplified functional forms. Analytical solutions of linear diffusion problems have been analyzed and compiled in Mikhailov and Ozisik (1984), where seven different classes of heat and mass diffusion formulations were systematically solved by the classical integral transform method. The obtained formal solutions are applicable to a very broad range of problems in heat and mass transfer. Later on, the classical integral transform approach gained a hybrid numerical-analytical implementation, referred to as the generalized integral transform technique (GITT) (Cotta and Ozisik 1986, Cotta 1990, 1993, 1998, Cotta and Mikhailov 1997, Cotta and Mikhailov 2006), offering more flexibility in handling nontransformable problems, including among others, the analysis of nonlinear diffusion and convection-diffusion problems. The methodology to be employed here forms the basis of the mixed symbolic-numerical computational code called ‘‘UNified Integral Transforms’’ (UNIT) (Sphaier et al. 2009), which was intended to bridge the gap between simple problems that allow for a straightforward analytical solution, and those more complex and involved situations that would otherwise require expensive commercial software systems. The open source UNIT code is then an implementation and development platform for researchers and engineers interested in the hybrid integral transform solutions of diffusion and convection-diffusion problems. Thus, the integral transformation approach discussed above becomes very attractive for
combined use with the Bayesian estimation procedure, since all steps in the method are determined analytically at once by symbolic computation, and the single numerical repetitive task is the solution of an algebraic matrix eigenvalue problem (Naveira-Cotta et al. 2009). Also, instead of seeking the function estimation in the form of a sequence of local values for the variable coefficients, an alternative path is followed based on the eigenfunction expansion of the coefficients to be estimated themselves (Naveira-Cotta et al. 2009, 2011a,b), and then seeking the estimation of the corresponding series coefficients. Another novel aspect in the present work is the alternative analysis of the inverse problem
in the transformed temperature field instead of employing the directly measured temperature data along the domain (Naveira-Cotta et al. 2011b). Thus, the experimental temperature
values of increasing order. This procedure is particularly advantageous when a substantial amount of experimental measurements are available, such as in thermographic sensors, permitting a remarkable data compression after the integral transformation process. Typical applications were selected to illustrate the robustness of the proposed combin-
ation of methodologies related to the estimation of thermal properties in two-phase dispersed media, complementing in scope what has been discussed in Chapters 1 and 2. Simulated experimental results were used in inverse analysis allowing for the inspection of the identification problem behavior in terms of the parameters to be estimated.
