Diego C. Knupp

Unified Integral Transforms Algorithm for Solving Multidimensional Nonlinear Convection-Diffusion Problems

The present work summarizes the theory and describes the algorithm related to an open-source mixed symbolic-numerical computational code named unified integral transforms (UNIT) that provides a computational environment for finding hybrid numerical-analytical solutions of linear and nonlinear partial differential systems via integral transforms. The reported research was performed by employing the well-established methodology known as the generalized integral transform technique (GITT), together with the symbolic and numerical computation tools provided by the Mathematica system. The main purpose of this study is to illustrate the robust precision-controlled simulation of multidimensional nonlinear transient convection-diffusion problems, while providing a brief introduction of this open source implementation. Test cases are selected based on nonlinear multidimensional formulations of Burgers’ equation, with the establishment of reference results for specific numerical values of the governing parameters. Special aspects in the computational behavior of the algorithm are then discussed, demonstrating the implemented possibilities within the present version of the UNIT code, including the proposition of a progressive filtering strategy and a combined criteria reordering scheme, not previously discussed in related works, both aimed at convergence acceleration of the eigenfunction expansions.

Heat Transfer in Microchannels with Upstream–Downstream Regions Coupling and Wall Conjugation Effects

Heat transfer in microchannels is analyzed, including the coupling between the regions upstream and downstream of the heat transfer section and taking into account the wall conjugation and axial diffusion effects which are often of relevance in microchannels. The methodology is based on a recently proposed single-domain formulation for modeling the heat transfer phenomena simultaneously at the fluid stream and the channel walls, and applying the generalized integral transform technique (GITT) to find a hybrid numerical–analytical solution to the unified partial differential energy equation. The proposed mathematical model involves coefficients represented as space-dependent functions, with abrupt transitions at the fluid–wall interfaces, which carry the information concerning the transition of the two domains, unifying the model into a single-domain formulation with variable coefficients. Convergence of the proposed eigenfunction expansions is thoroughly investigated and the physical analysis is focused on the effects of the coupling between the downstream and the upstream flow regions.

Experimental Identification of Thermophysical Properties in Heterogeneous Materials with Integral Transformation of Temperature Measurements from Infrared Thermography

This work deals with the experimental estimation of spatially variable thermal conductivity and diffusivity in heterogeneous media, with temperature measurements obtained via infrared thermography being used in the inverse analysis. The direct problem solution for a one-dimensional heat conduction experiment is analytically obtained via integral transforms, and the related eigenvalue problem is solved by the generalized integral transform technique. The inverse problem is handled by Bayesian inference through a Markov chain Monte Carlo algorithm. The functional representation and estimation is based on the eigenfunction expansion of the thermal conductivity and diffusivity themselves, and the unknown parameters become the corresponding expansion coefficients. The inverse analysis is performed on the transformed experimental temperature field instead of employing the actual local temperature measurements, thus promoting a significant data reduction through the integral transformation of the experimental measurements. A demonstration experiment is built involving partially heated thin plates made of bakelite and polystyrene, including a variable thickness plate to simulate spatially variable thermophysical properties.

Space-variable thermophysical properties identification in nanocomposites via integral transforms, Bayesian inference and infrared thermography

Simultaneous estimation of space-variable thermal conductivity and heat capacity in heterogeneous samples of nanocomposites is dealt with by employing a combination of the generalized integral transform technique (GITT), for the direct problem solution, Bayesian inference as implemented with the Markov chain Monte Carlo (MCMC) method, for the inverse analysis and infrared thermography, for the temperature measurements. Another aspect of the proposed approach is the integral transformation of the thermographic experimental data along the space variable, which allows for a significant data compression since the inverse analysis is undertaken within the transformed field. Results are presented for the covalidation of the experiment with a homogeneous polyester plate, as well as for a plate made of polyester–alumina nanoparticles composite with abrupt variation of the filler concentration.

Conjugated Heat Transfer in Heat Spreaders With Micro-Channels

This work is aimed at the experimental verification of a recently proposed single domain formulation of conjugated conduction-convection heat transfer problems, which are often of relevance in thermal micro-systems analysis. The single domain formulation simultaneously models the heat transfer phenomena at both the fluid streams and the channels walls by making use of coefficients represented as space variable functions with abrupt transitions occurring at the fluid-wall interfaces. The Generalized Integral Transform Technique (GITT) is then employed in the hybrid numerical-analytical solution of the resulting convection-diffusion problem with variable coefficients. The considered experimental investigation involves the determination of the temperature distribution over a heat spreader made of a nanocomposite plate with a longitudinally molded single micro-channel that exchanges heat with the plate by flowing hot water at an adjustable mass flow rate. The infrared thermography technique is employed to analyze the response of the heat spreader surface, aiming at the analysis of micro-systems that provide a thermal response from either their normal operation or due to a promoted stimulus for characterization purposes.Copyright

An inverse analysis of the radiative transfer in a two-layer heterogeneous medium

The radiative transfer in multi-layer composite media has numerous applications, for example, in regional and global climate models, Solar System bodies research, Earth remote sensing, and multi-layer clouds studies, among others. In this study, we focus on the inverse analysis of the radiative transfer problem in a two-layer plane-parallel medium. For the direct problem solution, we use the well-known Chandrasekhars discrete ordinates method combined with the finite difference method. We are interested in the estimation of the scattering and absorbing coefficients using the measured data of the emerging radiation at both boundary surfaces of the medium and also at the interface between the two layers. The inverse problem is implicitly formulated and the minimization of the defined objective function is achieved with the Levenberg–Marquardt method. The solutions obtained are investigated in the face of the sensitivity analysis.

Inverse Problem of an Anomalous Diffusion Model Employing Lightning Optimization Algorithm

The classical diffusion equation models the behavior of several physical phenomena related to dispersion processes with great success. However, in some cases this approach does not represent the actual physical behavior. In most published works dealing with this situation, the well-known second-order parabolic equation is assumed to be the basic equation of the dispersion process, but the anomalous diffusion effect is modeled with the introduction of fractional derivatives or the imposition of a convenient variation of the diffusion coefficient with time or concentration. Alternatively, Bevilacqua et al. (Process Ann Braz Acad Sci 83:1443–1464, 1977) developed a new analytical formulation for the simulation of the diffusion phenomena with retention. Its purpose is to reduce all retention diffusion processes to a unifying phenomenon that can properly simulate the retention effect. This model may have relevant applications in different areas, such as population dispersion with partial retention of the population to ensure territorial dominance, chemical reactions inducing adsorption processes, and multiphase flow through porous media. In this chapter we are interested in the formulation and solution of the inverse problem involving the anomalous diffusion phenomenon. A two-step solution strategy for the inverse problem is employed because of the correlation between the model parameters, and it is used the Lightning Optimization Algorithm (LOA), a new heuristic that makes an analogy to the phenomena of atmospheric discharges.

A Hybrid Estimation Scheme Based on the Sequential Importance Resampling Particle Filter and the Particle Swarm Optimization (PSO-SIR)

Particle filters are recursive Bayesian estimators, which are being applied to many areas of engineering in recent years to estimate states and parameters, regarding fire spread, tumors, oil pipelines, heat transfer, chemical reactors, etc. The key idea behind particle filters is that they use an initial distribution (sample), based on the previous state estimate, to calculate the best estimate for the current state, relying only on the current available measurements and the knowledge about the system. The greatest advantage of these methods is the easy computational implementation. However, setting the standard deviation for the initial distribution is very important for the success of the method. For this reason, standard formulation of these methods may not provide good results in problems with large discontinuities (or irregular/abrupt changes). For example, this would be the case of estimating step changes in the heat flux on a plate. Although several solutions have been proposed to improve the estimation performance, they still suffer from the curse of discontinuity. This occurs because particle filters proposed in the literature are not adaptive methods. In the example mentioned above, particle filters can have both a priori information and sample satisfactory before the change. However, after the change begins, the available information could be not enough to draw a suitable sample for the estimation. At this point, it is necessary to modify the standard deviation to broaden the particle search field or to move the a priori information to a new region where a new sample should be drawn. In this regard, the aim of this chapter is to propose a hybrid estimation scheme based on Particle Swarm Optimization (PSO) built into the particle filter Sampling Importance Resampling (SIR) to project the a priori information to a new search region, according to the current observation. To demonstrate the proposal, the problem of estimating step changes on the heat flux on a plate is taken into account, considering experimental measurements. The results allow to state that the scheme combining PSO and SIR provides good performance for this type of problem.

Transformação Integral e Problemas Inversos aplicados à Estimativa de Parâmetros em um Modelo 2D de Transporte de Sedimentos Não-Coesivos

Neste trabalho apresenta-se uma abordagem de solucao hibrida analitico-numerica com a Tecnica da Transformada Integral Generalizada (GITT) para solucao de um problema bidimensional transiente do transporte de sedimentos nao-coesivos. Foram aplicados filtros analiticos para homogeneizar os contornos e melhorar a convergencia. A equacao foi transformada em um sistema acoplado de equacoes diferenciais parciais unidimensional que foi resolvido numericamente empregando recursos do Software Wolfram Mathematicar 10. Os resultados obtidos pela GITT se mostraram coerentes e apresentaram ganhos computacionais, o que e desejavel para a solucao do problema inverso de identificacao de parâmetros hidrodinâmicos e de transporte, utilizando o metodo de Monte Carlo via Cadeias de Markov.

Radiative Properties Estimation and Construction of Confidence Regions with a Combination of the Differential Evolution Algorithm and the Likelihood Method.

This work is aimed at the combination of the Differential Evolution algorithm and the likelihood method for the estimation of radiative properties and construction of confidence regions of the parameters estimates. Two cases with different levels of measurement error are employed, and the results indicate that the approach is adequate for the construction of confidence regions in the radiative transfer inverse problem considered. The results also demonstrate that with increasing measurement errors the traditionally employed elliptical confidence region might lead to poor approximations in this problem.