Graduate Thesis Or Dissertation
 

Numerical Methods for Upscaling Complex Transport Phenomena

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  • Transport phenomena specially the ones that occur in the various engineering disciplines such as tissue engineering and natural environment are often complex to analyze mostly due to the dynamic and geometric complexities; as such, they have been the subject of an active and intense area of research in past decades. It is often necessary to consider the details of the geometric configuration as well as the time and length scales of the system being investigated in order to accurately analyze such complex systems. The recent exploding advancement in the machine learning algorithms along with exponential growth in computing power are allowing solutions to the complex problems we have not seen before. Today, machine learning algorithms are successfully tailored for clustering, classification, and regression, among other tasks, even for a high-dimensional input data. This dissertation is primarily about two main examples of transport in complex systems (one of which uses machine learning algorithms). (1) The micro-macro transport of reactive chemical species in tissues (and the related upscaling problem), and (2) the problem of near-initial-conditions mass transport in the Taylor tube. These are two representatives that have relevance to many chemical and biological engineering problems, and they serve as archetypes for a much broader class of problems. In the first example, we use deep learning methods to help solve a problem of chemical transport and nonlinear kinetic reaction in tissue engineering. To start, we use averaging methods to filter the transport and reaction process from small scales to larger scales. This defines a sequence of three scales: the cell scale (microscale), the scale of the averaging volume (support scale), and the scale of the application (the macroscale). Because the kinetic reaction rate term is non-linear, the average of the reaction rate term is not the same as the kinetic reaction but with the average concentrations substituted. However, this latter form is the one that is desirable for applications. To correct the errors that occur by using the more convenient representation, the rate term is multiplied by what is conventionally called an effectiveness or correction factor. The nonlinearity in the problem prevents the effectiveness factor from being computed using strictly analytical mathematical techniques. We adopt a deep learning network to compute the effectiveness factor as a function of the macroscale variables that influence it. To accomplish this, we generate a large number of examples to train the deep learning network by varying all of the parameters defining the problem (the parameter space), and computing the microscale reaction rate. This information is used to train the deep learning network. The result is an algorithm that can be used by others to predict the effectiveness factor for similar systems, but without having to solve the microscale problem in detail again. In the second example, we focus on the evolution of solute spreading at early times in a Taylor tube. In particular, for some kinds of initial conditions, the solute evolution may actually exhibit a second moment that decreases in time. Most classical approaches would predict a negative effective hydrodynamic dispersion coefficient for such a situation which is inconsistent with the physics of the problem. We outline a set of four desirable qualities in a well-structured theory of unsteady dispersion as follows: (i) positivity of the dispersion coefficient (ii) non-dependence upon initial conditions, (iii) superposabililty of solutions, and (iv) convergence of solutions to classical asymptotic results. We use averaging to develop an upscaled result that adheres to these qualities. We find that the upscaled equation contains a source term that accounts for the relaxation of the initial configuration. This term decreases exponentially fast in time, leading to correct asymptotic behavior while also accounting for early-time solute dynamics.
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  • Pending Publication
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  • 2021-04-09 to 2021-06-02

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