Mark A. Stadtherr
The focus of our research is on the development and application of strategies for reliable engineering computing. In many applications of interest in chemical engineering it is necessary to deal with nonlinear models of complex physical phenomena, on scales ranging from the macroscopic to the molecular. Frequently these are problems that require solving a nonlinear equation system (algebraic and/or differential) or finding the global optimum of a nonconvex function. Thus, the reliability with which these computations can be done is often an important issue. For example, if there are multiple solutions to the model, have all been located? If there are multiple local optima, has the global optimum been found? If there are uncertain parameters and/or initial conditions in a dynamic model, have the effects of these uncertainties been rigorously quantified. The goal is to develop the tools needed to resolve these issues with mathematical and computational certainty, thus providing a degree of problem-solving reliability not available when using standard methods, and to apply these tools to problems of interest.
In recent years, our group has shown that strategies based on the use of interval mathematics can be used to reliably solve a wide variety of global optimization and nonlinear equation solving problems of interest in chemical and biomolecular engineering, including: 1) Computation of fluid phase equilibrium from activity coefficient models, cubic equation-of state (EOS) models, and statistical associating fluid theory (SAFT); 2) Calculation of critical points from cubic EOS models; 3) Location of azeotropes and reactive azeotropes; 4) Computation of solid-fluid equilibrium; 5) Parameter estimation using the standard least squares and error-in-variables approaches; 6) Calculation of adsorption in nanoscale pores from lattice density function theory models. In each case, the interval mathematics approach provides a mathematical and computational guarantee either that all solutions have been located in a nonlinear equation solving problem or that the global optimum has been found in an optimization problem. Since, in some cases, this approach is computationally intense, strategies that take good advantage of parallel computing architectures are also of significant interest.
Some problems of current interest in the group include: 1) Verified solution of uncertain dynamic systems, i.e., problems in which there are uncertainties in model parameters and/or initial conditions, including applications in ecology, physiology, epidemiology and chemical engineering; 2) Parameter estimation in modeling of phase equilibrium, including the implications of using locally vs. globally optimal parameters in subsequent computations; 3) Location of equilibrium states and bifurcations of equilibria in ecosystem models used to assess the risk associated with the introduction of new chemicals into the environment; 4) Molecular modeling, including transition state analysis and the calculation of molecular conformations; and 5) Global optimization, including problems involving dynamic models. Also of special interest (in collaboration with Professor Joan Brennecke and the Notre Dame Energy Center) are modeling problems that arise in the development of sustainable, energy-efficient and environmentally-conscious processing technology, in particular the use of supercritical carbon dioxide and room-temperature ionic liquids as environmentally-benign replacements for traditional organic solvents.
Recent Publications -- Find here a list of recent publications. Also find here links to abstracts and preprints of selected recent publications.
Recent Presentations -- Find here a list of recent presentations. Also find here links to materials available from recent presentations (no downloadable material earlier than late-1997).
Courses -- Find here links to information about courses in current and future semesters.
E-Mail: markst (at) nd.edu
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