Welcome From The Chair

Message From The Director of Graduate Admissions

Faculty

Mark Stadtherr, Professor

Education
B.Ch.E., University of Minnesota, Minneapolis (1972)
Ph.D., Chemical Engineering, University of Wisconsin, Madison (1976)

Professional Experience
Assistant Professor, University of Illinois, Urbana (1976-1982)
Associate Professor, University of Illinois, Urbana (1983-1995)
Professor, University of Notre Dame (1996-present)

Research Interests

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? 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. Since, in some cases, this approach is computationally intense, strategies that take good advantage of parallel computing architectures are also of significant 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. Some problems of current interest in the group include: 1) Parameter estimation in modeling of phase equilibrium, including the implications of using locally vs. globally optimal parameters in subsequent computations; 2) Nonlinear dynamics, in particular the 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; 3) Molecular modeling, including transition state analysis and the calculation of molecular conformations; 4) Validated solution of systems of ordinary differential equations (ODEs); and 5) Global optimization for parameter estimation in dynamic models (ODE or DAE). Also of special interest (in collaboration with Professor Joan Brennecke) are modeling problems that arise in the development of sustainable, 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.

Publications

Gang Xu, William D. Haynes and Mark A. Stadtherr. Reliable Phase Stability Analysis for Asymmetric Models. Fluid Phase Equilibria, 235:152-165, 2005. view abstract // link A deterministic technique for reliable phase stability analysis is described for the case in which asymmetric modeling (different models for vapor and liquid phases) is used. In comparison to the symmetric modeling case, the use of multiple thermodynamic models in the asymmetric case adds an additional layer of complexity to the phase stability problem. To deal with this additional complexity we formulate the phase stability problem in terms of a new type of tangent plane distance function, which uses a binary variable to account for the presence of different liquid and vapor phase models. To then solve the problem deterministically, we use an approach based on interval analysis, which provides a mathematical and computational guarantee that the phase stability problem is correctly solved, and that thus the global minimum in the total Gibbs energy is found in the phase equilibrium problem. The new methodology is tested using several examples, involving as many as eight components, with NRTL as the liquid phase model and a cubic equation of state as the vapor phase model. In two cases, published phase equilibrium computations were found to be incorrect (not stable).

Youdong Lin, C. Ryan Gwaltney and Mark A. Stadtherr. Reliable Modeling and Optimization for Chemical Engineering Applications: Interval Analysis Approach. Reliable Computing, 12:427-450, 2006. view abstract 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 and/or finding the global optimum of a nonconvex function. Thus, the reliability with which these computations can be done is often an important issue. Interval analysis provides tools with which these reliability issues can be addressed, allowing such problems to be solved with complete certainty.

This paper will focus on three types of applications: 1) Parameter estimation in the modeling of phase equilibrium, including the implications of using locally vs. globally optimal parameters in subsequent computations; 2) Nonlinear dynamics, in particular the 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; 3) Molecular modeling, with focus on transition state analysis of the diffusion of a sorbate molecule in a zeolite.

C. Ryan Gwaltney, Mark P. Styczynski and Mark A. Stadtherr. Reliable Computation of Equilibrium States and Bifurcations in Food Chain Models. Computers & Chemical Engineering, 28:1981-1996, 2004. view abstract // link Food chains and webs in the environment can be modeled by systems of ordinary differential equations that approximate species or functional feeding group behavior with a variety of functional responses. We present here a new methodology for computing all equilibrium states and bifurcations of equilibria in food chain models. The methodology used is based on interval analysis, in particular an interval-Newton/generalized-bisection algorithm that provides a mathematical and computational guarantee that all roots of a nonlinear equation system are enclosed. The procedure is initialization-independent, and thus requires no a priori insights concerning the number of equilibrium states and bifurcations of equilibria or their approximate locations. The technique is tested using several example problems involving tritrophic food chains.

Gabriela I. Burgos-Solórzano, Joan F. Brennecke and Mark A. Stadtherr. Validated Computing Approach for High-Pressure Chemical and Multiphase Equilibrium. Fluid Phase Equilibria, 219:245-255, 2004. view abstract // link For the computation of chemical and phase equilibrium at constant temperature and pressure, there have been proposed a wide variety of problem formulations and numerical solution procedures, involving both direct minimization of the Gibbs energy and the solution of equivalent nonlinear equation systems. Still, with very few exceptions, these methodologies may fail to solve the chemical and phase equilibrium problem correctly. Nevertheless, there are many existing solution methods that are extremely reliable in general and fail only occasionally. To take good advantage of this wealth of available techniques, we demonstrate here an approach in which such techniques can be combined with procedures that have the power to validate results that are correct, and to identify results that are incorrect. Furthermore, in the latter case, corrective feedback can be provided until a result that can be validated as correct is found. The validation procedure is deterministic, and provides a mathematical and computational guarantee that the global minimum in the Gibbs energy has been found. To demonstrate this validated computing approach to the chemical and phase equilibrium problem, we present several examples involving reactive and nonreactive components at high pressure, using cubic equation-of-state models.

Youdong Lin and Mark A. Stadtherr. Deterministic Global Optimization for Parameter Estimation of Dynamic Systems. Industrial & Engineering Chemistry Research, in press view abstract // link A method is presented for deterministic global optimization in the estimation of parameters in models of dynamic systems. The method can be implemented as an -global algorithm or, by use of the interval-Newton method, as an exact algorithm. In the latter case, the method provides a mathematically guaranteed and computationally validated global optimum in the goodness-of-fit function. A key feature of the method is the use of a new validated solver for parametric ordinary differential equations (ODEs), which is used to produce guaranteed bounds on the solutions of dynamic systems with interval-valued parameters, as well as on the first- and second-order sensitivities of the state variables with respect to the parameters. The computational efficiency of the method is demonstrated using several benchmark problems.

Youdong Lin and Mark A. Stadtherr. Deterministic Global Optimization of Molecular Structures Using Interval Analysis. Journal of Computational Chemistry, 26:1413-1420, 2005. view abstract // link The search for the global minimum of a molecular potential energy surface is a challenging problem. The molecular structure corresponding to the global minimum is of particular importance because it usually dictates both the physical and chemical properties of the molecule. The existence of an extremely large number of local minima, the number of which may increase exponentially with the size of the molecule, makes this global minimization problem extremely difficult. A new strategy is described here for solving such global minimization problems deterministically. The methodology is based on interval analysis, and provides a mathematical and computational guarantee that the molecular structure with the global minimum potential energy will be found. The technique is demonstrated using two sets of example problems. The first set involves a relatively simple potential model, and problems with up to 40 atoms. The second set involves a more realistic potential energy function, representative of those in current use, and problems with up to 11 atoms.

Awards

Computing in Chemical Engineering Award

Given on November 17, 1998 by the American Institute of Chemical Engineers

Excellence in Teaching Award

Given on August 21, 1978 by the University of Illinois at Urbana-Champaign, School of Chemical Sciences

Xerox Award for Faculty Research

Given on May 14, 1982 by the University of Illinois at Urbana-Champaign, College of Engineering

Courses

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