Math Research Center Models Industrial Problems
SOCORRO, N.M. September 2, 2009 – A fledgling industrial mathematics research center is proving to be productive in publishing research papers and developing professional partnerships.
Math professor Dr. Ranis Ibragimov organized the Research and Support Center in Applied Mathematical Modeling, or RSCAMM, at New Mexico Tech in 2008, with support from top administrators.
The Center aims to serve international interests and cooperation in frontier areas of mathematical modeling in industrial areas and environmental sciences. The main goal of the Center is to collaborate on research projects forming “hotspots” at the interface between applied mathematics and industry.
|New Mexico Tech math professor Dr. Ranis Ibragimov explains the concepts behind mathematical modeling of tidal mixing. Thomas Guengerich/New Mexico Tech|
One of the Center’s fortes in mathematical modeling is based on an applications of Lie Group Analysis of differential equations. Dr. Ibragimov said Lie Group Analysis provides a universal tool for analytically solving complex differential equations where traditional means of integration fail.
Dr. Ibragimov said Lie group analysis provides a universal tool for solving analytically considerable number of differential equations for which other traditional means of integration fail. Traditional integration methods typically taught at universities rely essentially on linearity and constant coefficients. Group analysis deals equally easily with linear and nonlinear differential equations, as well as with equations involving variable coefficients, Dr. Ibragimov said.
“The research of the RSCAMM Center is numerically oriented and theoretical in nature,” Dr. Ibragimov said. “The key feature in our approach is the combination of the approximate forms of the governing equations in mathematical modeling with careful and precise analysis. The approximations are required to make any progress possible, while precision is demanded to make the progress meaningful.”
Already, Center-initiated research has led to eight publications in scholarly journals. In addition to Dr. Ibragimov, other Tech authors of papers include mechanical engineering professor Dr. Nadir Yilmaz, graduate student Akshin Bakhtiyarov and undergraduate student Margaret Snell.
This year, Dr. Ibragimov has been invited to collaborate in two industrial projects: enhanced oil recovery and self-healing metals.
New Mexico Tech scientists are collaborating with petroleum experts in Russia and Azerbaijan to research more efficient methods of petroleum recovery. Mechanical Engineering Department chairman Dr. Sayavur Bakhtiyarov, who is also a member of the Center, is the principal investigator of the project with support from the Global Technology Transfer Inc. in Tulsa, Okla.
This project has resulted in several publications related to mathematical modeling of enhanced oil recovery. Engineers know that more than 85 percent of a petroleum is unrecoverable via primary standard recoveries. To recover the remaining oil, petroleum companies use secondary and tertiary extraction methods based on injection of water or other fluids to force the crude oil to production wells.
Dr. Ibragimov said mathematical models describe chemical reactions within porous media needed for oil recovery is given in terms of nonlinear partial differential equations. The existing experimental and numerical methods are not precise enough to provide estimates of the efficiency of the oil recovery by the chemical reaction. The Center is promoting Lie Group Analysis as a method of obtaining analytically exact solutions of those partial differential equations.
“The goal was to optimize the chemical reaction between certain chemicals and the petroleum in porous media,” Dr. Ibragimov said. “The efficiency of the reaction was determined by means of analytical solutions that we have obtained by solving the differential equations. Our next step is to develop a simple numerical package designed to improve the secondary oil recovery.”
Another project is related to mathematical modeling of self-healing materials. Engineers are developing metals that are coated with a metallic-chemical compound. As the metal heats, the coating compound slowly melts and fills cracks, preserving the original strength of the metal.
Cracks, which are very common in metals and other materials, lead to corrosion and deterioration. Self-healing represents an important engineering breakthrough, particularly in systems that require high surface quality, such as vehicles, optical systems and windows, and in systems that require high reliability, like aircraft and nuclear storage systems.
Detecting and repairing cracked metal surfaces is a complex analytic problem. At present, no detailed mathematical models predict how the self-healing processes functions over time. The Center is currently involved in the mathematical modeling of healing rates in metals that will help engineers develop more efficient crack prevention of materials.
Dr. Ibragimov also initiated the global environmental project designed to model mixing processes caused by the interactions and dissipation of internal waves in the ocean. Internal waves play an important climatic role; they disperse pollutants, chemical and biological tracers.
The energy available for mixing processes in the ocean operates at large scales (few tens of kilometers) and then transfers across the internal energetic wave spectrum down to small dissipation scales (a few meters). The goal of the project is to understand better the dynamics of the ocean and to understand what effects internal waves have on our environment.
Dr. Ibragimov, who is spearheading these studies, invited his colleagues from McMaster and Waterloo universities in Canada and Kanazawa Institute of Technology in Japan to participate in this project. Dr. Ibragimov is advising doctoral students Gunter Leguy and Eric Ovaska, who are working with Los Alamos National Lab scientists on modeling oceanic internal waves and global circulation model, which is the part of that global project. The project has led to several publications related to mathematical modeling of mixing processes and energy exchange between internal waves in the ocean.
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By Thomas Guengerich/New Mexico Tech