Erin M. Kiley

Assistant Professor, Department of Mathematics
Massachusetts College of Liberal Arts

Projects

Although my formal training is in mathematics, my research is on the application of mathematics to solving physical problems. Since 2006, I have been working on problems related to computer modelling of microwave ovens, a broad topic that has led to several targeted inquiries.

Jump to: microwave sintering | mixture models for dielectric properties | modelling perforated metal sheets |

Microwave Sintering

Sintering is a phenomenon used in the manufacturing of materials from particulate substances, and is characterized by particle growth, movement, and densification driven by excess surface energy. The amount of free surface energy of particles is typically increased via pressurization, heating, and exposure to changing electric fields, and sintering conducted in microwave ovens shows great promise as a green manufacturing technology that may, in well-designed systems, be more efficient and expedient than sintering in conventional ovens.

My research focusses on mathematical and computer modelling of microwave sintering, which I find interesting for several reasons:

• Sintering is a multiscale phenomenon in two different ways, occurring on different spatial scales and at different time scales.
• Sintering is a multiphysics phenomenon, involving three different physical processes (to wit: electromagnetic, thermal, and mechanical).
• Each of the three physical processes involved is strongly coupled--that is, coupled in both directions--with each of the other two.

My previous studies of sintering are summarized in my Ph.D. dissertation. Current and future directions for this area of research include expansion to a three-dimensional computer model, and incorporating shape optimization for designing microwave cavities for specific sintering applications.

Mixture Models for Effective Dielectric Properties

Dielectric properties govern how materials interact with electric and magnetic fields. Theories for predicting the dielectric properties of mixtures of materials (each of whose dielectric properties is known) were well established centuries ago, but still may yield surprisingly useful and powerful results when applied to modern problems. One such problem is the estimation of the effective dielectric properties of mixtures of metal powders: whereas solid metal does not couple with microwaves, it is known that inside microwave ovens, some metal powders do behave like dielectric materials. I have used computer modelling to compare the efficacy of several long-established mixture formulas in predicting the effective properties of these mixtures of metal powders by comparing the results of my computer models to experimental measurements taken by my collaborators at the Laboratory for Advanced Material Processing of the Swiss Federal Laboratories for Materials Science and Technology (EMPA). Our findings can be seen in our joint papers here and here.

Another interesting, modern application of the classical mixture formulas is in their inverse formulation; that is, given several constituent materials whose dielectric properties are known, how can they be combined in order to create a resulting mixture with desired dielectric properties? In the final semester of my Ph.D. program, I worked closely with my supervisor and his undergraduate student on answering this problem; their resulting publication is here, and the code I wrote to implement the inverse three-component model we derived can be found here on github. I am currently working with Jacob Foley on a four-component inverse model.

Perforated Metal Sheets

During the course of an industrial research project that involved modelling domestic microwave ovens, I noticed that virtually all domestic microwave ovens on the market have segments of perforated metal, commonly used to let air into the microwave cavity, or to let a user see through the door. Our computer models simply ignored these perforated metal segments (assuming the walls were solid in their place), but I wondered whether that was a responsible thing to do: might they have an effect on the electric field inside the cavity after all, and might that effect be captured by modelling the perforated segments as solids, but characterized by an effective intrinsic impedance different from that of metal they were manufactured from? To answer this question, I turned to computer modelling and to the well-known theory of reflection and transmission in layered media. The results are described in detail in my M.Sc. thesis.

Students

I am currently advising Jacob Foley, an undergraduate student at the Massachusetts College of Liberal Arts, for a project entitled "Mixture Formula Inversion for Creation of Materials with Desired Dielectric Properties". The project proposal may be found here [PDF, 91kb].

I am happy to collaborate with new students on projects involving applied mathematics and mathematical modelling. Please do not hesitate to get in touch with me. Some ideas for projects include:

• Development of population models for honeybee colonies; background reading: 1 | 2 | 3 | 4 | 5
• Advancing the links between cognitive science and the mathematical optimization technique of space mapping; background reading: 1 | 2 | 3
• Applications of the space mapping technique to optimization problems; background reading: 1 | 2 | 3
• Something else related to applied mathematics, mathematical modelling, optimization, or microwaves?

Collaborators

An incomplete list of my collaborators over the years:

• Shawn Allan
• John Bandler
• Sergey Bogachev
• Didier Bouvard
• Stephen Demjanenko
• Mitrajit Dutta
• Morgana Fall
• Jacob Foley
• Dena Feldman
• Edward Hinson
• Andrew Holmes
• Kotaro Ishizaki
• Paweł Kopyt
• Ryan Northrup
• Katy Nowak
• Holly Shulman
• Yury Gennadievich Smirnov
• Hien Tran
• Sébastien Vaucher
• Suzanne L. Weekes
• Vadim V. Yakovlev
• Vicky Yang

Funding Agencies

I am grateful to have received funding for my research from the following agencies:

• AIRBUS SE, formerly the European Aeronautic Defence and Space Company NV (EADS)
• Calabazas Creek Research, Inc.
• The Chateaubriand Fellowship in STEM, given by the Embassy of France in the United States
• General Mills, Inc.
• National Science Foundation Graduate Student Fellowship Program
• ThinkSwiss, administered by the Swiss Embassy in Washington
• U.S. Department of State