Handling editor: Chad Topaz
The MEC Lab at the University of Delaware
John A. Pelesko, University of Delaware
Graduate student Regan Beckham and Charter School of Wilmington mathematics faculty Charles Biehl ponder wave motion during an outreach activity in the University of Delaware's MEC LAB.
Introduction
The Department of Mathematical Sciences at the
University of Delaware houses a unique hands-on laboratory known
simply as "The MEC LAB." The MEC LAB, founded in 2002, derives its
name from three key facets of applied mathematics - Modeling,
Experiment, and Computation. Partially inspired by similar labs at the
Courant Institute, the University of North Carolina, the University of
Arizona, Georgia Tech, and the New Jersey Institute of Technology, the
MEC LAB seeks to provide mathematics students at all levels with a
unique educational experience as well as serve as a home for faculty
and graduate student research. We accomplish this through a variety of
outreach activities, an active undergraduate research program, and an
innovative capstone course. In addition, we provide resources and a
home for experimentally based faculty research projects.
Research Activities
The laboratory is open for use by all faculty,
postdocs, and graduate students in the Department of Mathematical
Sciences. Rather than serving as the home to a classic focused
research "group," the lab combines the activities of many loosely
connected faculty led "mini-groups." These mini-groups typically
consist of one or more faculty members, postdoctoral associates,
graduate students, and undergraduate researchers. The common thread
linking these groups to each other and to the MEC LAB is a desire or a
need to add an experimental component to their traditional applied
mathematical research. Some recent research activities are described
below.
Soap films. Professor John A. Pelesko leads a group
consisting of two graduate students and several undergraduate
researchers that focuses on the study of the statics and dynamics of
soap films subjected to electric fields. In the MEC LAB, the group has
built several apparatuses for creating highly symmetric soap film
surfaces. They subject these films to high voltage electrostatic
fields and film the resulting motion using the lab's high speed
camera. The mathematical models of these systems typically consist of
coupled non-linear partial differential equations. Pelesko and his
group use asymptotic, numerical, and analytical methods to study these
systems and compare the theory with the experiment.
The collapse of a catenoid. Catenoids are a class of minimal surfaces, as proved by Euler in 1744. Filmed in the MEC LAB by graduate student Derek Moulton as part of his doctoral research on electrified soap films
Tear films. Professor Richard Braun leads a group
of several faculty members, two graduate students, and two
undergraduate researchers that studies tear films during human blink
cycles. Part of this work requires that the team quantify the motion
of an eyelid during the blink cycle. With the help of a computer
science graduate student, Professor Braun and several undergraduate
students used the MEC LAB's high speed camera to capture lid motion
during blinking. The group developed an automated system for
extracting the motion of the edge of the eyelid from this
video. Finally, they then incorporated this motion into their models
of tear film evolution during the blink cycle.
An image from the Braun group's study of tear films. The red lines indicate a polynomial fit to the edges of the eyelids. Using video, a spatial polynomial is generated for each frame and so the fit evolves in time. In turn, this boundary motion is incorporated into the team's blink cycle models.
Ant foraging. Professor Louis Rossi leads a team of
several undergraduate researchers that studies ant foraging trail
dynamics. This team collaborates closely with faculty from the
University of Delaware's Computer Science Department. In the lab,
Professor Rossi and his team built a self-contained ant habitat
equipped with a video camera and frame grabbing software. By placing a
food source far from the ant's home and by otherwise modifying the
ant's environment, they were able to obtain data on foraging dynamics
that could be directly compared with their partial differential
equation based mathematical model.
Close up view of a foraging ant from Professor Lou Rossi's ant table in the MEC LAB.
A Focus on Undergraduate Research
The lab's most active months are the summer
months. Typically there are six to twelve undergraduate students
conducting research full time for ten weeks. These students are drawn
largely from the pool of undergraduate mathematics majors, but
occasionally come from other departments, and even other
universities. In the summer of 2005, the MEC Lab had two students
visit for the summer from INSA Toulouse in France. Each student
conducting summer research is supervised directly by a faculty member,
but they often work in conjunction with graduate students and with
each other. All of the undergraduate researchers meet weekly for an
informal lunch and research update session. At the end of the summer,
the Department of Mathematical Sciences hosts an undergraduate
research symposium at which the
summer students speak. A best paper award is sponsored by the Sigma Xi
Honor Society and the winning student receives a $100 prize.
The range of summer projects is immense. However,
each project typically contains an experimental, a mathematical, and a
computational component. The basic idea is that students perform their
own experiments, gather real data, construct a mathematical model of
their system, and then analyze the model using analytical and
numerical techniques. In the past, students have electrified soap
films, studied falling chains, created chemical oscillators, self
assembled cubes, monitored phase transformations in freezing water,
and analyzed the random packing of chains of spheres. A typical
project is the one carried out by Professor Russell Luke and several
undergraduates over the summer of 2006 and the summer of 2007. For
this project the team built a diffraction optical bench and fabricated
simple diffraction gratings using a laser printer and transparency
film. Then, using software written by Professor Luke, students
explored the physical nature of the Fourier transform and the
mathematical nature of diffraction.
Image of the diffraction grating fabricated by Professor Luke and his students (left). On the right is the diffraction image obtained when a point source is viewed through the grating on the left.
Capstone Course
During the academic year, the MEC Lab is used by
many instructors for various math courses. Some instructors use
pre-made demonstrations that they can carry into the classroom, some
have students engage in honors or extra credit projects that involve
the lab, and others integrate lab experiences more fully into their
classes. Math 512: Contemporary Applications of Mathematics is the
main course that makes use of the lab during the semester. This course
is a capstone course and is required for all undergraduate
B.S. mathematics majors. The course typically attracts students from
engineering and the sciences as well. Students in Math 512 are
assigned to project teams at the start of the semester. Each team is
given an open-ended project that they will work on throughout the
semester. In the past, student teams have studied air bearings, domino
toppling, and thermal MEMS devices. As with the summer research
projects, each of these projects has an experimental, computational,
and modeling component. Students carry out experiments in the lab,
collect data, and compare their experimental results to the models
they develop. By the end of the semester, each team has written,
revised, and completed a journal style report on their project.
Students testing the load bearing capacity of a shaped air bearing in their capstone course. This project led to a connection with a local company, New Way Air Bearings, that manufactures air bearings. They brought a new air bearing problem to the Mathematical Problems in Industry workshop held at the University of Delaware in the summer of 2004.
Outreach
The MEC LAB provides a unique opportunity to engage
local high school teachers and students through various outreach
activities. Most prominent among these activities are the professional
development days for the mathematics faculty of local high
schools. Several such workshops have been run for the faculty of the
Charter School of Wilmington (CSW), a nationally recognized,
Delaware-based math and science charter school. During a typical
workshop, high school teachers will spend the day working in the MEC
LAB on projects that use inexpensive materials to illustrate high
school level mathematical concepts. The success of these workshops has
led the Charter School of Wilmington to begin development of their own
"CSW MEC LAB." Faculty from the Department of Mathematical Sciences at
the University of Delaware continue to work with CSW faculty in
developing projects and activities that can be carried out in the CSW
lab. Most recently, the MEC lab has extended its outreach activities
to other local high schools, last fall organizing a workshop for the
mathematics faculty of McKean High School located in Hockessin,
Delaware. Plans are in place to extend this activity to other local
Delaware high schools.
Acknowledgements
The activities of the MEC LAB are made possible
through the generous support of numerous funding agencies and
organizations. Initial funding for the lab was provided by the Unidel
Foundation. The Department of Mathematical Sciences at the University
of Delaware provides continuing financial support. The Center for
Teaching Effectiveness at the University of Delaware has sponsored
several MEC LAB based projects. Various faculty have been supported by
the National Science Foundation and used such funding to defray
equipment costs and support graduate and undergraduate students. Many
undergraduate researchers have been funded by the Science and
Engineering Scholars Program at the University of Delaware as well as
by the Howard Hughes Medical Institute.
References
[1] |
|
D. Moulton and J.A. Pelesko, "Theory and Experiment for Soap-film Bridge in an Electric Field," Journal of Colloid and Interface Science, in press. |
[2] |
A. Heryudono, R.J. Braun, T.A. Driscoll, L.P. Cook, K. Maki and P.E. King-Smith, "Single-Equation Models for the Tear Film in a Blink Cycle: Realistic Lid Motion," Math. Med. Biol. 24 (2007) pp. 347-377. |
[3] |
R.J. Braun and P.E. King-Smith, "Model Problems for the Tear Film in a Blink Cycle: Single Equation Models," J. Fluid Mech. 586 (2007) pp. 465-490. |
[4] |
K. Johnson and L.F. Rossi, "A mathematical and experimental study of ant foraging trail dynamics," Journal of Theoretical Biology, 241 (2), pp. 360-399. 2006. |