UpDate - Vol. 15, No. 4, Page 1
September 21, 1995
Math center receives $4.25M in federal funds

     The Center for the Mathematics of Waves in the University's
Department of Mathematical Sciences has been granted $4.25 million
over five years to develop the mathematical foundations for improved
numerical codes for electromagnetic problems of direct concern to the
U.S. Air Force. The grant was awarded by the Department of Defense
through its Multidisciplinary University Research Initiative (MURI).
     Led by Ralph Kleinman, professor of mathematical sciences, the
research program will be carried out in collaboration with industry
and several other universities (MIT, Virginia Tech, the University of
Arizona, Colorado School of Mines, the City University of New York and
the New Jersey Institute of Technology). The multidisciplinary team
includes scientists from the fields of mathematics, electrical
engineering and computer science.
     "The broad range of expertise represented by the team will enable
us to carry out an interactive research program designed to accelerate
the transition from mathematical modeling to practical numerical
solution," Kleinman said.
     "The Air Force's interest in electromagnetics goes back to the
beginning of radar during World War II," he explained. "The field has
played a major role in the design of stealth aircraft-a subject that
most people are familiar with-but interest in electromagnetics is
actually much broader."
     The team's studies will address a range of topics from
transmission and reception of signals to analysis of the received data
for tracking and identification, from the effects of such radiation on
biological tissue to the identification of targets in a cluttered
environment. "The mathematical methods are equally applicable to
determining the shape of an airplane, locating a cancer in the body or
identifying cracks in a structure," Kleinman said.
     While most of us now take for granted the technology that enables
detection of an object using radar, it is not a trivial problem. "We
know that each object scatters electromagnetic waves in a unique way,
creating its own 'signature,'" Kleinman said. "But, an aircraft like a
747 is about the size of a football field, while radar waves are
typically one to 10 centimeters in length. So the problem has to be
discretized: We have to break that huge 747 down into cells much
smaller than a wave length. Even with today's computers, that's a
monumental problem, so we're looking for ways to solve it rapidly and
accurately.
     "Improving accuracy is as critical as increasing speed," he
continued. "Problems like this involve lots of approximations, so even
a small error-when it's multiplied by the number of discrete cells
involved-can render an answer that's not very accurate."
     The work also will focus on "multi-criteria optimization," an
approach to resolving the dilemma involved in trying to achieve a
number of conflicting goals simultaneously. Numerical modeling can be
used to prioritize these variables in a way that will produce the most
advantageous tradeoffs.
     Another area of interest for the Air Force lies in the use of
this technology for nondestructive evaluation. "Just as we can
determine the shape of an unknown object from scattering measurements,
we can also measure the speed of sound or index of refraction within
an object from measurements made outside the object, alerting us to
the presence of invisible cracks, voids, delaminations or other
flaws," Kleinman said. "Improving this technology is particularly
important to the solution of difficult problems such as those
encountered in imaging inhomogeneous materials like composites and
human tissue."
     Kleinman emphasized that a great deal of work in computational
electromagnetics has been carried out in the past 50 years, but the
rate at which progress has been made is insufficient for the solution
of these more complex problems. Thus, the MURI funding-which went to
two centers, the other at the University of Illinois-was awarded based
on the premise that a focused multidisciplinary, multi-organizational
approach was needed if major breakthroughs were to be made.
     "The grant will enable the people who have been involved in this
type of work to continue their efforts and even to expand them,"
Kleinman said. "We hope to interact enough with the other team members
to spark some novel ideas." In addition to Kleinman, participants at
Delaware include Peter Monk, Thomas Angell, George Hsiao and Fadil
Santosa, all from the Department of Mathematical Sciences.
                                                      -Diane S. Kukich