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Selection of Research Accomplishments by IMA Industrial Postdocs
David
Dobson
Industrial
Postdoc - September 1990 - August 1992
Several
IMA industrial postdocs have been working with Honeywell on
design of diffractive optics tools, since 1990. The first postdoc,
David Dobson developed rigorous modeling tools for the direct
problem, that is, for fast and accurate solvers of the Maxwell
equations in diffractive optics problems, and then developed
optimal design tools using the reduced field equations (scalar
Helmholtz equation and the Fraunhofer approximation).
This
work has resulted in three computer codes, MAXFELM, PROFOPT,
and PHASEOPT, which have been transferred to Honeywell and have
become significant resources in diffractive optics work at Honeywell.
As an illustration of the application of Dobson's w ork, we
cite three examples where Honeywell is making extensive use
of the rigorous direct solver code, MAXFELM. In one case, it
has been used since 1991 on a classified Air Force contract
for the design and performance prediction of structures needed
fo r signature reduction from aircraft.
Secondly,
MAXFELM has been the workhorse design and modeling tool on a
large DARPA contract (DABT63-93-C0066, "LIGA-Based Tunable Optical
Filters for Multispectral IR Images and Gas Analyzers, "Dr.
Den Gabriel, Program Manager). Here, Honeywell demons trated
a micro-electro-mechanical (MEMS) tunable filter using LIGA
technology. The filter device is a linear grating having permalloy
"slats" separated by an air gap and configured so the period
and air gap can be varied with a magnetic actuator. The dev
ice exhibits low-pass spectral transmittance in one polarization
with the cutoff wavelength equal to twice the air gap width,
and thus it serves as a tunable filter. However, the device
presented a real challenge for a robust mathematical modeling
tool: The air gap width is subwavelength, the period is on the
order of the wavelength, the dielectric constant within the
unit cell varies from vacuum to highly conductive metal, and
weight: period ratio is on the order of 10:1 to give mechanical
strength. MA XFELM has accurately predicted the behavior of
the device in the region of cutoff, including the presence of
resonances, and is in good agreement with measurements. The
challenging nature of this modeling task prompted further work
by Dr. Dobson to impro ve convergence and more accurately treat
the case of conical diffraction. Honeywell has subsequently
used the revised code to treat a new filter design intended
for smaller wavelengths.
Finally,
Honeywell has been using MAXFELM on an internally-funded effort
to demonstrate the recently discovered guided-mode grating resonance
phenomenon which holds much promise for ultra-narrow bandwidth,
high reflectance filters needed in vertical c avity surface
emitting lasers (VCSELs) and other optoelectronic devices. The
device consists of a zero-order grating embedded in planar waveguide
and exhibits a strong resonance when the evanescent first order
of the grating is matched to a guided mode w avevector. This
structure can be modeled accurately only with a rigorous solver,
and thus far, Dobson's finite element method has proven to be
remarkably fast compared with so-called rigorous coupled-wave
theory and consistent with other published result s. Honeywell
is currently fabricating a number of devices based on a design
developed with MAXFELM. This work, important in its own right,
also suggested some exciting possibility for extension to nonlinear
materials, and that was subsequently pursued by Dr. Gang
Bao, another IMA industrial postdoc.
Dobson
has an academic position at Texas A & M, but continues to
do consulting work for Honeywell.
Michael
Kouritzin
Industrial
Postdoc - September 1995 - August 1997
Around
the Second World War mathematicians in both the United States
and Russia became interested in filtering, that is, in estimating
the current state of an aircraft, submarine, or other dynamical
systems based upon observations from sensors. The wa rtime efforts
culminated in rather revolutionary discoveries by two prominent
mathematicians: N. Wiener in the U.S. and A.N. Kolmogorov in
the Soviet Union. Their theory, discovered independently, enabled
only very slow computations; it was "too general" and did not
take into account the special features of the dynamical systems
of the moving target.
It
was not until the early sixties that the American Researchers
Kalman and Bucy introduced the dynamical system into the general
model. This resulted in a tractable and extremely efficient
filter for real time use on a computer. This Kalman filter ha
s since become the principal practical method of estimating
and predicting system state in military and civilian applications
alike and it is taught in major universities around the world.
Ironically, military applications are now being used to demonstra
te the shortcoming of the Kalman filter and to justify and development
of other implementable filtering strategies. Indeed, there exists
a growing realization that many physical systems and observation
processes are best modeled by non-linear equations w here the
Kalman filter becomes non-optimal. Moreover, many contemporary
filtering problems incorporate non-Gaussian noise in such a
manner that the Kalman filter again is unjustified. Fortunately,
computer technology and filtering strategies have advance d
significantly in the last thirty-five years and there is a real
opportunity to develop demonstrably better strategies than the
Kalman filter for these non-linear/non-Gaussian problems.
With
defense and air traffic management applications in mind, Lockheed
Martin and the Institute for Mathematics and its Applications
(IMA) initiated collaborative research in non-linear filtering,
which included several graduate students and one postd octorate,
Michael Kouritzin. In contrast to the Kalman and Bucy filter
which may be viewed as finite dimensional (it requires the solution
of a finite number of ordinary stochastic differential equations)
Kouritzin developed efficient infinite dimensiona l exact filters,
drawing from both new theoretical results that he obtained on
fundamental solutions of evolutionary equations and novel approximation
schemes. The tools he developed for Lockheed Martin enhanced
the capabilities of Lockheed Martin in the area of detecting
and tracking highly maneuverable low observable targets, such
as cruise missile, for example. The tools he developed apply
to radar, sonar, infrared and electrical optical sensors.
Kouritzin
has an academic position but continues to do consulting work
for Lockheed Martin.
Bingyu
Zhang
Industrial Postdoc - September 1992 - August 1994
Blaise
Morton of Honeywell worked with Bingyu
Zhang for a two-year period at the IMA while Prof. Zhang was
an industrial postdoc. The focus of their work was to derive
robus t stability results for an aircraft flight-control methodology
called dynamic inversion. The results obtained by Zhang during
this collaboration are the best (most general and most applicable
to real-world fighter aircraft) that the community has along
these lines. The basic idea of the result is that the flight-control
system is proved to be stable if the functions in the aerodynamic
model of the vehicle satisfy a simple set of inequalities (which
are valid in practical situations). The stability resu lt holds
true even in the presence of bounded uncertainty in the data,
and the size of the limiting bounds are estimated.
The practical benefit is that dynamic inversion control laws
are being selected for the next-generation fighter aircraft,
so these results will give engineers some confidence that the
control design will be sound. There are numerous side benefits
that will accrue during the development phase of the system
having to do with code debugging, gain scheduling and verification
and validation of the flight software. There is a real cost
savings associated with these theoretical results, hard to quantify,
bu t very important in flight control design. These are the
first truly practical robust stability results we have obtained
for nonlinear flight control laws.
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