Moving Boundaries and Interfaces
Moving Boundaries and Interfaces
Many physical problems involve moving boundaries. Dynamic
boundaries change position and shape in response to the
particular physics at work: examples are breaking waves in
the ocean, dancing flames in the fireplace, and milk
swirling in a cup of tea. Static boundaries, such as tumors
in medical scans and cartoon characters against a background
animation, can be just as perplexing: try finding edges in a
picture of a dalmatian lying on a rug with spots!
Surprisingly, many other interesting problems, such as
negotiating a robot around obstacles and finding the
shortest path over a mountain range, can also be cast as
evolving boundary problems.
The physics and chemistry that drive a boundary or
interface may be difficult to describe, but even when the
speed and direction of a moving interface are well
understood, following its shape can be difficult. The first
concern is what to do when sharp corners appear, as they do
in, for example, the intricate patterns of a snowflake.
Second, distant edges can blend together: the "edge" of a
forest fire changes as separate fires burn together and
sparks carried by the wind ignite distant regions. Finally,
in three dimensions (and higher), even finding a nice way to
represent---let alone move---an undulating boundary is a
challenge.
One technologically important example of interface
motion involves the manufacture of computer chips. In the
etching and deposition process, a layer of metal is
deposited on a silicon wafer, etched away, and then the
process is repeated numerous times until a final profile is
obtained. As device sizes get smaller and smaller, using
trial and error to obtain the correct design becomes
impractical. Instead, one would like to simulate these
processes as accurately as possible in order to test various
layering strategies and resulting device characteristics.
In recent years, the application of new mathematical and
numerical algorithms for interface motion has afforded real
breakthroughs in this area. Before these techniques, complex
problems involving the evolution of profiles in two
dimensions were difficult; now, fully three-dimensional
simulations involving a wide range of physical effects are
easily within grasp. The new algorithms have been
incorporated into the simulation packages at many major
semiconductor manufacturers in the United States, and are
part of the production environment in various chip lines
today.
These computational techniques, known as level set
methods and fast marching methods, rest on a fundamental
shift in how evolving fronts are viewed. Rather than focus
on the evolving front itself, these techniques discretize
the region in which the front moves. Each point in that
space keeps track of either its distance to the front or of
the time when the front passes over it; the accumulation of
all this information gives an accurate portrait of the
moving interface. The key is to define equations for the
time at which the front passes over each point and then to
solve these equations.
The equations which keep track of the front at each
grid point in the domain are variants of the Hamilton-Jacobi
equations; these equations have a long history in such areas
as optics, wave propagation, and control theory. While they
can be very complex, their derivatives bear a resemblance to
hyperbolic conservation laws and to the equations of fluid
mechanics, allowing use of the knowledge acquired in those
well-developed fields. The main breakthrough in modeling
interface motion was the realization that schemes from fluid
mechanics could be unleashed onto the equations of moving
fronts. The result is a wide range of computational tools
for tracking evolving interfaces with sharp corners and
cusps, with topological changes, and in the presence of
three-dimensional complications. These schemes have found
their way into a vast number of applications, including
fluid mechanics, dendrite solidification and the freezing of
materials, image processing, medical imaging, combustion,
and robotic navigation.
Some of the most complex interface applications appear
in simulating the manufacture of computer chips. To begin,
a single crystal ingot of silicon is extracted from molten
pure silicon. This silicon ingot is then sliced into
several hundred thin wafers, each of which is polished to a
smooth finish. A thin layer of crystalline silicon is
oxidized, a light-sensitive "photoresist" is applied, and
the wafer is covered with a pattern mask that shields part
of the photoresist. This pattern mask contains the layout of
the circuit itself. Under exposure to a light or an electron
beam, the unshielded photoresist polymerizes and hardens,
leaving an unexposed material that is etched away in a dry
etch process, revealing a bare silicon dioxide layer.
Ionized impurity atoms such as boron, phosphorus, and argon
are implanted into the pattern of the exposed silicon wafer,
and silicon dioxide is deposited at reduced pressure in a
plasma discharge from gas mixtures at a low temperature.
Finally, thin films like aluminum are deposited by processes
such as plasma sputtering, and contacts to the electrical
components and component interconnections are established.
The result is a device that carries the desired electrical
properties.
This sequence of events produces considerable changes
in the surface profile as it undergoes various processes of
etching and deposition. Describing these changes is known
as the "surface topography problem" in microfabrication and
requires an analysis of the effects of many factors, such as
the visibility of the etching/deposition source from each
point of the evolving profile, surface diffusion along the
front, complex flux laws that produce faceting, shocks and
rarefactions, material-dependent discontinuous etch rates,
and masking profiles. The physics and chemistry that
contribute to the motion of the interface are areas of
active research. Once empirical models are formulated, one
is left with the problem of tracking the evolving front.
Here is where level set methods and fast marching
methods come into play: they provide the means to follow the
evolving profile as it is shaped by the etching and
deposition process, and they capture some of the most subtle
effects. For example, visibility has a key role; if part of
the evolving surface causes a shadow zone that blocks the
effects of the etching or deposition beam, the motion is
reduced. Computing this shadow zone was formerly a very
expensive proposition; however, the fast marching method
yields an elegant and fast way to do it.
Another example is the complex manufacturing process
called ion-milling, in which a beam of reactive ions acts
like a sandblaster and etches away at a surface. The etching
rate depends on, among other things, the angle at which the
beam hits the surface. The most effective etching angle is
not always directly straight down; the "yield function"
relates how much material is removed to the incoming angle.
Interestingly enough, this process produces beveled, rounded
edges in some areas and sharp cusps in others. While these
are difficult problems to model, they are easily handled by
level set and fast marching methods.
4 Education
The importance of strong ties between mathematics and
science is self-evident from the examples presented---which,
we stress again, are only a tiny sample from a very large
pool. Unfortunately, there is a clear shortage of people
able to bridge the gap between mathematics and the sciences,
and one of the challenges that must be faced is how to
educate more.
It is obvious to us that students of mathematics should
be able to understand problems in science, and that students
of science should understand the power and roles of
mathematics. Each area of science has its own unique
features, but the different areas share common features that
are often of a mathematical nature.
The themes of modeling, computation, and problem
solving are especially relevant to education.
- Modeling. Students in science and mathematics need to
be educated in modeling far beyond the simple paradigm
exemplified by ``do this experiment, plot the data, and
observe that they lie almost on a straight line''. Given a
physical problem and/or data, students should learn to
construct a mathematical model, explain why the model is
appropriate, perform mathematical analysis or a
computational simulation, devise experiments to check the
accuracy of their model, and then improve the model and
repeat the process.
- Computation. The view that ``anyone can compute'' is
just as wrong as the statement that ``anyone can build a
telescope''. One has to learn how. Much of the current
teaching of computation is flawed; a ``cookbook'' strategy
of using canned programs without attention to fundamentals
is completely inadequate. At the other extreme, scientists
should not waste their time implementing outmoded methods or
reinventing known algorithms and data structures. Students
in science and mathematics need to be aware of the
intellectual content and principles of modern computer
science.
- Problem-solving. In traditional academic presentations
of scientific and mathematical problems, the context is
stripped away and simplified so that students can focus on
the essentials. But, especially when developing
mathematical insights, students must learn how to approach
ill-defined, poorly formulated problems---an area in which
education is lacking. There are no shortcuts; the only way
to learn is by direct experience.
We offer a number of recommendations for education in
mathematics and science. Our primary focus is education for
students who specialize in mathematics or science; we cannot
begin to address the national problem of mathematics and
science education for all.
1. Support curriculum development in areas that are
essential for connections between mathematics and science.
Every curriculum-related activity should include production
of Web-based materials.
(a) Create modeling courses for high school, undergraduate,
and graduate students. Unlike many other skills, modeling
can be taught (at an elementary level) to students in high
school. At the undergraduate level, there would be enormous
benefits if a one-year modeling course were part of the core
curriculum in science, engineering, mathematics, and
computer science. Graduate modeling courses would deepen
the scientific knowledge of mathematics students while
enriching the mathematical skills of science students.
(b) Support development of courses that tie core computer
science to science, engineering, and mathematics.
Programming, numerical analysis, data structures, and
algorithms---each of which is a topic with serious
mathematical content---should be part of the education of
every scientist and mathematician.
(c) Encourage experiments in activities (courses, summer or
short-term workshops) that teach scientific and mathematical
problem solving. Such programs could involve not only
techniques and direct experience of problem solving, but
also group projects that teach students how to work
collaboratively with others and how to present their work.
2. Encourage students to undertake programs of study, at
both undergraduate and graduate levels, which combine
mathematics and science. That this can be done at the
graduate level has been shown by the successful
Computational Science Graduate Fellowship program of the
Department of Energy, which requires students to undertake a
demanding interdisciplinary program in exchange for a
generous fellowship.
3. Support summer institutes in (i) mathematical topics
that address scientific applications and (ii) scientific
topics with mathematical content.
The NSF Research Experiences for Undergraduates (REU)
program has been extremely successful in exposing students
to research at an early stage. REU and other institutes
have become important for top undergraduates interested in
science and mathematics, and it is now common to prepare for
graduate school by attending a summer school or institute.
However, these programs are overwhelmingly devoted to highly
specialized subjects. In part this is understandable; the
organizers want to give the students a taste of research,
which is more easily done in a narrow area. But because
those summer institutes often determine the direction
students will take, NSF should ensure that there are high-
quality institute programs with a multidisciplinary emphasis
centered on connections between mathematics and science.
Certain emerging areas (such as mathematical biology) are
not yet widely covered in graduate programs. Carefully
designed summer institutes would help to broaden the
education of graduate students whose home institutions lack
offerings in such fields.
4. Fund research groups that include both (i) a genuine
collaboration between scientists and mathematicians, and
(ii) a strong educational program for graduate students,
postdoctoral fellows, and possibly undergraduates. To be
effective, such funding should be as long-term as possible;
if funding is only short-term, researchers are unlikely to
make the huge investment of time needed to develop group
structures that will sustain multidisciplinary
collaborations.
5. Fund postdoctoral fellowships in environments that
combine excellence in science with excellence in
mathematics. Efforts to create industrial postdoc programs
could be expanded to create joint university/national lab
postdoctoral fellowships, as well as short-term fellowships
for scientists in mathematics programs with a strong applied
component.
Beyond the postdoctoral level, there should be programs to
encourage and support faculty who would like to become
active in collaborations outside their own discipline. The
existing NSF program in this vein, Interdisciplinary Grants
in the Mathematical Sciences (IGMS), is small and imposes
relatively strict requirements on qualification and support
by the home department.
6. Develop a program of group grants for mathematics and
science departments that encourage the creation of new
courses, experimentation with instructional formats, and
coordinated programs of hands-on experiments, modeling, and
computation. Departments that receive such grants should
have substantial science requirements for undergraduate
degrees in mathematics, and substantial mathematics
requirements for undergraduate degrees in science. Many, if
not most, U.S. undergraduates in mathematics take no, or
almost no, science courses. In certain areas of science and
engineering, undergraduates take only minimal, and sometimes
outdated, mathematics courses; even worse, those courses may
give students no understanding of the ties between their
fields and mathematics. These unfortunate situations are
likely to be corrected only if there is an incentive for
departments to change their basic programs.
5 Conclusions
Strong ties between mathematics and the sciences exist and
are thriving, but there need to be many more. To enhance
scientific progress, such connections should become
pervasive, and it is sound scientific policy to foster them
actively.
It is especially important to make connections between
mathematics and the sciences more timely. Scientists and
engineers should have access to the most recent mathematical
tools, while mathematicians should be privy to the latest
thinking in the sciences. In an earlier era of small
science, Einstein could use the geometry of Levi-Civita
within a few years of its invention. With today's vastly
expanded scientific enterprise and increased specialization,
new discoveries in mathematics may remain unknown to
scientists and engineers for extended periods of time;
already the analytical and numerical methods used in several
scientific fields lag well behind current knowledge.
Similarly, collaborations with scientists are essential to
make mathematicians aware of important problems and
opportunities.
6 References and URLs
Combustion
[1] Information about Chemkin, a registered trademark of
Sandia National Laboratories:
http://stokes.lance.colostate.edu/CHEMKIN_Collection.html
http://www.sandia.gov/1100/CVDwww/chemkin.htm
http://www.sandia.gov/1100/CVDwww/theory.htm
Cosmology
[2] M. S. Turner and J. A. Tyson (1999), Cosmology at the
Millennium, working paper.
[3] Web sites about mathematical models and numerical
simulation:
http://star-www.dur.ac.uk/~frazerp/virgo/aims.html
http://phobos.astro.uwo.ca/~thacker/cosmology/
Finance
[4] I. Karatzas and S. E. Shreve (1998), Methods of
Mathematical Finance, Springer-Verlag, New York.
[5] T. F. Coleman (1999), An inverse problem in finance,
Newsletter of the SIAM Activity Group on Optimization.
Functional Magnetic Resonance Imaging
[6] W. F. Eddy (1997), Functional magnetic resonance imaging
is a team sport, Statistical Computing and Statistical
Graphics Newsletter, Volume 8, American Statistical
Association.
[7] Information about functional image analysis software:
http://www.stat.cmu.edu/~fiasco
Hybrid System Theory and Air Traffic Management
[8] C. Tomlin, G. J. Pappas, and S. Sastry (1998), Conflict
resolution for air traffic management: a case study in multi-
agent hybrid systems, IEEE Transactions on Automatic
Control, 43, 509---521.
Internet Analysis, Reliability, and Security
[9] Willinger and V.\ Paxson (1998), Where mathematics meets
the Internet, Notices of the American Mathematical Society
45, 961---970.
[10] The Web site of the Network Research Group, Lawrence
Berkeley Laboratory:
http://www-nrg.ee.lbl.gov
Materials Science
[11] Research trends in solid mechanics (G. J. Dvorak, ed),
United States National Committee on Theoretical and Applied
Mechanics, to appear in International Journal of Solids and
Structures, 1999.
[12] G. Friesecke and R. D. James (1999), A scheme for the
passage from atomic to continuum theory for thin films,
nanotubes and nanorods, preprint.
Mixing in the Oceans and Atmospheres
[13] P. S. Marcus (1993), Jupiter's great red spot and other
vortices, The Annual Review of Astronomy and Astrophysics
31, 523---573.
Physiology
[14] J. Keener and J. Sneyd (1998), Mathematical Physiology,
Springer-Verlag , Berlin.
[15] Details about modeling melanophore in the black tetra
(the home page of Eric Cyntrynbaum, the University of Utah):
http://www.math.utah.edu/~eric/research
Diagnosis Using Variational Probabilistic Inference
[16] T. S. Jaakkola, T. S. and M. I. Jordan (1999).
Variational methods and the QMR-DT database, submitted to
Journal of Artificial Intelligence Research.
[17] M. I. Jordan (1998), Learning in Graphical Models, MIT
Press, Cambridge, Massachusetts.
Iterative Control of Nuclear Spins
[18] R. Tycko, J. Guckenheimer, and A. Pines (1985), Fixed
point theory of iterative excitation schemes in NMR, J.
Chem. Phys. 83, 2775---2802.
[19] A. Lior, Z. Olejniczak, and A. Pines (1995), Coherent
isotropic averaging in zero-field NMR, J. Chem. Phys. 103,
3966---3997.
http://www.nsf.gov/pubs/2000/mps0001/mps0001.txt
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