Industrial Applied Mathematics
People
- Richard Braun, Professor
Nonlinear BVPs and free boundary problems in fluid mechanics and materials science
- Fioralba Cakoni, Professor
Boundary integral equations and coupling BEM-FEM. Applications in electromagnetic and elastic scattering problems
- David Colton, UNIDEL Professor
Scattering and inverse scattering
- L. Pamela Cook, Professor
Complex fluids, viscoelastic fluids: flows and modeling; transonic aerodynamics
- Tobin Driscoll, Professor
High-order and spectral methods for differential equations; radial basis functions; numerical conformal mapping
- David Edwards, Professor
Asymptotic and perturbation methods in biochemical, chemical engineering, and viscoelastic systems
- George C. Hsiao, Carl Rees Professor of Mathematics
Boundary integral equations, finite element methods. Applications in elasticity. Signal and image processing, computational optics, and inverse problems
- Peter Monk, UNIDEL Professor
Finite element methods, applications in computational electromagnetism and inverse problems
- David Olagunju, Professor
Viscoelastic flows, stability, bifurcations, dynamical systems and combustion
- John Pelesko, Professor and Chair
Mathematical modeling of real world micro- and nanoscale devices. Problems arising in an industrial setting that involve continuum mechanicss
- Rakesh, Professor
Scattering and inverse problems
- Louis Rossi, Professor
Neural networks, vortex methods, flow in porous media and fluid mechanics
- Gilberto Schleiniger, Associate Professor
Fluid flow: viscoelastic and transonic flows; mathematical finance
Research
There are a wide range of problems being solved in collaboration with industry and national labs. Research ranges from basic numerical analysis (finite element, boundary element and finite difference convergence theory) and fast methods (multigrid) to applications in materials science (foam evolution, phase transformations in crystalline alloys, approximation of microstructure, visco-elastic phenomena) and electromagnetism (scattering, inverse scattering and ferromagentism). Graduate students are welcome!
Seminars
A variety of application areas and mathematical methods are featured in the
Applied Mathematics seminar series. A
weekly seminar calendar gives up-to-date information on this and other seminars in the department
Graduate Studies
Students wishing to study applied mathematics usually enroll in the Applied Mathematics graduate program (see The Graduate Program web pages for more details). There are a number of courses for students interested in this area as well as topics courses. For example
Funding
Besides the usual university and departmental funding for TAs and fellows
some students are also supported directly by companies
Many of the projects listed here have been at least partially supported by the National Science Foundation under grant numbers 9722854, 9631287 and 0322583. Any opinions, findings, and conclusions or recommendations expressed in this material are those of the author(s) and do not necessarily reflect the views of the National Science Foundation.
Computing Resources
The department operates several computers in support of research computing. All faculty and graduate students involved with scientific computing have personal workstations.
Completed Projects
- Mathematical Problems in Industry Workshop 2004
- Richard Braun,
braun@math.udel.edu
- Philip Broadbridge
- Pam Cook
- Tobin Driscoll
- David Edwards, edwards@math.udel.edu
- John Pelesko
- Louis Rossi
- Gilberto Schleiniger
- Don Schwendeman, Rensselaer Polytechnic Institute
- Bogdan Vernescu, Worcester Polytechnic Institute
- Joseph Fehribach, Worcester Polytechnic Institute
- Tom Witelski, Duke University
- Burt Tilley, Olin College
- The Department of Mathematical Sciences hosted the
20th annual
Mathematical
Problems in Industry workshop, June 21-25, 2004.
- DuPont
- Capillary viscometry for polymer processing
- Bryan Wood, U of Delaware
- Pam Cook (PI),
cook@math.udel.edu
- Gilberto Schleiniger (PI), schleini@math.udel.edu
- Emeka Nwankwo, DuPont
- Bryan Wood completed his PhD working on the mathematical modeling of the flow of a
polymer solution in a small tube; this work is expected to help improve
the efficiency of some of DuPont's polymer processes. He was
supported by the GIG at U of Delaware and by support in kind from
DuPont. The project is being supervised by Pam Cook and Gilberto Schleiniger;
they were supported by an ATE grant from DuPont.
- AstroPower
- Crystal Growth on a Substrate
- Mikhail Khenner, U of Delaware
- Richard Braun (PI),
braun@math.udel.edu
- Michael Mauk, AstroPower
- Theory is being developed for crystal growth
of a not-necessarily thin film on a substrate with
a mask.
The fully nonlinear surface geometry was taken into
account with a geometric growth model including surface diffusion
and condensation from the vapor; the anistropic case was studied as well.
The anisotropic case is now being considered.
This started out as a follow-on project
from the MPI 1999 meeting held at UD.
Rich is partially supported by,
and Mikhail was fully supported by NSF GIG DMS-9631287 during his stay at UD.
The paper that appeared from this project is
"A Model for Anisotropic Epitaxial Lateral Overgrowth,"
M. Khenner, R.J. Braun and M.G. Mauk,
J. Crystal Growth 241, (2002) 330-346.
- Air Force
- On the Mathematical Basis of the Linear Sampling
Method
- David Colton (PI),
colton@math.udel.edu
- Fioralba Cakoni, U of Delaware
- The linear sampling method is an algorithm for solving the inverse
scattering problem for acoustic and electromagnetic waves.
The first version of the linear sampling method is more flexible,
being able to treat the aforementioned cases as well as partially
coated obstacles. However, its mathematical foundation is less well
established. In this paper we provide arguments giving
a mathematical justification of the first version of the linear
sampling method. Fioralba is supported by NSF GIG DMS-9631287; David
is supported by AFOSR.
- DuPont
- A Theoretical Formulation for Rapid, Unambiguous Polymer
Characterization using Flow-Referenced Capillary Viscometers
- Pam Cook (PI),
cook@math.udel.edu
- Emeka Nwankwo, DuPont
- Gilberto Schleiniger, U of Delaware (PI), schleini@math.udel.edu
- Bryan Wood, U of Delaware
- The behavior of a polymer solution in a small tube has
been studied and a paper has been submitted for publication.
Pam and Gilberto have been supported by ATE grants from DuPont
and Bryan is supported by DuPont and NSF GIG DMS-9631287.
- Air Force
- The direct and Inverse Scattering Problems for
Partially Coated Obstacles
- Peter Monk (PI),
monk@math.udel.edu
- Fioralba Cakoni, U of Delaware
- David Colton, U of Delaware
- The direct and inverse scattering problems for partially
coated obstacles is considered. The method of integral
equations of the first kind is used to solve a scattering problem for the
Helmholtz equation where the scattered field satisfies mixed
Dirichlet-impedance boundary conditions on the Lipschitz boundary
of the scatterer $D$. Then the inverse scattering problem of determining
$D$ from a knowledge of the far field pattern of the scattered field
is solved by using the linear sampling method. Numerical examples are
given showing the performance of the
linear sampling method in this case.
This work will appear in Inverse Problems; Peter is partially
by, and Fioralba is fully
supported by, NSF GIG DMS-9631287. Peter and David are supported by AFOSR.
- AstroPower
- Crystal Growth on a Substrate
- Mikhail Khenner, U of Delaware
- Richard Braun (PI),
braun@math.udel.edu
- Micheal Mauk, AstroPower
- Crystal growth
of a not-necessarily-thin film on a substrate with
a mask is studied.
The fully nonlinear surface geometry is taken into
account with an isotropic geometric growth model including surface diffusion
and condensation from the vapor. The results have been submitted
for publication. This is a follow-on project
from the MPI 1999 meeting held at UD.
Rich is partially supported by,
and Mikhail is fully supported by, NSF GIG DMS-9631287. The paper from this isotropic
model is "An Isotropic Model for Crystal Growth from Vapor
on a Patterned Substrate," M. Khenner, R.J. Braun and M.G. Mauk,
J. Crystal Growth, 235 (2002) 425-438.
- Dow Corning
- Bounded film evolution with variable surface properties
- Cynthia DeBisschop, now at Old Dominion University
- Richard Braun (PI),
braun@math.udel.edu
- Steven Snow, Dow Corning
- In this project, theory is being developed for a thin
film on a plate with nonlinear surface properties. The
results are of interest to Dow Corning and will allow us to tackle
a wider range of problems of interest to them.
Dow Corning is partially supporting Rich; Rich is partially supported by,
and Cyndi was fully supported by, NSF GIG DMS-9631287.
- Air Force
- The Linear Sampling Method for Anisotropic
Media
- David Colton (PI),
colton@math.udel.edu
- Fioralba Cakoni, U of Delaware
- Houssem Haddar, U of Delaware
- We consider the inverse scattering problem of determining the support of an
anisotropic inhomogeneous medium from a knowledge of the incident
and scattered time harmonic acoustic wave at fixed frequency. To
this end we extend the linear sampling method from the isotropic
case to the case of anisotropic media. In the case when the
coefficients are real we also show that the set of transmission
eigenvalues forms a discrete set.
This work will appear in Journal of Computational and
Applied Mathematics; Fioralba is supported by NSF GIG DMS-9631287;
David is supported by AFOSR.
- Air Force
- The Linear Sampling Method for Anisotropic
Media: Part II
- Fioralba Cakoni (PI), cakoni@math.udel.edu
- Houssem Haddar, U of Delaware
- We reconsider the linear sampling method for solving the inverse
scattering problem of determining the support of an anisotropic
inhomogeneous medium from a knowledge of the incident and
scattered time harmonic acoustic wave at fixed frequency. We
extend the results of a previous paper concerning
with the same problem to the case where the norm of the real
part of the matrix that describes the physical properties of the
medium is less than one.
This work is in Preprint No. 2001-026, Mathematical Sciences Research Institute,
University of California at Berkeley; Fioralba is supported by NSF GIG DMS-9631287.
- Dow Corning
- Vertical bounded film evolution with variable surface properties
- Homayoun Heidari, Sharif University of Technology, Iran (visiting RPI and UD)
- Richard Braun (PI),
braun@math.udel.edu
- Amir Hirsa, Rensselaer Polytechnic Institute
- Steven Snow, Dow Corning
- Shailesh Naire, University of Nottingham, UK
- In this project, theory is being developed for a thin
film on a vertical plate with nonlinear surface properties. The
results are of interest to Dow Corning and will allow us to make
very detailed comparison with experiments that have been developed
at RPI. Dow Corning is supporting Rich and Homayoun;
Rich is partially supported by NSF GIG DMS-9631287.
- SmithKline Beecham
- Summer 2000 Internship
- Ellen Phifer, U of Delaware
- Larry Greller, SmithKline Beecham
- Frank Tobin, SmithKline Beecham
- Ellen Phifer spent the summer analyzing and improving upon a variable
tension spline curve-fitting package. The resulting code was incorporated
into MATLAB and will be used by mathematical biologists at SmithKline
Beecham in their analysis of sparse data sets. Her studies were
directed by Larry Greller and Frank Tobin.
- Merck
- Summer 2000 Internship
- Chrissy Getman, U of Delaware
- Mathematical Problems in Industry Workshop 2000
- Richard Braun,
braun@math.udel.edu
- Pam Cook
- David Edwards, edwards@math.udel.edu
- Peter Monk
- Petr Plechac
- Gilberto Schleiniger
- Don Schwendeman, Rensselaer Polytechnic Institute
- The Department of Mathematical Sciences hosted the
16th annual
Mathematical
Problems in Industry workshop June 5-9, 2000.
Thank you to NIST for valuable support
once again!
- Mathematical Problems in Industry Workshop 1999
- Richard Braun,
braun@math.udel.edu
- Pam Cook
- David Edwards
- Peter Monk
- David Olagunju
- Gilberto Schleiniger
- Don Schwendeman, Rensselaer Polytechnic Institute
- The Department of Mathematical Sciences hosted the
15th annual
Mathematical
Problems in Industry workshop in early June 1999.
The MPI workshop has been
held at Rensselaer Polytechnic Institute for the last 13 years (with
one exception, when it was held at the University of New Mexico). We were
very excited about this week-long international workshop where industrial
problems are presented and studied. The workshop was partially
supported by a grant
from the Mathematical and Computational Sciences Division at the National
Institute of Standards and Technology; thank you NIST!
- Brooks Air Force Base
- Inverse problem for detection of underground objects
- Joe Coyle,
- Peter Monk,
monk@math.udel.edu
- David Colton (PI),
colton@math.udel.edu
- Richard Albanese, Brooks AFB
- Joe Coyle spent the summer of 1997 at Brooks AFB in San Antonio, TX, under
the supervision of Richard Albanese studying inverse problems related to the
detection of underground objects. This work has formed a substantial
part of his PhD thesis with Peter Monk.
- Dow Corning
- Gravitationally-driven drainage of a thick film with a tangentially-immobile surface
- Richard Braun (PI),
braun@math.udel.edu
- Steven Snow, Dow Corning
- Udo Pernisz, Dow Corning
- In this project, theory was necessary for an experiment that Steven Snow and Udo Pernisz
developed at Dow Corning. Their experiment can produce detailed information about the
dhape of a film suspended from a wire frame and draining vertically into a bath. In a limiting
regime of the experiments, the surface of the film can be made tangentially immobile. We have
have developed a theory for this case where we use singular pertubation and computational
methods to solve the problem.
From this work we have written a Dow Corning internal report
(I0000-1998-45018 by Braun, Snow and Pernisz) and a paper has appeared
(J. Colloid Interface Sci. 219 (1999) 225-240).
This material is based upon work
supported in part by the National Science Foundation under Grant No. 9631287.
- Dow Corning
- Limiting cases of gravitationally-driven drainage of thick film
- Shailesh Naire
- Richard Braun (PI),
braun@math.udel.edu
- Steven Snow, Dow Corning
- In this project, theory was necessary for an experiment that Steven Snow and Udo Pernisz
developed at Dow Corning. Their experiment can produce detailed information about the
shape of a film suspended from a wire frame and draining vertically into a bath.
Shailesh Naire has extended the work on the case of a tangentially-immobile film
(see below) to include more realistic
physical effects. Finite surface viscosity, the Marangoni effect, and surfactant
transport have been added. A "flat film" limit has been studied.
Shailesh has solutions from analytical and computional approaches that
bound the fastest and slowest draining limits of the experiments.
This work has appeared electronically in the SIAM Journal on Applied
Mathematics (61 (2000) 889-913),
and has resulted in the Dow Corning internal report (1999-I0000-41799).
This material is based upon work
supported in part by the National Science Foundation under Grant No. 9631287.
- Dow Corning
- Insoluble surfactant model for gravitationally-driven drainage of a thick film
- Shailesh Naire
- Richard Braun (PI),
braun@math.udel.edu
- Steven Snow, Dow Corning
- In this project, theory was necessary for an experiment that Steven Snow and Udo Pernisz
developed at Dow Corning. Their experiment can produce detailed information about the
shape of a film suspended from a wire frame and draining vertically into a bath.
Shailesh Naire has extended the work still further. Finite surface viscosity,
the Marangoni effect, and surfactant transport have been added, and
the case where the film is patched onto
the static meniscus has been studied. He also has found interesting
dynamics in the evolution of the film shape and other dependent variables.
These 1+1-d computations have appeared in the Journal of Colloid and Interface Science
(230 (2000) 91-106).
This material is based upon work
supported in part by the National Science Foundation under Grant No. 9631287.
- Dow Corning
- Gravitationally-driven drainage of a thick film in 3D
- Shailesh Naire, now going to the University of Nottingham, UK
- Richard Braun (PI),
braun@math.udel.edu
- Steven Snow, Dow Corning
- Dow Corning
- Gravitationally-driven vertical film
drainage with variable surface properties
- Shailesh Naire, now at Divsion of Theoretical Mechanics, University of
Nottingham, UK
- Richard Braun (PI),
braun@math.udel.edu
- Steven Snow, Dow Corning
- Shailesh added nonlinear surface properties to the one-dimensional film problem
and the results have appeared in print (Phys. Fluids 13, (2001)
2492-2502).
Dow Corning supported Shailesh and Rich; this material is based upon work
supported in part by the National Science Foundation under Grant No. 9631287.
- Shailesh completed a set of 2+1 dimensional computations and the results have been
submitted to the Journal of Fluid Mechanics.
Dow Corning supported Shailesh and Rich; this material is based upon work
supported in part by the National Science Foundation under Grant No. 9631287.
- National Institute of Standards and Technology
- Effect of initial conditions in phase-field modeling of dendritic growth
- Jie Zhang, now at Northwestern University
- Richard Braun, ,
braun@math.udel.edu
- Bruce Murray, SUNY Binghamton
- Jeff McFadden, NIST
- In this internship, Jie spent two months of the summer of 1996 at the NIST
Gaithersburg, MD, site. There he worked with Bruce Murray (then of NIST) and
Jeff McFadden in the Applied and Computational Mathematics Division of the
Information Technology Laboratory. Jie also attended a short course on diffuse-interface
modeling of phase change.
Jie studied the effect of different initial conditions on the subsequent evolution
Governed by a phase field model of solidification. Jie computed the solidification of an undercooled melt in a thermally-insulated
box with initial conditions for the thermal field that could be discontinuous, have only a
continuous temperature field, or have a continuous first derivative as well. He
found that the smoother the initial condition for the temperature field
was, the less likely it would be that unphysical spreading of the interfacial region
would occur. This is practically useful in computation of dendritic microstructures
and the results will soon be available in postscript form.
Jie did this project for MS thesis in the Math Sciences Department. He is now
attending the Northwestern Department of Engineering Sciences and Applied Mathematics
and is studying for his PhD there.
- Thomson CSF, France
- Inverse scattering of electromagnetic waves from obstacles
- Cristoph Labreuch, Thomson CSF
- Peter Monk,
monk@math.udel.edu
- David Colton (PI),
colton@math.udel.edu
- Drs Colton and Monk were visited for a year by Christoph Labreuch, an
employee of Thomson CSF, to study the identification of objects using
inverse scattering of electromagnetic waves. This work formed the bulk of
Christoph's PhD thesis.
- Equipment for the Lab
- Peter Monk,
monk@math.udel.edu
- Pam Cook,
cook@math.udel.edu
- Richard Braun,
braun@math.udel.edu
- An SGI Octane with an R10000 cpu, an SGI Indigo2 with an R10000 cpu, and
a Pentium III machine are in an office suite in Ewing Hall.
