GEMS
Groups Exploring the Mathematical
Sciences
GEMS
Groups Exploring the Mathematical Sciences
GEMSGroups Exploring the Mathematical Sciences
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In the summer of 2008 we began a project to build a toolbox for solving extremely large-scale nonlinear equations and variational principles for nonconvex and nonsmooth unconstrained optimization. A number of commercial toolboxes are available for this task, however the majority do not accommodate a reverse communication architecture wherein the user, not the optimization software, controls function evaluations. Our initial users are computational quantum chemists. The project involves an interplay between abstract variational analysis, numerical optimization, linear algebra, and computer science. Some of the work is a revision of conventional methods to the logic for reverse communication, but we are also developing new methods for which a comparison with conventional techniques needs to be made. Our team currently consists of one senior graduate student and an undergraduate mathematics major. The GEMS student will be the fourth person to complete the team. A successful applicant will have strong linear algebra skills and some knowledge of numerical analysis, including some programming experience in either Matlab or Fortran. Knowledge of optimization and/or variational analysis is a plus, but not expected.
For the past several years we have been studying the behavior of bubbles and droplets in the presence of electric and magnetic fields. This includes both mathematical modeling and experimental efforts carried out in the MEC Lab. Numerous undergraduate and graduate students have participated in these efforts, including two recent Ph.D. students, Derek Moulton and Regan Beckham. Some of their most recent work can be found on our team wiki at: http://capillaryteam.pbwiki.com/ . This summer, we plan to refresh our group with at least three new undergraduate students and at least one graduate student, hopefully a GEMS student. We plan to focus on three basic projects:
Electro-Capillarity - In this project, we study the interaction of electric fields with capillary surfaces. In particular, we look at how soap films are deformed in the presence of an electric field. Some photographs, taken using the high speed camera in the MEC Lab are shown on the left below. In this case, our goal was to understand both the static behavior of the film and the dynamic response of the film to an electric field. The mathematical model was formulated using variational calculus and resulted in a coupled systems of non-linear PDE's that were solved using asymptotic and numerical methods. Our goal for this summer is to extend this work and carry out a more careful comparison of experimental and theoretical results.
Magnetic Film Draining - The photographs on the right (above) are close up snapshots of a soap film draining under the action of gravity. However, in this system, magnetic nanoparticles have been added to the soap film to make it susceptible to the influence of a magnetic field. In the leftmost photograph, no nanoparticles have been added. The colors indicate thinning of the film and "normal" soap film drainage is seen to occur. In the photos on the right, nanoparticles have been added to the solution and the film drains not only under the influence of gravity, but also in the presence of a magnetic field. In the MEC Lab, we have demonstrated what we call "reverse draining," i.e., the upwards draining of a soap film against the action of gravity. This work will be featured at an upcoming meeting of the American Physical Society. A preliminary mathematical model of this system has been formulated and simplified using standard techniques from the theory of thin films. Our goal for this summer is to push both the theory and the experiment to the point where a quantitative comparison becomes possible.
Electro-Elastic Capillarity - The third project we will pursue this summer is the addition of an elastic substrate to the electro-capillarity problems discussed above. In essence, we want to understand how droplets sitting on an elastic substrate deform in the presence of an electric field. Our goal for this summer is to develop a first, simple, experimental version of this problem and to formulate the corresponding mathematical model.
All of the problems above involve using tools and techniques from variational calculus, ordinary differential equations, partial differential equations, asymptotic methods, and numerical analysis. The interested student is not expected to be familiar with all of this material, but should be willing to learn! A GEMS graduate student participating in this project will be expected to help supervise undergraduate researchers and to participate in laboratory work. No prior experimental expertise is expected, but again a willingness to learn is essential.
The chebfun system is a free matlab add-on package designed to allow
operations on functions that feel symbolic but run at numerical speed. With it,
one can compute with functions as easily as with numbers. The additional chebop
package uses the technology to solve boundary-value and eigenvalue differential
equations to full machine accuracy automatically, using only an operator
notation to describe the problem. The goal of this GEMS project will be to
develop an application of the technology to a novel research problem, for
example in the study of the stability of complex fluid flows. Basic knowledge of
matlab, differential equations, and linear algebra are essential for the
project, but deep understanding of numerical methods for such problems is not a
prerequisite.
Some examples of chebfun/chebop quickies:
What's
the integral of exp(-sqrt(x)) from 0 to 10?
>> x = chebfun('x',[0
10]); sum(exp(-sqrt(x)))
ans =
1.647628069579947
What are the characteristic vibration
frequencies of a string?
>> sqrt(eigs(A))/pi
ans
=
1.000000000000078
2.000000000000004
3.000000000000020
4.000000000000010
5.000000000000001
5.999999999999995
What is the deflection of a loaded, simply
supported beam?
>> [d,x]=domain(0,1); D = diff(d); A = -D^2;
A.bc='dirichlet';
>> [d,x]=domain(0,1); D= diff(d); I =
eye(d);
>> A = D^4; A.lbc(1)=I; A.lbc(2)=D^2; A.rbc(1)=I;
A.rbc(2)=D^2;
>> f = -exp(3*sin(pi*x));
>>
plot(A\f,'.-')
To approximate the solution of a partial differential equation (PDE) by the Finite Element Method (FEM), we decompose the domain of the problem into small subdomains or a grid. For better approximations, a family of grids is considered, and for each grid appropriate finite dimensional spaces of piecewise polynomial functions are build. The given PDE problem is restricted to the finite dimensional spaces and reduced to solving large but sparse linear systems. The following pictures show the grid and the plot of the numerical solution of the Laplace equation on the unit square obtained by a short MATLAB code with less than 100 lines.
For Stokes type systems on more complicated domains, special type of grids are needed and special type of iterative algorithms should be used for solving the corresponding non positive definite linear systems. There are various packages of FEM software available to researchers in this field, but they are not simple to use or easy to adapt to specific problems. The project proposes to develop simpler and more efficient algorithms and programs that will apply to solving Stokes type systems.
The objectives of the project are:
1) To learn the basic theory and the implementation aspects of the FEM.
2) To implement MATLAB code for solving elliptic PDEs.
3) To develop a simple and easy to read MATLAB package specifically tailored for solving Stokes type systems and based on existing MATLAB subroutines.
4) To apply the developed MATLAB package to approximating PDEs that arise from modeling incompressible fluid flow.
Prerequisite - Basic knowledge of partial differential equations, linear algebra, and MATLAB.
Note -The graduate student choosing to participate in this project, will help supervise an undergraduate student with whom will do collaborative work and programming in MATLAB.
Faculty member: Lou
Rossi
Project title:
Exploratory modeling of raphidophytes.
Keywords: Mathematical biology,
modeling.
Raphidophytes are a type of algae that are
capable of both photosynthesis and consuming simpler life forms (grazing). In
this image of raphidophyte Heterosigma
akashiwo, the chloroplasts appear blue and the nucleus appear green.
Photosynthesis is very efficient but requires carbon dioxide and light.
Grazing is an alternative if conditions for photosynthesis are not ideal.
Successful organisms strike the right balance depending upon environmental
conditions. Raphidophyte communities are thought to be an excellent
indicator of water quality because their fitness is so sensitive to
environmental factors. Raphidophytes have two flagella, and so they are
mobile. The project objective is to develop a descriptive mathematical
model of raphidophyte populations as a function of environmental inputs. A
participant need not know any biology but must be willing to learn and
collaborate with others. The problem is being studied by Prof. Kathy
Coyne in the College of Marine Sciences (on the beautiful Lewes campus) so
the participant will have access to expert biological insights and lots of data
in various forms to verify and refine models. This project is ideal for
someone interested in mathematical modeling, differential equations and
environmental studies.
Faculty member: Lou
Rossi
Project title: High performance computing and software verification
of vortex methods.
Keywords: Scientific computing, numerical analysis,
software verification.
With the advent of advanced multi-core
and multi-CPU architectures, a current hot topic in scientific computing is
software verification. Production codes that implement high accuracy
algorithms seldom achieve the designed accuracy. Often codes
that work as designed on single processors do not perform as designed when
scaled across many cores or CPUs. These types of software errors are not
typical software bugs that cause the program to halt. These are
mathematical and algorithmic bugs that cause numerical errors to pollute the
results. Complex problems require complex algorithms, so the challenge is
to develop automated tools that can check the mathematical correctness of
software. Recently, Prof.
Stephen Siegel and I collaborated to perform a verification of
study of BlobFlow. BlobFlow is a high order
vortex method implemented in C. A vortex method is a technique for solving
fluid flow problems using particles. BlobFlow achieves high accuracy by
using particles that deform. In the image above, the flow field on the
left has many fine details but was captured with a small number of deforming
particles. The boxed subregion is shown on the right with the particles
position and shapes that represent the solution. The code is publicly
available as an open source project. Our first effort was a hybrid
manual/automated verification of BlobFlow. Prof. Siegel wishes to automate
the entire process, and BlobFlow will be the test case.
The objective of
this project is to implement new algorithms for BlobFlow in serial and parallel
while collaborating with colleagues in CIS who will verify the implementation
using their new software tools. The automated verification software will
be developed and implemented by participants from the CIS department. The
participant will have use of our cluster (The WOPR) in the department and
possibly much larger machines at Department of Energy labs. This project
would be ideal for someone interested in scientific computing, numerical
analysis and parallel computing.
Every time you blink, a complex multi-layer thin film of fluid is established on the front of your eye. This thin film helps the visual function of the eye and helps protect its surface. Millions of people are affected by defects in the tear film leading to a family of conditions called dry eye; optometrists and ophthalmologists are interested in better understanding of the tear film with respect to those conditions as well as the normal function of the tear film. Observations of the tear film offer many striking phenomena to understand and explain; a sample of the motion of the outer surface of the tear film is shown below (by Ewen King-Smith and co-workers at Ohio State).
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This is an ongoing project to study the behavior of the tear film and the
mathematics required to do so; two PhD students will have finished their degrees
from it by summer 2009, and one MS student and two undergraduates have already
contributed. Experimental results on the tear film itself, which have a
significant influence on the choice of problems to study, are available via a
collaboration with the College of Optometry at Ohio State University.
A hierarchy of mathematical models is available to understand various aspects of the tear film's dynamics of its formation during blinking and its dynamics thereafter. Those mathematical models offer challenging partial differential equations to solve on moving or stationary domains with complex shapes. We have solved some models of the tear film already, with sample results shown below. On the left is a comparison for between experimental data from the front of an eye with numerical simulation from one-dimensional model (by Alfa Heryudono, PhD 2008 with T Driscoll); on the right is a two-dimensional result for the tear film thickness (4s) after a blink (by Kara Maki, PhD 2009 with R Braun).
One-dimensional computations: This summer, one appropriate project would be to solve a one-dimensional model of the tear film on a stationary and, if time and progress permit a moving domain, for a one dimensional equation for the tear film with evaporation. The purpose would be to model the blink cycle with evaporation. Another project would be to solve a system of two partial differential equations on a fixed or moving domain; one equation would be for the thickness of the tear film and the other for the concentration of a surfactant on its surface. Candidate numerical methods in both cases are finite difference, spectral and radial basis function discretization in space with an ode solver used in time; we have been successful using these approaches. The programming will be in Matlab.
Two-dimensional computations: A more advanced project is two dimensional simulations of one equation models for the tear film with a stationary boundary. These simulations would be in the Overture framework developed at LLNL by William Henshaw and coworkers.
This project is in collaboration with Tobin Driscoll and Pamela Cook.
Copyright 2008 University of Delaware