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Large-scale Nonlinear Equation Solvers and Numerical Optimization

**Faculty Member: Russell Luke**

**Project Description:** 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.

Bubbles and Drops in Electric and Magnetic Fields

**Faculty Member:** John. A. Pelesko

**Project Description: **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 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.

Applications of the chebfun system to differential equations

**Faculty Member:** Tobin A. Driscoll

**Project Description: **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))/pians = 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,'.-')

MATLAB Finite Element Implementation of Stokes Type Systems

**Faculty Member:** Constantin Bacuta

**Project Description: **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.

Exploratory Modeling of Raphidophytes

**Faculty Member: **Lou Rossi

**Project Description:** 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.

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