REU Projects
by
Mohamed B. Elgindi
Subject: Mathematical and Numerical Analysis of Physical and Engineering
Problems
Projects Goals: To introduce interested
undergraduate students to some problems of physical or engineering interests. After introducing some
analytical and numerical techniques, modeling of the problems under
consideration will be developed by the students with the guidance of the
faculty. Students will then be directed through the appropriate mathematical
analysis (e. g. set up of equations in appropriate function spaces, existence
and uniqueness, parameters dependence and qualitative behavior of solutions)
and / or numerical analysis (e.
g. considerations of
different numerical techniques, programming in Matlab)
of mathematical models developed. Interpretations of the analytical and
numerical results obtained will then be given.
Problems:
(1) Numerical computation
of temperature distribution and bulk temperature of an incompressible
non-Newtonian fluid flowing in circular pipe is important in engineering
applications such as food processing, polymer processing and other
biomechanical industries. The heat distribution is govern
by a singular non-linear parabolic partial differential equation. In this
project, we develop a suitable finite difference scheme for approximating the
temperature distribution and bulk temperature. To do this a special finite
difference scheme must be developed near the singularity. Comparison with the
results based on the finite element method will be given.
(2) Consider a thin-walled
elastic cylinder tethered by numerous nonlinear springs to an outer rigid
cylinder. Large deformations of such an elastic cylinder as an external
pressure increases past a critical value models the collapse of a blood vessel
embedded in soft tissue as the difference between external pressure and
internal pressure increases beyond some critical value. Previous studies
assumed that the springs are linear. However, the nonlinearity of biological
tissues is well known. It is also well known that the elastic properties of the surrounding tissue is the major factor in
resisting the collapse of a capillary vessel. The objectives of this project
are to investigate the available mathematical models for the elastic properties
of the soft tissues surrounding some blood vessels, and to use the Matlab functions to develop numerical
algorithms for calculating the
collapsed shapes of the blood vessel and the blood flow rate through them.
(3) Consider a uniform
beam-column of a finite length, supported by a nonlinear foundation G, with
hinged ends. We consider the deformation of the beam-column under a compressive
force P, and a lateral force F. Neglecting geometric and material
nonlinearities, the governing equation reduces to a fourth order non-
linear boundary value problem
involving several parameters. In engineering applications one is interested in
the behavior of the beam-column as the axial force P exceeds the smallest \
buckling load". Mathematically, this is a bifurcation problem; buckling
occurs because solutions bifurcate at the eigenvalues
of the linearized problem. The computation of the
post-buckling shape is, however, a nonlinear problem. Depending upon the
stiffness of G, the smallest eigenvalue may be simple
or double. The purpose of this project is to illustrate the qualitative
behavior of the solutions by giving perturbation expansions of the solution
curves in a neighborhood of the bifurcation point in each case.
(4) We consider an inclined membrane trough. We assume that there is a fluid flowing down the trough. We are interested in the flow rate, which varies for various trough shapes and inclinations. In particular, the following question is posed. What would be the maximum flow possible through the trough of given lateral perimeter? The objectives of this project are to approximate, both analytically and numerically, the shape of the cross section of the membrane trough that produces optimum flow rate. Since the flow rate depends on trough cross sectional area as well as the flow velocity profile, it is expected that the shape that produces optimum flow rate will differ from the opening that produces maximal cross sectional area. We plan to write a computer program using the Matlab software to (A) determine the cross sectional shape of the trough, (B) solve the flow equation through the computed cross section and determine the flow velocity profile, and (C) calculate the flow rate by integrating the computed velocity function over the computed cross section. Accurate perturbation approximations of a second order will be developed using the Maple software and an approximation for the optimum opening width for maximum flow will be given. Finally, we will validate the numerical results obtained by comparing them with the results obtained using the perturbation method.
(5) Analytical and Numerical Aspects for Computing Maximal Regular and Stable Subinterval Matrices A matrix whose entries are intervals is known as an interval matrix. Such an interval matrix is said to be regular if all of its point matrices are nonsingular, and stable if all the eigenvalues of all its point matrices have negative real parts. The goals of this project are:
(A) To establish the existence and uniqueness of a maximal regular (stable) subinterval matrix containing a given nonsingular (stable) point matrix of an interval matrix.
(B) To establish an iterative numerical algorithm for approximating such a maximal subinterval matrix. These problems are of interest in engineering and scientific applications that require solvability (stability) of a given linear system of algebraic equations (time-invariant system of differential equations) under data perturbation.