Research objectives

Methodology

Research personnel

Institutional commitment and sources of additional support

Student involvement and training opportunities in science and engineering

Methodology

Research personnel

Institutional commitment and sources of additional support

Student involvement and training opportunities in science and engineering

**1. Biomedical background and objectives**

The research in this proposal
has been motivated by a study of blood flow through compliant vessels
after endovascular repair. The PI and cardiologist Dr. Krajcer at
St. Luke's Episcopal Hospital in Houston have begun a study of
blood flow through the abdominal aorta after the insertion of a
prosthesis (a spring-like device called stent)
used to treat abdominal aneurysm (sac-like protrusion
of weakened sections of the aorta).
See Figure 1.
The procedure entails inserting a stent inside the aneurysm where it
serves to hold open the weakened artery and to exclude
the aneurysm from circulation.
The main goal of this collaborative study is to understand the behavior
of the prosthesis and of the walls of the aorta subject to the
pressures induced by pulsatile blood flow.

A sketch of abdominal aneurysm (left) and a sketch of a stent

inserted in the abdominal aorta (right), not showing the aneurysm sac

The model
equations are obtained using asymptotic reduction of the
axi-symmetric Navier-Stokes equations in narrow and long channels. The
equations have first been derived in [2],
and a mathematically rigorous asymptotic reduction is presented in [5].
The equations are in the form of a quasilinear hyperbolic system of
partial differential equations
describing conservation of mass and momentum

= | 0, | (1) | |

= | f(A,m). |
(2) |

Here

**2. Mathematical
background and
objectives**

In order to study discontinuities in the cross-sectional area we need
to be able to define a weak solution for a quasilinear system of
hyperbolic equations with discontinuous coefficients.
There is no general theory that deals with systems of hyperbolic
partial differential equations with discontinuous coefficients. The
literature is sparse even for hyperbolic equations in conservation form
allowing discontinuous coefficients.
There are various difficulties associated with the solutions
of such systems.
For example, physically reasonable solutions exhibit jump
discontinuities that are not compressive
because the jump discontinuity in the solution is induced by the
discontinuity in the coefficients,
and not by the compressive nature of the characteristics.
Deriving the ``correct'' entropy criterion for such jump
discontinuities,
and capturing such solutions numerically, is a problem.
In the case when the equations are not in conservation form, as in the
case of our model,
even the basic issue such as the meaning of the weak form of the
equations
in unclear. The main difficulty stems from the fact that the weak form
of the equations contains the products of the
Dirac measure by the Heaviside function which cannot be
defined in the sense of distributions.

In the case
when the equations are *linear* and in
conservation form
there is a theory developed in [9]
that deals with numerical approximation of one-dimensional
linear conservation equations with discontinuous coefficients.
Unfortunately, the techniques used in [9]
do not generalize to nonlinear problems. Quasilinear *scalar*
hyperbolic problems have been studied in [16, 17] where
variational methods were used
to study existence and uniqueness of solutions.
The methods heavily rely on the fact that there is only one hyperbolic
equation under consideration and it is unlikely that the approach can
be generalized to systems. Finally, the most relevant to this proposal
are the work of A.I Volpert [21]
and a
work by Dal Maso, P.G. LeFloch and F. Murat
[12] where issues
related to the
product of the Dirac measure by the Heaviside function are discussed.
Although these works do not deal with hyperbolic problems with
discontinuous coefficients,
the ideas about the meaning of the ambiguous products
are very much related to the problems that arise in hyperbolic
equations with discontinuous coefficients.
In the work proposed here one of the objectives
is to generalize those ideas
to the systems with discontinuous coefficients.

For the past
ten years the PI has had continuous research support by
the National Science Foundation to study properties of solutions to
multi-dimensional systems of conservation laws and to study uniqueness
and properties of viscosity solutions to one-dimensional conservation
laws
with smooth coefficients. The project described in this proposal opens
a new direction in the PI's research program.
Various aspects of this project, however, rely heavily on the knowledge
and the experience gained from the work supported by the NSF.

**1. Theoretical approach**

To study weak solutions of equations (1)
and (2)
we begin with the standard approach:
multiply the equations by a test function
and
integrate by parts to obtain

It was shown in [4] that at the points where the coefficient

which is not defined in the sense of distributions.

To get around
this difficulty we take the following approach.
Motivated by the underlying application in which the Young's modulus
coefficient
*G*_{0} is actually continuous, with high gradients at the
anchoring cites of the stent, we regularize the problem by smoothing
out the Dirac measure term *G*_{0}'.
We regularize Dirac measure by using, the so called, admissible
averaging kernels [21]

Consider the weak form that corresponds to the regularized coefficient

Following the ideas presented in [12, 21], we define (6) to be the

The PI has studied the performance of two numerical methods on the calculation of the solutions of equations (1) and (2) with discontinuous

In this
project the PI plans to investigate
the effects that need to be taken into account when hyperbolic
equations with discontinuous coefficients and singular source terms are
simulated in order to obtain
the physically reasonable solutions.
Error analysis will be performed by generalizing the ideas presented in
[10] to the
systems with
discontinuous coefficients.

**3. Schedule**

Although the development of a general existence and uniqueness
theory for quasilinear hyperbolic problems with
discontinuous coefficients is one of the goals of the PI's research,
it is unrealistic to expect that within the two years of the duration
of this
grant the PI and the students will be able to complete the
theory. A realistic goal is to study equations
(1) and (2)
and gain an insight into the properties
of the solutions that can be generalized to a larger class of
equations.

The PI on
the project is Canic.
She has extensive research experience in hyperbolic conservation
laws, in conservation laws that change type, and in modeling of
physical processes by conservation laws.
Her accomplishments have been recently recognized by the
UH 2001 Associate Professor Research and Scholarship Award.
The PI has experience in mentoring undergraduate students (3), graduate
students (2) and post-doctoral visitors (1). In addition, this summer
she ran a Research for Undergraduates program at UH.
One of the students (Burns)
who participated in the program is expected to graduate in the Fall
and continue working on a problem related to this project
as graduate student in Mathematics.
Another student (Sharma), currently a graduate student in Mechanical
Engineering,
is expected to change majors and begin working with the PI on the
numerical simulation and error analysis of the model equations with
discontinuous *G*_{0}.

**D.
Institutional
commitment and sources of
additional support**

The
Mathematics Department employs graduate students as Teaching
Fellows,
and this would be the normal source of support for students
engaged in the project.
The requested budget will partially support two graduate students
providing reduced teaching load to expedite the time to degree.
The students have already been recruited.
The PI has directed students who are now employed in academia
and in industry.
In addition,
the Mathematics Department is planning to submit a VIGRE
(Vertical Integration of Research and Education) proposal in Fall 2001.
If awarded, the VIGRE support would
provide supplemental funds for two graduate students and one
postdoctoral visitor.

The PI has external support which expires this summer. A renewal for the grant will be submitted in the Fall. If awarded, the grant will provide partial travel support and partial summer salary.

**E.
Student
involvement and training opportunities
in science and engineering**

From the
very beginning of this project students and postdoctoral
visitors
have been involved in numerical simulation, visualization,
and the development of a new theory to study the underlying model
equations.
Two undergraduate students graduated (Pritts, Roy) and continued their
studies at the first rate universities.
Post-doctoral visitor Kim is moving
to a tenure-track position to continue her career in teaching and
research.
In addition,
a group of the most talented undergraduate students from all over the
US has been involved this summer in the NSF-sponsored REU program.
The topics of their projects originated from the blood-flow problem
described here. All of them will continue their studies as graduate
students.
Student problems that cover a broad range of interests
have been formulated and offered by the PI to the graduate student
community at UH
(two students have already been recruited).

**Student project 1.** Analysis of
weak solutions to conservation
laws
with discontinuous coefficients and a study of stability of the related
singular products with the focus on the blood flow problem.

**Student project 2.**
Numerical simulation of equations (1)
and (2)
with discontinuous coefficient *G*_{0},
using different numerical methods. Analysis of the influence of the
artifical numerical effects on the calculation of the solution. Error
analysis.

Student project 3.
Modeling on the behavior of the prostheses.
Development of a more sophisticated model and comparison with the
measurements obtained by Dr. Chandar at UT Austin.
Inclusion of the porosity of the stent in the model equations.

Student project 4.
Development of a simplified two-dimensional
model through curved vessels.
This entails studying fluid-structure interactions between the
stent/aorta and the blood flow.

Student project 3.

Student project 4.