Introduction
We propose to develop advanced parallel algorithms and software
for simulating complex flows with dynamic interfaces. The development of
scalable, parallel high-accuracy algorithms for simulating such
flows
poses enormous challenges, particularly on systems with thousands of
processors. We will use the resulting tools for microstructural
simulation of blood flow in artificial heart devices. This application
provides an excellent testbed for the methods we develop:
simulation-based artificial organ design is extremely computationally
challenging and of critical societal importance.
Flows with dynamic interfaces arise in many fluid-solid and
fluid-fluid interaction problems, and are among the most difficult
computational problems in continuum mechanics. Examples abound in the
aerospace, automotive, biomedical, chemical, marine, materials, and
wind engineering sciences. These include large-amplitude vibrations of
such flexible aerodynamic components as high aspect ratio wings and
blades; flows of mixtures and slurries; wind-induced deformation of
towers, antennas, and lightweight bridges; hydrodynamic flows around
offshore structures; interaction of biofluids with elastic vessels;
and materials phase transition problems.
Blood Flow
Our specific interest is in modeling the flow of blood, which is a
mixture of interacting gel-filled solid cells and fluid
plasma. Current blood flow models are macroscopic, treating the
mixture as a homogeneous continuum. Microstructural models resolve
individual cell deformations and their interactions with the
surrounding fluid plasma. Because of the computational difficulties
of resolving tens of thousands of dynamically deforming cellular
interfaces, no one to date has simulated realistic blood flows at the
microstructural level. Yet such simulations are necessary in order to
gain a better understanding of blood damage - which is central to
improved artificial organ design - and for the development of more
rational macroscopic blood models.
Parallelism
Parallel flow solvers on fixed domains are reasonably well understood.
In contrast, simulating flows with dynamic interfaces is much more
difficult. The central challenges are to develop numerical algorithms
that stably and accurately couple the moving fluid and solid domains
and resolve the deforming interfaces, and geometric algorithms for
computing the resulting dynamic meshes. The associated dynamic
data structures are particularly troublesome on highly parallel
computers, which are made necessary by the complexity of many
applications. Most current methods approach the difficulties of
dynamic interfaces by computing the flow on a fixed, regular grid, and
incorporate the effect of the dynamic interface through some type of
constraint. Parallelizing these methods is relatively straightforward,
since the flow is computed on a fixed grid. However, the resulting
fixed resolution is a serious disadvantage if one wants to vary
resolution sharply within the grid. This is the case for example when
local interfacial dynamics are critical, as in blood flow or phase
change problems.
Our Approach
Our approach is radically different. We will treat the fluid and solid
domains as collections of grid points, with associated meshes, that
evolve over time, and devise numerical algorithms that couple the
fluid and solid together seamlessly. We will attack the difficulty of
generating and managing a constantly evolving mesh by creating
fundamentally new highly parallel and scalable algorithms for the
convex hull, Delaunay triangulation, meshing, and partitioning
components. Our preliminary 2D work demonstrates that the resulting
geometric computations can be made very cheap compared to numerical
computations. Despite the conventional wisdom on parallel dynamic
mesh methods, we believe that - with careful attention to fundamental
algorithmic issues - flow simulations on constantly evolving domains
can be made to scale to the thousands of processors that characterize
multi-teraflop systems.
Although artificial organs form the target applications, our models
and algorithms will be applicable to a wide variety of other equally
important treatments for disease. Since blood perfuses all of the
body's organs, accurate simulation of the transport and interactions
of blood cells will provide a foundation for better understanding of
vital organ function. This will strengthen our insight into, for
example, strokes, atherosclerosis and thrombosis, sickle cell disease,
and the development of blood additives to improve transport and treat
hypertension.
The research component in computer and computational science will
benefit a much wider community of scientists and engineers. The
computational algorithms and software we create will be more widely
applicable to a variety of fluid-solid and fluid-fluid interaction
problems. More generally, the core parallel computational geometry
kernels - convex hull, Delaunay triangulation, coarsening/refinement,
partitioning - provide generic support for the geometric
computations underlying many dynamic irregular problems. We will
create and publically distribute a portable library of efficient
implementations of these algorithms. Much as the PETSc library has
greatly simplified the task of programming parallel PDE solvers by
providing many of the necessary numerical kernels, we envision a
library of parallel geometric kernels being of great benefit
across a wide range of dynamic irregular scientific computing
problems.
Team
We propose a broad-based, interdisciplinary program integrating
research and education. The program brings together applied
mathematicians, biochemists, bioengineers, computational fluid
dynamicists, computer scientists, continuum mechanicists,
hemorheologists, numerical analysts, and transplant surgeons. The team
combines Carnegie Mellon's leadership in computer and computational
science with the University of Pittsburgh Medical Center's world-class
programs in artificial organs and transplant surgery. This
interdisciplinary environment will foster the education of 11 graduate
students and a group of undergraduates involved in the research. These
students will be part of a new program at CMU in Computational Science
and Engineering that we are in the process of establishing. The
proposed project will also be part of that program, and we believe
will serve as an archetype of how applications, computational,
computer, and mathematical scientists can work together to tackle
societal problems that cannot be addressed solely from the vantage of
any one discipline. Moreover, we intend to communicate our work to the
broader public (as we have done in the past), in the process
demonstrating how high end computing can contribute to improving the
health of our society.
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