Research (updated fall 2007)
My primary field of research is applied mathematics,
specifically the sub-disciplines of scientific computation and
numerical analysis. In the fast-developing field of scientific
computation, there are three main research thrusts that together
aim to allow mathematical insight and innovation to make a
difference in the physical, biological and engineering sciences:
the incorporation of increased realism into mathematical
modeling systems; the development of increasingly robust and
accurate numerical methods for solution of mathematical models;
and the development of computational algorithms to allow for
efficient solution of these problems on increasingly-larger
computational hardware.
I am an active member of the scientific computation community,
having organized a plenary panel discussion at the 2007 SIAM
Conference on Computational Science and Engineering on Research
Directions for the CS&E community [16]. In my own research,
I have contributed to each of the aforementioned challenges
through investigations on the algorithmic development and
computational solution of coupled multi-physics systems of
partial differential equations, arising in real-world
applications ranging from fusion energy to cosmological
astrophysics and materials science. In these areas, I strive to
explore three fundamental applied mathematics issues: (1) the
accurate modeling of physical systems involving disparate time
and space scales, (2) the development and use of highly-accurate
and efficient time evolution algorithms for stiff multi-rate
problems, and (3) the investigation of discretization and
solution methods that retain constraint-preserving properties of
PDE systems. I am pursuing these investigations in the context
of a number of highly collaborative physics applications: the
modeling of solid-state phase transformations and thermodynamics
in shape memory alloys, simulations of macroscopic stability and
refueling of fusion plasmas, calculations determining the
physical processes underlying core-collapse supernova phenomena,
and investigations of reionization physics within the early
universe.
MULTI-SCALE MATHEMATICAL MODELING AND SIMULATION
In my thesis research, I developed a new mathematical model for
simulating multi-scale thermodynamic phase transformation
processes in shape memory alloys (SMA), based on continuum-level
thermodynamic PDE modeling of their behavior. Such alloys are
already at the forefront of research in materials science and
engineering applications, as they may be used in the production
of microscopic devices with the ability to perform work without
the use of moving parts. Such behaviors are made possible by
the understanding and control of phase transformation processes
that occur at the atomistic scale of these materials, in which a
variety of small-scale crystalline transformations coordinate to
allow large-scale nonlinear thermodynamic actions, a diagram of
which is presented in Figure 1.
Figure 1:
Shape Memory Effect and Hysteresis: when cooled,
austenite transforms into twinned martensite, which is easily
deformed under stress. Upon heating, the deformed martensite
returns to the austenitic configuration. Hysteresis in these
materials provides a measure of the overall energy absorbed by
the material during a full loop of the forward and reverse phase
transformations.
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On the macroscopic continuum level, these phase transformations
may be modeled through the use of non-convex free energy
potentials, resulting in temperature-dependent hysteresis,
super-elasticity, and other nonlinear material behaviors.
Mathematically, the use of non-convex Helmholtz free energy
potentials in the macroscale model results in a PDE system of
the form
where
is the macroscopic material deformation,
is the material temperature,
 ,
is the material-dependent Helmholtz free energy density,
 ,
 ,
and
are various material-dependent parameters, and
and
provide external body forces and heat source interactions with
the surrounding environment. Of particular interest in such
models of shape memory materials is that unlike standard models
in linear or even nonlinear elasticity, the potential
is non-quasiconvex, giving rise to variational PDE formulations
that do not satisfy weak lower semicontinuity, thus precluding
the use of standard PDE existence theory and computational
solution techniques toward their solution. For simplified
one-dimensional model systems of these materials we developed
stochastic computational techniques for computing steady-state
averaged crystalline properties [1, 2, 3, 4], and I derived
physically accurate deterministic PDE models for SMA wires [12].
In addition to deriving qualitatively and quantitatively
accurate continuum-level models for SMA wires, I developed an
efficient and robust computational solution approach for
approximating solutions to the system (1), based on natural
parameter continuation approaches employing the material
viscosity
to regularize the modeling system, thereby allowing the
computational solution of the non-convex PDE model to
successfully reproduce material phase transitions and
time-dependent computations of thermodynamic SMA processes [6].
I then used the resulting numerical model to design techniques
for damping vibrational energy based on thermal actuation
[7, 8, 9].
While these advances have proven fruitful in the modeling and
design of one-dimensional SMA wires, there remains significant
work on extending such continuum-level descriptions to realistic
models for slabs and solids, as these two and three-dimensional
models require construction of very complex multi-dimensional
free-energy potentials. Therefore in addition to investigations
of free energy potentials for higher-dimensional models of SMA
materials, current research in this area attempts to span the
gap between macroscopic-scale systems and atomistic material
dynamics through the use of multi-scale modeling approaches. In
my future investigations in this area, I plan to follow this
second path of realistic material thermodynamics through the
development of multi-scale and multi-model approaches for SMA
thermodynamics, along with the accompanying
numerical techniques required for multi-scale computational
simulations.
ALGORITHMIC DESIGN AND SOFTWARE FOR ROBUST AND EFFICIENT
MULTI-PHYSICS PROBLEMS
As a post-doctoral researcher both at Lawrence Livermore
National Laboratory and at the University of California at San
Diego, I have been investigating the use of fully implicit
computational approaches for time evolution of large-scale,
multi-rate PDE systems. These efforts have been in the
context of resistive magnetohydrodynamics (MHD), arising in
studies of fusion plasma stability and refueling, and radiation
hydrodynamics (RHD), used to model core-collapse supernova
explosions. These applications involve the solution of coupled
PDE systems for modeling multiple interacting physical
processes. For example, the resistive MHD model couples the
compressible viscous Euler equations for modeling plasma
hydrodynamics,
with the low-frequency Maxwell equations that model the
evolution of the surrounding electromagnetic fields,
Here
is the density,
is the velocity,
is the magnetic induction,
is the electric current,
is the electric field,
is the total energy,
is the pressure,
is the plasma temperature,
is the plasma viscosity, and
and
correspond to the coefficients of viscosity and resistivity.
Similarly, in the model for radiation hydrodynamics, additional
PDE models for nonlinear radiation diffusion are coupled with
either the hydrodynamical system (2), or even the full MHD
system (2)-(3) for increased modeling accuracy.
A distinct feature of these type of multi-physics models is that
each variable or group of variables will often evolve on
drastically different time and space scales. As a result,
standard-practice explicit and operator-split time integration
techniques fail to efficiently and accurately track some of the
more slowly-evolving processes of interest, such as the
macroscopic stability of a fusion plasma or the energy
partitioning within a collapsing star. Moreover, for any
consistent mathematical system to successfully model problems
with large disparities in scale, it requires
spatial resolutions that may only be tractable on the world's
largest supercomputers, along with novel computational
algorithms that may efficiently operate on such large scale
machines.
In my postdoctoral work, I have investigated the use of
high-accuracy fully implicit approaches for solving the MHD and
RHD systems of equations. Due to the disparity of time scales
involved in such problems they are typically very stiff,
requiring advances in nonlinear solvers and preconditioning
approaches that allow the methods to efficiently `step over'
their stiff transient components while still accurately tracking
the more slowly-evolving quantities of interest. In
collaboration with computational physicists from Princeton
Plasma Physics Laboratory and computational mathematicians from
LLNL, I have successfully developed implicit simulations of
resistive MHD processes for stability and refueling studies [5,
15]. I have also introduced new preconditioning approaches for
these type of fully implicit methods, and am currently assessing
their efficacy and scalability on multi-physics problems of this
type [11, 13, 14]. Moreover, in collaboration with
astrophysicists from the State University of New York at Stony
Brook, we are nearing completion of the first fully implicit
approaches for RHD simulations of core-collapse supernova.
I plan to continue these investigations over the next 4 years
under a recently-awarded DOE SciDAC grant, moving these fields
ahead in both computational efficiency through scalable
computational approaches, and in the incorporation of additional
physical processes and constraint-handling properties of the
associated models. As exemplified in the MHD system (2)-(3),
many multi-physics PDE models involve both evolution and
constraint equations that should be simultaneously satisfied
for their accurate solution. In many numerical models these
constraints fail at the discrete level, though certain
algorithms retain such quantities given a satisfactory initial
state. I intend to examine these issues in detail, specifically
in regards to the development of preconditioning techniques that
retain such desirable algorithmic properties, while also
allowing efficient solution at large computational scale.
MULTI-SCALE, MULTI-PHYSICS MODELING AND PETASCALE SCIENTIFIC
COMPUTING
For the last two years I have also actively collaborated with
researchers in the Center for Astrophysics and Space Sciences at
UCSD, addressing fundamental questions regarding reionization of
the early universe following the Big Bang. Modern observational
astronomy has offered some glimpses into the earliest precursors
of galactic formation, giving rise to debate as to the
underlying physical processes that formed our universe. For
these studies, we are developing coupled
radiation-hydrodynamic-chemical kinetics simulations that will
attempt to test some of these theories against experimental
observation. As with the energy research described above, these
models involve the coupling of a large number of physical
processes, including the compressible Euler equations for gas
motion, nonlinear radiation diffusion equations for
multi-frequency radiation transport, chemical kinetics models
for tracking the ionization state of elemental species, as well
as a model for gravitational acceleration due to star clustering
and galactic formation. This problem therefore combines all of
the aforementioned challenges facing scientific computation:
consistent modeling of the coupled physical processes present in
cosmic reionization, analysis of the well-posedness of the
resulting PDE modeling system, the use of high-fidelity
numerical methods for accurately approximating processes at
large (cosmic) and ``small'' (solar) scales, and the invention
of solution strategies for efficiently solving the resulting PDE
model system on some of the largest NSF supercomputers.
In this effort I have been working on nearly all of these
fronts: deriving a coupled PDE modeling system that will both
capture the lowest-order physical processes while lending itself
to efficient numerical solution; deriving an accurate
time-evolution technique for the coupled radiation transport and
chemical kinetics processes occurring on multiple time and space
scales, and implementing computational algorithms based on
these approaches designed to utilize next-generation petascale
computing hardware. Progress is still under way in this
endeavor, with initial results described in [10], and I hope to
continue this research for the next four years under a
recently-proposed NSF grant.
REFERENCES
[1] D.D. Cox, P. Kloucek, and D.R. Reynolds. The computational
modeling of crystalline materials using a stochastic variational
principle. Lecture Notes in Computer Science, 2330:461--469, 2002.
[2] D.D. Cox, P. Kloucek, and D.R. Reynolds. A
subgrid projection method for relaxation
of non-attainable differential inclusions. In Proceedings: ENUMATH
2001 European Conference on Numerical Mathematics, Berlin, 2002.
Springer-Verlag.
[3] D.D. Cox, P. Kloucek, and D.R. Reynolds. On the asymptotically
stochastic computational modeling of microstructures. Future
Generation Computer Systems, 20:409--424, 2004.
[4] D.D. Cox, P. Kloucek, D.R. Reynolds, and P. Solin.
Stochastic relaxation of variational integrals with non-attainable
infima. In Proceedings: ENUMATH 2003 European Conference on
Numerical Mathematics, Berlin, 2004. Springer-Verlag.
[5] D.E. Keyes, D.R. Reynolds, and C.S. Woodward. Implicit solvers
for large-scale nonlinear problems. Journal of Physics: Conference
Series, 46:433--442, 2006.
[6] P.Kloucek and D.R. Reynolds. On the modeling of nonlinear
thermodynamics in SMA wires. Computer Methods in Applied Mechanics
and Engineering, 196:180--191, 2006.
[7] P. Kloucek, D.R. Reynolds, and T.I. Seidman. On thermodynamic
active control of shape memory alloy wires. Systems
& Control Letters, 48, 2003.
[8] P. Kloucek, D.R. Reynolds, and T.I. Seidman. Thermal
stabilization of shape memory alloy wires. R.C. Smith, editor, In
Smart Structures and Materials 2003: Modeling, Signal Processing,
and Control, volume 5049 of Proc. of SPIE, 2003.
[9] P. Kloucek, D.R. Reynolds, and T.I. Seidman. Computational
modeling of vibration damping in SMA wires. Continuum Mechanics
and Thermodynamics, 16:495--514, 2004.
[10] M.L. Norman, G.L.Bryan, R. Harkness, J. Bordner,
D. Reynolds, B. O'Shea, and R. Wagner. Petascale Computing:
Algorithms and Applications, chapter Simulating cosmological
evolution with Enzo. CRC Press, 2007.
[11] D.R. Reynolds. On the improvement of splitting methods for
fully implicit systems of equations. (in preparation).
[12] D.R. Reynolds. A Nonlinear Thermodynamic Model for Phase
Transitions in Shape Memory Alloy Wires. PhD thesis, Rice
University Dept. of Computational and Applied Mathematics, 2003.
[13] D.R. Reynolds, R. Samtaney, and C.S. Woodward. Operator-based
preconditioning of stiff hyperbolic magnetohydrodynamics
systems. (submitted).
[14] D.R. Reynolds, R. Samtaney, and C.S. Woodward. Physics-based
preconditioning of resistive MHD systems. (in preparation).
[15] D.R. Reynolds, R. Samtaney, and C.S. Woodward. A fully implicit
numerical method for single-fluid resistive magnetohydrodynamics.
Journal of Computational Physics, 219:144--162, 2006.
[16] D.R. Reynolds and R. Szypowski. Panel discusses research
directions and enabling technologies for the future of CS&E.
SIAM News, May 2007.
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