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Multiscale Analysis
My research in multiscale analysis primarily focuses on the development and analysis of
peridynamics as a multiscale material model. Peridynamics is a nonlocal formulation of continuum
mechanics. As a multiscale material model, it sidesteps many of the difficulties encountered
in coupling nonlocal atomistic models with local continuum models.
Peridynamics
The peridynamic theory of continuum mechanics belongs to the class of
microcontinuum theories defined by generalizing the local force
assumption to allow force at a distance, which necessarily
introducs a length-scale. In the classical
continuum context, a local force means that only continuum
points in direct contact can exert a force on each other.
In contrast, peridynamics employs an integral operator to sum forces avoiding
the use of stress/strain fields in its equation of motion. Instead,
the material behavior in peridynamics is specified by nonlocal force interactions,
assumed to be a function of the positions of the continuum
points. No assumption are made on the continuity or differentiability
of the displacement field. Because the displacement field is
not assumed even weakly differentiable, peridynamics can be employed for
deformation that does not satisfy the smoothness assumptions of
classical continuum mechanics, e.g., fracture or fragmentation.
For more on the theory of peridynamics, visit the webpage of its inventor,
Stewart Silling.
A particular discretization of the peridynamic model has the same computational structure as
classical molecular dynamics. I am the principal author of the peridynamic model implemented within Sandia's
massively parallel molecular dynamics code, LAMMPS. Visit my
software page for more information.
Atomistic-to-Continuum Coupling
The deformation and failure of many engineering materials are inherently multiscale processes.
Models for such processes frequently call for decomposition of the material domain into atomistic and continuum
subdomains, where the continuum subdomain is modeled via a finite element analysis.
This coupling enables a continuum calculation to
be performed over the majority of a domain while limiting
the more expensive atomistic simulation to some small subset of the domain.
The treatment of the boundary between these subdomains, or handshake region,
is frequently what distinguishes one atomistic-to-continuum coupling method from another. In this transition
region, approximations are made such as treating finite element nodes as atoms, or vice-versa, to accommodate
the incompatibility between a non-local atomistic description and a local finite element description.
This work was funded by a three year DOE award under the Office of Science's Multiscale
Mathematics program.
The project is a collaborative, multi-institutional
effort led by the Sandians
Rich Lehoucq and
Pavel Bochev.
Other participating personnel are:
The Institute for Computational Engineering and Sciences (ICES) at The University of Texas at Austin hosted the
second Workshop on Atomistic-to-Continuum (AtC) Coupling Methods, April 2-3, 2007.
The first AtC Workshop was organized by the
Computer Science Research Institute (CSRI) at Sandia National Laboratories on March 20-21, 2006, in Alburquerque, NM.
Iterative Methods
My research in iterative methods focuses primarily on the development of robust
iterative solvers for ill-conditioned linear systems, especially long
sequences of such systems.
Krylov Subspace Recycling
Many problems in engineering and physics require the solution of a large sequence of linear systems.
We can reduce the cost of solving subsequent systems in the sequence by recycling information from
previous systems. The solvers GCRO-DR and GCROT accomplish this. For some problems, the iteration count
required to solve a linear system can be cut by a factor of two.
For a Matlab version of GCRO-DR, see my software page. I am currently developing recycling solvers
within the Trilinos framework.
Domain Decomposition
My research in domain decomposition methods includes generalized domain bridging and
the Finite Element Tearing and Interconnecting (FETI) family of methods.
Mesh-Tying
In the case where two domains sharing a common curved interface are meshed independently,
the domains will generally have an inconsistent description of that boundary. A minimal
requirement for any proposed mechanism to tie these two meshes together is that the resulting
finite element formulation pass a first-order patch test, whether or not the two discretizations
of the shared boundary coincide. Along with Pavel Bochev and Louis Romero, I have developed a
novel computationally efficient Lagrange-multiplier method for tying together independently
meshed subdomains with
non-coincident contact boundaries in two dimensions.
KKT Preconditioners for FETI Methods
Preconditioners for KKT (Karush-Kuhn-Tucker) linear systems have been studied extensively.
The one-level finite element tearing and interconnecting method produces
a linear system of this form. In the fourth chapter of my
Ph.D. dissertation,
I show new connections between
recently proposed KKT preconditioners and solvers and the one-level FETI method. These
connections provide a new perspective on the analysis of FETI preconditioners by leveraging
work for KKT systems. In particular, they provide a means of bounding the eigenvalues
of preconditioned FETI systems, and thus the rate of convergence of an iterative solver.
This theoretical framework gives a means to analyze the usefulness of improvements to
FETI preconditioners.
Chromatography
Chromatography is a family of analytical chemistry techniques for the separation
of mixtures. In gas chromatography, a chemical sample separates into its constituent components
as it travels along a long thin column. In a traditional chromatograph, the column has a
circular cross section. With the advent of MEMS technology, columns can be miniaturized
to fit on a single chip. Unfortunately, these columns cannot be manufactured to have a
circular cross-section. With Louis Romero, Joshua Whiting, and Joe Simonson, I am working
to analyze the effects of the cross-section geometry on performance, and to develop optimized
designs within the constraints of MEMS manufacturing limitations.
Previous Research
As a masters student in the Department of Computer Science at Virginia Tech, my work was interdisciplinary
between the departments of physics and computer science.
As an undergraduate at Virginia Tech earning dual degrees in the departments of Computer Science and Physics, I participated
in undergraduate research in both departments.
- Undergraduate Thesis: The Construction and Analysis of Factorial Experiments: Application to
Tribochemical Vapor Deposition (1998)
- Tribochemical Vapor Deposition - A New Deposition Technique: Poster at the 1997 Gordon Research
Conference on Solid State Studies in Ceramics (with
Jimmy Ritter) (1997)
- Virginia Tech Physics Department: Tribochemical Vapor Deposition (TCVD) Experiment (1996-98)
- Virginia Tech Computer Science Department:
Learning in Networked Communities Project on Collaborative Education (1996)
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