Title: Numerical Simulation and Control of Sublimation Growth of SiC Bulk Single Crystals: Modeling, Finite Volume Method, Analysis and Results
Abstract:
A transient mathematical model for the sublimation growth of silicon
carbide single crystals (SiC) by physical vapor transport is
presented. Continuous mixture theory is used to obtain balance
equations for energy, mass, and momentum inside the gas phase. In
particular, reaction-diffusion equations are deduced. Heat conduction
is treated inside solid materials. Heat transport by radiation is
modeled via the net radiation method for diffuse-gray radiation to
allow for radiative heat transfer between the surfaces of
cavities. The model includes the semi-transparency of the single
crystal via a band approximation. Induction heating is modeled by an
axisymmetric complex-valued magnetic scalar potential that is
determined as the solution of an elliptic problem. The resulting heat
source distribution is calculated from the magnetic potential.
The transient heat problem is discretized in time by the implicit
Euler method and in space by the finite volume method. Existence and
uniqueness of the discrete solution is shown as well as a maximum
principle in a simplified case. A control problem for the optimization
of the gradient in the gas phase is considered, as this is relevant to
the crystal growth process. Based on a known existence result for the
semilinear, but still nonlocal, case, the existence of an optimal
solution as well as necessary optimality conditions.
The presented numerical simulations are conducted in an axisymmetric
setting. They constitute transient investigations of control
parameters affecting the temperature evolution during the heating of
the growth apparatus. It is studied how the temperature difference
between source and seed, which is highly relevant to the growth
process, is related to the measurable temperature difference between
bottom and top. Results concerning the time lack between the heating
of the surface of the source powder and the heating of its interior
are considered. Finally, numerical optimization is used to determine
the control parameters frequency, power, and coil position for the
radio frequency (RF) induction heating with the objective to minimize
a functional, tuning the radial temperature gradient on the single
crystal surface as well as the vertical temperature gradient between
SiC source and seed, both being crucial for high-quality growth.
Title: Hybridized globally divergence-free LDG methods for the Stokes problem
Abstract:
We devise and analyze a new local discontinuous Galerkin (LDG)
method for the Stokes equations of incompressible fluid flow.
One of the main features of the method is that it has the
smallest number of degrees of freedom among all other DG
methods for the Stokes problem and yet it is locally
conservative and optimally convergent. Moreover, it yields
globally divergence-free approximations to the velocity. It
is obtained by applying an LDG method to a vorticity-velocity
formulation of the Stokes equations and then by hybridizing
the resulting formulation.
Title: Wave Confinement: Modeling Short Acoustic Pulses as Nonlinear Solitary Waves
Abstract:
A new computational method, Wave Confinement, is described. The
method has been shown to efficiently treat thin acoustic pulses
in complex time domain wave equation problems, allowing them to
be propagated over arbitrarily long distances with no spreading
due to numerical errors. The method involves only a fixed, Eulerian
computational grid with no Lagrangian markers. On this grid,
the pulses are only 2-3 cells thick, and are treated as a type
of weak solution which obey a nonlinear difference equation
derived from the wave equation, as opposed to a conventional
finite difference approximation. As such, the pulses are,
essentially, nonlinear solitary waves that live on the lattice.
An important feature is that, in spite of the non-linearity, the
computed pulses can pass through each other with no phase shift
or amplitude exchange. This is necessary because the equation
that is being simulated is the linear wave equation.
Title: Multiscale Curve and Strip Constructions with Applications
Abstract:
We present a fast algorithm for detecting and characterizing a
cloud of points that are concentrated around a curve in a
D-dimensional Euclidean plane. The algorithm characterizes the
cloud data by detecting the underlying curve, separating between
a stable set and a deviating set (outliers) and estimating the
local variances of the stable set around the underlying curve.
We have adapted this algorithm to analyze DNA array data from
ChIP-chip experiments as well as expression profiling microarray
data. We use the algorithm for both purposes of normalization
and for ranking and identifying enriched sites (or differentially
expressed genes). Our methods accommodate the unique
characteristics of ChIP-chip data, where the set of
immunoprecipitation-enriched segments is large, asymmetric and
whose proportion to the whole data varies locally.
We establish some estimates for the performance of our
algorithm and exemplify its efficiency with high-dimensional
data by applying it to pixel neighborhoods of various images.
Here, the ``deviating points'' detected by the algorithm
correspond to edges in the original image.
This is a joint work with Joseph McQuown and Bud Mishra.
The ChIP-chip analysis is also joint with Alexandre Blais and
Brian David Dynlacht.
Title: Fast Sweeping Methods for Static Hamilton-Jacobi Equations
Abstract:
Hamilton-Jacobi equations arise in many applications such as geometrical optics,
crystal growth, path planning, and seismology. Viscosity solutions of these
nonlinear differential equations usually develop singularities in their derivatives.
In this talk, we will present several fast sweeping methods which are based on the
Godunov Hamiltonian or the Lax-Friedrichs Hamiltonian to approximate the viscosity
solution of convex or arbitrary static Hamilton-Jacobi equations in any number of
spatial dimensions. We solve for the value of a specific grid point in terms of its
neighbors, so that a Gauss-Seidel type nonlinear iterative method can be utilized.
Furthermore, by incorporating a group-wise causality principle into the Gauss-Seidel
iteration by following a finite group of characteristics, we have an
easy-to-implement, sweeping-type, and fast convergent numerical method. For the
sweeping methods based on Lax-Friedrichs Hamiltonian, unlike other methods based on
the Godunov numerical Hamiltonian, some computational boundary conditions are needed
in the implementation. We give a simple recipe which enforces a version of discrete
min-max principle. Some convergence analysis is done for the one-dimensional eikonal
equation. Extensive 2-D and 3-D numerical examples illustrate the efficiency and
accuracy of the new approaches.
Title: Computational Modeling of Fiber-Reinforced and Particulate Composite Materials
Abstract:
This presentation discusses the development of a computational
basis for modeling micro- and macroscopic behavior of composite
materials. A significant feature of the work is the capability
to directly simulate the microstructure of micro-porous and
composite materials by numerous non-overlapping inclusions
(fibers or particles) with interphases and holes (pores) of
arbitrary size.
In our model the fibers are idealized as uniform, parallel,
infinite circular cylindrical inclusions (two-dimensional case)
and the particles are assumed to be spherical. In general,
the fibers and particles can be distributed randomly, they
can all have different elastic properties, and the bond
between them and the surrounding material matrix can be
imperfect. For example, the inclusions may be connected to
the material matrix through arbitrarily thin interphase layers.
Our approach allows one to incorporate in a model the effect
of a free boundary as well as some nonlinear effects due to
localized slip and/or separation of the interfaces between
the fibers and the material matrix, and cracking inside the
matrix or reinforcements. Time-dependent effects due to
transient heat conduction with thermal stresses or to viscoelastic
behavior of the material can be also considered.
Computational realization of the model includes the use of
fast methods (based on fast multipole acceleration) that make
it possible to study detailed interactions among thousands
of particles (or fibers) and pores. The method allows for
accurate calculation of the displacement, stress, and temperature
fields anywhere within the material, including the inclusions
and interphases. The overall properties of an equivalent
homogeneous material can be found directly from the properties
of microstructural elements.
Computer simulations provide a means to test and compare
major effective-medium theories and will allow us to make more
realistic evaluations of the microscopic behavior and overall
properties of micro-porous and composite materials than is
currently feasible. Such simulations could enhance understanding
of failure mechanisms in materials and provide valuable information
necessary to design composite materials with specified physical
properties.
Title: The human eye tear film: Modeling and analysis
Abstract:
Problem formulation, modeling, and resultant fits to data are
presented of a single layer thin complex fluid film representing
the human eye tear film. The model parameters are fit to
viscometric shear data of extracted tear fluid and the resultant
system is analyzed under drainage conditions. The lubrication
approximation is used and its validity is analyzed. Asymptotic,
analytic and computational results are presented. The work is
important for understanding the behavior of the tear film in both
normal and "dry" eyes.
Title: Bifurcation and asymptotic stability of free boundary problems
Abstract:
I shall consider several examples of free boundary problems, with
spherical free boundary. The examples include electrically charged
liquid droplets, and several tumor models. I will state results
on existence of bifurcation branches of solutions which break
the symmetric structure of the spherical solutions. I shall
also discuss the asymptotic stability of the spherical and
non-spherical solutions.
Title: Parallel Simulations of the Linear Boltzmann Equation for Models in Microelectronics Manufacturing
Abstract:
Production steps in the manufacturing of microelectronic devices
involve gas flow at a wide range of pressures. We develop a
kinetic transport and reaction model (KTRM) based on a system of
time-dependent linear Boltzmann equations. These kinetic equations
have the property that velocity appears as an independent variable,
in addition to position and time. A deterministic numerical solution
for realistic three-dimensional application problems requires
the discretization of the three-dimensional velocity space,
the three-dimensional position space, and time.
We design a spectral Galerkin method to discretize the velocity space
by specially chosen basis functions. The basis functions in the
expansion lead to a system of hyperbolic conservation laws with
constant diagonal coefficient matrices for each of the linear
Boltzmann equations. These systems of conservation laws are solved
using the discontinuous Galerkin finite element method.
As an application example, we simulate chemical vapor deposition at
the feature scale in two and three spatial dimensions and analyze
the effect of pressure.
Finally, we present parallel performance results which indicate that
the implementation of the method possesses excellent scalability
on a Beowulf cluster with a high-performance Myrinet interconnect.
Title: A Dual Method for Total Variation-Based Image Restoration
Abstract:
This talk will describe a computational method for the
inverse problem of edge-preserving image restoration. Total Variation (TV)
regularization removes noise from an image while retaining
its edges. We solve an equivalent dual version of the TV problem.
Images restored using this dual approach will have crisp edges
(discontinuities), whereas images recovered under earlier primal
methods may contain blurred edges. Joint work with Tony F. Chan, Pep Mulet,
and Lieven Vandenberghe.
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Title: Elliptic Variational Problems on Constricted Networks with Applications to Ginzburg-Landau Theory
Abstract:
I will analyze variational problems set on a network of thin constricted
tubes. In the asymptotic regime where the tubes collapse to a graph, one
can identify a one dimensional variational problem with interesting
natural boundary conditions at the nodes which in particular encourage
jumps in the (limit of) minimizers across the various branches of the
graph. An application to tunneling across weak links in a superconductor
will be discussed.
Title: A multiscale model for the dynamics of solids
Abstract:
We present a multiscale method for coupling atomistic and continuum models
of solids. Both models are formulated in the form of physical conservation
laws, and the coupling is achieved through balancing the fluxes. Error
estimate is provided, which is of great help in choosing the size of the
macro grid and microscale system. Finally we shall show some applications
including phase transformation, and dynamic fracture mechanics.
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Title: On the Rate of Convergence of Finite-difference Approximations for Bellman Equations with Constant Coefficients
Abstract:
We consider elliptic Bellman equations with coefficients independent
of the variable x. Error bounds for certain types of finite-difference
schemes are obtained. These estimates are sharper than those of earlier results.
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