epshteyn (at) math.utah.edu
)August 19
Welcome Back Meeting
September 9
Speaker:
Masashi Mizuno, Department of Mathematics, Nihon University, Japan
Title: Grain boundary motion with time-dependent misorientation and mobility effects
Abstract: In my talk, I will present a model of grain boundary motion with
time-dependent misorientation and mobility effects. Dynamics of grain
boundaries play an important role in defining the materials properties of
polycrystals.
The model, considered in the talk, is given by the system of partial
differential equations, and is derived using energetic variational
principle. Next, I will present the analysis of the model for the grain
boundary motion. The key ingredient of the analysis is the so-called
monotonicity formula of Huisken's type. Moreover, to treat the
time-dependent mobility, we use the modified Gauss kernel to the
monotonicity formula.
September 23
Speaker:
Tom Alberts, Department of Mathematics, University of Utah
Title: Random Matrix Theory for Homogenization of Composites on Graphs
Abstract: I will discuss the random matrix theory behind two-component random
resistor networks on general graphs. This involves random submatrices of
the graph's Gamma projection operator, with the particular realization of
the submatrix determined by the disorder of the conductances. Certain
combinations of graph symmetries together with different models for the
random conductances lead to exactly computable spectral statistics. Our
recent results lead to exact spectral statistics for the uniform spanning
tree model on a diamond hierarchical lattice. Joint work with Ken Golden,
Elena Cherkaev, Ben Murphy, Han Le.
September 30
Speaker:
Andrejs Treibergs, Department of Mathematics, University of Utah
Title: Compatibility Conditions for Discrete Planar Structures
Abstract: The nonlinear and linearized equations for the state of a planar material with
prescribed strain are overdetermined PDE's. To be solvable, the strains must satisfy
the necessary compatibility conditions. Approximations of the material by discrete
network satisfy overdetermined algebraic equations whose compatibility conditions
are less well understood.
For generic networks, the number of compatibility conditions is given by the Maxwell
number which is a measure of the resilience of a network to damage. It can be easily
computed for triangulated networks. We will give a geometric description for the
compatibility conditions, discuss which networks are generic, explain why the
discrete problems approximate the continuous ones and consider boundary integrals
over regions satisfying the compatibility conditions.
October 21
Speaker:
Graeme Milton, Department of Mathematics, University of Utah
Title: The spider web problem: guiding stress with discrete Networks
Abstract: Pentamode materials are a class of materials that are useful for guiding
stress. In particular, they have been proposed for acoustic cloaking by
guiding stress around objects, and have been physically constructed.
A key feature of pentamode materials is that each vertex in the material is
the junction of 4 double cone elements. Thus the tension in one element
determines the tension in the other elements, and by extension uniquely determines
the stress in the entire metamaterial. Here we show how this key
feature can be extended to discrete wire networks, supporting forces at the
terminal nodes and which may have internal nodes where no forces are applied.
In usual wire or cable networks, such as in a bridge or bicycle wheel,
one distributes the forces by adjusting the tension in the wires. Here our discrete
networks provide an alternative way of distributing the forces through
the geometry of the network. In particular the network can be chosen so
it is uniloadable, i.e. supports only one set of forces at the terminal nodes.
Such uniloadable networks provide the natural generalization of pentamode materials
to discrete networks.
This is joint work with G.Bouchitte, O. Mattei, and Pierre
Seppecher, and is summarized in the paper "On the forces that cable webs under
tension can support and how to design cable webs to channel stresses", Proceedings
of the Royal Society A: 475 (2019), 20180781.
October 28. Joint Applied Math/SCI Seminar
Speaker:
Tan Bui-Thanh, Department of Aerospace Engineering and
Engineering Mechanics, The University of Texas Austin
Title: An Upwind Hybridized Discontinuous Galerkin Framework: Theory, Algorithms, and
Applications
Abstract: We will present several new developments on the
emerging Hybridized Discontinuous Galerkin (HDG) method. First, starting either from the Godunov upwind idea or from
the Rankine- Hugoniot condition we derive a unified HDG framework for linear PDEs
that allows one to uncover new HDG methods and recover most of the existing ones for
a large class of PDE including the Friedrichs' systems. Analysis and numerical
results for the unified framework will be presented. Second, we will present an IMEX
scheme for the HDG method that: 1) facilitates high-order solutions both in time and
space; 2) avoids overly small time-step sizes; 3) requires only one linear system
solve per time stage. Third, we will present our recent developments on multilevel
and multigrid as scalable solvers and preconditioners for HDG trace systems. Various
numerical results in atmospheric flows and magnetohydrodynamics will be presented to
validate the developments.
October 29. Note day is Tuesday and Time is 3pm - 3:50pm. Room
is LCB 225.
Speaker:
Jeffrey Rickman, Department of Materials Science and Engineering,
Lehigh University
Title: Using Materials Informatics to Quantify Complex Correlations Linking
Structure, Properties and Processing in Materials
Abstract: I will present several examples in which materials informatics can be used
to elucidate and quantify complex correlations linking structure,
properties and processing of materials. In the first example, I consider
the case of high-entropy (HE) (or multi-principal element) alloys,
typically comprising five or more elements. The study of these alloys is a
relatively new area of materials research that has attracted intense
interest in recent years as, in many cases, these systems possess
unexpected and superior mechanical (and other) properties relative to those
of conventional alloys. However, the identification of promising HE alloys
presents a daunting challenge given the associated vastness of the
chemistry/composition space. I will describe a supervised learning
strategy for the efficient screening of HE alloys that combines two
complementary tools, namely: (1) a canonical-correlation analysis (CCA) and
(2) a genetic algorithm (GA) with a CCA-inspired fitness function. In
the second example, I consider the ubiquitous phenomenon of grain
abnormality in a microstructure that involves the unusually rapid growth of
a minority of constituent grains, with the resulting bimodal structure
often having a deleterious impact on the thermomechanical properties of a
system. In this context, I will describe the use of CCA in conjunction
with the formalism of extreme-value statistics to correlate abnormal grain
growth with powder processing and chemistry for specialty ceramics.
Finally, I will outline the use of detrended correlation analyses to
interpret time series data associated with ceramic powder processing.
November 4
Speaker (Invited by Graeme Milton):
Yekaterina Epshteyn, Department of Mathematics, University of Utah
Title: Grain Structure, Grain Growth and Evolution of the Grain Boundary
Network
Abstract: Cellular networks are ubiquitous in nature. Most technologically useful
materials arise as polycrystalline microstructures, composed of a myriad
of small monocrystalline cells or grains, separated by interfaces, or
grain boundaries. Grain boundaries play an essential role in determining
the properties of materials across a wide range of scales. During grain
growth (also termed coarsening), an initially random grain boundary
arrangement reaches a steady state that is strongly correlated to the
interfacial energy density. In this presentation, we will discuss open
questions and recent progress
on modeling, simulation, experiments and analysis of the evolution of the grain boundary network in
polycrystalline materials.
Some of the most recent results that will be discussed as a part
of the talk are the joint work with Patrick Bardsley, Katayun Barmak (Columbia
University), Elias Clark (U of U), Lajos Horvath (U of U), Chun Liu (IIT)
and Masashi Mizuno (Nihon University).
November 7
Speaker:
David A. Kopriva, Department of Mathematics, Florida State University/Computational Science Research Center, San Diego State University
Title: Summation by Parts, a Discrete Integral Calculus and How We Design
Spectral Element Methods that Work
Abstract: Spectral element methods (SEMs) including Discontinuous Galerkin
versions (DGSEMs) have advanced to the point of having open source
solvers available that can compute solutions to industrial grade
problems. For advection or advection dominated problems (e.g.
compressible flows, shallow water flows, electromagnetic wave
propagation, etc.) it is also known that the DGSEMs can go unstable,
especially for under-resolved simulations like those that occur in
turbulence computations. We have developed a discrete integral calculus
framework for the analysis of DGSEMs that starts with summation by
parts. The framework allows us to identify when approximations are
stable or not, and has enabled us to derive provably stable methods for
both linear and nonlinear systems of equations approximated in the open
source solvers.
November 20. Time 4pm - 5pm, Room LCB 219. Joint Statistics/Stochastics/Applied Math Seminar.
Speaker:
Mireille Boutin, School of Electrical and Computer Engineering,
Purdue University
Title: Reconstructing a Room from Echoes and other Unlabeled Distance
Geometry Problems
Abstract: Suppose that some microphones are placed on a drone
inside a room with planar walls/floors/ceilings. A loudspeaker emits a sound impulse and
the microphones receive several delayed responses corresponding to the
sound bouncing back from each planar surface. These are the first-order
echoes. We are interested in reconstructing the shape of the room from the
first-order echoes. The time delay for each echo determines the distance
from the microphone to a mirror image of the source reflected across a
wall. Since we do not know which echo corresponds to which wall, the
distances are unlabeled. The problem is to figure out under which
circumstances, and how, one can find out the correct distance-wall
assignments and reconstruct the wall positions. This is one example of an
unlabeled distance geometry problem. Another example is the problem of
reconstructing a point configuration, up to a rigid motion, from the
multi-set of its pairwise (unlabeled) distances. We show that under some
mild genericity assumptions, these problems are well-posed. Extensions to
the case where the measurements are noisy will also be discussed. This is
joint work with Gregor Kemper (TU Munich).
November 25
Speaker:
Junshan Lin, Department of Mathematics, Auburn University
Title: Resonances Through Subwavelength Holes and Their Applications in
Sensing and Imaging
Abstract: The so-called extraordinary optical transmission (EOT) through
metallic nanoholes has triggered extensive research in modern plasmonics,
due to its significant applications in bio-sensing, imaging, etc. The
mechanisms contributing to the EOT phenomenon can be complicated due to the
multiscale nature of the underlying structure. In this talk, I will focus
on mechanisms induced by scattering resonances.
In the first part of the talk, based upon the layer potential technique,
asymptotic analysis and the homogenization theory, I will present rigorous
mathematical analysis to investigate the scattering resonances for several
typical two-dimensional structures, these include Fabry-Perot resonance,
Fano resonance, spoof surface plasmon, etc. In the second part of the talk,
preliminary mathematical studies for their applications in sensing and
super-resolution imaging will be given. I will focus on the resonance
frequency sensitivity analysis and how one can achieve super-resolution by
using plasmonic nanohole structures.
epshteyn (at) math.utah.edu
).
Past lectures: Spring 2019, Fall 2018, Spring 2018, Fall 2017, Spring 2017, Spring 2016, Fall 2015, Spring 2015, Fall 2014, Spring 2014, Fall 2013, Spring 2013, Fall 2012, Spring 2012, Fall 2011, Spring 2011, Fall 2010, Spring 2010, Fall 2009, Spring 2009, Fall 2008, Spring 2008, Fall 2007, Spring 2007, Fall 2006, Spring 2006, Fall 2005, Spring 2005, Fall 2004, Spring 2004, Fall 2003, Spring 2003, Fall 2002, Spring 2002, Fall 2001, Spring 2001, Fall 2000, Spring 2000, Fall 1999, Spring 1999, Fall 1998, Spring 1998, Winter 1998, Fall 1997, Spring 1997, Winter 1997, Fall 1996, Spring 1996, Winter 1996, Fall 1995.