7,931 research outputs found
Lecture Notes: The Galerkin Method
These lecture notes introduce the Galerkin method to approximate solutions to
partial differential and integral equations. We begin with some analysis
background to introduce this method in a Hilbert Space setting, and
subsequently illustrate some computational examples with the help of a sample
matlab code. Finally, we use the Galerkin method to prove the existence of
solutions of a nonlinear boundary value problem
Periodic Orbits of Gross Pitaevskii in the Disc with Vortices Following Point Vortex Flow
We prove the existence of non-constant time periodic vortex solutions to the
Gross-Pitaevskii equations for small but \textit{fixed} The
vortices of these solutions follow periodic orbits to the point vortex system
of ordinary differential equations \textit{for all time}. The construction uses
two approaches-- constrained minimization techniques adapted from \cite{GS} and
topological minimax techniques adapted from \cite{LinMinMax}, applied to a
formulation of the problem within a rotational ansatz.Comment: 36 pages. Final version- exposition was substantially streamlined
thanks to a detailed referee report. To appear in Calc. Var. & PD
Conormal Varieties on the Cominuscule Grassmannian
Let be a simply connected, almost simple group over an algebraically
closed field , and a maximal parabolic subgroup corresponding to
omitting a cominuscule root. We construct a compactification
, where is a Schubert variety corresponding
to the loop group . Let be the conormal variety of
some Schubert variety in ; hence we obtain that the closure of
in is a -stable compactification of . We
further show that this compactification is a Schubert subvariety of if
and only if is smooth, where is the longest element
in the Weyl group of . This result is applied to compute the conormal fibre
at the zero matrix in any determinantal variety
Cross-linked polymers in strain: Structure and anisotropic stress
Molecular dynamic simulation enables one to correlate the evolution of the
micro-structure with anisotropic stress when a material is subject to strain.
The anisotropic stress due to a constant strain-rate load in a cross-linked
polymer is primarily dependent on the mean-square bond length and mean-square
bond angle. Excluded volume interactions due to chain stacking and spatial
distribution also has a bearing on the stress response. The bond length
distribution along the chain is not uniform. Rather, the bond lengths at the
end of the chains are larger and uniformly decrease towards the middle of the
chain from both ends. The effect is due to the presence of cross-linkers. As
with linear polymers, at high density values, changes in mean-square bond
length dominates over changes in radius of gyration and end-to-end length. That
is, bond deformations dominate over changes in size and shape. A large change
in the mean-square bond length reflects in a jump in the stress response.
Short-chain polymers more or less behave like rigid molecules. Temperature has
a peculiar effect on the response in the sense that even though bond lengths
increase with temperature, stress response decreases with increasing
temperature. This is due to the dominance of excluded volume effects which
result in lower stresses at higher temperatures. At low strain rates, some
relaxation in the bond stretch is observed from to
. At high strain rates, internal deformation of the chains
dominate over their uncoiling leading to a rise in the stress levels.Comment: 30 pages, 29 figure, 1 tabl
Structure, molecular dynamics, and stress in a linear polymer under dynamic strain
The structural properties of a linear polymer and its evolution in time have
a strong bearing on its anisotropic stress response. The mean-square bond
length and mean bond angle are the critical parameters that influence the
time-varying stress developed in the polymer. The bond length distribution
along the chain is uniform without any abrupt changes at the ends. Among the
externally set parameters such as density, temperature, strain rate, and chain
length, the density as well as the chain length of the polymer have a
significant effect on the stress. At high density values, changes in
mean-square bond length dominates over changes in radius of gyration and
end-to-end length. In other words, bond deformations dominate as opposed to
changes in size and shape. Also, there is a large change in the mean-square
bond length that is reflected as a jump in the stress. Beyond a particular
value of the chain length, , called the entanglement length,
stress-response is found to have distinctly different behavior which we
attribute to the entanglement effects. Short chain polymers more or less behave
like rigid molecules. There is no significant change in their internal
structure when loaded. Further, temperature and rate of loading have a very
mild effect on the stress. Besides these new results, we can now explain well
known polymeric mechanical behavior under dynamic loading from the point of
view of the evolution of the molecular dynamics and the derived structural
properties. This could possibly lead to polymer synthesis with desired
mechanical behavior.Comment: 25 pages, 33 figures, 1 tabl
A Smart Meter Data-driven Distribution Utility Rate Model for Networks with Prosumers
Distribution grids across the world are undergoing profound changes due to
advances in energy technologies. Electrification of the transportation sector
and the integration of Distributed Energy Resources (DERs), such as
photo-voltaic panels and energy storage devices, have gained substantial
momentum, especially at the grid edge. Transformation in the technological
aspects of the grid could directly conflict with existing distribution utility
retail tariff structures. We propose a smart meter data-driven rate model to
recover distribution network-related charges, where the implementation of these
grid-edge technologies is aligned with the interest of the various stakeholders
in the electricity ecosystem. The model envisions a shift from charging
end-users based on their KWh volumetric consumption, towards charging them a
"grid access fee" that approximates the impact of end-users' time-varying
demand on their local distribution network. The proposed rate incorporates two
cost metrics affecting distribution utilities (DUs), namely 'magnitude' and
'variability' of customer demand. The proposed rate can be applied to prosumers
and conventional consumers without DERs.Comment: Accepted to Utilities Policy Journal, to appear in 2021
(https://www.sciencedirect.com/journal/utilities-policy
A Ginzburg-Landau type problem for highly anisotropic nematic liquid crystals
We carry out an asymptotic analysis of a thin nematic liquid crystal in which
one elastic constant dominates over the others, namely \begin{align}
\label{energyab} \inf E_\varepsilon(u)\quad\mbox{where}\quad E_\varepsilon(u)
:= \frac{1}{2}\int_\Omega \left\{\varepsilon\,|\nabla u|^2 +
\frac{1}{\varepsilon} \,(|u|^2 - 1)^2 + L \,(\mathrm{div}\,u)^2\right\} \,dx.
\end{align} Here is a vector field, is a small parameter, and is a fixed constant,
independent of . We derive the -limit , which is a
sum of a bulk term penalizing divergence and an Aviles-Giga type wall energy
involving the cube of the jump in the tangential component of the
-valued order parameter. We then derive criticality conditions
for and analyze minimization of both rigorously and numerically for
various domains and a variety of Dirichlet boundary conditions
Effect of particle size and inter-particle spacing on dislocation behaviour of Nickel based super alloys
Ni-based superalloys have been the subject of enormous usage in scenarios
where the loading is heavy and often occurs at elevated temperatures. The
strengthening mechanisms that come into play within the metallic lattice have
been studied extensively as micromechanical MMC models. These continuum
formulations suffer from several limitations. The underlying mechanisms at the
atomistic scale have not yet been well understood. The report attempts to model
the interaction of moving dislocation with cuboidal precipitates and explain
the strengthening effect. The effect of particle size and inter-particle
distance on the strength are evaluated. Several physically meaningful results
have also been interpreted and shown.Comment: 6 page
Dynamic Co-Simulation Methods for Combined Transmission-Distribution System and Integration Time Step Impact on Convergence
Combined Transmission and Distribution Systems (CoTDS) simulation for power
systems requires development of algorithms and software that are numerically
stable and at the same time accurately simulate dynamic events that can occur
in practical systems. The dynamic behavior of transmission and distribution
systems are vastly different, especially with the increased deployment of
distribution generation. The time scales of simulation can be orders of
magnitude apart making the combined simulation extremely challenging. This has
led to increased research in applying co-simulation techniques for integrated
simulation of the two systems. In this paper, a rigorous mathematical analysis
on convergence of numerical methods in co-simulation is presented. Two methods
for co-simulation of CoTDS are proposed using parallel and series computation
of the transmission system and distribution systems. Both these co-simulation
methods are validated against total system simulation in a single time-domain
simulation environment. The series computation co-simulation method is shown to
have better numerical stability at larger integration time steps. The series
computation co-simulation method is additionally validated against commercial
EMTP software and the results show remarkable correspondence.Comment: 10 page
Double Antisymmetry and the Rotation-Reversal Space Groups
Rotation-reversal symmetry was recently introduced to generalize the symmetry
classification of rigid static rotations in crystals such as tilted octahedra
in perovskite structures and tilted tetrahedral in silica structures. This
operation has important implications for crystallographic group theory, namely
that new symmetry groups are necessary to properly describe observations of
rotation-reversal symmetry in crystals. When both rotation-reversal symmetry
and time-reversal symmetry are considered in conjunction with space group
symmetry, it is found that there are 17,803 types of symmetry, called double
antisymmetry, which a crystal structure can exhibit. These symmetry groups have
the potential to advance understanding of polyhedral rotations in crystals, the
magnetic structure of crystals, and the coupling thereof. The full listing of
the double antisymmetry space groups can be found in the supplemental materials
of the present work and online at our website:
http://sites.psu.edu/gopalan/research/symmetry
- β¦