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} $\varepsilon > 0.$ 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 $G$ be a simply connected, almost simple group over an algebraically closed field $\mathbf k$, and $P$ a maximal parabolic subgroup corresponding to omitting a cominuscule root. We construct a compactification $\phi:T^*G/P\rightarrow X(u)$, where $X(u)$ is a Schubert variety corresponding to the loop group $LG$. Let $N^*X(w)\subset T^*G/P$ be the conormal variety of some Schubert variety $X(w)$ in $G/P$; hence we obtain that the closure of $\phi(N^*X(w))$ in $X(u)$ is a $B$-stable compactification of $N^*X(w)$. We further show that this compactification is a Schubert subvariety of $X(u)$ if and only if $X(w_0w)\subset G/P$ is smooth, where $w_0$ is the longest element in the Weyl group of $G$. 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 $\epsilon=0.2$ to $\epsilon=0.5$. 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, $n = 50$, 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 $u: \Omega \to \mathbb R^2$ is a vector field, $0 < \varepsilon \ll 1$ is a small parameter, and $L > 0$ is a fixed constant, independent of $\varepsilon$. We derive the $\Gamma$-limit $E_0$, 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 $\mathbb{S}^1$-valued order parameter. We then derive criticality conditions for $E_0$ and analyze minimization of $E_0$ both rigorously and numerically for various domains $\Omega$ 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