Newton's method: A link between continuous and discrete solutions of nonlinear problems

Newton's method for nonlinear mechanics problems replaces the governing nonlinear equations by an iterative sequence of linear equations. When the linear equations are linear differential equations, the equations are usually solved by numerical methods. The iterative sequence in Newton's method can exhibit poor convergence properties when the nonlinear problem has multiple solutions for a fixed set of parameters, unless the iterative sequences are aimed at solving for each solution separately. The theory of the linear differential operators is often a better guide for solution strategies in applying Newton's method than the theory of linear algebra associated with the numerical analogs of the differential operators. In fact, the theory for the differential operators can suggest the choice of numerical linear operators. In this paper the method of variation of parameters from the theory of linear ordinary differential equations is examined in detail in the context of Newton's method to demonstrate how it might be used as a guide for numerical solutions

Buckling of imperfect cylinders under axial compression

Donnell equations, Newton method, and numerical solution applied to buckling of imperfect cylinders under axial compressio

The stability of shallow spherical shells under concentrated load

Effect of load area on deformation of clamped spherical cap and behavior of transition from axisymmetric to asymmetric deflection shape

Buckling of cylindrical shell end closures by internal pressure

Buckling of cylindrical shell end closures by internal pressur

Solution of the symmetric eigenproblem AX=lambda BX by delayed division

Delayed division is an iterative method for solving the linear eigenvalue problem AX = lambda BX for a limited number of small eigenvalues and their corresponding eigenvectors. The distinctive feature of the method is the reduction of the problem to an approximate triangular form by systematically dropping quadratic terms in the eigenvalue lambda. The report describes the pivoting strategy in the reduction and the method for preserving symmetry in submatrices at each reduction step. Along with the approximate triangular reduction, the report extends some techniques used in the method of inverse subspace iteration. Examples are included for problems of varying complexity

A Note on Real Tunneling Geometries

In the Hartle-Hawking ``no boundary'' approach to quantum cosmology, a real tunneling geometry is a configuration that represents a transition from a compact Riemannian spacetime to a Lorentzian universe. I complete an earlier proof that in three spacetime dimensions, such a transition is ``probable,'' in the sense that the required Riemannian geometry yields a genuine maximum of the semiclassical wave function.Comment: 5 page

Learning SO(3) Equivariant Representations with Spherical CNNs

We address the problem of 3D rotation equivariance in convolutional neural networks. 3D rotations have been a challenging nuisance in 3D classification tasks requiring higher capacity and extended data augmentation in order to tackle it. We model 3D data with multi-valued spherical functions and we propose a novel spherical convolutional network that implements exact convolutions on the sphere by realizing them in the spherical harmonic domain. Resulting filters have local symmetry and are localized by enforcing smooth spectra. We apply a novel pooling on the spectral domain and our operations are independent of the underlying spherical resolution throughout the network. We show that networks with much lower capacity and without requiring data augmentation can exhibit performance comparable to the state of the art in standard retrieval and classification benchmarks.Comment: Camera-ready. Accepted to ECCV'18 as oral presentatio

Cell replacement strategies for lithium ion battery packs

The economic value of high-capacity battery systems, being used in a wide variety of automotive and energy storage applications, is strongly affected by the duration of their service lifetime. Because many battery systems now feature a very large number of individual cells, it is necessary to understand how cell-to-cell interactions can affect durability, and how to best replace poorly performing cells to extend the lifetime of the entire battery pack. This paper first examines the baseline results of aging individual cells, then aging of cells in a representative 3S3P battery pack, and compares them to the results of repaired packs. The baseline results indicate nearly the same rate of capacity fade for single cells and those aged in a pack; however, the capacity variation due to a few degrees changes in room temperature (\u27±3 ◦C) is significant (\u27±1.5% of capacity of new cell) compared to the percent change of capacity over the battery life cycle in primary applications (\u2720–30%). The cell replacement strategies investigation considers two scenarios: early life failure, where one cell in a pack fails prematurely, and building a pack from used cells for less demanding applications. Early life failure replacement found that, despite mismatches in impedance and capacity, a new cell can perform adequately within a pack of moderately aged cells. The second scenario for reuse of lithium ion battery packs examines the problem of assembling a pack for less-demanding applications from a set of aged cells, which exhibit more variation in capacity and impedance than their new counterparts. The cells used in the aging comparison part of the study were deeply discharged, recovered, assembled in a new pack, and cycled. We discuss the criteria for selecting the aged cells for building a secondary pack and compare the performance and coulombic efficiency of the secondary pack to the pack built from new cells and the repaired pack. The pack that employed aged cells performed well, but its efficiency was reduced

Radiation exposure awareness from patients undergoing nuclear medicine diagnostic 99mTc-MDP bone scans and 2-deoxy-2-(18F) fluoro-D-glucose PET/computed tomography scans

INTRODUCTION: Medical imaging is on average the largest source of artificial radiation exposure worldwide. This study seeks to understand patient's awareness of radiation exposure derived from nuclear medicine diagnostic scans and assess if current information provided by leaflets is adequate. METHODS: Single-centre cross-sectional questionnaire study applied to bone scan and FDG PET/computed tomography patients, at a nuclear medicine and PET/computed tomography department over a 15-week period in 2018. Questionnaires on dose comparators were designed in collaboration with patients, public, and experts in radiation exposure. Qualitative data were analysed using thematic analysis and quantitative data using SPSS (V. 24). RESULTS: A total of 102 questionnaires were completed (bone scan = 50; FDG PET/computed tomography = 52). Across both groups, 33/102 (32.4%) patients reported having a reasonable understanding of nuclear medicine and 21/102 (20.6%) reported a reasonable knowledge of ionising radiations. When asked to compare the exposure dose of respective scans with common comparators 8/50 (16%) of bone scan patients and 11/52 (21.2%) FDG PET/computed tomography answered correctly. On leaflet information, 15/85 (17.6%) patients reported the leaflets do not provide enough information on radiation exposure and of these 10/15 (66.7%) commented the leaflets should incorporate more information on radiation exposure dose. CONCLUSION: More observational and qualitative studies in collaboration with patients are warranted to evaluate patients' understanding and preferences in communication of radiation exposure from nuclear medicine imaging. This will ensure communication tools and guidelines developed to comply with ionising radiation (medical exposure) regulation 2017 are according to patients needs and preferences