There is, I think, something ethereal about i —the square root of minus one. I remember first hearing about it at school. It seemed an odd beast at that time—an intruder hovering on the edge of reality.

Usually familiarity dulls this sense of the bizarre, but in the case of i it was the reverse: over the years the sense of its surreal nature intensified. It seemed that it was impossible to write mathematics that described the real world in …

I and i

Three parallel algorithms for classical molecular dynamics are presented. The first assigns each processor a fixed subset of atoms; the second assigns each a fixed subset of inter-atomic forces to compute; the third assigns each a fixed spatial region. The algorithms are suitable for molecular dynamics models which can be difficult to parallelize efficiently—those with short-range forces where the neighbors of each atom change rapidly. They can be implemented on any distributed-memory parallel machine which allows for message-passing of data between independently executing processors. The algorithms are tested on a standard Lennard-Jones benchmark problem for system sizes ranging from 500 to 100,000,000 atoms on several parallel supercomputers--the nCUBE 2, Intel iPSC/860 and Paragon, and Cray T3D. Comparing the results to the fastest reported vectorized Cray Y-MP and C90 algorithm shows that the current generation of parallel machines is competitive with conventional vector supercomputers even for small problems. For large problems, the spatial algorithm achieves parallel efficiencies of 90% and a 1840-node Intel Paragon performs up to 165 faster than a single Cray C9O processor. Trade-offs between the three algorithms and guidelines for adapting them to more complex molecular dynamics simulations are also discussed.

Fast parallel algorithms for short-range molecular dynamics

Probability Distributions.- Linear Models for Regression.- Linear Models for Classification.- Neural Networks.- Kernel Methods.- Sparse Kernel Machines.- Graphical Models.- Mixture Models and EM.- Approximate Inference.- Sampling Methods.- Continuous Latent Variables.- Sequential Data.- Combining Models.

Pattern Recognition and Machine Learning

In this chapter, we will restrict our attention to the ferrites and a few other closely related materials. The great interest in ferrites stems from their unique combination of a spontaneous magnetization and a high electrical resistivity. The observed magnetization results from the difference in the magnetizations of two non-equivalent sub-lattices of the magnetic ions in the crystal structure. Materials of this type should strictly be designated as “ferrimagnetic” and in some respects are more closely related to anti-ferromagnetic substances than they are to ferromagnetics in which the magnetization results from the parallel alignment of all the magnetic moments present. We shall not adhere to this special nomenclature except to emphasize effects, which are due to the existence of the sub-lattices.

Ferromagnetic materials

The stability of two-dimensional (2D) layers and membranes is subject of a long standing theoretical debate. According to the so called Mermin-Wagner theorem, long wavelength fluctuations destroy the long-range order for 2D crystals. Similarly, 2D membranes embedded in a 3D space have a tendency to be crumpled. These dangerous fluctuations can, however, be suppressed by anharmonic coupling between bending and stretching modes making that a two-dimensional membrane can exist but should present strong height fluctuations. The discovery of graphene, the first truly 2D crystal and the recent experimental observation of ripples in freely hanging graphene makes these issues especially important. Beside the academic interest, understanding the mechanisms of stability of graphene is crucial for understanding electronic transport in this material that is attracting so much interest for its unusual Dirac spectrum and electronic properties. Here we address the nature of these height fluctuations by means of straightforward atomistic Monte Carlo simulations based on a very accurate many-body interatomic potential for carbon. We find that ripples spontaneously appear due to thermal fluctuations with a size distribution peaked around 70 \AA which is compatible with experimental findings (50-100 \AA) but not with the current understanding of stability of flexible membranes. This unexpected result seems to be due to the multiplicity of chemical bonding in carbon.

Intrinsic ripples in graphene

Elastic field of an isotropic matrix with a nanoscale elliptical inhomogeneity

Two-dimensional elastic field of a nanoscale circular hole/inhomogeneity in an infinite matrix under arbitrary remote loading and a uniform eigenstrain in the inhomogeneity is investigated. The Gurtin-Murdoch surface/interface elasticity model is applied to take into account the surface/interface stress effects. A closed-form analytical solution is obtained by using the complex potential function method of Muskhelishvili. Selected numerical results are presented to investigate the size dependency of the elastic field and the effects of surface elastic moduli and residual surface stress. Stress state is found to depend on the radius of the inhomogeneity/hole, surface elastic constants, surface residual stress, and magnitude of far-field loading.

Analytical Solution for Size-Dependent Elastic Field of a Nanoscale Circular Inhomogeneity

The dynamic response of an elastic pile subjected to transient torsional and axial loading is considered. The pile is embedded in a multi-layered elastic soil. The stress field in a soil is simplified by following the existing solutions for a pile subjected to static torsional and axial loading. The dynamic equilibrium equation of soil is solved in the Laplace domain by using analytical techniques. The soil resistance is coupled into a one-dimensional governing equation of a pile segment and an analytical solution is presented. An impedance matrix can then be derived for a pile segment relating end stress resultants to the displacements. Non-homogeneous initial conditions of the pile are considered. The impedance matrices of pile segments are assembled by following the concepts of finite element method to analyse a pile embedded in a multi-layered soil. The treatment of soil base response is also discussed. Time domain solutions are obtained by using a numerical Laplace inversion procedure. The extension ...

Dynamic response of a pile in a multi-layered soil to transient torsional and axial loading

This paper presents a comprehensive analytical treatment of the three-dimensional response of a transversely isotropic elastic half space subjected to time-harmonic excitations. General solutions for equations of equilibrium expressed in terms of displacements are derived by applying Fourier expansion with respect to the circumferential coordinate and Hankel integral transforms with respect to the radial coordinate. The general solutions are used to derive the explicit solutions for Green’s functions (displacements and stresses) corresponding to a set of time-harmonic circular ring loads acting inside a half space. The circumferential variation of the ring loads are assumed to be cos \Im\Nθ\N for loadings in the vertical and radial directions and sin \Im\Nθ\N for the loading in the circumferential direction. These Green’s functions can be used as the kernel functions of the boundary-integral-equation method and in the development of solutions for a variety of elastodynamic boundary value problems. Comparisons with existing numerical solutions for an isotropic half space are presented to confirm the accuracy of the present solutions. Selected numerical results for displacements and stresses are presented to portray the dependence of the response of the half space on the frequency of excitation and the degree of anisotropy of the medium.

Green's Functions for Transversely Isotropic Elastic Half Space

Nanoscale beams are commonly found in nanomechanical and nanoelectromechanical systems (NEMS) and other nanotechnology-based devices. Surface energy has a significant effect on nanoscale structures and is associated with their size-dependent behavior. In this paper, a general mechanistic model based on the Gurtin-Murdoch continuum theory accounting for surface energy effects is presented to analyze thick and thin nanoscale beams with an arbitrary cross section. The main contributions of this paper are a set of closed-form analytical solutions for the static response of thin and thick beams under different loading (point and uniformly distributed) and boundary conditions (simply-supported, cantilevered, and clamped ends), as well as the solution of the free vibration characteristics of such beams. Selected numerical results are presented for aluminum and silicon beams to demonstrate their salient response features. It is shown that classical beam theory is not accurate in situations where the surface residual stress and/or surface elastic constants are relatively large. An intrinsic length scale for beams is identified that depends on beam surface properties and cross-sectional shape. The present work provides a convenient set of analytical tools for researchers working on NEMS design and fabrication to understand the static and dynamic behavior of nanoscale beams including their size-dependent behavior and the effects of common boundary conditions.

R. K. N. D. Rajapakse

Papers

Elastic field of an isotropic matrix with a nanoscale elliptical inhomogeneity

Analytical Solution for Size-Dependent Elastic Field of a Nanoscale Circular Inhomogeneity

Dynamic response of a pile in a multi-layered soil to transient torsional and axial loading

Green's Functions for Transversely Isotropic Elastic Half Space

Continuum Models Incorporating Surface Energy for Static and Dynamic Response of Nanoscale Beams