On partitions into four cubes

We use partially ordered sets (posets) and graphs in order to obtain a formula for the number of partitions of a positive integer n into four cubes with two of them equal

Distributed Finite Element Analysis Using a Transputer Network

The principal objective of this research effort was to demonstrate the extraordinarily cost effective acceleration of finite element structural analysis problems using a transputer-based parallel processing network. This objective was accomplished in the form of a commercially viable parallel processing workstation. The workstation is a desktop size, low-maintenance computing unit capable of supercomputer performance yet costs two orders of magnitude less. To achieve the principal research objective, a transputer based structural analysis workstation termed XPFEM was implemented with linear static structural analysis capabilities resembling commercially available NASTRAN. Finite element model files, generated using the on-line preprocessing module or external preprocessing packages, are downloaded to a network of 32 transputers for accelerated solution. The system currently executes at about one third Cray X-MP24 speed but additional acceleration appears likely. For the NASA selected demonstration problem of a Space Shuttle main engine turbine blade model with about 1500 nodes and 4500 independent degrees of freedom, the Cray X-MP24 required 23.9 seconds to obtain a solution while the transputer network, operated from an IBM PC-AT compatible host computer, required 71.7 seconds. Consequently, the $80,000 transputer network demonstrated a cost-performance ratio about 60 times better than the$15,000,000 Cray X-MP24 system

A fault tolerant grid generation technique

Automatic and parallel mesh generation has been highlighted as a bottleneck for large scale automated Computational Fluid Dynamics analysis. The desire for large scale automated CFD is driven by the growing computational capabilities in large scale supercomputers. Unfortunately, as compute clusters grow in size, they also suffer more failures. Left unchecked, the increased frequency of failures may stymie any efforts to fully utilize these machines. This work aims to tackle one component required for automated large scale engineering analysis by developing a fault tolerant mesh generator. The mesh generator uses a novel com- munication layer written using the transport layer ZeroMQ and is made fault tolerant through an integrated in-memory checkpoint and recovery strategy. Benefits of using in-memory checkpoints vs traditional in-disk checkpoints are discussed. By relying on in-memory checkpointing, it is demonstrated that the mesh generator to be capable of generating Cartesian meshes in parallel. The generator continues to operate even while the compute cluster it is running suffers failures. The generator is shown to be high performing, including being capable of generating an 8.6 billion element mesh in just over 1 minute while creating multiple in-memory checkpoints

Molecular Dynamics Simulation in Arbitrary Geometries for Nanoscale Fluid Mechanics

Simulations of nanoscale systems where fluid mechanics plays an important role are required to help design and understand nano-devices and biological systems. A simulation method which hybridises molecular dynamics (MD) and continuum computational fluid dynamics (CFD) is demonstrated to be able to accurately represent the relevant physical phenomena and be computationally tractable. An MD code has been written to perform MD simulations in systems where the geometry is described by a mesh of unstructured arbitrary polyhedral cells that have been spatially decomposed into irregular portions for parallel processing. The MD code that has been developed may be used for simulations on its own, or may serve as the MD component of a hybrid method. The code has been implemented using OpenFOAM, an open source C++ CFD toolbox (www.openfoam.org). Two key enabling components are described in detail. 1) Parallel generation of initial configurations of molecules in arbitrary geometries. 2) Calculation of intermolecular pair forces, including between molecules that lie on mesh portions assigned to different, and possibly non-neighbouring processors. To calculate intermolecular forces, the spatial relationship of mesh cells is calculated once at the start of the simulation and only the molecules contained in cells that have part of their surface closer than a cut-off distance are required to interact. Interprocessor force calculations are carried out by creating local copies of molecules from other processors in a layer around the processor in question. The process of creating these copied molecules is described in detail. A case study of flow in a realistic nanoscale mixing channel, where the geometry is drawn and meshed using engineering CAD tools, is simulated to demonstrate the capabilities of the code for complex simulations

Determination of the structure of y-alumina using empirical and first principle calculations combined with supporting experiments

Aluminas have had some form of chemical and industrial use throughout history. For little over a century corundum (α-Al2O3) has been the most widely used and known of the aluminas. The emerging metastable aluminas, including the γ, δ, η, θ, κ, β, and χ polymorphs, have been growing in importance. In particular, γ-Al2O3 has received wide attention, with established use as a catalyst and catalyst support, and growing application in wear abrasives, structural composites, and as part of burner systems in miniature power supplies. It is also growing in importance as part of the feedstock for aluminium production in order to affect both the adsorption of hydrogen fluoride and the feedstock solubility in the electrolytic solution. However, much ambiguity surrounds the precise structure of γ-Al2O3. Without proper knowledge of the structure, understanding the properties, dynamics and applications will always be less than optimal. The aim of this research was to contribute towards settling this ambiguity. This work was achieved through extensive computer simulations of the structure, based on interatomic potentials with refinements of promising structures using density functional theory (DFT), and a wide range of supporting experiments. In addition to providing a more realistic representation of the structure, this research has also served to advance knowledge of the evolution of the structure with changing temperature and make new insights regarding the location of hydrogen in γ-Al2O3.Both the molecular modelling and Rietveld refinements of neutron diffraction data showed that the traditional cubic spinel-based structure models, based on m Fd3 space group symmetry, do not accurately describe the defect structure of γ-Al2O3. A more accurate description of the structure was provided using supercells of the cubic and tetragonal unit cells with a significant number of cations on c symmetry positions. These c symmetry based structures exhibited diffraction patterns that were characteristic of γ-Al2O3. The first three chapters of this Thesis provide a review of the literature. Chapter One provides a general introduction, describing the uses and importance of the aluminas and the problems associated with determining the structure of γ-Al2O3. Chapter Two details the research that has been conducted on the structure of vi γ-Al2O3 historically. Chapter Three describes the major principles behind the computational methods employed in this research. In Chapter Four, the specific experimental and computational techniques used to investigate the structure of γ-Al2O3 are described. All preparation conditions and parameters used are provided. Chapter Five describes the methodology employed in computational and experimental research. The examination of the ~ 1.47 billion spinel-based structural possibilities of γ-Al2O3, described using supercells, and the selection of ~ 122,000 candidates for computer simulation, is detailed. This chapter also contains a case study of the structure of κ-Al2O3, used to investigate the applicability of applying interatomic potentials to solving complex structures, where many possibilities are involved, and to develop a systematic procedure of computational investigation that could be applied to γ-Al2O3. Chapters Six to Nine present and discuss the results from the experimental studies.Preliminary heating trials, performed to determine the appropriate preparation conditions for obtaining a highly crystalline boehmite precursor and an appropriate calcination procedure for the systematic study of γ-Al2O3, were presented in Chapter Six. Chapter Seven details the investigation of the structure from a singletemperature case. Several known structural models were investigated, including the possibility of a dual-phase model and the inclusion of hydrogen in the structure. It was demonstrated that an accurate structural model cannot be achieved for γ-Al2O3 if the cations are restricted to spinel positions. It was also found that electron diffraction patterns, typical for γ-Al2O3, could be indexed according to the I41/amd space group, which is a maximal subgroup of m Fd3 . Two models were presented which describe the structure more accurately; Cubic-16c, which describes cubic γ-Al2O3 and Tetragonal-8c, which describes tetragonal γ-Al2O3. The latter model was found to be a better description for the γ-Al2O3 samples studied. Chapter Eight describes the evolution of the structure with changing calcination temperature. Tetragonal γ-Al2O3 was found to be present between 450 and 750 °C. The structure showed a reduction in the tetragonal distortion with increasing temperature but at no stage was cubic γ-Al2O3 obtained. Examination of the progress of cation migration indicates the reduction in the tetragonal nature is due to ordering within inter-skeletal oxygen layers of the unit cell, left over from the breakdown of the hydroxide layers of boehmite when the transformation to γ-Al2O3 occurred. Above 750 °C, δ-Al2O3 was not observed, but a new phase was identified and designated γ.-Al2O3.The structure of this phase was determined to be a triple cell of γ-Al2O3 and is herein described using the 2 4m P space group. Chapter Nine investigates the presence of hydrogen in the structure of γ-Al2O3. It was concluded that γ-Al2O3 derived from highly crystalline boehmite has a relatively well ordered bulk crystalline structure which contains no interstitial hydrogen and that hydrogen-containing species are located at the surface and within amorphous regions, which are located in the vicinity of pores. Expectedly, the specific surface area was found to decrease with increasing calcination temperature. This trend occurred concurrently with an increase in the mean pore and crystallite size and a reduction in the amount of hydrogen-containing species within the structure. It was also demonstrated that γ-Al2O3 derived from highly crystalline boehmite has a significantly higher surface area than expected, attributed to the presence of nano-pores and closed porosity. The results from the computational study are presented and discussed in Chapter Ten. Optimisation of the spinel-based structural models showed that structures with some non-spinel site occupancy were more energetically favourable. However, none of the structural models exhibited a configuration close to those determined from the experimental studies. Nor did any of the theoretical structures yield a diffraction pattern that was characteristic of γ-Al2O3. This discrepancy between the simulated and real structures means that the spinel-based starting structure models are not close enough to the true structure of γ-Al2O3 to facilitate the derivation of its representative configuration.Large numbers of structures demonstrate migration of cations to c symmetry positions, providing strong evidence that c symmetry positions are inherent in the structure. This supports the Cubic-16c and Tetragonal-8c structure models presented in Chapter Seven and suggests that these models are universal for crystalline γ-Al2O3. Optimisation of c symmetry based structures, with starting configurations based on the experimental findings, resulted in simulated diffraction patterns that were characteristic of γ-Al2O3

Mesh generation for large-scale and complex computational simulation.

This thesis presents work in the area of mesh generation for large scale and complex computational simulation. The work covers two areas of great interest within the field of mesh generation; anisotropic mesh generation and parallel large scale mesh generation. Examples of anisotropic Delaunay mesh generation are presented with application to fluid dynamics and computational electromagnetic scattering simulations. Results are shown with reference to simulation accuracy and computational efficiency. Research into parallel mesh generation is presented and a method of parallel Delaunay mesh generation suitable for use on distributed and shared memory parallel computers is described. Results are shown with reference to computational efficiency, memory usage and finale mesh quality. Examples of meshes generated in parallel are shown for both computational fluid dynamics simulations on simple aeronautical geometries to full aircraft and computational electromagnetic scattering simulations on full aircraft. The meshes range in size from a few thousand tetrahedral elements to a mesh for a computational electromagnetic simulation containing approximately one billion tetrahedral elements

Decomposing 40 Billion Integers by Four Tetrahedral Numbers

[[abstract]]Based upon a computer search performed on a massively parallel supercomputer, we found that any integer n less than 40 billion (40B) but greater than 343, 867 can be written as a sum of four or fewer tetrahedral numbers. This result has established a new upper bound for a conjecture compared to an older one, 1B, obtained a year earlier. It also gives more accurate asymptotic forms for partitioning. All this improvement is a direct result of algorithmic advances in efficient memory and cpu utilizations. The heuristic complexity of the new algorithm is O(n) compared with that of the old, O(n5/3 log n)

Resonant and high resolution photoemission of rare-earth cobalt oxides

LnBaCo2O5+δ (Ln-112) where Ln = lanthanide element; 0 ≤ δ ≤1 and LnBaCo4O7+δ (Ln-114) are highly correlated cobalt oxides. Synchrotron photoemission spectroscopy of LnBaCo2O5+δ (Ln = Gd, Dy, Dy1-xTbx) and LnBaCo4O7 (Ln = Yb) has been undertaken at the UK Synchrotron Radiation Source (Daresbury Laboratory).During the photoemission experiments, the samples were observed to be contaminated due to residual gases inside the main vacuum chamber. The surface degradation of the samples is studied using the difference spectra generated from the valence band spectra of freshly scraped and contaminated samples and the nature of contaminated species on these samples is identified in the light of the reviewed literature. High-resolution photoemission is carried out to study the metal-insulator (MI) transition in double perovskites LnBaCo2O5+δ (Ln = Gd, Dy, Dy1-xTbx - Ln-112; 0 ≤ δ ≤ 1) as a function of temperature. The high-resolution photoemission results of single crystal samples of GdBaCo2O5.5, DyBaCo2O5.5 and Dy1-xTbxBaCo2O5+δ show that the temperature-based MI transitions in these compounds occur in the 300-400 K temperature range. A post-growth oxygen annealing treatment for as-grown single crystals of Ln-112 is necessary, achieving oxygen contents close to 5.50, to observe a marked nonmetal-to-metal transition. Resonant photoemission is used to identify the atomic parentage of the valence band states. A comparison of the electronic structure of LnBaCo2O5+δ (Ln = Gd, Dy, Dy1-xTbx - Ln-112; 0 ≤ δ ≤ 1) and LnBaCo4O7 (Ln = Yb - Ln-114) single-crystal surfaces is made using synchrotron photoemission spectroscopy. In both cases, the states close to the Fermi energy are found to be of mixed Co 3d/O 2p character, and the comparison allows identification of states due to low spin Co³⁺ in octahedral environments. The contributions from Ln elements to the valence band are found at higher binding energies.EThOS - Electronic Theses Online ServiceGBUnited Kingdo