A treatise on infinitesimal calculus, containing differential and integral calculus, calculus of variations, applications to algebra and geometry, and analytical mechanics.

Vol. 3 and 4 form the author's Treatise on analytical mechanics.

Modeling of adsorption and nucleation in infinite cylindrical pores by two-dimensional density functional theory

In this paper, we present an analysis of argon adsorption in cylindrical pores having amorphous silica structure by means of a nonlocal density functional theory (NLDFT). In the modeling, we account for the radial and longitudinal density distributions, which allow us to consider the interface between the liquidlike and vaporlike fluids separated by a hemispherical meniscus in the canonical ensemble. The Helmholtz free energy of the meniscus was determined as a function of pore diameter. The canonical NLDFT simulations show the details of density rearrangement at the vaporlike and liquidlike spinodal points. The limits of stability of the smallest bridge and the smallest bubble were also determined with the canonical NLDFT. The energy of nucleation as a function of the bulk pressure and the pore diameter was determined with the grand canonical NLDFT using an additional external potential field. It was shown that the experimentally observed reversibility of argon adsorption isotherms at its boiling point up to the pore diameter of 4 nm is possible if the potential barrier of 22kT is overcome due to density fluctuations.

The influence of cutting tool geometry upon aspects of chip flow and tool wear:a theoretical, three dimensional examination of the cutting geometry and the shape of the twist drill

This work is undertaken in the attempt to understand the processes at work at the cutting edge of the twist drill. Extensive drill life testing performed by the University has reinforced a survey of previously published information. This work demonstrated that there are two specific aspects of drilling which have not previously been explained comprehensively. The first concerns the interrelating of process data between differing drilling situations, There is no method currently available which allows the cutting geometry of drilling to be defined numerically so that such comparisons, where made, are purely subjective. Section one examines this problem by taking as an example a 4.5mm drill suitable for use with aluminium. This drill is examined using a prototype solid modelling program to explore how the required numerical information may be generated. The second aspect is the analysis of drill stiffness. What aspects of drill stiffness provide the very great difference in performance between short flute length, medium flute length and long flute length drills? These differences exist between drills of identical point geometry and the practical superiority of short drills has been known to shop floor drilling operatives since drilling was first introduced. This problem has been dismissed repeatedly as over complicated but section two provides a first approximation and shows that at least for smaller drills of 4. 5mm the effects are highly significant. Once the cutting action of the twist drill is defined geometrically there is a huge body of machinability data that becomes applicable to the drilling process. Work remains to interpret the very high inclination angles of the drill cutting process in terms of cutting forces and tool wear but aspects of drill design may already be looked at in new ways with the prospect of a more analytical approach rather than the present mix of experience and trial and error. Other problems are specific to the twist drill, such as the behaviour of the chips in the flute. It is now possible to predict the initial direction of chip flow leaving the drill cutting edge. For the future the parameters of further chip behaviour may also be explored within this geometric model.

Monte Carlo Random Walk Simulations Based on Distributed Order Differential Equations with Applications to Cell Biology

Mathematics Subject Classification: 65C05, 60G50, 39A10, 92C37

Existence Results for Fractional Functional Differential Inclusions with Infinite Delay and Applications to Control Theory

Mathematics Subject Classification: 26A33, 34A60, 34K40, 93B05

Nonlinear soil-structure interaction for buried pressure pipes under differential ground motion

Pipelines extend thousands of kilometers across wide geographic areas as a network to provide essential services for modern life. It is inevitable that pipelines must pass through unfavorable ground conditions, which are susceptible to natural disasters. This thesis investigates the behaviour of buried pressure pipelines experiencing ground distortions induced by normal faulting. A recent large database of physical modelling observations on buried pipes of different stiffness relative to the surrounding soil subjected to normal faults provided a unique opportunity to calibrate numerical tools. Three-dimensional finite element models were developed to enable the complex soil-structure interaction phenomena to be further understood, especially on the subjects of gap formation beneath the pipe and the trench effect associated with the interaction between backfill and native soils. Benchmarked numerical tools were then used to perform parametric analysis regarding project geometry, backfill material, relative pipe-soil stiffness and pipe diameter. Seismic loading produces a soil displacement profile that can be expressed by isoil, the distance between the peak curvature and the point of contraflexure. A simplified design framework based on this length scale (i.e., the Kappa method) was developed, which features estimates of longitudinal bending moments of buried pipes using a characteristic length, ipipe, the distance from peak to zero curvature. Recent studies indicated that empirical soil springs that were calibrated against rigid pipes are not suitable for analyzing flexible pipes, since they lead to excessive conservatism (for design). A large-scale split-box normal fault simulator was therefore assembled to produce experimental data for flexible PVC pipe responses to a normal fault. Digital image correlation (DIC) was employed to analyze the soil displacement field, and both optical fibres and conventional strain gauges were used to measure pipe strains. A refinement to the Kappa method was introduced to enable the calculation of axial strains as a function of pipe elongation induced by flexure and an approximation of the longitudinal ground deformations. A closed-form Winkler solution of flexural response was also derived to account for the distributed normal fault pattern. Finally, these two analytical solutions were evaluated against the pipe responses observed in the large-scale laboratory tests.

Some Inverse Problems in Analysis and Geometry

Thesis (Ph.D.)--University of Washington, 2016-08

Magnetoresistance in a High Mobility Two-Dimensional Electron System as a Function of Sample Geometry

In a high mobility two-dimensional electron gas (2DEG) realized in a GaAs/Al0.3Ga0.7As quantum well we observe changes in the Shubnikov-de Haas oscillations (SdHO) and in the Hall resistance for different sample geometries. We observe for each sample geometry a strong negative magnetoresistance around zero magnetic field which consists of a peak around zero magnetic field and of a huge magnetoresistance at larger fields. The peak around zero magnetic field is left unchanged for different geometries.

Conservation laws, integrability, and transport in one-dimensional quantum systems

In integrable one-dimensional quantum systems an infinite set of local conserved quantities exists which can prevent a current from decaying completely. For cases like the spin current in the XXZ model at zero magnetic field or the charge current in the attractive Hubbard model at half filling, however, the current operator does not have overlap with any of the local conserved quantities. We show that in these situations transport at finite temperatures is dominated by a diffusive contribution with the Drude weight being either small or even zero. For the XXZ model we discuss in detail the relation between our results, the phenomenological theory of spin diffusion, and measurements of the spin-lattice relaxation rate in spin chain compounds. Furthermore, we study the Haldane-Shastry model where a conserved spin current exists.

Continuous structural transitions in quasi-one-dimensional classical Wigner crystals

We study the structural phase transitions in confined systems of strongly interacting particles. We consider infinite quasi-one-dimensional systems with different pairwise repulsive interactions in the presence of an external confinement following a power law. Within the framework of Landau's theory, we find the necessary conditions to observe continuous transitions and demonstrate that the only allowed continuous transition is between the single-and the double-chain configurations and that it only takes place when the confinement is parabolic. We determine analytically the behavior of the system at the transition point and calculate the critical exponents. Furthermore, we perform Monte Carlo simulations and find a perfect agreement between theory and numerics.

A hybrid-mixed finite element formulation for the geometrically exact analysis of three-dimensional framed structures

This paper addresses the development of a hybrid-mixed finite element formulation for the quasi-static geometrically exact analysis of three-dimensional framed structures with linear elastic behavior. The formulation is based on a modified principle of stationary total complementary energy, involving, as independent variables, the generalized vectors of stress-resultants and displacements and, in addition, a set of Lagrange multipliers defined on the element boundaries. The finite element discretization scheme adopted within the framework of the proposed formulation leads to numerical solutions that strongly satisfy the equilibrium differential equations in the elements, as well as the equilibrium boundary conditions. This formulation consists, therefore, in a true equilibrium formulation for large displacements and rotations in space. Furthermore, this formulation is objective, as it ensures invariance of the strain measures under superposed rigid body rotations, and is not affected by the so-called shear-locking phenomenon. Also, the proposed formulation produces numerical solutions which are independent of the path of deformation. To validate and assess the accuracy of the proposed formulation, some benchmark problems are analyzed and their solutions compared with those obtained using the standard two-node displacement/ rotation-based formulation.

A transient three-dimensional heat transfer model of the human body

The objective of this work is to develop an improved model of the human thermal system. The features included are important to solve real problems: 3D heat conduction, the use of elliptical cylinders to adequately approximate body geometry, the careful representation of tissues and important organs, and the flexibility of the computational implementation. Focus is on the passive system, which is composed by 15 cylindrical elements and it includes heat transfer between large arteries and veins. The results of thermal neutrality and transient simulations are in excellent agreement with experimental data, indicating that the model represents adequately the behavior of the human thermal system. (C) 2009 Elsevier Ltd. All rights reserved.

Effects of mechanical properties, residual stress and indenter tip geometry on instrumented indentation data in thin films

In this work, an axisymmetric two-dimensional finite element model was developed to simulate instrumented indentation testing of thin ceramic films deposited onto hard steel substrates. The level of film residual stress (sigma(r)), the film elastic modulus (E) and the film work hardening exponent (n) were varied to analyze their effects on indentation data. These numerical results were used to analyze experimental data that were obtained with titanium nitride coated specimens, in which the substrate bias applied during deposition was modified to obtain films with different levels of sigma(r). Good qualitative correlation was obtained when numerical and experimental results were compared, as long as all film properties are considered in the analyses, and not only sigma(r). The numerical analyses were also used to further understand the effect of sigma(r) on the mechanical properties calculated based on instrumented indentation data. In this case, the hardness values obtained based on real or calculated contact areas are similar only when sink-in occurs, i.e. with high n or high ratio VIE, where Y is the yield strength of the film. In an additional analysis, four ratios (R/h(max)) between indenter tip radius and maximum penetration depth were simulated to analyze the combined effects of R and sigma(r) on the indentation load-displacement curves. In this case, or did not significantly affect the load curve exponent, which was affected only by the indenter tip radius. On the other hand, the proportional curvature coefficient was significantly affected by sigma(r) and n. (C) 2010 Elsevier B.V. All rights reserved.

Ladder operator for the one-dimensional Hubbard model

The one-dimensional Hubbard model is integrable in the sense that it has an infinite family of conserved currents. We explicitly construct a ladder operator which can be used to iteratively generate all of the conserved current operators. This construction is different from that used for Lorentz invariant systems such as the Heisenberg model. The Hubbard model is not Lorentz invariant, due to the separation of spin and charge excitations. The ladder operator is obtained by a very general formalism which is applicable to any model that can be derived from a solution of the Yang-Baxter equation.

Two dimensional computational fluid dynamic models for waste stabilisation ponds

Traditional waste stabilisation pond (WSP) models encounter problems predicting pond performance because they cannot account for the influence of pond features, such as inlet structure or pond geometry, on fluid hydrodynamics. In this study, two dimensional (2-D) computational fluid dynamics (CFD) models were compared to experimental residence time distributions (RTD) from literature. In one of the-three geometries simulated, the 2-D CFD model successfully predicted the experimental RTD. However, flow patterns in the other two geometries were not well described due to the difficulty of representing the three dimensional (3-D) experimental inlet in the 2-D CFD model, and the sensitivity of the model results to the assumptions used to characterise the inlet. Neither a velocity similarity nor geometric similarity approach to inlet representation in 2-D gave results correlating with experimental data. However. it was shown that 2-D CFD models were not affected by changes in values of model parameters which are difficult to predict, particularly the turbulent inlet conditions. This work suggests that 2-D CFD models cannot be used a priori to give an adequate description of the hydrodynamic patterns in WSP. (C) 1998 Elsevier Science Ltd. All rights reserved.