Robustness of Tube Formula Based Confidence Bands

Abstract

Simultaneous confidence bands of a regression curve may be used to quantify the uncertainty of an estimate of the curve. The tube formula for volumes of tubular neighborhoods of a manifold provides a very powerful method for obtaining such bands at a prescribed level, when errors are Gaussian. This article studies robustness of the tube formula for non-Gaussian errors. The formula holds without modification for an error vector with a spherically symmetric distribution. Simulations are used for a variety of independent non-Gaussian error distributions. The results are acceptable for contaminated and heavy tailed error distributions. The formula can break down in some extreme cases for discrete and highly skewed errors. Computational issues involved in applying the tube formula are also discussed.     

Analysis of ray tube area in geometrical shock dynamics

We provide a rigorous analysis of structure of a ray tube in geometrical shock dynamics. Our aim is to shed more lights on the cross-sectional area function of a ray tube. We have shown that for a given ray tube and a given initial value of cross-sectional area, then its cross-sectional area can be determined uniquely everywhere. We give a definition of cross-sectional area of a ray tube in precise mathematical terms and from that we derive a set of relations, each describing the cross-sectional area for an arbitrary ray tube in geometrical shock dynamics. We have shown that from our results one can deduce Whitham’s area function relation as a partial differential equation from our general formulations. Some applications are discussed.     

Theoretical and Experimental Investigations of Piezoelectrically Excited Travelling Waves in Cylindrical Tubes

Flügge's equations of motion for cylindrical shells are used to analyze the generation of travelling waves in a tube of an ultrasonic travelling wave transport system. The vibration-shape of a semi-infinite acrylic tube with structural damping excited by a piezoelectric ring at the end is calculated and the influence of the driving frequency is shown. Starting with an exponential ansatz the frequency dependent wavenumbers can be determined. Three frequency bands with different combinations of imaginary, real and complex wavenumbers can be found. Supposing a semi-infinite tube with given boundary conditions at the end of excitation the vibrations of the tube can be calculated depending on the frequency. Simulation agrees well with experimental results. (© 2011 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Collisionless Flow in Vacuum Traps

Molecules stream in collisionless flow out of a reservoir intoan axially symmetric tube consisting of two equal and oppositelyinclined conical frustra—a shape used in industrial vacuumtraps. In the general case there is a baffle in the centre ofthe tube. Some of the molecules strike the wall of the tube,some the baffle, and are reflected; eventually they either flyback into the reservoir or escape out of the tube into the vacuumbeyond. Other particles stream directly from the reservoir intothe vacuum without any collisions with a surface. The impact density of molecules along the wall in the steadystate is determined by a system of integral equations, fourin all when the baffle is present, otherwise two. These numberscan be reduced by half by utilizing certain combinations ofthe equations, suggested by the physical system, that yieldanalytical relations between the impact densities. The remainingintegral equations are solved numerically. One of the kernelshas an indeterminacy corresponding to the kink in the longitudinalcross-section of the tube, but the existence and uniquenessof the solution is established by analytical means. The resultsof calculations of the molecular impact density on the surfacesand the conductance of the tube with baffle for a number oftypical systems are given in graphical form.     

二相平面渗流问题的不变流管解法及其理论分析

本文总结和改进了工程上广泛应用的求解二相平面渗流问题的不变流管近似方法.对其核心部份即给定压差时一维变截面流管中的二相驱替问题作了全面的考察,证明了解的存在唯一性,给出了准确解、数值解及其收敛性和稳定性分析.     

Waves in an Inhomogeneous Tube with a Flowing Two-Phase Fluid

Results of theoretical and mathematical justification of the problem on a pulsating flow of a two-phase barotropic bubbly fluid enclosed in an elastic semi-infinite cylindrical tube inhomogeneous along its length are presented. Linear one-dimensional equations are used. It is assumed that the tube is rigidly attached to the surrounding medium and therefore its displacement in the axial direction is absent. At infinity, the tube material is assumed to be homogeneous. To describe the pressure, flow rate, and displacement of the fluid, a pulsating pressure is given at the tube end. The problem stated is reduced to a singular Sturm-Liouville boundary-value problem, which in turn is reduced to a Volterra-type integral equation. This equation is solved by the method of successive approximations. By assuming that the corresponding potential is integrable, it is proved that these approximations converge to the exact solution of the problem. It is shown that this assumption also covers the very important practical case of piecewise inhomogeneity. For numerical realization, we consider a homogeneous tube with flowing water containing a small amount of bubbles. The effect of the volume content of bubbles on wave characteristics is revealed. In particular, it is stated that, for the oscillation regime selected, an increased bubble volume content decreases the wave velocity and considerably increases the flow speed (rate).     

Optimization methodology assessment for the inlet velocity profile of a hydraulic turbine draft tube: part I—computer optimization techniques

In recent years, numerical and experimental investigations on the draft tube performance have confirmed the importance of the inlet swirling flow created by the runner vanes. The results indicate that it is still a challenge to get the optimal flow distribution at the draft tube inlet which gives the best machine performance over a range of operation points. Consequently, there is a need to adjust the runner-draft tube coupling to minimize the losses arising from the inlet flow distribution. This paper focus on establishing an optimization methodology for maximizing the draft tube performance as a function of the inlet velocity profile. The overall work is divides into two parts: The part one establish the inlet velocity parametrization, the numerical optimization set-up and the objective function definition. The part two validate the numerical CFD draft tube model. These steps are represented by the coupling of the commercial softwares MATLAB, FLUENT and iSIGHT. It is considered that this proved methodology will help to find a inlet velocity profile shape which will be able to suppress or mitigate the undesirable draft tube flow characteristics.     

Inflation-Extension and Eversion of a Tube of Incompressible Isotropic Elastic Material

Adopting a form of strain-energy function for highly deformableincompressible elastic materials recently proposed by Ogden,we study in this paper two deformations of a circular cylindricaltube. In the first deformation the tube is inflated by increasingits internal radius by an assigned factor, and in the secondthe tube is turned inside out. In each case it is assumed thatthe tube is free of longitudinal constraint to the extent thatthe normal tractions on the end-faces have zero resultant. Inflationand eversion are therefore accompanied by axial shortening orelongation. Detailed numerical results are presented and comparedwith the limited experimental data to hand. Consideration isalso given to the existence and uniqueness of deformations whichpreserve the shape of the tube and comply with the assumed boundaryconditions. Restrictions are obtained on the material constantsappearing in the strain-energy function which ensure the existenceof precisely one such solution to the problem of combined inflationand extension, and sufficient conditions for the existence ofat least one solution to the eversion problem are also found.These requirements agree with criteria for "physically realisticresponse" which have been established elsewhere.     

充液弹性毛细管低温相变的力学分析

充液弹性毛细管广泛存在于生物体(如毛细血管、植物导管等)和工程领域(如微流控冰阀门、制冷系统热管、MEMS微通道谐振器等).低温工作环境中,充液弹性毛细管内部的液柱会发生相变并引发冻胀效应,从而导致管壁的变形、损伤乃至断裂.该文建立并求解了考虑温度梯度、界面张力及液体冻胀作用的弹性毛细管平衡方程,分析了液柱低温相变过程中毛细管壁的径向和环向应力,发现管壁应力分布受热毛细弹性数和冻毛细弹性数的影响,且影响大小跟壁厚相关.该研究不仅有助于理解生物体内充液弹性毛细管冻胀失效机制,还可为MEMS微流控芯片的抗冻胀失效设计提供理论指导.     

Thermal buckling and post-buckling analysis of geometrically imperfect FGM clamped tubes on nonlinear elastic foundation

Present research deals with the thermal buckling and post-buckling analysis of the geometrically imperfect functionally graded tubes on nonlinear elastic foundation. Imperfect FGM tube with immovable clamped–clamped end conditions is subjected to thermal environments. Tube under different types of thermal loads, such as heat conduction, linear temperature change, and uniform temperature rise is analyzed. Material properties of the FGM tube are assumed to be temperature dependent and are distributed through the radial direction. Displacement field satisfies the tangential traction free boundary conditions on the inner and outer surfaces of the FGM tube. The nonlinear governing equations of the FGM tube are obtained by means of the virtual displacement principle. The equilibrium equations are based on the nonlinear von Kármán assumption and higher order shear deformation circular tube theory. These coupled differential equations are solved using the two-step perturbation method. Approximate solutions are provided to estimate the thermal post-buckling response of the perfect/imperfect FGM tube as explicit functions of the various thermal loads. Numerical results are provided to explore the effects of different geometrical parameters of the FGM tube subjected to different types of thermal loads. The effects of power law index, springs stiffness of elastic foundation, and geometrical imperfection parameter of tube are also included.     

Optimization methodology assessment for the inlet velocity profile of a hydraulic turbine draft tube: part II—performance evaluation of draft tube model

Computational Fluids Dynamics (CFD) tools guide engineers and designers to estimate the performance of new designs. However, a CFD analysis can be very time-consuming depending mainly on the grid size and domain complexity. Thus, this paper aims to describe the tools used to evaluate and compare the performance of different 3D draft tube models for reducing the time-effort needed in an optimization procedure. The results presented here, are the second part of an overall research to establish a global optimization methodology to improve the performance of an hydraulic draft tube through the inlet velocity profile. Previously, three steps of optimization methodology to minimize the energy losses were studied: the inlet velocity profile parameterization, the numerical optimization set-up and the objective function validation. In the latter step, a global optimization method called Multi Island Genetic Algorithm (MIGA) was considered, which requires a large number of iterations before producing a reliable result. This step is able to identify an efficient inlet velocity profile to minimize the energy losses through the draft tube model. However, each iteration is expensive in terms of computational time due to the need for 3D Navier–Stokes (NS) computations to evaluate each profile’s fitness. Thus, in this work the methodology attempts to accelerate the optimization process with accurate results. In order to achieve the goal, the grid size of the 3D draft tube model was minimized, resulting in a much lower computational cost. Specifically, the draft tube calculations were performed on a sequence of five different grids each having approximately twice the number of elements compared to the previous. The measurements of the sensitivity of the draft tube performance quantities to the change of the inlet velocity parameters during the process showed that, in spite of the numerical difference between its performance, the results have the same tendency. Consequently, the 3D draft tube numerical model with a minimal grid size, is reliable and left record of its capabilities for being integrated in the optimization process.     

The tube wave generated by a point source located outside a borehole

The excitation of the tube wave by a point source situated inside an elastic medium is considered. The amplitude of the tube wave as a function of the radial distance is given for media with high and low velocies. Two models of sources are considered: a point vertical source and a point horizontal source. It is shown that the radial source is an efficient tool for excitation of a tube wave in high-and low-velocity media. Bibliography: 17 titles. __________ Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 332, 2006, pp. 99–122.     

Geometric polyhedral models for nanotubes comprising hexagonal lattices

Two new models for the geometric structure of nanotubes comprising hexagonal lattices are described. The existing models for nanotubes typically involve rolled up planar sheets and ignore discrepancies due to curvature. The first of the models presented here assumes that all atomic locations are equidistant from the tube axis which applies for single species nanotubes such as carbon nanotubes. This model assumes that all bond angles and all bond lengths are equal in the cylindrical state, and that all atoms are equidistant from the tube axis, and from these three assumptions, expressions are given for the major geometric parameters. The second model extends this notion to tubes where all the atomic locations are not equidistant from the tube axis, which may be employed to model nanotubes comprising two chemical species that bond into a hexagonal lattice such as boron nitride nanotubes. In the second model, all bond lengths are taken to be equal and the atoms of the same species are taken to be equidistant from the tube axis, and the nanotube is assumed to comprise two species and thus there may be two radii. Fundamental to both models is the determination of a solution of a transcendental equation. Here we present a new formal Lagrange expansion of the solution. Previously given asymptotic series expansions of the exact formulae for both models lead to the conventional expressions as the leading order term. Although the correction terms are typically small, knowledge of the precise structure may be critical to comprehending many nanoscale phenomena. The new models also give rise to an expression for the wall thickness, an important geometric parameter for which at present no reliable information is available.     

Large Eddy Simulation of a flow through circular tube bundle

The Large Eddy Simulation approach is used to perform a numerical simulation of a flow around a circular tube bundle. The model uses the immersed boundaries technique to represent tube geometry in the Cartesian grid. Simulation is conducted in an area away from the boundaries, where the flow is considered homogenous and, thus, the three Cartesian directions are considered periodic. The results are compared with experimental data in terms of mean velocity and turbulence quantities, showing good agreement in general.     

Frequencies of Free Acoustic Oscillations in a Fluid Separating Rigid and Elastic Tube Walls of Finite Length

A solution of the problem of determining the frequencies and mode shapes of free nonsymmetric oscillations in an annular volume filled with an ideal compressible fluid is constructed. The inner tube and the end plane walls are ideally rigid. A thin elastic shell with edges clamped to the end walls is located on the outer tube boundary. A phenomenon of a decrease in the fundamental frequency as the thickness of a fluid layer adjacent to the elastic wall decreases is confirmed. Bibliography: 8 titles.     

Resonant three-wave interactions in non-linear hyper-elastic fluid-filled tubes

A multiple-scales method is used to derive the Three-Wave Interaction (TWI) equations describing resonantly interacting triads in nonlinear hyper-elastic fluid-filled tubes. The tube wall is assumed to be an axially-tethered nonlinear membraneous cylindrical shell for which the resultant stresses can be determined by a strain-energy functional. The fluid within the tube is assumed to be two-dimensional, axi-symmetric and inviscid. We show that small-but-finite amplitude strongly dispersive pressure wave packets can continuously exchange energy in a resonant triad while conserving total energy. For a Mooney-Rivlin shell wall the theory presented predicts a short wavelength cutoff on the order of the tube radius. Thus pressure pulses containing wavelengths on the order of the tube radius and longer may contain resonantly interacting modes. Special solutions are presented: temporally developing modes, pump-wave approximations and explosively unstable steadily-traveling wave packets.     

Digital quantum batteries: Energy and information storage in nanovacuum tube arrays

Dielectric material between capacitor electrodes increases the capacitance. However, when the electric field exceeds a threshold, electric breakdown in the dielectric discharges the capacitor suddenly and the stored energy is lost. We show that nanovacuum tubes do not have this problem because (i) electric breakdown can be suppressed with quantization phenomena, and (ii) the capacitance is large at small gap sizes. We find that the energy density and power density in nanovacuum tubes are large compared to lithium batteries and electrochemical capacitors. The electric field in a nanovacuum tube can be sensed with MOSFETs in the insulating walls. Random access arrays of nanovacuum tubes with an energy gate, to charge the tube, and an information gate attached to the MOSFET, to sense the electric field in the tube, can be used to store both energy and information. © 2010 Wiley Periodicals, Inc. Complexity 2010     

Stresses in elastic conical tubes of transversely isotropic materials with spherical anisotropies under temperature and force loadings

Analytic solutions are proposed for a number of new problems on determining the state of stress of a transversely-isotropic hollow cone with spherical anisotropy. An exact solution of the problem of the axisymmetric deformation of a long conical tube (or continuous cone) from an elastic transversely-isotropic material with spherical anisotropy subjected to an axial force is obtained in a spherical coordinate system R, , θ, the material axis of symmetry is directed along the spherical radius R. A rigorous solution is given of the problem of the uniform heating of a conical tube of transversely-isotropic material with spherical anisotropy for particular values of Poisson's ratios; the material axis of symmetry is directed along the θ-axis. For arbitrary Poisson's ratios an asymptotic solution is found for the temperature problem for a tube with small conicity.     

Kinematic-wave analysis of particle settling in tube centrifuges

The kinematic-wave theory of particle settling in tube centrifuges is subject to controversy. G. Anestis and W. Schneider [1] made the common assumption that the flow in a rotating tube of large length/diameter ratio can be treated in one-dimensional approximation. Their results have been found in agreement with measurements [2], [3]. However, M. Ungarish [4] performed an analysis of the two-dimensional settling process in the limit of vanishing particle concentration and obtained results that are not in accord with the one-dimensional flow approximation, irrespective of the length/diameter ratio. It is the aim of the present investigation to gain a better understanding of the process. According to this analysis, quasi-one-dimensional kinematic waves are embedded in a two-dimensional bulk flow that is governed by the boundary conditions at the walls of the tube. The results are compared with those of the one-dimensional flow approximation. (© 2014 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)