Three-dimensional Half-space Research Articles

Boundary layers of the Boltzmann equation in three-dimensional half-space

We consider the nonlinear boundary layers of the Boltzmann equation on a three-dimensional half-space by perturbing around a Maxwellian under the assumption that the Mach number of the Maxwellian satisfies M∞<−1. In earlier works by S. Ukai, T. Yang, S.-H. Yu, nonlinear boundary layers of the Boltzmann equation in a half-line are considered, in a half-line are considered, the stationary solutions were obtained and nonlinear stability was confirmed. We establish the existence and uniqueness of stationary and time-periodic solutions for the three-dimensional half-space model and show that these solutions are asymptotically stable.

Wave propagation in a three-dimensional half-space with semi-infinite irregularities

ABSTRACT Dynamic analysis of problems with complex geometries requires utilization of numerical methods. To completely capture the effects of seismic wave propagation in a system, one must consider the structure or irregularity within its encompassing half-space. Correct consideration of half-space in a numerical model is important specially when it comes to cases where the half-space contains semi-infinite irregularities. In this study, a generalized numerical methodology is presented for dynamic analysis of a half-space with semi-infinite irregularities. The methodology is first verified through comparison with analytical solution of known problems. Then the method is employed to solve the dynamic analysis of an arbitrary-shaped existing valley within a half-space under propagating plane harmonic SH wave with different angles of incidence. Results indicate complete capability of the methodology in solving wave propagation problems for complex systems without the usual restrictions of common existing methods.

Reconstructing a point source from diffusion fluxes to narrow windows in three dimensions

We develop a computational approach to locate the source of a steady-state gradient of diffusing particles from the fluxes through narrow windows distributed either on the boundary of a three-dimensional half-space or on a sphere. This approach is based on solving the mixed boundary stationary diffusion equation with Neumann–Green’s function. The method of matched asymptotic expansions enables the computation of the probability fluxes. To explore the range of validity of this expansion, we develop a fast analytical-Brownian numerical scheme. This scheme accelerates the simulation time by avoiding the explicit computation of Brownian trajectories in the infinite domain. The results obtained from our derived analytical formulae and the fast numerical simulation scheme agree on a large range of parameters. Using the analytical representation of the particle fluxes, we show how to reconstruct the location of the point source. Furthermore, we investigate the uncertainty in the source reconstruction due to additive fluctuations present in the fluxes. We also study the influence of various window configurations: clustered versus uniform distributions on recovering the source position. Finally, we discuss possible applications for cell navigation in biology.

Scattering of Plane P1 Wave by an Inclusion in a Three-Dimension Poroelastic Half-Space

Local inclusion topography has significant influence on seismic wave propagation, and the propagation characteristics of seismic waves in poroelastic soils are obviously different from those in single-phase media. Based on Biot’s theory, the scattering of plane P1 wave by inclusion in a three-dimensional poroelastic half-space is studied by using the indirect boundary element method (IBEM). The scattering field is constructed by introducing a virtual wave near the interface between inclusion and half-space and the surface of half-space, and the virtual wave density is obtained by establishing boundary integral equation based on the boundary conditions. The effects of the depth, geometric characteristics, boundary permeability, porosity, incident frequency, and incident angle of the inclusion on elastic wave scattering are systematically analyzed. The results show that due to the soil skeleton-pore water coupling effect, when the porosity is n = 0.3, the surface displacement amplitude of dry soil is larger than that of poroelastic soil. When the porosity is n = 0.36, the surface displacement amplitude of poroelastic soil is larger than that of dry soil. The surface displacement amplitude of poroelastic-drained condition is slightly larger than that of undrained condition. With the increase of inclusion depth, the scattering of elastic wave by inclusion decreases gradually. When P1 wave is incident, the surface displacement amplitude at the depth of H = 0.5 can be increased up to three times as much as that at the depth of H = 1.5. As the inclusion becomes narrower and flatter, the scattering of elastic waves by inclusion decreases gradually. When the ratio between height and length is S = 2/5, the surface displacement magnitude can reach up to 9.5.

Born series for the photon diffusion equation perturbing the Robin boundary condition

The photon diffusion equation is solved making use of the Born series for the Robin boundary condition. We develop a general theory for arbitrary domains with smooth enough boundaries and explore the convergence. The proposed Born series is validated by numerical calculation in the three-dimensional half space. It is shown that in this case, the Born series converges regardless the value of the impedance term in the Robin boundary condition.

Application of shakedown theory in track substructure design

Tracks require high performance of earth structures against excessive residual settlement under repeated train loads. In the current design standards, the quality of soil materials and compaction control are strictly prescribed based on empirical knowledge and scale tests; however, construction costs become relatively high accordingly. A more rational design method based on shakedown theory is presented here. This method makes use of the lower-bound shakedown theorem and models the soils as Mohr–Coulomb materials. By calculating the analytical load-induced dynamic elastic stress field in a three-dimensional half-space and introducing a self-equilibrated residual stress field, the maximum magnitude of surface pressure on the soil foundation (i.e. shakedown limit) against long-term residual settlement is calculated through an optimisation programme. It is found the shakedown limit is dependent on the ratio of train velocity to shear wave velocity for a given earth structure. The maximum admissible train loads for the cases with different design velocities can be predicted by modifying the quasi-static shakedown limit with an attenuation factor.

Kinetics of deposition in the diffusion-controlled limit.

The adsorption of particles diffusing in a half space bounded by the substrate and irreversibly sticking to the substrate upon contacts is investigated. We show that when absorbing particles are planar disks diffusing in the three-dimensional half space, the coverage approaches its saturated "jamming" value as t^{-1} in the large time limit (generally as t^{-1/(d-1)} when the substrate is d dimensional and d>1, and as e^{-t/ln(t)} when d=1). We also analyze the asymptotic behavior when particles are spherical and when particles are planar aligned squares.

Formation of Slow Electron Pairs in an External Coulomb Field

The wave equation for two electrons in an external Coulomb field (helium-like atoms) has been shown to be a problem in a three-dimensional half-space. The wave-equation becomes quasi-separable in inertial coordinates. This allows to work out the electron motion in the frame of principal inertia axes. We find that non-adiabatic coupling terms constitute a fictitious force and lead to a deformation of the static potential surface. Incoming and outgoing modes of electron pairs are studied in detail, and applied to the threshold ionization of hydrogen-like atoms by electrons. Our analysis confirms the classical work by Wannier. However, we go beyond Wannier and present bending and stretch vibrations of electron pairs. The bending vibration has no influence onto the total ionization cross-section. The pair formation below threshold destroys the existence of high double Rydberg resonances. Finally, we describe the propagation of an electron pair through a linear chain of Rydberg atoms.

Charged-particle energy loss by the excitation of surface magnetoplasmons in a structure with two- and three-dimensional plasmas

Electron energy loss by the excitation of surface magnetoplasma oscillations by an electron moving along a static magnetic field in vacuum over a two-dimensional plasma layer on the surface of three-dimensional plasma half-space has been studied theoretically. Electron energy loss by the excitation of surface magnetoplasmons has been calculated in the electrostatic approximation. It has been shown that the type of the dispersion law of electrons in such a plasma (quadratic for a two-dimensional Drude gas or linear for graphene) can be determined from the qualitative character of the dependence of the maximum of the spectral density of this loss on the electron density in the two-dimensional plasma.

A Liouville Theorem for the Planer Navier-Stokes Equations with the No-Slip Boundary Condition and Its Application to a Geometric Regularity Criterion

We establish a Liouville type result for a backward global solution to the Navier-Stokes equations in the half plane with the no-slip boundary condition. No assumptions on spatial decay for the vorticity nor the velocity field are imposed. We study the vorticity equations instead of the original Navier-Stokes equations. As an application, we extend the geometric regularity criterion for the Navier-Stokes equations in the three-dimensional half space under the no-slip boundary condition.

Transient Disturbance in a Three-Dimensional Thermo-elastic Half-Space Under Green–Naghdi Theory

A three-dimensional problem for a homogeneous isotropic thermo-elastic half-space subjected to a time-dependent heat source on the boundary of the space, which is traction free, is considered in the context of the Green and Naghdi model II and III (without energy dissipation and with energy dissipation, respectively) of generalized thermo-elasticity. Normal mode analysis and eigenvalue approach techniques are used to solve the resulting non-dimensional coupled field equations. Numerical results for the real parts of the temperature, thermal stresses, and displacement distributions are presented graphically and we analyze the obtained results in various cases—(i) for different values of the space variables; (ii) for different values of the time instant, and (iii) for different values of the Green–Naghdi parameter. Comparisons are made with the results for the real parts of the temperature, thermal stresses, and displacement distributions predicted by the Green–Naghdi model II and Green–Naghdi model III in these cases also.

Half Space Acoustic Problems Analysis by Fast Multipole Boundary Face Method

Abstract: In this paper, a half space adaptive fast multipole boundary facemethod (FMBFM) is presented for solving the three-dimensional half space ex-terior acoustic problems. In the presented method, the Burton-Miller equationbased on the conventional boundary integral equation (CBIE) and its hyper-singularboundary integral equation (HBIE) is used to deal with the fictitious eigenfrequen-cies problem. The half space Green’s function is employed, thus the tree structurein the fast multipole method can be used only for the real domain. The higher orderelements and an adaptive tree structure are used to improve the efficiency of theFMBFM. This half space adaptive FMBFM for half space acoustic problems is anextension of the adaptive FMBFM for full space acoustic problems developed bythe authors. Numerical examples for half space acoustic problems in this paperdemonstrate the efficiency and validity of this method.Keywords: fast multipole boundary face method, Burton-Miller equation, acous-tic problems, half space, modified hyper-singular boundary integral equation1 IntroductionAcoustic problems in half space are a major subject in engineering. Many researchworks have been done on this aspect (Seybert et al., 1988; Secgin et al., 2010; Caoet al., 2010). The half space Green’s function (Seybert et al., 1988) can be appliedby adding the source solution at the image point to the original full-space Green’sfunction for dealing with half space problems. Sound source localization and iden-tification of a refrigerator were accomplished by using the half space boundaryelement method (BEM) (Secgin et al., 2010). Using the half space Green’s func-tion, the discretization of the rigid infinite plane is removed and only the boundaries

Three-dimensional transport imaging for the spatially resolved determination of carrier diffusion length in bulk materials

A contact-free optical technique is developed to enable a spatially resolved measurement of minority carrier diffusion length and the associated mobility-lifetime (μτ) product in bulk semiconductor materials. A scanning electron microscope is used in combination with an internal optical microscope and imaging charge-coupled device (CCD) to image the bulk luminescence from minority carrier recombination associated with one-dimensional excess carrier generation. Using a Green's function to model steady-state minority carrier diffusion in a three-dimensional half space, non-linear least squares analysis is then applied to extract values of carrier diffusion length and surface recombination velocity. The approach enables measurement of spatial variations in the μτ product with a high degree of spatial resolution.

Thermal field diffusion in one, two and three-dimensional half space

The diffusion of suddenly occurring local high temperature in homogeneous half-infinite space is studied in the cases of one, two and three-dimensional half space. Comparison of the three cases is made. Applications of theoretically analyzed models are suggested. Errors induced by assumptions are evaluated.

Application of transition matrix to scattering of elastic waves in half-space by a surface scatterer

A transition matrix formulism for the scattering problems of elastic waves using a surface scatterer of three-dimensional half-space is derived by applying Betti's third identity and orthogonality conditions. The basis functions and their corresponding regular parts are derived from Lamb's singular solutions. The new orthogonal functions are constructed based on the linear transform. The intrinsic properties of the scattering and transition matrices derived in this paper are investigated. Numerical results for verification are presented and discussed.

Omni-directional Rayleigh, Stoneley and Schölte waves with general time dependence

Just as uni-directional Rayleigh waves at the traction-free surface of a transversely isotropic elastic half-space and Stoneley waves at the interface between two such media may have arbitrary waveform and may be represented in terms of a single function harmonic in a half-plane, it is shown that surface-guided waves travelling simultaneously in all directions parallel to the surface may be represented, at each instant, in terms of a single function satisfying Laplace’s equation in a three-dimensional half-space. That harmonic function is determined so that its normal derivative at the surface equals the normal displacement of the surface (or interface). It is shown, moreover, that the time evolution of that normal displacement may be any solution to the membrane equation with wave speed being equal to that of classical, uni-directional, time-harmonic Rayleigh or Stoneley waves. A similar representation is also shown to exist for Schölte waves at a fluid–solid interface, in the non-evanescent case. Thus, every surface- or interface-guided disturbance in media having rotational symmetry about the surface normal is governed by the membrane equation with appropriate wave speed, provided that the combination of materials allows uni-directional, time-harmonic waves that are non-evanescent. Conversely, each solution to the membrane equation may be used to construct a representation of either a Rayleigh wave, a Stoneley wave or a (non-evanescent) Schölte wave. In each case, the disturbance at all depths may be represented at each instant in terms of a single function harmonic in a half-space.

Convergence Rate Toward Planar Stationary Waves for Compressible Viscous Fluid in Multidimensional Half Space

The present paper concerns a large-time behavior of motion of isentropic and compressible viscous fluid in a two- or three-dimensional half space. We obtain a convergence rate of a solution toward a planar stationary wave for an outflow problem, where fluid flows out through a boundary. For a supersonic flow at spatial infinity, we obtain an algebraic or an exponential decay rate provided that an initial perturbation decays, with respect to a normal direction, with the algebraic or the exponential rate, respectively. The algebraic convergence rate is also obtained for a transonic flow. Owing to a degenerate property of the transonic flow, the convergence rate is worse than that for the supersonic flow. The proofs are based on deriving a priori estimates of the perturbation from the stationary wave by using a time and space weighted energy method.

A semi-analytical model in time domain for moving loads

A model for investigations of ground motions due to continuously moving loads with constant and time-varying amplitudes is presented. The vertical displacements excited by moving load areas are obtained at a fixed observation point at the surface of the three-dimensional halfspace in time domain. The load is moving along a straight line with constant speed. To solve this nonaxisymmetric, initial boundary value problem a semi-analytical, discretized model is developed. It is based on Green's functions for a suddenly applied, stationary surface point load with Heaviside time dependency. These functions, also called influence functions of the halfspace, are valid for any homogeneous, isotropic and linear-elastic medium. The principle of superposition is used. Results are shown for the transient and the steady-state ground motions, and they are compared with analytical solutions. The load speed is varied in the subcritical range up to the propagation velocity of Rayleigh-waves.

Dynamic rupture on a planar fault in three-dimensional half space - I. Theory

In this study, we extended the boundary integral equation method (BIEM) to include the effect of free surface, which is crucial for the interpretation of near-field observations of large earthquakes on dipping faults, which are buried shallower or rupture to the ground. On the basis of the Green's function in 3-D half space, the boundary integral equations (BIEs) for dynamic rupture propagation on a planar fault embedded in 3-D elastic half space are derived. Since the Green's function can only be expressed as double integration, rather than analytic closed form as in full space, computation of the BIEs for rupture dynamics in half space is rather complicated. Consequently, serious difficulties arise, including the existence of hypersingularities in the kernels and the computation of rapid oscillatory multiple integrals. To resolve them, we first systematically presented the effective regularization procedures, which consists of the generalized Apsel-Luco correction and the regularization algorithm proposed by Karami & Derakhshan to remove the hypersingularities, then proposed a simple while efficient algorithm to speed up the computation of integrals involved in the kernels of BIEs. Finally, the BIEM algorithm proposed in this study for modelling rupture dynamics of a fault with arbitrary dip angle in 3-D half space becomes exercisable, and provides a powerful tool for investigating the physics of earthquake dynamics.

Three-dimensional halfspace constructive solid geometry tree construction from implicit boundary representations

This paper presents a new method to compute constructive solid geometry (CSG) tree representations of an object whose faces consist of planar and non-planar surfaces. The algorithm described accepts as input a valid boundary representation of an object consisting of piecewise implicit surfaces, and computes a halfspace CSG representation of the object. A class of objects that are describable by the surfaces bounding them are valid input for the algorithm of this work, although methods currently exist to compute the additional information necessary to process non-describable quadric objects as well. This work builds on and complements the other work in this area, in which dominating halfspaces are used to simplify the b-rep to CSG conversion process. We include factored faces to enable the factorization of dominating halfspaces throughout the algorithm. Thus, an efficient disjoint decomposition of the solid is obtained as a matter of course in the algorithm, so that CSG minimization is generally not necessary.This work is motivated by reverse engineering of mechanical parts, in which a model of a part is recovered from information obtained by some sort of sensing technique (e.g. CAT scanning, laser range finding). The recovery of a valid CSG-tree description of an object from a boundary representation of it can provide useful information to an engineer in the area of reverse engineering and in other areas related to solid modeling as well. The CSG tree also provides a relatively neutral representation that can enhance form feature recognition and translation.