Half-space Problem Research Articles

Application of low-order potential solutions to higher-order vertical traction boundary problems in an elastic half-space.

New solutions of potential functions for the bilinear vertical traction boundary condition are derived and presented. The discretization and interpolation of higher-order tractions and the superposition of the bilinear solutions provide a method of forming approximate and continuous solutions for the equilibrium state of a homogeneous and isotropic elastic half-space subjected to arbitrary normal surface tractions. Past experimental measurements of contact pressure distributions in granular media are reviewed in conjunction with the application of the proposed solution method to analysis of elastic settlement in shallow foundations. A numerical example is presented for an empirical ‘saddle-shaped’ traction distribution at the contact interface between a rigid square footing and a supporting soil medium. Non-dimensional soil resistance is computed as the reciprocal of normalized surface displacements under this empirical traction boundary condition, and the resulting internal stresses are compared to classical solutions to uniform traction boundary conditions.

Radiative albedo from a linearly fibered half-space

.A growing acceptance of fiber-reinforced composite materials imparts some relevance to exploring the effects which a predominantly linear scattering lattice may have upon interior radiative transport. Indeed, a central feature of electromagnetic wave propagation within such a lattice, if sufficiently dilute, is ray confinement to cones whose half-angles are set by that between lattice and the incident ray. When such propagation is subordinated to a viewpoint of an unpolarized intensity transport, one arrives at a somewhat simplified variant of the Boltzmann equation with spherical scattering demoted to its cylindrical counterpart. With a view to initiating a hopefully wider discussion of such phenomena, we follow through in detail the half-space albedo problem. This is done first along canonical lines that harness the Wiener-Hopf technique, and then once more in a discrete ordinates setting via flux decomposition along the eigenbasis of the underlying attenuation/scattering matrix. Good agreement is seen to prevail. We further suggest that the Case singular eigenfunction apparatus could likewise be evolved here in close analogy to its original, spherical scattering model. A cursory contact with related problems in the astrophysical literature suggests, in addition, that the basic physical fidelity of our scalar radiative transfer equation (RTE) remains open to improvement by passage to a (4×1) Stokes vector, (4×4) matricial setting.

When fast and slow interfaces grow together: Connection to the half-space problem of the Kardar-Parisi-Zhang class.

We study height fluctuations of interfaces in the (1+1)-dimensional Kardar-Parisi-Zhang (KPZ) class, growing at different speeds in the left half and the right half of space. Carrying out simulations of the discrete polynuclear growth model with two different growth rates, combined with the standard setting for the droplet, flat, and stationary geometries, we find that the fluctuation properties at and near the boundary are described by the KPZ half-space problem developed in the theoretical literature. In particular, in the droplet case, the distribution at the boundary is given by the largest-eigenvalue distribution of random matrices in the Gaussian symplectic ensemble, often called the GSE Tracy-Widom distribution. We also characterize crossover from the full-space statistics to the half-space one, which arises when the difference between the two growth speeds is small.

Initial-boundary value problem of a parabolic–hyperbolic system arising from tumor angiogenesis

In this paper, we consider the half-space problem for a parabolic–hyperbolic system arising from tumor angiogenesis. Under a mixed type boundary condition, we construct the Green's function for half-space problem by relating it with fundamental solution for an initial value problem. After differential equation method for the boundary operator and proper estimates for additional exponential factors generated by the asymmetry of characteristics, one reduces the Green's function (for initial-boundary value problem) into fundamental solution (for Cauchy problem) which could be estimated by singularity removal, long wave-short wave decomposition and weighted energy method outside cone. We finally obtain the pointwise estimate for Green's function which results in the pointwise convergence rate for nonlinear problem. The Green's function constructed is precise enough to avoid a priori estimate for derivatives by energy method. The half space problem studied here is a main ingredient for future study of shock profile with presence of boundary.

Solution of the radiative transfer equation for Rayleigh scattering using the infinite medium Green’s function

The infinite medium Green’s function is used to solve the half-space albedo, slab albedo and Milne problems for the unpolarized Rayleigh scattering case; these problems are the most classical problems of radiative transfer theory. The numerical results are obtained and are compared with previous ones.

Multi-dimensional Wave Propagation Over a Burgers Shock Profile

We study the stability of viscous shock profiles under multi-dimensional perturbation in order to understand the propagation ofmulti-dimensional dispersionwaves over the compressive shock profile. Our analysis is based on the new algorithm of explicitly constructing the Green’s function for the system linearized around the shock profile. We first construct Green’s function for the system linearized around the inviscid profile by matching the two half space problems. The Green’s function around the shock profile is then constructed by iterations. Our approach is of a general nature. We carry out the approach for a simple model possessing the Burgers shocks and wave structure similar to acoustic cones in multi-dimensional gas flows. The approach exhibits a rich phenomenon of wave propagation along and dispersing away from the shock profile.

Multidimensional Analogs of the Cauchy–Riemann System and Representations of Their Solutions via Harmonic Functions

At present, there are numerous multidimensional generalizations of holomorphic vectors. The most general of these is the four-dimensional generalization of the Cauchy–Riemann system. In the present work, by introducing two quaternion functions and the notion of quaternion differentiation, we obtain, for the first time, a five-dimensional generalization of holomorphic vectors. By using the representation of holomorphic vectors via the quaternion harmonic function and its derivatives, we consider the Riemann–Hilbert problem and one problem in a layer. A new solution of the Riemann–Hilbert problem in the five-dimensional half space is obtained.

Parallel Integral Equation Based Nonoverlapping DDM for Solving Electrically-Large Scattering Problems in Half Space

In this paper, a parallel nonoverlapping non-conformal domain decomposition method using combined-field integral equation is proposed for fast and accurate analysis of electrically large objects situated in half space. This method decomposes the whole model into several easily solvable closed subdomains in a flexible way, and an explicit boundary condition is applied to ensure the continuity of electric currents across the boundary. The adaptive direction partitioning parallelization scheme for the oct-tree of a multilevel fast multipole algorithm is adopted to accelerate matrix-vector multiplications of subdomains as well as the coupling between them. The introducing of real image source to account for far interaction matrix greatly accelerates the process and reduces memory requirement with precision guaranteed. Numerical examples demonstrate that the proposed method is able to simulate realistic problems with a maximum dimension greater than 1000 wavelengths.

Contact Problem for a Viscoelastic Half-Space and a Rough Rigid Cylinder under the Conditions of Hydrodynamic Lubrication

A model has been proposed for use in studying the combined effect of the roughness of a rigid punch and the viscous properties of a base separated by a thin lubricant layer exerted on the characteristics of contact interactions and the sliding friction force. The problem of the motion of a thin a lubricant layer between a fixed rigid cylinder with a regular relief, as well as the surface of a moving viscoelastic half-space, the rheological properties of which are described by an integral operator with an exponential creep kernel, has been considered. The pressure and thickness of the lubricating layer, as well as the deformation component of the frictional force depending on the sliding velocity, have been analyzed. A comparison of the results of solutions of contact problems for viscoelastic and elastic rough bodies in the presence of a lubricant has been presented.

Semi-analytical research on acoustic-structure coupling calculation of partially submerged cylindrical shell

Vibroacoustic analysis of a partially submerged cylindrical shell-liquid coupling system is a typical acoustic-structure interaction problem in an acoustic half-space. Generally, the vibration and acoustic solutions of this problem almost depend on numerical method in previous studies. However, whether from the aspect of verifying the numerical methods or from the aspect of revealing the vibroacoustic mechanism of the acoustic-structure interaction system, the development of an analytical or semi-analytical method is indispensable. In this study, a semi-analytical method is proposed to address the vibroacoustic response of a horizontal cylindrical shell partially immersed in water The acoustic coordinate system is established on the free surface, and the sine series function is introduced to express the sound pressure to meet the pressure release boundary condition on the free surface automatically. Then based on the two-dimensional Flgge shell theory, the motion equation of the shell-liquid coupling system is established with using the center of the shell as the origin of the coordinate system. By using the Galerkin method, the velocity continuous condition on the fluid-structure interface is established. Then the relation matrix between the acoustic pressure amplitude and the shell displacement amplitude is derived after converting the sound pressure from different coordinate systems into that in the same coordinate system by using the geometry relationship of two coordinate systems. Then the vibration and sound radiation of this coupling system are predicted by solving the coupled matrix equations, and the coupled free vibration can also be addressed by solving the characteristic equation after setting the determinant of the coefficient matrix to be zero. To verify the accuracy and reliability of the present method, the coupled natural frequencies, the vibration response and the distribution of the sound pressure obtained from the present method are compared with those obtained from the finite element method, showing that they are in good agreement with each other. Then the shapes of the first four order modes for the coupled system are presented and compared with those of the system submerged in unbounded fluid field, and the couple between different modes, and the couple between the symmetric and antisymmetric modes are observed due to the effect of the free surface. The characteristics of the root mean square velocity of the shell with different immersed depths are discussed, which reveals that the peaks of response curves shift to the lower frequencies with increasing immersed depth. The characteristics of far field sound pressure directivity are presented and explained by the image method in detail. This study provides a novel method to analytically predict the vibroacoustic response of an elastic structure partially coupling with the fluid field when bounded in a sound half space.

On the Alternative Vector Potential Formulation of the Sommerfeld Half-Space Problem

We address the non-uniqueness of the vector potential for the Sommerfeld half-space problem with a horizontal Hertzian dipole. In contrast to the classical solution, which utilizes horizontal and vertical components of the vector potential, the formulation presented here employs two horizontal components. In this approach, unlike in the traditional one, the scalar potentials of single-point charges associated with the vertical and horizontal Hertzian dipoles are identical.

Fractional order generalized thermoelastic response in a half space due to a periodically varying heat source

PurposeThe present work is concerned with the solution of a fractional-order thermoelastic problem of a two-dimensional infinite half space under axisymmetric distributions in which lower surface is traction free and subjected to a periodically varying heat source. The thermoelastic displacement, stresses and temperature are determined within the context of fractional-order thermoelastic theory. To observe the variations of displacement, temperature and stress inside the half space, the authors compute the numerical values of the field variables for copper material by utilizing Gaver-Stehfast algorithm for numerical inversion of Laplace transform. The effects of fractional-order parameter on the variations of field variables inside the medium are analyzed graphically. The paper aims to discuss these issues.Design/methodology/approachIntegral transform technique and Gaver-Stehfast algorithm are applied to prepare the mathematical model by considering the periodically varying heat source in cylindrical co-ordinates.FindingsThis paper studies a problem on thermoelastic interactions in an isotropic and homogeneous elastic medium under fractional-order theory of thermoelasticity proposed by Sherief (Ezzat and El-Karamany, 2011b). The analytic solutions are found in Laplace transform domain. Gaver-Stehfast algorithm (Ezzat and El-Karamany, 2011d; Ezzat, 2012; Ezzat, El Karamany, Ezzat, 2012) is used for numerical inversion of the Laplace transform. All the integrals were evaluated using Romberg’s integration technique (El-Karamany et al., 2011) with variable step size. A mathematical model is prepared for copper material and the results are presented graphically with the discussion on the effects of fractional-order parameter.Research limitations/implicationsConstructed purely on theoretical mathematical model by considering different parameters and the functions.Practical implicationsThe system of equations in this paper may prove to be useful in studying the thermal characteristics of various bodies in real-life engineering problems by considering the time fractional derivative in the field equations.Originality/valueIn this problem, the authors have used the time fractional-order theory of thermoelasticity to solve the problem for a half space with a periodically varying heat source to control the speed of wave propagation in terms of heat and elastic waves for different conductivity like weak conductivity, moderate conductivity and super conductivity which is a new and novel contribution.

Acoustical Green’s Function and Boundary Element Techniques for 3D Half-Space Problems

This paper presents a review of basic concepts of the boundary element method (BEM) for solving 3D half-space problems in a homogeneous medium and in frequency domain. The usual BEM for exterior problems can be extended easily for half-space problems only if the infinite plane is either rigid or soft, since the necessary tailored Green’s function is available. The difficulties arise when the infinite plane has finite impedance. Numerous expressions for the Green’s function have been found which need to be computed numerically. The practical implementation of some of these formulas shows that their application depends on the type of impedance of the plane. In this work, several formulas in frequency domain are discussed. Some of them have been implemented in a BEM formulation and results of their application in specific numerical examples are summarized. As a complement, two formulas of the Green’s function in time domain are presented. These formulas have been computed numerically and after the application of the Fourier Transformation compared with the frequency domain formulas and with a FEM calculation.

On the Sommerfeld half-space problem: appraisal of approximate solutions with extensions to plasmonics

The classical Sommerfeld problem of a vertical electric dipole radiating over an imperfectly conducting half-space is revisited. Two variants of the modified saddle-point integration method are employed to derive new, closed-form, second-order approximations of the far-zone electromagnetic fields. The new formulas are applied to compute radiation patterns and surface fields for sample problems including both ordinary and plasmonic media, and their accuracy is assessed by comparison with the results of rigorous numerical evaluation of the Sommerfeld integrals.

Elastic Sinusoidal Wavy Surface Contact Under Full Stick Conditions

A purely normal contact problem of an elastic half-space with a three-dimensional periodic sinusoidal wavy surface and a rigid flat under the full stick condition is studied. The contacting points from mating surfaces have zero relative tangential displacement under the full stick condition. The scope of this study is restricted to a special case where the entire contact interface is in contact (referred to as complete contact) under the full stick condition. Complete contact is defined as when there are no gaps remaining between the surfaces. The corresponding state of stress of the half-space is derived analytically. According to the state of stress, we find (1) an analytical solution for the average pressure required to cause complete contact, (2) the location of the global maxima of the von Mises stress and (3) the critical magnitude of the waviness amplitude below which the plastic yielding of the half-space will never occur before the initiation of complete contact. The results are also compared with the solution under the perfect slip condition. We find that the location of the maximum von Mises stress may occur either on the contact interface or beneath it depending on the value of Poisson’s ratio.

Investigation on effects of stochastic loading at the boundary under thermoelasticity with two relaxation parameters

• A problem of half space with stochastic loading at the boundary is solved. • Stochastic mechanical distribution is applied at the boundary plane. • Integral transform technique and stochastic simulation are applied. • The effects of stochastic type boundary condition are highlighted. • Results are compared with the results for deterministic cases. The main objective of the present work is to study the responses of stochastic type mechanical distribution at the boundary of an elastic half space in the context of generalized thermoelasticity. In order to compare the results under the stochastic mechanical distribution, we have also considered the case of deterministic mechanical distribution prescribed at the boundary. The stochastic mechanical distribution is considered in such a way that it reduces to a deterministic type distribution as a special case. Laplace transform technique is used to solve the problem and its inversion is carried out by using asymptotic expansions valid for short times to obtain the solution of all the physical field variables like, stress and temperature distributions in the physical domain. Numerical results are found out to compare the effects of stochastic and deterministic load prescribed at the boundary of the elastic half space.

Half space problem for Euler equations with damping in 3-D

In this paper, we consider the half space problem for Euler equations with damping in 3-D. We restudy the fundamental solution for the Cauchy problem to obtain an exponentially sharp pointwise structure and a clear decomposition of the singular-regular components. Later, both Green's function for initial boundary value problem and fundamental solutions for Cauchy problems are investigated in the transformed domain after Laplace transform. The symbols are obtained and a connection between Green's function and fundamental solutions are established for the pointwise space–time structure of Green's function. Finally, the sharp estimates for Green's function together with a priori estimates from the energy method for high order derivatives result in the nonlinear stability of the solution and also the decaying rates.

A convergent method for linear half-space kinetic equations

We give a unified proof for the well-posedness of a class of linear half-space equations with general incoming data and construct a Galerkin method to numerically resolve this type of equations in a systematic way. Our main strategy in both analysis and numerics includes three steps: adding damping terms to the original half-space equation, using an inf-sup argument and even-odd decomposition to establish the well-posedness of the damped equation, and then recovering solutions to the original half-space equation. The proposed numerical methods for the damped equation is shown to be quasi-optimal and the numerical error of approximations to the original equation is controlled by that of the damped equation. This efficient solution to the half-space problem is useful for kinetic-fluid coupling simulations.

Existence and asymptotic stability of stationary waves for symmetric hyperbolic–parabolic systems in half-line

In this paper, we consider a one-dimensional half-space problem for a system of viscous conservation laws which is deduced to a symmetric hyperbolic–parabolic system under assuming that the system has a strictly convex entropy function. We firstly prove existence of a stationary solution by assuming that a boundary strength is sufficiently small. The existence of the stationary solution is characterized by the number of negative characteristics. In the case where one characteristic speed is zero at spatial asymptotic state [Formula: see text], we assume that the characteristic field corresponding to the characteristic speed 0 is genuinely nonlinear in order to show existence of a degenerate stationary solution with the aid of a center manifold theory. We next prove that the stationary solution is time asymptotically stable under a smallness assumption on an initial perturbation in the Sobolev space. The key to proof is to derive the uniform a priori estimates by using the energy method, where the stability condition of Shizuta–Kawashima type plays an essential role.

On Boussinesq’s problem for a cracked halfspace

This paper examines the axisymmetric elastostatic problem that deals with the action of a concentrated normal force on the surface of an isotropic elastic halfspace containing a penny-shaped crack. The mathematical formulation of the elasticity problem should take into consideration the sense of action of the concentrated force. The paper presents the development of Fredholm integral equations of the second-kind that are associated with this category of problem and indicates the numerical technique that is adopted for their solution. The numerical results are presented for the stress intensity factors generated at the penny-shaped crack experiencing either opening or closure.