Half-space Problem Research Articles

An analytical solution versus half space BEM formulation for acoustic radiation and scattering from a rigid sphere

A half space problem in acoustics is described by introducing an infinite plane boundary that reflects the wave coming into the plane. A numerical solution using Boundary Element Method (BEM) has been known which is formulated using a modified Green's function in the Helmholtz Integral Formulation, which eliminates the discretization over the infinite plane. Hence, the discretization are confined to the body or obstacle in question only. This feature constitutes the main advantage of the BEM formulation for half space problems. However, no general analytical solution is available to verify the BEM results for half space problems. This paper is aimed to propose an analytical solution for the BEM to compare with, hence to verify the BEM calculation. This analytical approach is currently developed for a half space problem involving radiation and scattering of acoustic waves from a rigid sphere. The image of sphere as well as the image of the field point are defined with respect to the infinite plane. Then, an ad hoc solution is assumed involving a constant and the distance from the center of the sphere to the field point and the distance from the center of the image of the sphere to the field point. The constant is determined by imposing the boundary conditions. Test cases were run with several configuration involving the location of field points and the sphere. Comparison of the analytical solution with BEM calculations shows a good agreement between the two results..

Nonstationary heat transfer in anisotropic half-space under the conditions of heat exchange with the environment having a specified temperature

An analytical solution to the problem of heating anisotropic half-space by the environment with a spatially and temporally variable temperature (boundary conditions of the third kind at an anisotropic body) has been obtained for the first time based on the construction of the boundary influence function (the Green’s function), which is determined using the Fourier and Laplace integral transforms. Nonstationary temperature fields in anisotropic blunt bodies have been found under the conditions of aero-gas-dynamic heating of hypersonic aircrafts with different heat-transfer coefficients and incoming-flow temperatures. The solution obtained is recommended for determining the state of thermal protection fabricated from composites which are generally anisotropic.

Plane problem of filtration consolidation for an elastic half-space with discontinuous initial conditions

The plane consolidation problem for an elastic half-space under an instantaneously applied vertical load is solved in the general case assuming that the Terzaghi hypothesis holds.

Discrete Velocity Models for Mixtures Without Nonphysical Collision Invariants

An important aspect of constructing discrete velocity models (DVMs) for the Boltzmann equation is to obtain the right number of collision invariants. It is a well-known fact that DVMs can also have extra collision invariants, so called spurious collision invariants, in plus to the physical ones. A DVM with only physical collision invariants, and so without spurious ones, is called normal. For binary mixtures also the concept of supernormal DVMs was introduced, meaning that in addition to the DVM being normal, the restriction of the DVM to any single species also is normal. Here we introduce generalizations of this concept to DVMs for multicomponent mixtures. We also present some general algorithms for constructing such models and give some concrete examples of such constructions. One of our main results is that for any given number of species, and any given rational mass ratios we can construct a supernormal DVM. The DVMs are constructed in such a way that for half-space problems, as the Milne and Kramers problems, but also nonlinear ones, we obtain similar structures as for the classical discrete Boltzmann equation for one species, and therefore we can apply obtained results for the classical Boltzmann equation.

Half space albedo problem for the nonconservative vector equation of transfer with a combination of Rayleigh and isotropic scattering

The half-space albedo problem has been solved for a combination of Rayleigh and isotropic scattering using $H_{N}$ method which is developed for the neutron transport studies. The numerical results are compared with exact values obtained using variational method and Chandrasekhar’s equation for the $\mathbf{H}$ -matrix. The analytical solutions of $H_{N}$ method are easy to handle in comparison with the other methods. The numerical results are in good agreement with previous works in literature.

A two-dimensional generalized thermoelastic diffusion problem for a half-space

In this work, we consider the two-dimensional problem of a thermoelastic half-space with a permeating substance in contact with the bounding plane in the context of the theory of generalized thermoelastic diffusion with one relaxation time. The bounding surface of the half-space is taken to be traction free and subjected to a time-dependent thermal shock. Laplace and Fourier transform techniques are used. The solution in the transformed domain is obtained by a direct approach. Numerical inversion techniques are used to obtain the inverse double transforms. Numerical results are represented graphically and discussed.

Analytical solution of a two-dimensional thermoelastic problem subjected to laser pulse

In this article, the problem of a two-dimensional thermoelastic half-space are studied using mathematical methods under the purview of the generalized thermoelastic theory with one relaxation time is studied. The surface of the half-space is taken to be thermally insulated and traction free. Accordingly, the variations of physical quantities due to by laser pulse given by the heat input. The nonhomogeneous governing equations have been written in the form of a vector-matrix differential equation, which is then solved by the eigenvalue approach. The analytical solutions are obtained for the temperature, the components of displacement and stresses. The resulting quantities are depicted graphically for different values of thermal relaxation time. The result provides a motivation to investigate the effect of the thermal relaxation time on the physical quantities.

An investigation on thermoelastic interactions under an exact heat conduction model with a delay term

ABSTRACTThe present work is concerned with a very recently proposed heat conduction model: an exact heat conduction model with a single delay term. A generalized thermoelasticity theory was proposed by Roy Choudhuri based on the heat conduction law with three-phase-lag effects for the purpose of considering the delayed response in time due to the microstructural interactions in the heat transport mechanism. However, the model defines an ill-posed problem in Hadamard sense. Quintanilla has recently proposed to reformulate this heat conduction model as an alternative heat conduction theory with a single delay term and subsequently, Leseduarte and Quintanilla investigated the spatial behavior of the solutions for this theory and they extended the results to a thermoelasticity theory by considering the Taylor series approximation of the equation of heat conduction with one delay term. In the present work, we consider the thermoelasticity theory based on this newly proposed heat conduction model and investigate a problem of thermoelastic interactions. State-space approach is used to formulate the problem and the formulation is then applied to a problem of an isotropic elastic half-space with its plane boundary subjected to sudden increase in temperature and zero stress. The integral transform method is applied to obtain the solution of the problem. A detailed analysis of analytical results is provided by finding the short-time approximated solutions of different field variables analytically and comparing the results of the present model with the corresponding results reported for other existing theories. An attempt has also been made to illustrate the problem and numerical values of field variables are obtained for a particular material. Results are analyzed with different graphs. To the best of the author\ extquoteright s knowledge, this thermoelastic model is not yet investigated by any researcher in this direction.

Solution of a multidimensional impact deformation problem for an elastic half-space with curved boundary on the basis of a modified ray method

A generalization of the method for constructing approximate solutions of boundary value problems of impact deformation dynamics in the form of ray expansions for two-dimensional plane deformation problems is presented. For each shock wave, the solution near its front is determined on the basis of ray coordinates consistent with this wave. The nonlinear divergence of curvilinear rays is taken into account. A mechanism of transformation from one ray coordinate system to another, which is crucially important in the ray method, is described. The developed technique is illustrated by solving the impact deformation problem for a half-space with boundary of nonzero curvature.

Asymptotic solutions for generalized thermoelasticity with variable thermal material properties

In this paper, a unified generalized thermoelastic solution with variable thermal material properties is proposed in the context of different generalized models of thermoelasticity, including thermoelasticity with one thermal relaxation time (LS theory), thermoelasticity with two thermal relaxation times (GL theory) and thermoelasticity without energy dissipation (GN theory). The unified form of governing equations is presented by introducing unifier parameters. The unified formulations are derived and given for isotropic homogenous materials with variable thermal material properties. The Laplace transform techniques and the Kirchhoff’s transformation are used to obtain general solutions for any set of boundary conditions in the physical domain. Asymptotic solutions for a specific problem of an elastic half-space with variable thermal conductivity and a specific heat, whose boundary is subjected to a thermal shock, are derived by means of the limit theorem of Laplace transform. In the context of these asymptotic solutions, some generalized thermoelastic phenomena are observed. Especially, the jumps at the wavefronts induced by the propagation of finite signal speed for the heat are clearly noticed. In addition, the effect of variable characteristics of material properties on thermoelastic behaviors is revealed by a comparison with the results obtained in the case of constant material properties.

Interaction of punches on orthotropic half-space

An integral equation of the three-dimensional contact problem for an orthotropic half-space (9 independent elastic parameters in Hooke’s law) is obtained where its kernel does not include integrals, but it depends on the solution of a characteristic binary cubic. The interaction between two identical symmetrically embedded punches is considered for the case of the elliptic paraboloids. Galanov’s method of nonlinear boundary integral equations is used for solving the problem with an unknown contact domain that makes it possible to determine simultaneously the contact domain and the contact pressure. The exact solution to one elliptical punch is used for debugging the computer program. Contact pressures, contact zones and pressing forces are calculated for various orthotropic materials at the specified settlement, base forms of the punches, and relative distances between the punches. The orthotropic body model is applicable for describing lots of materials which are in-demand in the machinery and industry: sulfur, Rochelle salt, wolframite, barite, and various wood species.

Interactions due to hall current and rotation in a magneto-micropolar fractional order thermoelastic half-space subjected to ramp-type heating

Purpose – The purpose of this paper is to investigate a two dimensional problem in magneto-micropolar thermoelastic half-space with fractional order derivative in the presence of combined effects of hall current and rotation subjected to ramp-type heating. Design/methodology/approach – The fractional order theory of thermoelasticity with one relaxation time derived by Sherief et al. (2010) has been used to investigate the problem. Laplace and Fourier transform technique has been used to solve the resulting non-dimensional coupled field equations to obtain displacement, stress components and temperature distribution. A numerical inversion technique has been applied to obtain the solution in the physical domain. Findings – Numerical computed results of all the considered variables have been shown graphically to depict the combined effect of hall current and rotation. Some particular cases of interest are also deduced from the present study. Originality/value – Comparison are made in the presence and absence of hall current and rotation in a magneto-micropolar thermoelastic solid with fractional order derivative.

Stochastic thermal shock problem and study of wave propagation in the theory of generalized thermoelastic diffusion

A one-dimensional problem for a thermoelastic half-space is considered within the context of the theory of generalized thermoelastic diffusion with one relaxation time. The bounding surface is traction free and subjected to a time dependent thermal shock. Two types of boundary conditions are considered, deterministic and stochastic. A permeating substance is considered in contact with the bounding surface. Laplace transform technique is used to obtain the solution in the transformed domain by using a direct approach. Analysis of wave propagation in the medium is presented. The solution in the physical domain for temperature, displacement, stress, concentration and chemical potential are obtained in an approximate manner. Numerical results are carried out and represented graphically.

Inverse contact problem for an elastic half-space

This paper presents a system of integral equations for the determination of contact stresses on a part of the boundary of elastic half-space by measured data of displacements on the rest of the stress-free boundary. Inverse problems like this are refereed to as conditionally ill-posed with pronounced dependence of the solution from small perturbations in measured data. The 3D problem formulation is based on spatial harmonic functions. It is proposed to use a Trefftz-type method for the sought harmonic functions based on the radial basis functions to solve the system of integral equations. A synthetic example is presented to illustrate the proposed approach.

Analysis of numerical solutions to Sommerfeld integral relation of the half-space radiator problem

Sommerfeld integrals relate a spherical wave from a point source to a convolution set of plane and cylindrical waves. This relation does not have analytical solutions but it submits to a solution by numerical integration. Among others, it is significant for theoretical studies of many optical and radiation phenomena involving surfaces. This approach is preferred over discretized computational models of the surface because of the many orders of increased computations involved in the latter. One of the most widely used and accurate methods to compute these solutions is the numerical integration of the Sommerfeld integrand over a complex contour. We have analyzed the numerical advantages offered by this method, and have justified the optimality of the preferred contour of integration and the choice of two eigenfunctions used. In addition to this, we have also analyzed four other approximate methods to compute the Sommerfeld integral and have identified their regions of validity, and numerical advantages, if any. These include the high relative permittivity approximation, the short distance approximation, the exact image theory and Fourier expansion of the reflection coefficient. We also finally compare these five methods in terms of their computational cost.

Steady sliding frictional contact problem for a 2d elastic half-space with a discontinuous friction coefficient and related stress singularities

The steady sliding frictional contact problem between a moving rigid indentor of arbitrary shape and an isotropic homogeneous elastic half-space in plane strain is extensively analysed. The case where the friction coefficient is a step function (with respect to the space variable), that is, where there are jumps in the friction coefficient, is considered. The problem is put under the form of a variational inequality which is proved to always have a solution which, in addition, is unique in some cases. The solutions exhibit different kinds of universal singularities that are explicitly given. In particular, it is shown that the nature of the universal stress singularity at a jump of the friction coefficient is different depending on the sign of the jump.

A solution of the monoenergetic neutral particle transport equation for adjacent half-spaces with anisotropic scattering

We present highly accurate solutions to the neutral particle transport equation in a half-space. While our initial motivation was in response to a recently published solution based on Chandrasekhar's H-function, the presentation to follow has taken on a more comprehensive tone. The solution by H-functions certainly did achieved high accuracy but was limited to isotropic scattering and emission from spatially uniform and linear sources. Moreover, the overly complicated nature of the H-function approach strongly suggests that its extension to anisotropic scattering and general sources is not at all practical. For this reason, an all encompassing theory for the determination of highly precise benchmarks, including anisotropic scattering for a variety of spatial source distributions, is presented for particle transport in a half-space. We illustrate the approach via a collection of cases including tables of 7-place flux benchmarks to guide transport methods developers. The solution presented can be applied to a considerable number of one and two half-space transport problems with variable sources and represents a state-of-the-art benchmark solution.

A Brief Note on Generalized Thermoelastic Response in a Half Space due to a Periodically Varying Heat Source under Axisymmetric Distribution

A two dimensional problem for an infinite thermoelastic half space is formulated, to study the thermoelastic response due to a periodically varying heat source within the context of Lord-Shulman theory. The bounding surface is traction free and subjected to a known temperature distribution. Integral transform technique is developed to find the analytic solution in the transform domain by using direct approach. Inversion of transforms is done employing a numerical scheme. Mathematical model is prepared for Copper material and numerical results for temperature, displacements and stresses are obtained and illustrated graphically.

Random vibration of an elastic half-space subjected to a moving stochastic load

In this paper, a non-stationary random vibration problem of an elastic half-space under a moving stochastic load modelling the dynamic characteristic of the train load is studied. An analytical method combining the pseudo-excitation method (PEM) and Fourier transformation method (FTM) is proposed. Because the load is moving and random, the responses at fixed locations in the half-space have characteristics of evolutionary non-stationary randomness. By means of the FEM, the non-stationary random vibration analysis is transformed into a conventional moving pseudo harmonic load problem. A closed-form solution of random vibration responses is derived in an integral form, which avoids step-by-step integration in time domain and thus leads to great saving in computational time. An adaptive quadrature method is adopted in order to obtain the desired numerical results of the singular integrand with non-stationary power spectral density. Through numerical examples, the proposed method is first validated, and then the dynamics characteristics of the non-stationary random responses of the half-space in all subsonic, transonic and supersonic cases are analysed.

A variational approach to the Brown–Ravenhall operator for the relativistic one-electron atoms

We study the Brown–Ravenhall operator (the projection of the Dirac operator perturbed by a potential V∈Lw3) using the Foldy–Wouthuysen (unitary) transformation and its realization as an elliptic problem in the 4-dim half space R+4 with Neumann boundary condition. Using this approach we give a characterization of its eigenvalues and eigenfunctions alternative to the classical Rayleigh quotients.