Three-dimensional fundamental solutions for transient heat transfer by conduction in an unbounded medium,half-space,slab and layered media

This paper presents analytical Green's functions for the transient heat transfer phenomena by conduction, for an unbounded medium, half-space, slab and layered formation when subjected to a point heat source. The transient heat responses generated by a spherical heat source are computed as Bessel integrals, following the transformations proposed by Sommerfeld [Sommerfeld A. Mechanics of deformable bodies. New York: Academic Press; 1950; Ewing WM, Jardetzky WS, Press F. Elastic waves in layered media. New York: McGraw-Hill; 1957]. The integrals can be modelled as discrete summations, assuming a set of sources equally spaced along the vertical direction. The expressions presented here allow the heat field inside a layered formation to be computed without fully discretizing the interior domain or boundary interfaces.The final Green's functions describe the conduction phenomenon throughout the domain, for a half-space and a slab. They can be expressed as the sum of the heat source and the surface terms. The surface terms need to satisfy the boundary conditions at the surfaces, which can be of two types: null normal fluxes or null temperatures. The Green's functions for a layered formation are obtained by adding the heat source terms and a set of surface terms, generated within each solid layer and at each interface. These surface terms are defined so as to guarantee the required boundary conditions, which are: continuity of temperatures and normal heat fluxes between layers.This formulation is verified by comparing the frequency responses obtained from the proposed approach with those where a double-space Fourier transformation along the horizontal directions [Tadeu A, António J, Simões N. 2.5D Green's functions in the frequency domain for heat conduction problems in unbounded, half-space, slab and layered media. CMES: Computer Model Eng Sci 2004;6(1):43–58] is used. In addition, time domain solutions were compared with the analytical solutions that are known for the case of an unbounded medium, a half-space and a slab.     

Benchmark Solutions for Three-Dimensional Transient Heat Transfer in Two-Dimensional Environments Via the Time Fourier Transform

The evaluation of heat propagation in the time domain generated by transient heat sources placed in the presence of three-dimensional media requires the use of computationally demanding numerical schemes. The implementation of numerical 3D solutions may benefit from the existence of benchmark solutions to test the accuracy of approximate schemes.
With this purpose inmind, this article presents analyticalnumerical solutions to evaluate the heat-field elicited by monopole heat sources in the presence of three different inclusions, namely, a cylindrical circular solid inclusion, a cylindrical circular cavity with null fluxes and a cavity with null temperatures prescribed along its boundary, buried in an unbounded medium. The problem is first subjected to a time and space Fourier Transform, which allows the solution to be obtained in the frequency domain as summation of 2D solutions for different spatial wavenumbers. Then, using the inverse Fourier transforms in the wavenumber and frequency domains, the 3D time responses are synthesized. Complex frequencies are used to avoid the aliasing phenomena.
This methodology is first validated calculating the fundamental time solutions for one, two and three dimensions in an unbounded medium. Simulation analyses of these idealized models are then used to study the patterns of heat propagation in the vicinity of the inclusions.     

Boundary element method analyses of transient heat conduction in an unbounded solid layer containing inclusions

The boundary element method (BEM) is used to compute the three-dimensional transient heat conduction through an unbounded solid layer that may contain heterogeneities, when a pointwise heat source placed at some point in the media is excited. Analytical solutions for the steady-state response of this solid layer when subjected to a spatially sinusoidal harmonic heat line source are presented when the solid layer has no inclusions. These solutions are incorporated into a BEM formulation as Greens functions to avoid the discretization of flat media interfaces. The solution is obtained in the frequency domain, and time responses are computed by applying inverse (Fast) Fourier Transforms. Complex frequencies are used to prevent the aliasing phenomena. The results provided by the proposed Greens functions and BEM formulation are implemented and compared with those computed by a BEM code that uses the Greens functions for an unbounded media which requires the discretization of all solid interfaces with boundary elements. The proposed BEM model is then used to evaluate the temperature field evolution through an unbounded solid layer that contains cylindrical inclusions with different thermal properties, when illuminated by a plane heat source. In this model zero initial conditions are assumed. Different simulation analyses using this model are then performed to evaluate the importance of the thermal properties of the inclusions on transient heat conduction through the solid layer.     

Transient Disturbance in a Half Space Under Thermoelasticity with Two Relaxation Times Due to Moving Internal Heat Source

In this paper transient waves caused by a line heat source moving with a uniform velocity inside isotropic homogeneous thermoelastic half-space are studied under the GL model of generalized thermoelasticity. The problem is reduced to the solution of three differential equations by introducing the elastic vector potential and the thermoelastic scalar potential. Using Laplace and Fourier transforms solutions are obtained in transforms domain. Applying inverse transforms approximate solutions of the displacement at the boundary valid in the small time range are given. Also the approximate region valid for the solutions is given and two special cases, (i) the source is motionless and (ii) the relaxation times vanish, are studied. Numerical evaluations are presented for the medium of copper.     

Wave propagation in cracked elastic slabs and half-space domains—TBEM and MFS approaches

In this paper, the traction boundary element method (TBEM) and the method of fundamental solutions (MFS), formulated in the frequency domain, are used to evaluate the 3D scattered wave field generated by 2D empty cracks embedded in an elastic slab and a half-space. Both models overcome the thin-body difficulty posed when the classical BEM is applied.The crack exhibits arbitrary cross section geometry and null thickness. In neither model are the horizontal formation surfaces discretized, since appropriate fundamental solutions are used to take them into consideration.The TBEM models the crack as a single line. The singular and hypersingular integrals that arise during the TBEM model's implementation are computed analytically, which overcomes one of the drawbacks of this formulation. The results provided by the proposed TBEM model are verified against responses provided by the classical BEM models derived for the case of an empty cylindrical circular cavity.The MFS solution is approximated in terms of a linear combination of fundamental solutions, generated by a set of virtual sources simulating the scattered field produced by the crack, using a domain decomposition technique. To avoid singularities, these fictitious sources are not placed close to the crack, and the use of an enriched function to model the displacement jumps across the crack is unnecessary.The performances of the proposed models are compared and their limitations are shown by solving the case of a C-shaped crack embedded in an elastic slab and a half-space domain.The applicability of these formulations is illustrated by presenting snapshots from computer animations in the time domain for an elastic slab containing an S-shaped crack, after applying an inverse Fourier transformation to the frequency domain computations.     

The thin‐layer method in a cross‐anisotropic 3D space

This article presents a generalization of the thin‐layer method to three dimensions, a tool that allows assessing layered media subjected to loads eliciting nonsymmetrical wave fields. It is based on a formulation that fully couples the three components of motion, and allows finding effective solutions to either stationary or moving loads of arbitrary shape that act on (or within) horizontally layered media. In particular, it is an ideal tool for finding analytical solutions to the so‐called 2.5D problem, which entails loads with arbitrary distribution in one horizontal coordinate direction together with a harmonic (sinusoidal) variation in the other. Inasmuch as the Green's functions for the latter case are found explicitly in the spatial domain without recourse to numerical integration, it allows using such functions — most likely in the context of the boundary element method — as fundamental solutions for problems of soil–structure interaction where the structure is invariant in one horizontal direction, such as a railroad track resting on an embankment. The method entails solving at each frequency of interest two uncoupled eigenvalue problems for generalized SH and SVP waves (i.e. horizontally and vertically polarized shear and pressure waves), after which the fundamental solutions are obtained in closed‐form at any desired point in space. Inasmuch as the proposed technique dispenses with at least one of the two inverse Fourier transforms into the spatial domain, in due time the methodology presented is likely to become the preferred tool for a wide class of problems. The technique is first benchmarked against the known solution for a point load and then applied to a rectangular and triangular load distribution. Copyright © 2011 John Wiley & Sons, Ltd.     

Fundamental solution for a layered porous half space subject to a vertical point force or a point fluid source

The focus of this contribution is to develop a transmission and reflection matrices (TRM) method for a layered porous half-space subject to a point force or a fluid point source. Applying the Hankel and the Fourier transformation, the general solutions for the displacements, stresses and pore pressure are derived from the potentials for the solid skeleton and the pore fluid as well as the governing equations of Biots theory. The transmission and reflection matrices (TRM) for each interface are obtained by using the general solutions as well as the continuity conditions at the interface. The TRM method for the layered porous medium is developed on the basis of the transmission and reflection matrices (TRM) and the boundary conditions as well as the source terms for the point force or the fluid point source. The fundamental solutions of the point force and the point fluid source in both the frequency domain and the time domain are obtained by using the proposed TRM method. Some numerical examples are given in the paper.     

Green's functions for composite media

A method is given on construction of Green's functions for finding field excited by a point (or line) source in a layered composite medium. The concept of Green's function for a composite medium as a whole is introduced in a rigorous manner by showing that it possesses all properties of the Green's function for an uniform medium. The appropriate Green's function for the whole composite medium is constructed through two solutions of the corresponding homogeneous equation over the whole multiregion. The method enables one to write down, directly, the explicit expression for the field in an arbitrary region produced by a point (or line) source located in another arbitrary region without resort to images, Fourier-Bessel integrals, integral transforms of quasi-orthogonal functions. It is an extension of the existing method, which constructs the Green's functions for uniform media through the generalized Fourier expansions and two solutions of the corresponding homogeneous equations, to cases heretofore beyond its scope.     

Three-dimensional wave scattering by a fixed cylindrical inclusion submerged in a fluid medium

This paper presents the solution for a fixed cylindrical irregular cavity of infinite length submerged in a homogeneous fluid medium, and subjected to dilatational point sources placed at some point in the fluid. The solution is first computed for a wide range of frequencies and wavenumbers, which are then used to obtain time-series by means of (fast) inverse Fourier transforms into space–time.The method and the expressions presented are implemented and validated by applying them to a fixed cylindrical circular cavity submerged in an infinite homogeneous fluid medium subjected to a point pressure source for which the solution is calculated in closed form.The boundary elements method is then used to evaluate the wave-field elicited by a point pressure source in the presence of fixed rigid cylindrical cavities, with different cross-sections, submerged in an unbounded fluid medium and in a half-space. Simulation analyses with this idealized model are then used to study the patterns of wave propagation in the vicinity of these inclusions. The amplitude of the wavefield in the frequency vs axial-wavenumber domain is presented, allowing the recognition, identification, and physical interpretation of the variation of the wavefield.     

流体饱和半空间中埋置球面P1、P2和SV波源动力格林函数

基于Biot流体饱和孔隙介质理论,采用Hankel积分变换方法,在频域内求解了流体饱和半空间中埋置球面P1、P2和SV波源的动力格林函数。首先由Hankel积分变换将空间域内球面波展开为波数域内柱面波的叠加;然后在半空间表面对称位置虚拟放置一同样大小的球面波源,这样对于球面膨胀波源(P1和P2波源),地表剪应力为零,但存在非零正应力和孔隙水压,对于球面剪切波源(SV波源),地表正应力和孔隙水压为零,但存在非零剪应力;最后叠加球面波源、虚拟波源和残余半空间表面应力产生的动力响应,即可求得流体饱和半空间中埋置球面波源波数域内的动力响应,空间域内埋置球面波源的动力格林影响函数则由Hankel逆变换求得。该文给出的球面波源动力格林函数,为建立以球面P1、P2和SV波动力格林函数为基本解的间接边界元方法,求解饱和多孔介质中三维轴对称弹性波散射问题奠定了基础。     

Accurate H-shaped absorbing boundary condition in frequency domain for scalar wave propagation in layered half-space

An accurate absorbing boundary condition (ABC) is developed in frequency domain for finite element analysis of scalar wave propagation in unbounded layered half-space. The proposed ABC is H-shaped line that consists of two parts: a new ABC at horizontal bottom boundary of finite domain to replace semiinfinite strip below horizontal boundary and between two vertical boundaries, and a general consistent ABC at vertical lateral boundary to replace semiinfinite layered half-space outside vertical boundary. The key point for constructing the ABC is that a new continued fraction (CF) is presented to expand dynamic stiffness of underlying half-space, and the CF-based stress-displacement relationship is then transformed into an auxiliary variable system with square of horizontal wavenumber. The ABC has only one undetermined real parameter that is the CF-order independent of frequency and incidence angle of propagating outgoing waves. The parameter can be chosen relatively small value to achieve an accurate ABC. Moreover, the ABC can couple seamlessly with finite element method of finite domain. The finite domain can be chosen very small size due to high accuracy of the ABC. Numerical examples are finally given to demonstrate the effectiveness of the ABC.     

Three-dimensional Green’s functions for a steady point heat source in a functionally graded half-space and some related problems

Three-dimensional Green’s functions are derived for a steady point heat source in a functionally graded half-space where the thermal conductivity varies exponentially along an arbitrary direction. We first introduce an auxiliary function which satisfies an inhomogeneous Helmholtz equation. Then by virtue of the image method which was first proposed by Sommerfeld for the homogeneous half-space Green’s function of a steady point heat source, we arrive at an explicit expression for this function. Finally with this auxiliary function, we derive the three-dimensional Green’s functions due to a steady point heat source in a functionally graded half-space. Also investigated in this paper are the temperature field induced by a point heat source moving at a constant speed in a functionally graded full-space; the electric potential due to a static point electric charge in a dielectric full-space with electric field gradient effects; and the two-dimensional time-harmonic dynamic Green’s function for homogeneous and functionally graded materials with strain gradient effects.     

A two-dimensional generalized thermoelastic diffusion problem for a half-space subjected to harmonically varying heating

A two-dimensional problem for a thermoelastic half-space is considered within the context of the theory of generalized thermoelastic diffusion with one relaxation time. The upper surface of the half-space is taken to be traction free and subjected to harmonically varying heating with constant angular frequency of thermal vibration. 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 discussed and represented graphically.     

A relation between the solutions of the half-space Dirichlet problems for Helmholtz's equation in ℝ n and Laplace's equation in ℝ n+1

Summary Multiple Fourier transforms are used to derive the solutions of the half-space Dirichlet problems for Helmholtz's equation in ℝ ⁿ and Laplace's equation in ℝ ⁿ⁺¹ and to exhibit the relation between the two solutions.     

On Analytical Solutions During Damped Wave Conduction and Relaxation in a Finite Slab Subject to the Convective Boundary Condition

This article describes the use of the final condition in the time domain to obtain bounded and physically reasonable solutions for the convective boundary condition for the case of a finite slab. The temperature overshoot problem is revisited for the convective boundary condition. The use of a physically realistic time condition is shown to remove the overshoot and lead to bounded solutions within Clausius’s inequality. The ramifications of these findings are discussed. The method of separation of variables was used to obtain the analytical solution for the wave temperature. The governing equation for temperature, a hyperbolic partial differential equation (PDE) is multiplied by exp(τ/2) that results in a hyperbolic PDE less the damping component. The wave temperature can be used to better understand the transient phenomena of heat conduction. For materials with large relaxation times, 4h{\tau_{\rm r} > \frac{\rho C_p}{4h}} , the temperature can be expected to undergo subcritical damped oscillations. The analytical solution is presented as an infinite Fourier series solution. The solution was found to be bifurcated. For materials with a small relaxation time, the time domain part of the solution was found to be a decaying exponential and for materials with large relaxation times the time domain part of the solution was found to be cosinuous. Analytical solutions for the average temperature of the finite slab were also obtained. The thermal time constant of the material was found from the solution. The average temperature versus time was found to exhibit convex curvature for systems with large Biot numbers and the average temperature versus time was found to exhibit concave curvature for systems with small Biot numbers. The thermal time constant for the finite slab at different Biot numbers were found and tabulated. The thermal time constant versus Biot number was found to exhibit a maxima. When Fourier parabolic equations are used, the thermal constant decreases monotonically with an increase in Biot number.     

Plane strain deformation in an orthotropic micropolar thermoelastic solid with a heat source

The problem of plane strain deformation in an orthotropic micropolar generalized thermoelastic half-space subjected to an arbitrary point heat source is solved. Closed-form solutions for spatial distributions of displacements, stresses, and temperature are derived by using the Fourier transform. A numerical inversion technique has been applied to obtain the solution in the physical domain. Numerical results are obtained and presented graphically along with a comparison of the ones for concentrated and distributed, as well as mechanical and thermal, sources.     

列车移动荷载作用下饱和地基的地面振动特性分析

基于饱和土的Biot波动方程和边界条件,利用Fourier变换和Galerkin法推导出频域内的u-w格式的2.5维有限元方程。把轨道视为饱和地基上的Euler梁,通过沿轨道方向的波数变换将三维空间问题降为平面应变问题。通过解方程首先获得饱和土体频域-波数域的位移解答,然后通过快速Fourier逆变换求得三维时域-空间域内的位移解答。通过计算实例验证了计算模型。建立轨道-地基模型,对饱和地基上列车运行引起的地面振动进行了分析,详细讨论了动力渗透系数、孔隙率、土骨架密度和剪切波速等参数对地面振动传播与衰减的影响规律分析。研究表明;在垂直轨道方向,随着与轨道中心距离增加,加速度幅值和主频迅速衰减;饱和土体动力渗透系数、孔隙率、剪切波速和土骨架密度对地表位移幅值均有较大的影响     

层状饱和半空间中沉积谷地对斜入射平面P1波的三维散射

在作者给出层状饱和场地三维精确动力刚度矩阵和层状饱和半空间中移动荷载动力格林函数基础上,采用间接边界元方法在频域内求解了层状流体饱和场地中沉积谷地对斜入射平面P1波的三维散射问题。该方法的特点在于虚拟移动均布荷载和斜线孔隙水压可以直接施加在沉积与层状饱和半空间交界面而不存在奇异性。该文通过与已有结果的比较验证了方法的正确性,并以均匀饱和半空间和弹性基岩上单一饱和土层中沉积谷地为例进行了数值计算分析。研究表明,沉积谷地对平面P1波的三维散射与二维散射之间存在本质差别,入射角度、孔隙率、饱和土层刚度和饱和土层厚度等参数对沉积谷地附近动力响应有着显著影响。     

Convergence of a three-dimensional quantum lattice Boltzmann scheme towards solutions of the Dirac equation

We investigate the convergence properties of a three-dimensional quantum lattice Boltzmann scheme for the Dirac equation. These schemes were constructed as discretizations of the Dirac equation based on operator splitting to separate the streaming along the three coordinate axes, but their output has previously only been compared against solutions of the Schr?dinger equation. The Schr?dinger equation arises as the non-relativistic limit of the Dirac equation, describing solutions that vary slowly compared with the Compton frequency. We demonstrate first-order convergence towards solutions of the Dirac equation obtained by an independent numerical method based on fast Fourier transforms and matrix exponentiation.     

Dynamic Green’s functions for a poroelastic half-space

The dynamic responses of a poroelastic half-space to an internal point load and fluid source are investigated in the frequency domain in this paper. By virtue of a method of displacement potentials, the 3D general solutions of homogeneous wave equations and fundamental singular solutions of inhomogeneous wave equations are derived, respectively, in the frequency domain. The mirror-image technique is then applied to construct the dynamic Green’s functions for a poroelastic half-space. Explicit analytical solutions for displacement fields and pore pressure are obtained in terms of semi-infinite Hankel-type integrals with respect to the horizontal wavenumber. In two limiting cases, the solutions presented in this study are shown to reduce to known counterparts of elastodynamics and those of Lamb’s problem, thus ensuring the validity of our result.