Non-axisymmetric consolidation of poroelastic multilayered materials with anisotropic permeability and compressible constituents

This paper presents an analytical layer-element solution to non-axisymmetric consolidation of multilayered poroelastic materials with anisotropic permeability and compressible constituents. By applying Fourier expansions, Hankel transforms and Laplace transforms to the state variables involved in the governing equations of poroelasticity with respect to the circumferential, radial and time coordinates, respectively, the analytical layer-element (i.e. a symmetric stiffness matrix) is derived, which describes the relationship between the transformed generalized stresses and displacements at the surface (z = 0) and those at an arbitrary depth z, considering the corresponding boundary and continuity conditions at the layer interfaces, the global stiffness matrix of a multilayered system is assembled in the transformed domain. The actual solutions in the physical domain are acquired by applying numerical quadrature schemes for the inversion of the Laplace–Hankel transform. Finally, numerical calculation is presented to investigate the influence of layering and poroelastic material parameters on consolidation process.     

Green's function of the Brinkman equation in a 2D anisotropic case

The purpose of this paper is to obtain the Green's functionof the Brinkman equation in a 2D case of hydrodynamic anisotropywith respect to the permeability. The anisotropic nature ofthe permeability is assumed to be not space or time dependent.We use the method of Fourier transform which reduces the computationof the Green's function to the computation of the fundamentalsolution of a fourth-order partial differential equation. Thisresearch work has several applications in engineering and medicineto the motion of bodies in anisotropic porous media.     

The stochastic fluid–fluid model: A stochastic fluid model driven by an uncountable-state process,which is a stochastic fluid model itself

We introduce the Stochastic Fluid–Fluid Model, which offers powerful modeling ability for a wide range of real-life systems of significance. We first derive the infinitesimal generator, with respect to time, of the driving stochastic fluid model. We then use this to derive the infinitesimal generator of a particular Laplace–Stieltjes transform of the model, which is the foundation of our analysis. We develop expressions for the Laplace–Stieltjes transforms of various performance measures for the transient and limiting analysis of the model. This work is the first direct analysis of a stochastic fluid model that is Markovian on a continuous state space.     

The hitting time for a Cox risk process

This paper investigates the hitting time of a Cox risk process. The relationship between the hitting time of the Cox risk process and the classical risk process is established and an explicit expression of the Laplace–Stieltjes transform of the hitting time is derived by the probability method. Similarly, we derive the explicit expression of the Laplace–Stieltjes transform of the last exit time. Further, we study the situation when the intensity process is an n$n$-state Markov process.     

Equations of anisotropic elastodynamics in 3D quasicrystals as a symmetric hyperbolic system: Deriving the time-dependent fundamental solutions

The fundamental solution (FS) of the time-dependent differential equations of anisotropic elasticity in 3D quasicrystals are studied in the paper. Equations of the time-dependent differential equations of anisotropic elasticity in 3D quasicrystals are written in the form of a symmetric hyperbolic system of the first order. Using the Fourier transform with respect to the space variables and matrix transformations we obtain explicit formulae for Fourier images of the FS columns; finally, the FS is computed by the inverse Fourier transform. As a computational example applying the suggested approach FS components are computed for icosahedral QCs.     

Weighted Fourier–Laplace transforms in reproducing kernel Hilbert spaces on the sphere

We study the action of a weighted Fourier–Laplace transform on the functions in the reproducing kernel Hilbert space (RKHS) associated with a positive definite kernel on the sphere. After defining a notion of smoothness implied by the transform, we show that smoothness of the kernel implies the same smoothness for the generating elements (spherical harmonics) in the Mercer expansion of the kernel. We prove a reproducing property for the weighted Fourier–Laplace transform of the functions in the RKHS and embed the RKHS into spaces of smooth functions. Some relevant properties of the embedding are considered, including compactness and boundedness. The approach taken in the paper includes two important notions of differentiability characterized by weighted Fourier–Laplace transforms: fractional derivatives and Laplace–Beltrami derivatives.     

A two dimensional thermoviscoelastic problem due to instantaneous point heat source

A dynamic problem of thermoviscoelasticity in the presence of heat source is studied here. The solution in Laplace–Fourier transformed domain is obtained using eigen value approach. Stress distributions are computed numerically.     

The solution by fast Fourier transforms of Laplace's equation in a toroidal region with a rectangular cross-section

The fast Fourier transform method is described for Laplace's equation in a toroidal region using the 9-point difference approximation to the Laplacian operator. Numerical results are given which indicate the efficiency and accuracy of the method. Accurate difference approximations are also derived for the determination of the electrostatic field in a toroidal region.     

Rate of consolidation of stone column-improved ground considering change in permeability and compressibility during consolidation

In the available mathematical formulations to determine the rate of radial consolidation of stone column-improved ground, the soil properties especially permeability and compressibility are assumed to be constant during consolidation process. However, permeability changes with void ratio (void ratio also changes with consolidation) and compressibility of soil varies along the consolidation curve (compressibility is a function of effective stress and void ratio). Thus, these properties are not constant during consolidation period. In the present paper, mathematical formulation is developed to determine the rate of consolidation of stone column-improved ground due to radial flow considering change in permeability and compressibility of soft soil during consolidation period. Equal strain approach has been considered in the analysis. The parabolic variation in permeability and compressibility within smear zone are incorporated in the formulation. It is observed that the variation of degree of consolidation due to change in stress concentration ratio and diameter ratio reduces when variable soil properties are considered. The difference between the degree of consolidation obtained considering variable and constant soil properties is almost constant due to variation of smear zone parameters. The time required to achieve 90% degree of consolidation increases or decreases (depending on the properties of soil) by around 50%–100% or up to 25%, respectively, when change in soil properties during consolidation is considered.     

Queue lengths and workloads in polling systems

For a single-server multi-station polling system, we focus on the generating function and Laplace–Stieltjes transform of the time-stationary joint queue length and workload distributions, respectively, under no further assumptions on the service discipline. We express these quantities as expressions involving the generating functions of the joint queue length distribution at visit beginnings and visit completions at the various stations. The latter is known for a broad variety of cases. Finally, we identify a workload decomposition result.     

Performance analysis of MAP/G/1 queue with working vacations and vacation interruption

We consider the MAP/G/1 queue with working vacations and vacation interruption. We obtain the queue length distribution with the method of supplementary variable, combined with the matrix-analytic method and censoring technique. We also obtain the system size distribution at pre-arrival epoch and the Laplace–Stieltjes transform (LST) of waiting time.     

Combined Fourier-wavelet transforms for studying dynamic response of anisotropic multi-layered flexible pavement with linear-gradual interlayers

The objective of this study is to develop an analytical solution for studying dynamic response of anisotropic multi-layered flexible pavement with linear-gradual interlayers under a harmonic moving load. In this study, the flexible pavement structure is simplified as one multi-layered flexible system, which is assumed to be a semi-infinite medium. A new approach combining the Fourier transform with the wavelet transform for solving the dynamic analytical solution. The wavelet transforms for solving inverse Fourier transform, in solving the solution in the physical domain, is superior to the conventional inverse Fourier transform. The linear-gradual interlayer between the adjacent layers is defined using the shear spring model, and the anisotropic property is simplified as transverse isotropy. Also, in the dynamic analytical solving processes, the motion in the transversely isotropic medium is decoupled into in-plane motion and out-of-plane motion because of the propagation of the waves in a transversely isotropic medium with coupling phenomena. The corresponding analytical solution is entered into a MATLAB-based computer program, which can compute the dynamic responses of an anisotropic multi-layered medium at different interlayer conditions. The accuracy of this program is confirmed through comparison with the results from the examples from the references. The influence analyses of linear-gradual interlayers and anisotropic properties of structural layers are illustrated. It is concluded that the proposed analytical solution-based computer program could be used in the multi-layered flexible pavement structural design and risk management in civil engineering.     

Hybrid Laplace transform-finite element method to a generalized electromagneto-thermoelastic problem

A finite element method based on the Laplace transform technique is developed for a two-dimensional problem in electromagneto-thermoelasticity. The problem is in the context of the following generalized thermoelasticity theories: Lord–Shulman’s, Green–Lindsay’s, the Chandrasekharaiah–Tzou, as well as the dynamic coupled theory. The Laplace transform method is applied to the time domain and the resulting equations are discretized using the finite element method. The inversion process is carried out using a numerical method based on a Fourier series expansions. Numerical results compared with those given in literature prove the good performance of the used method. It is demonstrated that the Chandrasekharaiah–Tzou theory can be considered as an extension of Lord–Shulman’s, and the generalized heat conduction mechanism is completely different from the classical Fourier’s in essence.     

Direct and iterative solution of the generalized Dirichlet–Neumann map for elliptic PDEs on square domains

In this work we derive the structural properties of the Collocation coefficient matrix associated with the Dirichlet–Neumann map for Laplace’s equation on a square domain. The analysis is independent of the choice of basis functions and includes the case involving the same type of boundary conditions on all sides, as well as the case where different boundary conditions are used on each side of the square domain. Taking advantage of said properties, we present efficient implementations of direct factorization and iterative methods, including classical SOR-type and Krylov subspace (Bi-CGSTAB and GMRES) methods appropriately preconditioned, for both Sine and Chebyshev basis functions. Numerical experimentation, to verify our results, is also included.     

定荷载和双面排水条件下非饱和土一维固结的解析解

基于多孔介质弹性理论,结合粒间吸应力表示的有效应力原理,建立了非饱和土固结的耦合偏微分控制方程.考虑一维问题,采用Laplace积分变换,得到了定荷载和双面排水条件下非饱和土固结的解析解答.通过数值算例,分析了土体饱和度对超孔隙水压力、有效应力以及土层沉降的影响规律.结果表明,土体的初始饱和度越高,则孔隙水压力消散得越快,有效应力增加越快.     

Analyzing a multiserver bulk-service finite-buffer queue

This paper analyzes a finite-buffer multiserver bulk-service queueing system in which the interarrival and service times are, respectively, arbitrarily and exponentially distributed. Using the supplementary variable and the imbedded Markov chain techniques, we obtain the queue-length distributions at prearrival and arbitrary epochs. We also present Laplace–Stiltjes transform of the actual waiting-time distribution in the queue. Finally, several performance measures and a variety of numerical results in the form of tables and graphs are discussed.     

Computation of the fundamental solution of electrodynamics for anisotropic materials

A new method for computation of the fundamental solution of electrodynamics for general anisotropic nondispersive materials is suggested. It consists of several steps: equations for each column of the fundamental matrix are reduced to a symmetric hyperbolic system; using the Fourier transform with respect to space variables and matrix transformations, formulae for Fourier images of the fundamental matrix columns are obtained; finally, the fundamental solution is computed by the inverse Fourier transform. Applying the suggested approach, the fundamental solution components are computed in general anisotropic media. Computational examples confirm robustness of the suggested method.     

任意荷载下成层粘弹性地基的一维固结

针对成层粘弹性地基模型,运用Laplace变换及矩阵传递法求解了任意荷载下成层粘弹性地基一维变形问题,得到了频域内的通解,通过Laplace逆变换,即可计算成层粘弹性地基在任意荷载下的一维变形.Terzaghi一维固结理论解是本文的一个特例.结合三层地基的算例,可以看到粘弹性地基的固结相对于弹性地基有个滞后过程,但随时间最终趋于一致;循环荷载下粘弹性多层地基固结时,其有效应力和变形都呈振荡增长,且不与荷载同步,而要相对滞后.此外,通过一工程实例,对该方法的可靠性进行论证,以证明该法确能指导工程实践.     

轴对称饱和地基竖向振动分析

基于Biot动力固结方程,考虑了土体和水体的惯性力以及水土之间的耦合作用,采用Laplace-Hankel积分变换求解耦联合方程组,得到动荷载下饱和地基振动问题的解答．根据下边界为不透水基岩的边界条件,获得了地基表面作用圆形轴对称任意荷载时土层应力、位移等的一般积分形式解．研究表明,激振频率对饱和地基的竖向振动有很大影响,地基表面的竖向位移与施加荷载之间存在相位差．此外,动力渗透系数在荷载施加的初期对结果有较大影响,随着荷载趋于稳定,其影响变得很小．     

Vibration of a pre-stressed plate on a transversely isotropic multilayered half-plane due to a moving load

This paper presents an analytical research on the dynamic interaction problem between a pre-stressed plate and a transversely isotropic multilayered half-plane subjected to a moving load. The pre-stressed plate is governed by the Kirchhoff plate theory, and the transversely isotropic multilayered half-plane is solved by the analytical layer-element method. Combining the frictionless contact and displacement compatibility conditions between the plate and the soil, the contact stress and the deflection of plate in the Fourier transform domain are derived. With the aid of the inverse Fourier transform, the actual solutions can be further achieved. Numerical examples are given to illustrate the influence of load speed, the rigidity of plate, the axial force applied on the plate and the stratified character of the soil.