三维水质预测问题的Laplace变换边界元法及其推广

本文用边界元法解决非稳定系统中水质预测的计算问题,建立了三维水质预测问题的边界元法计算模型,为简化计算手续、提高计算精度,讨论了Laplace变换边界元法,推广了Laplace变换边界元法中的Schapery方法。     

A direct boundary element method for plane strain poroelasticity

This paper presents a direct boundary element method (BEM), formulated in the Laplace transform space, for plane strain poroelasticity. The paper expands on work by Cheng and Liggett by recasting the theoretical foundation of BEM within the framework of Rice and Cleary's formulation of the Biot theory of poroelasticity. Furthermore, the numerical algorithm is generalized to deal with both interior and exterior domain problems, and a method for indirectly calculating the Cauchy principal value of the singular integrals is presented. Formulae for the stress and flux inside the domain are also derived. Finally, the algorithm is validated by comparing the numerical results with the analytical solution of a borehole subject to a far-field deviatoric stress (exterior domain) and with the solution of Mandel's problem (interior domain). These two examples provide a critical test of the algorithm.     

饱和土表面在水平集中荷载作用下的瞬态反应

分析了水平集中荷载作用在半空间饱和土表面时的瞬态问题。利用Laplace-Hankel变换对非轴对称Biot固结方程进行解耦，利用Laplace-Hankel数值逆变换得到半空间饱和土在时域内的数值解。退化到线弹性中的解与文献中的结果进行比较，验证了文中结果的正确性和数值逆变换的可靠性.可以用于研究地震工程中地震波的传播以及饱和土与结构之间相互作用等问题。     

Modeling pressure pipe embedded in two-layer soil by a half-plane BEM

In this paper, an elastostatic half-plane boundary element method (BEM) formulation was applied to analyze the stress behavior of underground pressure pipes, embedded in two-layer soils. In the use of this method, only the boundary of pipe and interfaces were required to be discretized. In this regard, first, a computer code was prepared based on a multi-region substructuring process in the BEM scheme. Then, the efficiency and applicability of the method as well as the prepared algorithm were verified by solving some practical examples and comparing the results with those of the published works. Finally, a parametric study was done to evaluate the effect of pipe depth and determine the soil stress distribution. The studies showed that the half-plane BEM was in good agreement with the existing solutions and its capability was very favorable for elastostatic problems including semi-infinite domain. It is obvious that this method can be practically used to analyze the geotechnical underground buildings in substituting the full-plane BEM formulation.     

横观各向同性含液饱和多孔介质黏弹性瞬态反应

在考虑横观各向同性含液饱和多孔介质固体骨架和流体可压缩性以及固体骨架的黏弹性特征下,基于横观各向同性含液饱和多孔介质u-w形式的三维动力控制方程,以固相位移u、液相相对位移w为基本未知量,综合运用Laplace变换、双重Fourier变换等方法,在直角坐标系下通过引入中间变量,将六元2阶动力控制方程组化为两组各含4个未知变量的常微分方程组,给出了直角坐标系下横观各向同性含液饱和多孔介质三维黏弹性动力反应的积分形式一般解;作为理论推导的验证,通过引入初始条件和边界条件,对横观各向同性含液饱和多孔介质半空间黏弹性瞬态反应问题进行了求解。解答的退化验证表明,所推导的理论解是正确的。     

Boundary integral analysis of transient thermoelasticity

A simple and efficient method for the solution of uncoupled transient thermoelastic problems using boundary integral techniques is presented. The method employs a Laplace transformation to remove temporarily the time dependence of the governing equations. Numerical analysis is then carried out in the transform space, and results in the time-position space are found by numerical inversion of the Laplace transform. The method has the advantage that it avoids time-stepping and the costly evaluation of domain integrals. Boundary element and analytic solutions are compared, and the effect of cooling on the stresses around a deep underground opening is examined.     

瞬变弹性动力问题的一个边界元法

本文给出了瞬变弹性动力问题的一个边界元法。该法是利用威尔逊——θ法的差分公式,把运动方程化为椭圆型微分方程。根据贝蒂定理和动力点荷载的特解,可获得动力问题的边界积分方程。这个解法是在真实时间域内逐步求解的,不需要使用拉氏变换。一个应力波传播的数值算例证实了该方法使用方便,且解答精度较高。     

An indirect boundary integral equation method for poroelasticity

This paper presents an indirect boundary integral equation method for analysis of quasi-static, time-harmonic and transient boundary value problems related to infinite and semi-infinite poroelastic domains. The present analysis is based on Biot's theory for poroelastodynamics with fluid viscous dissipation. The solution to a given boundary value problem is reduced to the determination of intensities of forces and fluid sources applied on an auxiliary surface defined interior to the surface on which the boundary conditions are specified. A coupled set of integral equations is established to determine the intensities of forces and fluid sources applied on the auxiliary surface. The integral equations are solved numerically in the Laplace domain for quasi-static and transient problems, and in the frequency domain for time-harmonic excitations. The kernel functions of the integral equation correspond to appropriate Green's functions for a poroelastic full space or half-space. The convergence and numerical stability of the present scheme are established by considering a number of bench mark problems. The versatility of the present method is demonstrated by studying the quasi-static response of a rigid spheroidal anchor, and time-harmonic and transient response of a rigid semi-circular tunnel.     

Elasto-dynamic Problem

A boundary element method formulation in the real time domain in the transient elastodynamic problems is presented. The equation motion of the linear isotropic elastiic bodare transformed into the elliptic partial differential eqnations by means of the differential formulations of the Wilson-θ method. By utilizing the particular solution corresponding to a dynamic point force in an infinite medium and the Betti’s theorem the bundary integral equations may be obtain. As this approach formulates and solves the problem in real time domain in conjunction with a time-stepping algorithm, it does not require a nimerical inversion.Which transformes the results from the Laplace or Fourier ransformed domain to the time domain. A numerical example was shown in order to demonstrate the applicability of the present method to the practical problems.     

The fundamental solution of poroelastic plate saturated by fluid and its applications

In this paper, the numerical model of the transverse vibrations of a thin poroelastic plate saturated by a fluid was proposed. Two coupled dynamic equations of equilibrium related to the plate deflection and the equivalent moment were established for an isotropic porous medium with uniform porosity. The fundamental solutions for a porous plate were derived both in the Laplace transform domain and in the time domain. A meshless method was developed and demonstrated in the Laplace transform domain for solving two coupled dynamic equations. Numerical examples demonstrated the accuracy of the method of the fundamental solutions and comparisons were made with analytical solutions. The proposed meshless method was shown to be simple to implement and gave satisfactory results for a poroelastic plate dynamic analysis. Copyright © 2009 John Wiley & Sons, Ltd.     

Three‐dimensional transient thermo‐hydro‐mechanical fundamental solutions of unsaturated soils

The governing differential equations of unsaturated soils considering the thermo‐poro‐mechanical behaviour consist of equilibrium, moisture air and heat transfer equations. In this paper at first, following some necessary simplifications, the thermal three‐dimensional fundamental solution for an unsaturated deformable porous medium with linear elastic behaviour in Laplace transform domain is presented. Subsequently, the closed‐form time domain fundamental solutions are derived by analytical inversion of the Laplace transform domain solutions. Then a set of numerical results are presented, which demonstrate the accuracies and some salient features of the derived analytical transient fundamental solutions. Finally, the closed‐form time domain fundamental solution will be verified mathematically by comparison with the previously introduced corresponding fundamental solution. Copyright © 2009 John Wiley & Sons, Ltd.     

A transient finite element analysis of linear viscoelastic material model

Since the attenulation of propagating waves through soil/rock is related to the localized material properties as well as the strain developed, the commonly used Rayleigh-type damping model and its variations are not suitable for dynamic finite element analysis of such materials. A linear viscoelastic material model based on the concept of the relaxation spectrum is manipualted in place of the damping model in this paper. The method proposed by Day and Minster¹¹ to transform the convolutional form of the stress–strain relationship to a set of differential operators using the Pade approximant method is generalized to non-scalar waves and implemented for transient finite element analyses. A time-marching scheme is also proposed to incorporate the resultant differential operators into the governing equation of motion. The accuracy related to the Pade approximant method and the time-marching scheme is investigated by critically analysing some scalar wave propagation problems. The proposed technique is further verified using two one-dimensional stress wave propagation problems and a two-dimensional transient propagating wave through an unbounded linear viscoelastic medium. Some encouraging results have been obtained using the proposed technique and guidelines for using this technique are also presented. Comparisons of analytical solutions obtained by Fourier synthesis and numerical results have been provided.     

Dynamic analysis of a cylindrical casing–cement structure in a poroelastic stratum

The transient response of a cylindrical casing–cement structure in a poroelastic stratum under dynamic radial tractions is one of the significant issues during the analysis of downhole operations and the selection of safe material. Based on the Biot theory and general elastic mechanics, this paper gives a set of exact solutions for radial displacement, stresses for the casing–cement system and the pore pressure of the infinite surrounding poroelastic stratum in the Laplace transform space. Solutions are presented for three different types of transient radial loadings acting on the surface of casing, i.e., suddenly applied constant load, gradually applied step load and triangular pulse load. Time domain solutions are obtained using a reliable numerical method of inverse Laplace transforms. A detailed parametric study about the transient response is presented both at the casing–cement interface and the cement–stratum interface, and the distributions of the pore pressure and the effective stresses in the stratum are also examined. Copyright © 2017 John Wiley & Sons, Ltd.     

Numerical modelling of transient contaminant migration problems in infinite porous fractured media using finite/infinite element technique. Part I: Theory

The leakage effect in porous fissured media has been considered in a general sense by introducing a new expression of the leakage term in this paper. The double porosity concept is employed and the related expressions are formulated using the upwind finite element approach. Considering the infinite extension of the problem domain, a mapped transient infinite element has been presented to simulate the far field of the infinite medium. Since the mass transfer function of the present mapped transient infinite element is dependent on both space and time variables, the mechanism of transient contaminant migration problems in infinite porous fractured media can be rigorously simulated because the property matrices of the element are evaluated at any time of interest. By comparing the current numerical results with the analytical ones, the accuracy, correctness and effectiveness of the present method have been established. Three different time discretization schemes were examined and it was found that either the central difference or the backward difference approximation is suitable for the upwind finite element simulation of transient contaminant migration problems.     

关于瞬变电磁法2.5维正演中的几个问题

这里讨论了瞬变电磁法2.5维正演模拟中的两个积分变换:傅里叶逆变换和拉普拉斯逆变换。针对发收距为零的中心回线方式瞬变电磁法,提出了在傅氏域中考察傅氏变换函数随波数的变化规律,进而根据曲线首尾支渐近线来划定波数覆盖的范围,然后同解析解对比,确定出最少个数的傅氏域波数。另外,介绍了只需对较少的拉氏变换变量作纯实数运算的拉普拉斯数值反演计算方法。通过对均匀半空间表面上垂直磁偶极子源形成的瞬变电磁场进行正演模拟,结果表明,关于傅氏域波数的范围划定原则及个数选取方案是合理的、拉氏逆变换算法也是切实可行的。     

An approximate spring–dashpot artificial boundary for transient wave analysis of fluid-saturated porous media

Practical civil engineering problems are usually formulated in an infinite half-space domain, and a selected finite domain is required to analyze the dynamic responses of a fluid-saturated porous medium by the finite element method (FEM). Devising a method to deal with the boundaries of the finite domain is the key issue for this open system. In this paper, a two-dimensional spring–dashpot artificial boundary (SDAB) for transient analysis in a fluid-saturated porous media is developed. Based on Biot’s dynamic theory of fluid-saturated porous media, the normal and tangential boundary stress formulae are deduced for out-going cylindrical body waves. The boundary stress is proportional to displacement and velocity, thus continuously distributed dashpots and springs can be placed on the artificial boundaries in the normal and tangential directions to simulate the energy absorption of the infinite media outside of the finite domain for the interior distributed source problems. In this paper, the input seismic motion can be realized by applying an equivalent load on the SDAB for the seismic scattering problems of exterior distributed sources. Numerical examples are given and the analyzed results show that the SDAB and the method of wave motion input have good stability and acceptable accuracy.     

层状地基中基于Laplace变换的桩基横向振动阻抗计算

基于Laplace变换,对层状地基中桩土横向振动阻抗计算问题进行了研究。考虑土层天然分层的特性及桩顶轴向力的参与作用,结合频域内桩-土动力文克尔理论,采用传递矩阵法并通过拉普拉斯变换,将振动微分方程变成代数方程以求解桩的横向振动响应参数,并导出了单桩横向振动阻抗。基于所得解,进一步计算出桩-土-桩水平动力相互作用因子。通过实例分析对比,验证其有效性和可行性。该方法计算工作量小,易于理解,计算结果与已有结果具有良好的一致性,并能保证解的连续性,对桩-土动力相互作用的研究具有一定的实用意义。     

非饱和土中圆柱形衬砌隧道的瞬态动力响应

在以往关于圆柱形衬砌隧道的瞬态动力响应中,衬砌周围土体大多假定为弹性介质或饱和介质。然而,自然界中的土体大多为非饱和介质。考虑土体与衬砌结构的动力相互作用及动荷载引起的附加质量密度的影响,研究了瞬态荷载作用下非饱和土中无限长深埋圆柱形衬砌隧道的动力响应。基于多孔介质混合物理论和连续介质力学理论,建立了非饱和土中圆柱形衬砌隧道受到瞬态荷载作用时衬砌及周围土体的控制方程,利用Durbin数值反演法得到了衬砌及土体在时间域的动力响应。数值分析了饱和度对瞬态荷载下径向位移、径向应力、环向应力和孔隙水压力的影响。结果表明：饱和度对衬砌及周围土体的瞬态响应影响显著;饱和度对径向位移沿径向的衰减影响较小,对环向应力和孔隙压力沿径向的衰减影响较大。     

爆炸荷载作用下深埋圆形隧洞的饱和黏弹性土-衬砌系统动力响应

假定土骨架服从标准线性固体黏弹性本构关系,研究了深埋圆形隧洞的饱和黏弹性土-弹性衬砌耦合系统在轴对称爆炸作用下的瞬态动力响应。首先,基于饱和土的Biot模型和衬砌的弹性理论,通过引入势函数和Laplace变换,利用弹性衬砌和饱和黏弹性土界面处的连续性条件以及边界条件,得到饱和黏弹性土体和弹性衬砌位移、应力和孔隙水压力等在Laplace变换域中的解析解。其次,利用Laplace数值Crump逆变换得到耦合系统在时间域的动力响应,数值分析了不同土体模型下土体-衬砌耦合系统的径向位移和环向应力以及土体孔隙水压力等。结果表明：对不同土体模型的土体-衬砌耦合系统,其在爆炸载荷作用下的动力响应性态基本一致,但动力响应的振动周期和幅值等具有明显的差异。同时,对于饱和黏弹性土-弹性衬砌系统,土体黏性参数对土体径向位移和孔隙水压力有明显的影响,但对土体环向应力影响较小。     

Dynamic Green's functions for an anisotropic poroelastic half-space

The dynamic responses of an anisotropic poroelastic half-space under an internal point load and fluid source are investigated in the frequency domain in this paper. By virtue of Fourier transform and Stroh formalism, the three-dimensional (3D) general solutions of the anisotropic Biot's coupling dynamics equations are derived in the frequency domain. Considering the two surface conditions, permeable and impermeable, the analytical solutions for displacement fields and pore pressure in half-space under a point source (point load or a fluid source) are obtained. When the material properties are isotropic, the numerical results of the poroelastic half-space are in excellent agreement with the existing analytical solutions. For anisotropic half-space cases, numerical results show the strong dependence of the dynamic Green's functions on the material properties.