A 2.5-D dynamic model for a saturated porous medium: Part I. Green’s function

Based on Biot’s theory, the dynamic 2.5-D Green’s function for a saturated porous medium is obtained using the Fourier transform and the potential decomposition methods. The 2.5-D Green’s function corresponds to the solutions for the following two problems: the point force applied to the solid skeleton, and the dilatation source applied within the pore fluid. By performing the Fourier transform on the governing equations for the 3-D Green’s function, the governing differential equations for the two parts of the 2.5-D Green’s function are established and then solved to obtain the dynamic 2.5-D Green’s function. The derived 2.5-D Green’s function for saturated porous media is verified through comparison with the existing solution for 2.5-D Green’s function for the elastodynamic case and the closed-form 3-D Green’s function for saturated porous media. It is further demonstrated that a simple form 2-D Green’s function for saturated porous media can be been obtained using the potential decomposition method.     

Decoupling coefficients of dilatational wave for Biot’s dynamic equation and its Green’s functions in frequency domain

Green’s functions for Biot’s dynamic equation in the frequency domain can be a highly useful tool for the investigation of dynamic responses of a saturated porous medium. Its applications are found in soil dynamics, seismology, earthquake engineering, rock mechanics, geophysics, and acoustics. However, the mathematical work for deriving it can be daunting. Green’s functions have been presented utilizing an analogy between the dynamic thermoelasticity and the dynamic poroelasticity in the frequency domain using the u-p formulation. In this work, a special term “decoupling coefficient” for the decomposition of the fast and slow dilatational waves is proposed and expressed to present a new methodology for deriving the poroelastodynamic Green’s functions. The correctness of the solution is demonstrated by numerically comparing the current solution with Cheng’s previous solution. The separation of the two waves in the present methodology allows the more accurate evaluation of Green’s functions, particularly the solution of the slow dilatational wave. This can be advantageous for the numerical implementation of the boundary element method (BEM) and other applications.     

NON-AXISYMMETRIC MIXED BOUNDARY VALUE PROBLEM FOR DYNAMIC INTERACTION BETWEEN SATURATED POROUS FOUNDATION AND RECTANGULAR PLATE$^{\bf 1)}$

在同一界面的不同区域具有多种边界条件, 称之为混合边界, 这是一个熟知的力学问题. 对这类问题进行精确分析时, 必须要进行混合边值问题的求解. 而对于一般的三维非轴对称情形, 混合边值问题的求解往往存在数学困难. 本文利用Hilbert定理和双重Fourier变换, 给出了一种求解三维非轴对称混合边值问题的解析方法, 利用该方法对具有混合透水边界的饱和多孔地基上矩形板的振动弯曲进行了解析研究(板与地基接触面为不透水边界, 其余为透水边界). 首先, 基于Kirchhoff理论和Biot多孔介质理论建立矩形板与饱和多孔地基的动力控制方程, 进行耦合求解. 针对板土接触面和非接触面的混合边值问题, 采用双重Fourier变换构造出两对二维对偶积分方程, 以接触应力和接触面孔隙压力为基本未知量, 用Jacobi正交多项式将未知量展开, 再利用Schmidt法对二维对偶积分方程完成求解, 最终推导出板土系统在动力作用下的位移和应力解析式. 通过将本文计算模型退化为单一弹性地基, 与已有研究结果进行对比, 验证了本文方法的正确性和有效性. 最后, 通过数值算例, 对饱和多孔地基上矩形板的动力响应及参数影响做出分析和讨论. 此外, 本文提出的解析法具有一般性, 可广泛应用于复杂接触问题和多场耦合问题的求解.     

饱和多孔地基与矩形板动力相互作用的非轴对称混合边值问题

在同一界面的不同区域具有多种边界条件, 称之为混合边界, 这是一个熟知的力学问题. 对这类问题进行精确分析时, 必须要进行混合边值问题的求解. 而对于一般的三维非轴对称情形, 混合边值问题的求解往往存在数学困难. 本文利用Hilbert定理和双重Fourier变换, 给出了一种求解三维非轴对称混合边值问题的解析方法, 利用该方法对具有混合透水边界的饱和多孔地基上矩形板的振动弯曲进行了解析研究(板与地基接触面为不透水边界, 其余为透水边界). 首先, 基于Kirchhoff理论和Biot多孔介质理论建立矩形板与饱和多孔地基的动力控制方程, 进行耦合求解. 针对板土接触面和非接触面的混合边值问题, 采用双重Fourier变换构造出两对二维对偶积分方程, 以接触应力和接触面孔隙压力为基本未知量, 用Jacobi正交多项式将未知量展开, 再利用Schmidt法对二维对偶积分方程完成求解, 最终推导出板土系统在动力作用下的位移和应力解析式. 通过将本文计算模型退化为单一弹性地基, 与已有研究结果进行对比, 验证了本文方法的正确性和有效性. 最后, 通过数值算例, 对饱和多孔地基上矩形板的动力响应及参数影响做出分析和讨论. 此外, 本文提出的解析法具有一般性, 可广泛应用于复杂接触问题和多场耦合问题的求解.      

GREEN QUASIFUNCTION METHOD FOR FREE VIBRATION OF CLAMPED THIN PLATES

The Green quasifunction method is employed to solve the free vibration problem of clamped thin plates.A Green quasifunction is established by using the fundamental solution and boundary equation of the...     

GREEN QUASIFUNCTION METHOD FOR FREE VIBRATION OF SIMPLY-SUPPORTED TRAPEZOIDAL SHALLOW SPHERICAL SHELL ON WINKLER FOUNDATION

The idea of Green quasifunction method is clarified in detail by considering a free vibration problem of simply-supported trapezoidal shallow spherical shell on Winkler foundation.A Green quasifunction is established by using the fundamental solution and boundary equation of the problem.This function satisfies the homogeneous boundary condition of the problem.The mode shape differential equation of the free vibration problem of simply-supported trapezoidal shallow spherical shell on Winkler foundation is reduced to two simultaneous Fredholm integral equations of the second kind by Green formula.There are multiple choices for the normalized boundary equation.Based on a chosen normalized boundary equation, the irregularity of the kernel of integral equations is avoided.Finally, natural frequency is obtained by the condition that there exists a nontrivial solution in the numerically discrete algebraic equations derived from the integral equations.Numerical results show high accuracy of the Green quasifunction method.     

多孔饱和半空间上弹性圆板的动力分析

用解析方法研究多孔饱和半空间上弹性圆板的低垂直振动，首先用Ｈａｎｋｅｌ变换求解多孔饱和介质动力问题控制方程，然后按混合边值条件建立多孔饱和半空间上弹性板的垂直振动的对偶积分方程，用Ａｂｅｌ变换化对偶积分方程为第二类Ｆｒｅｄｈｏｌｍ积分方程，并给出了数值算例。     

简支梯形底扁球壳弯曲问题的准格林函数方法

以简支梯形底扁球壳的弯曲问题为例,详细阐明了准格林函数方法的思想.即利用问题的基本解和边界方程构造一个准格林函数,这个函数满足了问题的齐次边界条件,采用格林公式将简支扁球壳弯曲问题的控制微分方程化为两个互相耦合的第二类Fredholm积分方程.边界方程有多种选择,在选定一种边界方程的基础上,可以通过建立一个新的边界方程...     

饱和土–隧道动力响应的2.5维有限元–边界元耦合模型

针对饱和土中异形隧道的三维动力响应问题,建立了2.5维有限元与边界元耦合模型.将隧道结构视为弹性体,采用2.5维有限元建立隧道模型;将地基土视为饱和多孔介质,采用2.5维边界元建立饱和土体模型.借助组合辅助问题基本解消除了边界积分方程的奇异性.利用饱和土与隧道接触面的位移、面力连续和完全透水或完全不透水边界条件,实现2.5维有限元和边界元模型的耦合求解.模型具有计算效率高、适用范围广的优点.通过与完全透水和完全不透水边界条件下轴对称问题的半解析解以及单相介质的2.5维有限元与边界元耦合模型对比,验证了本文模型的正确性.最后利用该模型计算了饱和土体中类矩形隧道在移动载荷作用下的三维动力响应,分析了不同土体渗透性下位移及孔隙水压力沿隧道轴向、环向和深度的分布规律.结果表明:(1)孔隙水压力随土体渗透性增大而显著减小,位移受土体渗透性影响小;(2)位移及孔隙水压力在隧道环向主要分布在距载荷作用点两侧约2 m的范围内;(3)孔隙水压力沿深度的衰减比土体位移快,且孔隙水压力和轴向位移沿深度的分布受土体渗透性影响大.     

Green function on two-phase saturated medium by concentrated force in two-dimensional displacement field

The Green function on two-phase saturated medium by concentrated force has a broad and important use in seismology, seismic engineering, soil mechanics, geophysics, dynamic foundation theory and so on. According to the Green function on two-phase saturated medium by concentrated force in three-dimentional displacement field obtained by Ding Bo-yang et al. , it gives out the Green function in two-dimensional displacement field by infinite integral method along x3 -direction derived by De Hoop and Manolis. The method adopted in the thesis is simpler. The result will be simplified to the boundary element method of dynamic problem.     

Green quasifunction method for vibration of simply-supported thin polygonic plates on Pasternak foundation

A new numerical method—Green quasifunction is proposed.The idea of Green quasifunction method is clarified in detail by considering a vibration problem of simply-supported thin polygonic plates on Pasternak foundation.A Green quasifunction is established by using the fundamental solution and boundary equation of the problem. This function satisfies the homogeneous boundary condition of the problem.The mode shape differential equation of the vibration problem of simply-supported thin plates on Pasternak foundation is reduced to two simultaneous Fredholm integral equations of the second kind by Green formula.There are multiple choices for the normalized boundary equation.Based on a chosen normalized boundary equation,a new normalized boundary equation can be established such that the irregularity of the kernel of integral equations is overcome.Finally,natural frequency is obtained by the condition that there exists a nontrivial solution in the numerically discrete algebraic equations derived from the integral equations.Numerical results show high accuracy of the Green quasifunction method.     

层状横观各向同性饱和土的非轴对称动力响应

通过方位角的Fourier变换,将圆柱坐标系下横观各向同性饱和土的Biot非轴对称波动方 程转化为一组一阶常微分方程组. 然后基于径向Hankel变换,建立问题的状态方程;求解状态方程后,得到传递矩阵. 进而利用传递矩阵,结合饱和层状地基的边界条件、排水条件及层间接触和连续条件,求解 了任意震源力作用下层状横观各向同性饱和地基频域动力响应问题. 时域解可通过频率的Fourier积分得到.     

The boundary contour method for magneto-electro-elastic media with quadratic boundary elements

This paper presents a development of the boundary contour method (BCM) for magneto-electro-elastic media. First, the divergence-free of the integrand of the magneto-electro-elastic boundary element is proved. Second, the boundary contour method formulations are obtained by introducing quadratic shape functions and Green’s functions [Ding, H.J., Jiang, A.M., 2004. A boundary integral formulation and solution for 2D problems in magneto-electro-elastic media. Computers and Structures, 82 (20–21), 1599–1607] for magneto-electro-elastic media and using the rigid body motion solution to regularize the BCM and avoid computation of the corner tensor. The BCM is applied to the problem of magneto-electro-elastic media. Finally, numerical solutions for illustrative examples are compared with exact ones. The numerical results of the BCM coincide very well with the exact solution, and the feasibility and efficiency of the method are verified.     

集中力作用下的两相饱和介质位移场Green函数

以复模两相饱和介质Biot动力学方程为基础,根据该方程D'Alembert解的Fourier变换所属的Homholtz方程特性,由Biot方程解的相容性条件及δ函数性质较好地解决了快、慢纵波位势的耦合问题.较为简便地得到了两相饱和介质在集中力作用下低频(ω＜ωc)时的频域和时域的Green函数.     

二维介质参数的大扰动反演方法

对非均匀介质参数反演问题进行了研究，并提出了用于反演二维介质参数的广义射线近似方法．利用参考场量和扰动变量对声波方程中的介质参数进行处理，并利用Ｇｒｅｅｎ函数理论得到扰动参数比的积分方程．基于非均匀介质中波函数的局部理论和射线理论，引入了全波场的广义射线近似形式，通过定义介质参数函数，把反演目标归结为其第一类Ｆｒｅｄｈｏｌｍ积分方程．利用积分变换方法得到二维介质的介质参数函数，从而得到介质参数，在Ｂｏｒｎ近似方法中，反演的介质参数扰动不能超过２０％，但是在本文中介绍的方法能够有效地反演其扰动比不超过５０％的变化情况     

敷设多孔吸声材料声腔特征值分析的径向积分边界元法

由于Helmholtz方程的基本解是频率的函数,因此传统边界元法在处理声场特征值问题时具有天生的缺陷。本文采用Laplace方程基本解生成积分方程,通过径向积分法将在此过程中产生的域积分项转化为边界积分。此方法克服了传统边界元法系数矩阵对频率的依赖,同时克服了特解积分法对特解的依赖,并通过对表面声导纳的多项式逼近,将敷设多孔吸声材料声腔特征值问题转化为矩阵多项式,从而避免了复杂的非线性求解。通过数值算例验证了算法的有效性。     

Quasi-Green＇s function method for free vibration of simply-supported trapezoidal shallow spherical shell

The idea of quasi-Green＇s function method is clarified by considering a free vibration problem of the simply-supported trapezoidal shallow spherical shell. A quasi- Green＇s function is established by using the fundamental solution and boundary equation of the problem. This function satisfies the homogeneous boundary condition of the prob- lem. The mode shape differential equations of the free vibration problem of a simply- supported trapezoidal shallow spherical shell are reduced to two simultaneous Fredholm integral equations of the second kind by the Green formula. There are multiple choices for the normalized boundary equation. Based on a chosen normalized boundary equa- tion, a new normalized boundary equation can be established such that the irregularity of the kernel of integral equations is overcome. Finally, natural frequency is obtained by the condition that there exists a nontrivial solution to the numerically discrete algebraic equations derived from the integral equations. Numerical results show high accuracy of the quasi-Green＇s function method.     

Laplace transform-boundary element coupling method for viscous fluid-structure impact analysis

Based on linearized 2-D Navier-Stokes equation, a Laplace transform-boundary element coupling method for viscous fluid-structure impact analysis is proposed. Under assumption of incompressibility for the fluid, the corresponding equivalent boundary integral equation in terms of the potential function and stream function is first established by Lamb's transform in the Laplace transform domain. It enables us to solve impact water problems in frequency domain by the boundary element method, in which the effect of viscous flow on the dynamic response can be taken into account. Then a complete solution of the problem under consideration in time domain is obtained by means of Durbin's formulas for the numerical inversion of the Laplace transform. Finally, a practical example is given to confirm the validity of the present method. Project supported by the National Defence Foundation of Science & Technology of China (No. J14. 8. 1. JW0515).     

Quasi-Green’s function method for free vibration of simply-supported trapezoidal shallow spherical shell

The idea of quasi-Green’s function method is clarified by considering a free vibration problem of the simply-supported trapezoidal shallow spherical shell. A quasi-Green’s function is established by using the fundamental solution and boundary equation of the problem. This function satisfies the homogeneous boundary condition of the prob-lem. The mode shape differential equations of the free vibration problem of a simply-supported trapezoidal shallow spherical shell are reduced to two simultaneous Fredholm integral equations of the second kind by the Green formula. There are multiple choices for the normalized boundary equation. Based on a chosen normalized boundary equa-tion, a new normalized boundary equation can be established such that the irregularity of the kernel of integral equations is overcome. Finally, natural frequency is obtained by the condition that there exists a nontrivial solution to the numerically discrete algebraic equations derived from the integral equations. Numerical results show high accuracy of the quasi-Green’s function method.     

Boundary Integral Procedures for Unsaturated Flow Problems

Abstract. A novel numerical scheme based on the singular integral theory of the boundary element method. (BEM) is presented for the solution of transient unsaturated flow in porous media. The effort in the present paper is directed in facilitating the application of the boundary integral theory to the solution of the highly non-linear equations that govern unsaturated flow. The resulting algorithm known as the Green element method (GEM) presents a robust attractive method in the state-of -the-art application of the boundary element methodology. Three GEM models based on their different methods of handling the non-linear diffusivity, illustrate the suitability and robustness of this approach for solving highly non-linear 1-D and 2-D flows which would have proved cumbersome or too difficult to implement with the classical BEM approach.