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1.
This paper identifies imbalanced columns (or rows) as a significant source of ill‐conditioning in the preconditioned coefficient matrix using the standard Jacobi preconditioner, for finite element solution of Biot's consolidation equations. A simple and heuristic preconditioner is proposed to reduce this source of ill‐conditioning. The proposed preconditioner modifies the standard Jacobi preconditioner by scaling the excess pore pressure degree‐of‐freedoms in the standard Jacobi preconditioner with appropriate factors. The performance of such preconditioner is examined using the symmetric quasi‐minimal residual method. To alleviate storage requirements, element‐by‐element iterative strategies are implemented. Numerical experiment results show that the proposed preconditioner reduces both the number of iteration and CPU execution time significantly as compared with the standard Jacobi preconditioner. Copyright © 2001 John Wiley & Sons, Ltd.  相似文献   

2.
In view of rapid developments in iterative solvers, it is timely to re‐examine the merits of using mixed formulation for incompressible problems. This paper presents extensive numerical studies to compare the accuracy of undrained solutions resulting from the standard displacement formulation with a penalty term and the two‐field mixed formulation. The standard displacement and two‐field mixed formulations are solved using both direct and iterative approaches to assess if it is cost‐effective to achieve more accurate solutions. Numerical studies of a simple footing problem show that the mixed formulation is able to solve the incompressible problem ‘exactly’, does not create pressure and stress instabilities, and obviate the need for an ad hoc penalty number. In addition, for large‐scale problems where it is not possible to perform direct solutions entirely within available random access memory, it turns out that the larger system of equations from mixed formulation also can be solved much more efficiently than the smaller system of equations arising from standard formulation by using the symmetric quasi‐minimal residual (SQMR) method with the generalized Jacobi (GJ) preconditioner. Iterative solution by SQMR with GJ preconditioning also is more elegant, faster, and more accurate than the popular Uzawa method. Copyright © 2003 John Wiley & Sons, Ltd.  相似文献   

3.
A numerically efficient and stable method is developed to analyze Biot's consolidation of multilayered soils subjected to non‐axisymmetric loading in arbitrary depth. By the application of a Laplace–Hankel transform and a Fourier expansion, the governing equations are solved analytically. Then, the analytical layer‐element (i.e. a symmetric stiffness matrix) describing the relationship between generalized displacements and stresses of a layer is exactly derived in the transformed domain. Considering the continuity conditions between adjacent layers, the global stiffness matrix of multilayered soils is obtained by assembling the inter‐related layer‐elements. Once the solution in the Laplace–Hankel transformed domain that satisfies the boundary conditions has been obtained, the actual solution can be derived by the inversion of the Laplace–Hankel transform. Finally, numerical examples are presented to verify the theory and to study the influence of the layered soil properties and time history on the consolidation behavior. Copyright © 2011 John Wiley & Sons, Ltd.  相似文献   

4.
Two formulations for calculating dynamic response of a cylindrical cavity in cross‐anisotropic porous media based on complex functions theory are presented. The basis of the method is the solution of Biot's consolidation equations in the complex plane. Employing two groups of potential functions for solid skeleton and pore fluid (each group includes three functions), the u–w formulation of Biot's equations are solved. Difference of these two solutions refers to use of two various potential functions. Equations for calculating stress, displacement and pore pressure fields of the medium are mentioned based on each two formulations. Copyright © 2009 John Wiley & Sons, Ltd.  相似文献   

5.
Nonphysical pressure oscillations are observed in finite element calculations of Biot's poroelastic equations in low‐permeable media. These pressure oscillations may be understood as a failure of compatibility between the finite element spaces, rather than elastic locking. We present evidence to support this view by comparing and contrasting the pressure oscillations in low‐permeable porous media with those in low‐compressible porous media. As a consequence, it is possible to use established families of stable mixed elements as candidates for choosing finite element spaces for Biot's equations. Copyright © 2011 John Wiley & Sons, Ltd.  相似文献   

6.
In this study, the dynamic response of a poroelastic half‐space to a point fluid sink is investigated using Biot's dynamic theory of poroelasticity. Based on Biot's theory, the governing field equations are re‐formulated in frequency domain with solid displacement and pore pressure. In a cylindrical coordinate system, a method of displacement potentials for axisymmetric displacement field is proposed to decouple the Biot's field equations to three scalar Helmholtz equations, and then the general solution to axisymmetric problems are obtained. The full‐space fundamental singular solution for a point sink is also derived using potential methods. The mirror‐image method is finally applied to construct the fundamental solution for a point sink buried in a poroelastic half‐space. Furthermore, a numerical study is conducted for a rock, that is, Berea sandstone, as a representative example. Copyright © 2013 John Wiley & Sons, Ltd.  相似文献   

7.
Non‐associated flow rule is essential when the popular Mohr–Coulomb model is used to model nonlinear behavior of soil. The global tangent stiffness matrix in nonlinear finite element analysis becomes non‐symmetric when this non‐associated flow rule is applied. Efficient solution of this large‐scale non‐symmetric linear system is of practical importance. The standard Krylov solver for a non‐symmetric solver is Bi‐CGSTAB. The Induced Dimension Reduction [IDR(s)] solver was proposed in the scientific computing literature relatively recently. Numerical studies of a drained strip footing problem on homogenous soil layer show that IDR(s = 6) is more efficient than Bi‐CGSTAB when the preconditioner is the incomplete factorization with zero fill‐in of global stiffness matrix Kep (ILU(0)‐Kep). Iteration time is reduced by 40% by using IDR(s = 6) with ILU(0)‐Kep. To further reduce computational cost, the global stiffness matrix Kep is divided into two parts. The first part is the linear elastic stiffness matrix Ke, which is formed only once at the beginning of solution step. The second part is a low‐rank matrix Δ, which is re‐formed at each Newton–Raphson iteration. Numerical studies show that IDR(s = 6) with this ILU(0)‐Ke preconditioner is more time effective than IDR(s = 6) with ILU(0)‐Kep when the percentage of yielded Gauss points in the mesh is less than 15%. The total computation time is reduced by 60% when all the recommended optimizing methods are used. Copyright © 2013 John Wiley & Sons, Ltd.  相似文献   

8.
Biot's dynamic equations of motion for one-dimensional wave propagation in a fluid-saturated linear elastic isotropic soil are solved using Laplace transformation followed by numerical inversion and the results compared with a direct finite element formulation. A soil column of finite dimension subjected to velocity boundary conditions is analysed, allowing for reflection of waves from boundaries. Comparison of time histories at given points along the column shows that the finite element solution gives good agreement with the Laplace transform solution for low as well as high drag.  相似文献   

9.
The purpose of this paper is to develop the macroscopic model of hydro‐mechanical coupling for the case of a porous medium containing isolated cracks or/and vugs. In the development, we apply the asymptotic expansion homogenization method. It is shown that the general structure of Biot's model is the same as in the case of homogeneous medium, but the poro‐elastic parameters are modified. Two numerical examples are presented. They concern the computations of Biot's parameters in isotropic and anisotropic cases. It can also be seen how the presence of near‐zero‐volume cracks influences Biot's parameters of the porous matrix. It can significantly affect the coupled hydro‐mechanical behaviour of damaged porous medium. Copyright © 2012 John Wiley & Sons, Ltd.  相似文献   

10.
In recent years, the authors have proposed a new double‐node zero‐thickness interface element for diffusion analysis via the finite element method (FEM) (Int. J. Numer. Anal. Meth. Geomech. 2004; 28 (9): 947–962). In the present paper, that formulation is combined with an existing mechanical formulation in order to obtain a fully coupled hydro‐mechanical (or HM) model applicable to fractured/fracturing geomaterials. Each element (continuum or interface) is formulated in terms of the displacements (u) and the fluid pressure (p) at the nodes. After assembly, a particular expression of the traditional ‘u–p’ system of coupled equations is obtained, which is highly non‐linear due to the strong dependence between the permeability and the aperture of discontinuities. The formulation is valid for both pre‐existing and developing discontinuities by using the appropriate constitutive model that relates effective stresses to relative displacements in the interface. The system of coupled equations is solved following two different numerical approaches: staggered and fully coupled. In the latter, the Newton–Raphson method is used, and it is shown that the Jacobian matrix becomes non‐symmetric due to the dependence of the discontinuity permeability on the aperture. In the part II companion paper (Int. J. Numer. Anal. Meth. Geomech. 2008; DOI: 10.1002/nag.730 ), the formulation proposed is verified and illustrated with some application examples. Copyright © 2008 John Wiley & Sons, Ltd.  相似文献   

11.
Accurate prediction of the interactions between the nonlinear soil skeleton and the pore fluid under loading plays a vital role in many geotechnical applications. It is therefore important to develop a numerical method that can effectively capture this nonlinear soil‐pore fluid coupling effect. This paper presents the implementation of a new finite volume method code of poro‐elasto‐plasticity soil model. The model is formulated on the basis of Biot's consolidation theory and combined with a perfect plasticity Mohr‐Coulomb constitutive relation. The governing equation system is discretized in a segregated manner, namely, those conventional linear and uncoupled terms are treated implicitly, while those nonlinear and coupled terms are treated explicitly by using any available values from previous time or iteration step. The implicit–explicit discretization leads to a linearized and decoupled algebraic system, which is solved using the fixed‐point iteration method. Upon the convergence of the iterative method, fully nonlinear coupled solutions are obtained. Also explored in this paper is the special way of treating traction boundary in finite volume method compared with FEM. Finally, three numerical test cases are simulated to verify the implementation procedure. It is shown in the simulation results that the implemented solver is capable of and efficient at predicting reasonable soil responses with pore pressure coupling under different loading situations. Copyright © 2015 John Wiley & Sons, Ltd.  相似文献   

12.
The propagation characteristic of Rayleigh waves in a fluid‐saturated non‐homogeneous poroelastic half‐plane is addressed. Based on Biot's theory for fluid‐saturated media, which takes the inertia, fluid viscosity, mechanical coupling, compressibility of solid grains, and fluid into account, the dispersion equations of Rayleigh waves in fluid‐saturated non‐homogeneous soils/rocks are established. By considering the shear modulus of solid skeleton variation with depth exponentially, a small parameter, which reflects the relative change of shear modulus, is introduced. The asymptotic solution of the dispersion equation expressing the relationship between the phase velocity and wave number is obtained by using the perturbation method. In order to analyze the effects of non‐homogeneity on the propagation characteristic of Rayleigh waves, the variation of the phase velocity with the wave number is presented graphically and discussed through numerical examples. Copyright © 2016 John Wiley & Sons, Ltd.  相似文献   

13.
The motions of fluid and solid phases in saturated porous media are coupled by inertial, viscous and mechanical interactions as described by Biot's equations. A one-dimensional exact analytical solution of the Biot's equations for the completely general solution of the transient problem in saturated, linear, elastic, porous media is presented. The problem is solved by using the Fourier series. The transient response of porous media is shown for typical material properties of a natural granular deposit and for different degrees of viscous coupling. The analytical results show the mechanics of dispersive wave propagation in saturated porous media and they should provide a useful comparison term for the existing numerical solution methods.  相似文献   

14.
An analytical solution is presented in this paper to study the time‐dependent settlement behaviour of a rigid foundation resting on a transversely isotropic saturated soil layer. The governing equations for a transversely isotropic saturated soil, within Biot's poroelasticity framework, are solved by means of Laplace and Hankel transforms. The problem is subsequently formulated in the Laplace transform domain in terms of a set of dual integral equations that are further reduced to a Fredholm integral equation of the second kind and solved numerically. The developed analytical solution is validated via comparison with the existing analytical solution for an isotropic saturated soil case, and adopted as a benchmark to examine the sensitivities of the mesh refinement and the locations of truncation boundaries in the finite element simulations using ABAQUS. Particular attention is paid to the influences of the degree of soil anisotropy, boundary drainage condition, and the soil layer thickness on the consolidation settlement and contact stress of the rigid foundation. Copyright © 2009 John Wiley & Sons, Ltd.  相似文献   

15.
This paper presents a novel analytical solution to the transient, z‐dependent, and asymmetric problem of an infinite wellbore drilled into a fluid‐saturated porous medium. The formulations are based on Biot's linear theory of poroelasticity, in which the dependency of poroelastic field variables to spatial coordinates as well as time domain is considered in the most general form. This gives flexibility to the solution in cases that cannot be analyzed using the conventional plane strain or symmetric models. One such case is when calculating the stress variations around an inclined wellbore where the far‐field stresses are acting over a finite vertical section. The results of our solution to this case with a three‐dimensional state of far‐field stress are used to analyze the stability of inclined wellbores passing through abnormally stressed formations. The presented solution is capable of finding expressions for fundamental solutions with stress or flow boundary conditions at the wellbore. These solutions are here adopted to analyze the pressure disturbances generated by multiprobe formation tester, a standard wireline device that is designed for downhole fluid sampling as well as estimating the directional permeabilities of subsurface earth formations. A comparison with the conventional solution for the relevant pressure diffusion equation indicates that the poroelastic effect is relatively significant in relation to the transient response of the pore pressure. Further, it is shown that the finite dimensions of sink probe would, to a great extent, contribute to the formation's pore pressure variations at its immediate proximity. Copyright © 2013 John Wiley & Sons, Ltd.  相似文献   

16.
This paper presents an analytical solution for the lateral dynamic response of a pipe pile in a saturated soil layer. The wave propagations in the saturated soil and the pipe pile are simulated by Biot's three‐dimensional poroelastic theory and one‐dimensional elastic theory, respectively. The governing equations of soil are solved directly without introducing potential functions. The displacement response and dynamic impedances of the pipe pile are obtained based on the continuous conditions between the pipe pile and both the outer and inner soil. A comparison with an existing solution is performed to verify the proposed solution. Selected numerical results for the lateral dynamic responses and impedances of the pipe pile are presented to reveal the lateral vibration characteristics of the pile‐soil system. Copyright © 2015 John Wiley & Sons, Ltd.  相似文献   

17.
In this paper a bifurcation analysis of boreholes in deep rock formations under uniform stress at infinity is presented. Rock is described by the constitutive equations of a deformation theory of plasticity for rigidplastic, incompressible and cohesive-frictional material. The corresponding bifurcation problem is solved numerically by the finite element method. The evidence gained from the numerical solution is utilized to establish a simplified borehole stability analysis that combines Biot's hodograph method with a surface instability analysis.  相似文献   

18.
In this paper, we used a theoretical model for the variation of Eulerian porosity, which takes into account the adsorption process known to be the main mechanism of production or sequestration of gas in many reservoir of coal. This process is classically modeled using Langmuir's isotherm. After implementation in Code_Aster, a fully coupled thermo‐hydro‐mechanical analysis code for structures calculations, we used numerical simulations to investigate the influence of coal's hydro‐mechanical properties (Biot's coefficient, bulk modulus), Langmuir's adsorption parameters, and the initial liquid pressure in rock mass during CO2 injection in coal. These simulations showed that the increase in the values of Langmuir's parameters and Biot's coefficient promotes a reduction in porosity because of the adsorption process when the gas pressure increases. Low values of bulk modulus increase the positive effect (i.e., increase) of hydro‐mechanical coupling on the porosity evolution. The presence of high initial liquid pressure in the rock mass prevents the progression of injected gas pressure when CO2 dissolution in water is taken into account. Copyright © 2014 John Wiley & Sons, Ltd.  相似文献   

19.
The dynamic response of a soft soil layer of finite thickness under the mutual effects of flow and periodical disturbance at the free surface is discussed in this work. The homogeneous water is governed by potential theory and the soil layer obeys Biot's theory of poroelasticity. The boundary‐value problem is solved by an analytical algorithm, in which the wave number is found first. Secondly, the closed form solutions are found by a two‐parameter perturbation method with the boundary‐layer correction. The results are also compared with those of the poroelastic soil layer of infinite thickness to show the impermeable rigid boundary effect. Copyright © 2003 John Wiley & Sons, Ltd.  相似文献   

20.
After describing the kinematics of a generic rigid block subjected to large rotations and displacements, the Udwadia's General Principle of Mechanics is applied to the dynamics of a rigid block with frictional constraints to show that the reaction forces and moments are indeterminate. Thus, the paper presents an incremental‐iterative algorithm for analysing general failure modes of rock blocks subject to generic forces, including non‐conservative forces such as water forces. Consistent stiffness matrices have been developed that fully exploit the quadratic convergence of the adopted Newton–Raphson iterative scheme. The algorithm takes into account large block displacements and rotations, which together with non‐conservative forces make the stiffness matrix non‐symmetric. Also included in the algorithm are in situ stress and fracture dilatancy, which introduces non‐symmetric rank‐one modifications to the stiffness matrix. Progressive failure is captured by the algorithm, which has proven capable of detecting numerically challenging failure modes, such as rotations about only one point. Failure modes may originate from a limit point or from dynamic instability (divergence or flutter); equilibrium paths emanating from bifurcation points are followed by the algorithm. The algorithm identifies both static and dynamic failure modes. The calculation of the factor of safety comes with no overhead. Examples show the equilibrium path of a rock block that undergoes slumping failure must first pass through a bifurcation point, unless the block is laterally constrained. Rock blocks subjected to water forces (or other non‐conservative forces) may undergo flutter failure before reaching a limit point. Copyright © 2007 John Wiley & Sons, Ltd.  相似文献   

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