Biot's Consolidation Equations Research Articles

Effective block diagonal preconditioners for Biot's consolidation equations in piled‐raft foundations

SUMMARYThe finite element (FE) simulation of large‐scale soil–structure interaction problems (e.g. piled‐raft, tunnelling, and excavation) typically involves structural and geomaterials with significant differences in stiffness and permeability. The symmetric quasi‐minimal residual solver coupled with recently developed generalized Jacobi, modified symmetric successive over‐relaxation (MSSOR), or standard incomplete LU factorization (ILU) preconditioners can be ineffective for this class of problems. Inexact block diagonal preconditioners that are inexpensive approximations of the theoretical form are systematically evaluated for mitigating the coupled adverse effects because of such heterogeneous material properties (stiffness and permeability) and because of the percentage of the structural component in the system in piled‐raft foundations. Such mitigation led the proposed preconditioners to offer a significant saving in runtime (up to more than 10 times faster) in comparison with generalized Jacobi, modified symmetric successive over‐relaxation, and ILU preconditioners in simulating piled‐raft foundations. Copyright © 2012 John Wiley & Sons, Ltd.

Solute transport in partially-saturated deformable porous media: Application to a landfill clay liner

Based on the one-dimensional Biot consolidation equations, this paper developed an advection–diffusion equation that incorporates saturation, compressibility of the pore fluid and longitudinal dispersivity of the solute transport in an unsaturated, deforming porous medium. A simplified model was proposed for the case of a landfill liner. Numerical results demonstrated that the longitudinal dispersivity and compressibility of the pore fluid can be significant. Furthermore, the degree of soil saturation and loading rate of the waste surcharge affect significantly the contamination advective emission, namely the cumulative contaminant mass outflow per unit area from compacted clay liner (CCL) due to advective flow.

Two Dimensional Parametric Analysis of Time Variation Characteristics of Raft Foundation on Saturated Subsoil

For further discussing time variation characteristics of raft foundation on saturated subsoil due to consolidation, parameterized numerical analyses are completed based on Biot consolidation equations by developing elasto-plastic coupled FE model in plane strain state, together with adopting Mohr-Coulomb yield criterion. Computational results show that internal force and deformation of raft foundation display obvious non-monotonic features, which is associated with dissipation of excess pore water pressure. The disequilibrium of dissipation of excess pore water pressure displays more intensive between subsoil areas near raft center and the border due to less Poission’s ratio of subsoil, which leads to more dissipation time and regulation magnitude of distribution of stress and deformation in subsoil. When subsoil is of low permeability coefficients and Poission’s ratios, the non-monotonic features become more clearly and sustain with longer term variation characteristics.

An integrated model of wave-seabed-structure interactions

The phenomenon of wave-seabed-structure interactions have occurred in the marine environment with a coastal permeable (or partially permeable) structure, which has attracted great attention from coastal engineers involved in the design of marine structures. In this study, a porous seabed model based on Biot's consolidation equation and pore-elastic theory is integrated into the COBRAS wave-structure model. This integrated model is verified by the laboratory experiments. Some dominant factors affecting the wave motion and its induced seabed response are examined by using this model, including wave parameters and structure properties.

Two formulations for dynamic response of a cylindrical cavity in cross‐anisotropic porous media

AbstractTwo 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.

Wave scattering in cross-anisotropic porous media around the cavities and inclusions

A method is presented for evaluating dynamic response of a cylindrical cavity or inclusion in an infinite cross-anisotropic porous medium. The basis of the method is the 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. The stress, displacement, and pore pressure fields are evaluated in the vicinity of the cylinder in complex plane. Also the effects of many parameters ( k h / k v , E h / E v , G v / E v , ν hh ) on the responses are evaluated by numerical examples.

Comparison between iterative solution of symmetric and non‐symmetric forms of Biot's FEM equations using the generalized Jacobi preconditioner

AbstractFinite element discretization of Biot's consolidation equations can produce a symmetric indefinite system (commonly used in geomechanics) or a non‐symmetric system. While this difference appears to be minor, however, it will require the selection of entirely different Krylov subspace solvers with potentially significant impact on solution efficiency. The former is solved using the symmetric quasi‐minimal residual whereas the latter is solved using the popular bi‐conjugate gradient stabilized. This paper presents an extensive comparison of the symmetric and non‐symmetric forms by varying the time step, size of the spatial domain, choice of physical units, and left versus left–right preconditioning. The generalized Jacobi (GJ) preconditioner is able to handle the non‐symmetric version of Biot's finite element method equation, although there are no practical incentives to do so. The convergence behaviour of GJ‐preconditioned systems and its relation to the spectral condition number or the complete spectrum are studied to clarify the concept of ill‐conditioning within the context of iteration solvers. Copyright © 2007 John Wiley & Sons, Ltd.

State-space solution of seabed–geotextile systems subjected to cyclic wave loadings

Analytical solutions for dynamic responses of seabed–geotextile systems subjected to cyclic wave loadings are presented in this paper, which contains the solutions of the transient and harmonic responses. The theory is based on the Biot consolidation equations in which the pore fluid as well as the soil skeleton is considered compressible and the flow in the porous seabed is assumed governed by Darcy's law. The present analysis is completely based on the state-space formulations, which is very effective for laminated systems analysis. Together with Laplace–Fourier transform techniques, state-space methods are used to solve the governing equations. Responses of seabed–geotextile systems can be calculated by using the matrix theory, boundary conditions and inverting integral transform. As illustrative examples, laboratory experiments, which conducted at the Oregon State University Wave Research Facility in USA by McDougal [1981. Ocean wave–soil–geotextile interaction. Ph.D. Dissertation, Oregon State University], are analysed. It is shown that the numerical results are in good agreement with those obtained from laboratory experiments, and the distinction between the transient and harmonic response should be taken into account for design of marine geosynthetic systems. Under the transient condition, the seabed is apt to liquefy. Seabed stability may be increased by placing geotextile beneath an armour layer. The numerical evaluations of the solution in the seabed–geotextile systems can be easily achieved with high efficiency and accuracy.

A modified SSOR preconditioner for sparse symmetric indefinite linear systems of equations

AbstractThe standard SSOR preconditioner is ineffective for the iterative solution of the symmetric indefinite linear systems arising from finite element discretization of the Biot's consolidation equations. In this paper, a modified block SSOR preconditioner combined with the Eisenstat‐trick implementation is proposed. For actual implementation, a pointwise variant of this modified block SSOR preconditioner is highly recommended to obtain a compromise between simplicity and effectiveness. Numerical experiments show that the proposed modified SSOR preconditioned symmetric QMR solver can achieve faster convergence than several effective preconditioners published in the recent literature in terms of total runtime. Moreover, the proposed modified SSOR preconditioners can be generalized to non‐symmetric Biot's systems. Copyright © 2005 John Wiley & Sons, Ltd.

Block preconditioners for symmetric indefinite linear systems

AbstractThis paper presents a systematic theoretical and numerical evaluation of three common block preconditioners in a Krylov subspace method for solving symmetric indefinite linear systems. The focus is on large‐scale real world problems where block approximations are a practical necessity. The main illustration is the performance of the block diagonal, constrained, and lower triangular preconditioners over a range of block approximations for the symmetric indefinite system arising from large‐scale finite element discretization of Biot's consolidation equations. This system of equations is of fundamental importance to geomechanics. Numerical studies show that simple diagonal approximations to the (1,1) block K and inexpensive approximations to the Schur complement matrix S may not always produce the most spectacular time savings when K is explicitly available, but is able to deliver reasonably good results on a consistent basis. In addition, the block diagonal preconditioner with a negative (2,2) block appears to be reasonably competitive when compared to the more complicated ones. These observation are expected to remain valid for coefficient matrices whereby the (1,1) block is sparse, diagonally significant (a notion weaker than diagonal dominance), moderately well‐conditioned, and has a much larger block size than the (2,2) block. Copyright © 2004 John Wiley & Sons, Ltd.

An efficient diagonal preconditioner for finite element solution of Biot's consolidation equations

AbstractFinite element simulations of very large‐scale soil–structure interaction problems (e.g. excavations, tunnelling, pile‐rafts, etc.) typically involve the solution of a very large, ill‐conditioned, and indefinite Biot system of equations. The traditional preconditioned conjugate gradient solver coupled with the standard Jacobi (SJ) preconditioner can be very inefficient for this class of problems. This paper presents a robust generalized Jacobi (GJ) preconditioner that is extremely effective for solving very large‐scale Biot's finite element equations using the symmetric quasi‐minimal residual method. The GJ preconditioner can be formed, inverted, and implemented within an ‘element‐by‐element’ framework as readily as the SJ preconditioner. It was derived as a diagonal approximation to a theoretical form, which can be proven mathematically to possess an attractive eigenvalue clustering property. The effectiveness of the GJ preconditioner over a wide range of soil stiffness and permeability was demonstrated numerically using a simple three‐dimensional footing problem. This paper casts a new perspective on the potentialities of the simple diagonal preconditioner, which has been commonly perceived as being useful only in situations where it can serve as an approximate inverse to a diagonally dominant coefficient matrix. Copyright © 2002 John Wiley & Sons, Ltd.

A modified Jacobi preconditioner for solving ill‐conditioned Biot's consolidation equations using symmetric quasi‐minimal residual method

A modified Jacobi preconditioner for solving ill‐conditioned Biot's consolidation equations using symmetric quasi‐minimal residual method

A modified Jacobi preconditioner for solving ill‐conditioned Biot's consolidation equations using symmetric quasi‐minimal residual method

AbstractThis 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.

Solutions of pore pressure build up due to progressive waves

AbstractThe analytical solution of soil pore pressure accumulations due to a progressive wave is examined in detail. First of all, the errors contained in a published analytical solution for wave‐induced pore pressure accumulation are addressed, and the correct solution is presented in a more general form. The behaviour of the solution under different soil conditions is then investigated. It is found that the solution for deep soil conditions is sensitive to the soil shear stress in the top thin layer of the soil. However the solution is significantly influenced by the shear stress in the thin layer of soil near the impermeable base, for shallow and finite depth soil conditions. It is also found that a small error in the soil shear stress can lead to a large error in the accumulated pore pressure. An error analysis reveals the relationships between the accuracy of the pore pressure accumulation and the accuracy of the soil shear stress. A numerical solution to the simplified Biot consolidation equation is also developed. It is shown that the error analysis is of significant value for the numerical modelling of pore pressure buildup in marine soils. Both analytical and numerical examples are given to validate the error estimation method proposed in the present paper. Copyright © 2001 John Wiley & Sons, Ltd.

Von Neumann stability analysis of Biot's general two-dimensional theory of consolidation

Von Neumann stability analysis is performed for a Galerkin finite element formulation of Biot's consolidation equations on two-dimensional bilinear elements. Two dimensionless groups—the Time Factor and Void Factor—are identified and these quantities, along with the time-integration weighting, are used to explore the stability implications for variations in physical property and discretization parameters. The results show that the presence and persistence of stable spurious oscillations in the pore pressure are influenced by the ratio of time-step size to the square of the space-step for fixed time-integration weightings and physical property selections. In general, increasing the time-step or decreasing the mesh spacing has a smoothing effect on the discrete solution, however, special cases exist that violate this generality which can be readily identified through the Von Neumann approach. The analysis also reveals that explicitly dominated schemes are not stable for saturated media and only become possible through a decoupling of the equilibrium and continuity equations. In the case of unsaturated media, a break down in the Von Neumann results has been shown to occur due to the influence of boundary conditions on stability. © 1998 John Wiley & Sons, Ltd.

Wave-induced porewater pressures and effective stresses around breakwater

A numerical model is developed to examine a global aspect of wave-induced porewater pressures and effective stresses in a sand seabed as well as those in a rubble mound foundation of a composite caisson-type breakwater and in rubble mound breakwater, where the Biot's consolidation equations are employed. We perform the calculation for the compositev caisson-type breakwater and the rubble mound breakwater under the action of standing waves. From the calculated results the following are summarized. (1) When the permeability of the rubble mound foundation under the caisson of the composite breakwater is large, the excess porewater pressures penetrate instantly, and the time variation of uplift force at any position is in phase with the wave pressure variation at the front toe of the caisson. When the permeability of the rubble mound foundation is small, there appears a phase difference in the uplift force variation depending on the location, and the shape of spatial distribution of uplift force at a given phase is not usually triangular. The larger the shear modulus of the mound foundation is, the smaller the effective stresses are in the mound foundation and in the sand layer. (2) The internal seepage flow in the rubble mound breakwater promotes the flow induced by waves as wave run-up and downrush. At the phases when the internal seepage flow moves downward, the solids are moved upward. In order to reduce the solid displacements, a compacted core is required. The wave-induced excess porewater pressures decrease exponentially toward the inside of the rubble mound breakwater. The wave-induced vertical effective stresses become as large as the wave pressure of the incident wave, but the horizontal effective stresses and the shear stresses are small.

APPLICABILITY OF SOLUTIONS FOR TRANSIENT WAVE-INDUCED POREWATER PRESSURES IN SEABED AND LIQUEFACTION CONDITIONS OF SEABED

There exist three solutions for the transient wave-induced porewater pressures in seabed, the seepage flow solution, the exact solution and the boundary region approximation solution of Biot's consolidation equations. Applicability of three solutions is shown graphically in terms of non-dimensional parameter for surf zone conditions. One of the important parameters is the ratio of the shear modulus of solid skelton to the effective bulk modulus of porewater. Another is a non-dimensional quantity proportional to the permeability coefficient and the shear modulus. As the former becomes large and the latter becomes small, the seepage flow solution can not be used but the boundary region approximation solution becomes applicable. Under such situation, the possibility of the vertical effective stress of solid skelton being zero under the wave trough increases. The maximum depth of zero stress is about one half of the wave height.

Analytical Approaches for Biot's Consolidation Equations Considering Material Non-Linearities by Finite Element Methods

Analytical Approaches for Biot's Consolidation Equations Considering Material Non-Linearities by Finite Element Methods

Geotextile stabilization of seabeds: theory

Geotextiles are synthetic fabrics which may be substituted for graded aggregate to protect ocean and coastal structures from erosion and soil instability. Geotextiles are commonly used as a structural membrane and as a filter between an undisturbed sediment surface below and an erosion resistant coarse aggregate placed above. The dynamic response of a horizontally-layered soil system including a geotextile is examined analytically using the quasi-static Biot consolidation equations, the storage equation, and modelling the geotextile as a permeable elastic membrane. The influence of geotextile properties on soil stability are found to be rather small. The coupling of soil layer properties has a significant influence on soil stability. Maximum failure potential exists for specific combinations of the relative permeabilities and thicknesses of the soils.

The application of the finite element method to solving Biot's consolidation equation

Biot's theory of consolidation of saturated soil, regards the consolidation process as a coupling problem between stress of elastic body and flow of fluid existing in pores[1]. It can more precisely reflect the mechanism of consolidation than Terzhigi's theory[2]. In this article, we obtain the general Biot's finite element equations of consolidation with classical variational principles. The equations have clear physical meaning and have been applied to analysing the consolidation of Bajiazui earth dam. The computational results are in accord with engineering practice.