A general viscous‐spring transmitting boundary for dynamic analysis of saturated poroelastic media

A time‐domain viscous‐spring transmitting boundary is presented for transient dynamic analysis of saturated poroelastic media with linear elastic and isotropic properties. The u–U formulation of Biot equation in cylindrical coordinate is adopted in the derivation. By this general viscous‐spring boundary, the effective stress and pore fluid pressure on the truncated boundary of the computational area are replaced by a set of continuously distributed spring and dashpot elements, of which the parameters are defined assuming an infinite permeability and considering the two dilatational waves. Numerical examples demonstrate good absorption of both the two cylindrical dilatational waves by the proposed ‘drained’ boundary. For general two‐dimensional wave propagation problems, acceptable accuracy can still be achieved by setting the proposed boundary relatively far away from the scatter. Numerical comparison shows that the results obtained by using this boundary are more accurate for all permeability values than those by the traditional viscous‐spring or viscous boundaries established for u–U formulation. Copyright © 2015 John Wiley & Sons, Ltd.     

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.     

A continuum mechanics‐based framework for optimizing boundary and finite element meshes associated with underground excavations‐framework

Many field problems, from stress analysis, heat transfer to contaminant transport, deal with disturbances in a continuum caused by a ‘source’ (defined by its discrete geometry) and a ‘region of interest’ (where a solution is sought). Depending on the location of ‘regions of interest’ in relation to the ‘sources’, the level of geometric detail necessary to represent the ‘sources’ in a model can vary considerably. A practical application of stress analysis in mining is the evaluation of the effects of continuous excavation on the states of stress around mine openings. Labour intensive model preparation and lengthy computation coupled with the interpretation of analysis results can have considerable impact on the successful operation of an underground mine, where stope failures can cost tens of millions of dollars and possibly lead to closure of the mine. A framework is proposed based on continuum mechanics principles to automatically optimize the level of geometric detail required for an analysis by simplifying the model geometry using expanded and modified algorithms that originated in computer graphics. This reduction in model size directly translates to savings in computational time. The results obtained from an optimized model have accuracy comparable to the uncertainty in input data (e.g. rock mass properties, geology, etc.). This first paper defines the optimization framework, while a companion paper investigates its efficiency and application to practical mining and excavation‐related problems. Copyright © 2005 John Wiley & Sons, Ltd.     

Non-linear analysis of geomechanical problems using coupled finite and infinite elements

In modeling of many geomechanics problems such as underground openings, soil-foundation structure interaction problems, and in wave propagation problems through semi-infinite soil medium the soil is represented as a region of either infinite or semi-infinite extent. Numerical modeling of such problems using conventional finite elements involves a truncation of the far field in which the infinite boundary is terminated at a finite distance. In these problems, appropriate boundary conditions are introduced to approximate the solution of the infinite or semi-infinite boundaries as closely as possible. However, the task of positioning the finite boundary in conventional finite element discretization and the definition of the boundary and its conditions is very delicate and depends on the modeller's skill and intuition. Moreover, such a choice is influenced by the size of the domain to be discretized. Consequently, the dimensions of the global matrices and the time required for solution of the problem will increase considerably and also selection of the arbitrary location of truncated boundary may lead to erroneous result. In order to over come these problems, mapped infinite elements have been developed by earlier researchers (Simoni and Schrefier, 1987). In the present work the applicability of infinite element technique is examined for different geomechanics problems. A computer program INFEMEP is developed based on the conventional finite element and mapped infinite element technique. It is then validated using selected problems such as strip footing and circular footing. CPU time taken to obtain solutions using finite element approach and infinite element approach was estimated and presented to show the capability of coupled modeling in improving the computational efficiency. Mesh configurations of different sizes were used to explore the enhancement of both computational economy and solution accuracy achieved by incorporation of infinite elements to solve elastic and elasto-plastic problems in semi-infinite/finite domain as applied to geotechnical engineering. © Rapid Science Ltd. 1998     

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。     

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.     

An artificial boundary approach for unbounded transient seepage problems

A new artificial boundary approach for transient seepage problems in unbounded domain is presented. The artificial boundary condition at the truncated boundary is derived from the analytical solutions for transient seepage problems in one dimension, including solutions, respectively, for flow in one‐dimensional infinite space and for radial flow in an infinite layer, and then it is tentatively applied for some two dimensional problems in addition to the one‐dimensional problems mentioned above. The boundary conditions derived relate the time‐dependent boundary flux with the time derivative of the hydraulic head at the truncated boundary, which makes the implementation much easier compared with the infinite element method. The accuracy and efficiency of the artificial boundary are validated by several numerical examples, which shows that the proposed boundary can give very good results for one‐dimensional transient seepage problems, as expected, whereas reasonable results can be also obtained for two‐dimensional problems, such as two‐dimensional axisymmetric flow and flow in an infinite plane. Copyright © 2014 John Wiley & Sons, Ltd.     

A novel finite element method for seepage analysis

A novel finite element method has been proposed in this paper for the solution of seepage problems economically and accurately. In this method the governing equation and the prescribed boundary conditions are transformed so that they refer to a suitable logarithmically condensed ‘image’ space; the physical problem domain is also mapped into the image space. The transformed equation is then solved in the image space using standard finite elements, subject to the transformed boundary conditions. Because physical space is logarithmically condensed in the image space, the proposed method is capable of dealing with large or very large aspect ratio seepage problems economically and accurately. The validity of the method has been demonstrated by means of a number of examples including anisotropy and non-linearity. In all cases an excellent degree of accuracy was achieved, efficiently and economically.     

The effect of Poisson's ratio and the far-field boundary conditions on the accuracy of finite element calculations

It is shown that a finite element calculation which approximates an ‘infinite medium’; problem by a mesh with finite boundaries will yield greater accuracy when stress boundary conditions are applied on the far-field boundary than is obtainable with displacement boundary conditions. In particular, with Poisson's ratio close to 0.5, the accuracy of the latter model is severely impaired, whereas the stress boundary condition model is unaffected for Poisson's ratio of 0.49 and a reasonable mesh. The eight-node quadratic isoparametric element displays superb accuracy for the axisymmetric thick cylinder with either type of boundary condition.     

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.     

Analysis of A New Model for Unsaturated Flow in Porous Media Including Hysteresis and Dynamic Effects

A new model for unsaturated flow in porous media, including capillary hysteresis and dynamic capillary effects, is analyzed. Existence and uniqueness of solutions are established and qualitative and quantitative properties of (particular) solutions are analyzed. Some results of numerical computations are given. The model under consideration incorporates simple ‘play’-type hysteresis and a dynamic term (time-derivative with respect to water content) in the capillary relation. Given an initial water content distribution, the model determines which parts of the flow domain are in drainage and which parts are in imbibition. The governing equations can be recast into an elliptic problem for fluid pressure and an evolution equation for water content. Standard methods are used to obtain numerical results. A comparison is given between J.R. Philip's semi-explicit similarity solution for horizontal redistribution in an infinite one-dimensional domain and solutions of the new model.     

Semi‐analytical elastostatic analysis of unbounded two‐dimensional domains

Unbounded plane stress and plane strain domains subjected to static loading undergo infinite displacements, even when the zero displacement boundary condition at infinity is enforced. However, the stress and strain fields are well behaved, and are of practical interest. This causes significant difficulty when analysis is attempted using displacement‐based numerical methods, such as the finite‐element method. To circumvent this difficulty problems of this nature are often changed subtly before analysis to limit the displacements to finite values. Such a process is unsatisfactory, as it distorts the solution in some way, and may lead to a stiffness matrix that is nearly singular. In this paper, the semi‐analytical scaled boundary finite‐element method is extended to permit the analysis of such problems without requiring any modification of the problem itself. This is possible because the governing differential equations are solved analytically in the radial direction. The displacement solutions so obtained include an infinite component, but relative motion between any two points in the unbounded domain can be computed accurately. No small arbitrary constants are introduced, no arbitrary truncation of the domain is performed, and no ill‐conditioned matrices are inverted. Copyright © 2002 John Wiley & Sons, Ltd.     

Infinite elements in a poroelastodynamic FEM

An infinite element is presented to treat wave propagation problems in unbounded saturated porous media. The porous media is modeled by Biot's theory. Conventional finite elements are used to model the near field, whereas infinite elements are used to represent the behavior of the far field. They are constructed in such a way that the Sommerfeld radiation condition is fulfilled, i.e. the waves decay with distance and are not reflected at infinity. To provide the wave information the infinite elements are formulated in Laplace domain. The time domain solution is obtained by using the convolution quadrature method as the inverse Laplace transformation. The temporal behavior of the near field is calculated using standard time integration schemes, e.g. the Newmark method. Finally, the near and far field are combined using a substructure technique for any time step. The accuracy as well as the necessity of the proposed infinite elements, when unbounded domains are considered, is demonstrated by different examples. Copyright © 2010 John Wiley & Sons, Ltd.     

Influence of construction technique on the performance of a braced excavation in marine clay

Finite element analyses are used to quantify the effects of construction technique on the performance of braced excavations. It is first shown that the computations can be continued to ‘collapse’ and that the results agree with limit plasticity solutions. A case study involving stratified deposits of marine clay and sand is used to carry out a Class C1 ‘prediction’ of field performance. The influence of construction technique on deformations and strut loads is shown. The results of the computations are in agreement with field observations, assuming a relatively ‘flexible’ construction technique. However, even if much ‘stiffer’ techniques had been used, large field deformations would have been unavoidable under certain circumstances.     

Incrementally non‐linear plasticity applied to rock joint modelling

Rock joint constitutive modelling is discussed through two new rock joint constitutive relations and a discrete numerical model. Regarding the constitutive relations, we emphasise the number of ‘tensorial zones’, that is, domains of constitutive incremental linearity; they involve four zones for the first (called ‘quadrilinear’) and an infinite number for the second one (called ‘incrementally nonlinear’). Using these formulations, a large class of loading paths can be considered. Hardening through shearing and relations between the normal and tangential directions of the joint (e.g., dilatancy) can be described. Their predictive abilities are checked. Plastic features are included even if the relations are defined outside the elasto‐plastic formalism. These relations obey, hence, the physical evidence as the plastic limit criterion and flow rule. The flow rule is nonassociated, and the corresponding link with the nonsymmetry of the constitutive matrix is examined. Comparisons between the two relations and the discrete numerical model, that is, a direct numerical simulation, which is fundamentally different, also are discussed within the context of infilled rock joints. Copyright © 2011 John Wiley & Sons, Ltd.     

A method of time-dependent analysis using elastic solutions for nonlinear materials

The formulation of viscoelastic solutions from elastic equations using the ‘correspondence principle’ and an inverse Laplace transform has been discussed extensively in the literature. Because this method has been developed, many time-dependent solutions can be obtained from closed form elastic solutions and conditions have been delineated in which the ‘quasi-elastic’ approximation of the viscoelastic solution is within acceptable tolerance. This communication shows the feasibility of the application of these methods to formulate approximate nonlinear viscoelastic solutions with nonlinear stress-strain materials, and for want of a specific nonlinear model to demonstrate this, the hyperbolic model was selected. The ‘power law’ is used to model the relaxation modulus of the viscoelastic materials. There are five related development that are discussed here using a simple numerical example to illustrate each of them and they are: (1) a linear elastic solution, (2) a linear viscoelastic solution, (3) a nonlinear elastic solution, (4) a nonlinear viscoelastic solution and finally, (5) a ‘regression’ approximation of the nonlinear viscoelastic solution which is suggested by the series form of the elastic solution. All of these are related to one another and each provides an acceptably accurate solution of the problem it addresses. The latter is of particular practical interest since it can be used to provide answers to problems involving nonlinear viscoelastic materials while requiring only very small calculation times. The problem used as an example is the calculation of the displacement of a circular hole in an infinite plate made of a material with a nonlinear time-dependent stress-strain relationship. The nonlinear elastic form of the solution was developed by matching results from nonlinear finite element analysis.     

Analytical solution of the harmonic waves diffraction by a cylindrical lined cavity in poroelastic saturated medium

This paper presents a model for the analysis of plane waves diffraction at a cavity in an infinite homogeneous poroelastic saturated medium, lined by a lining composed of four equal segments. An elastic boundary layer is placed between the cavity lining and the infinite porous medium. The boundary layer is simulated by ‘elastic boundary conditions’ in which the bulk matrix stress is proportional to the relative displacement between the lining and the surrounding medium matrix boundary. In addition, fluid impermeability through the intermediate layer is assumed. For the frequencies, that differ from the pseudoresonanse frequencies, the problem was reduced to the problem of an ideal elastic medium. A closed‐form analytical solution of the problem was obtained using Fourier–Bessel series, the convergence of which was proven. It was shown that the number of series terms required to obtain a desired level of accuracy can be determined in advance. The influence of the medium porosity on the medium dynamic stress concentration was studied. Copyright © 2006 John Wiley & Sons, Ltd.     

A New Viscous Boundary Condition in the Two-Dimensional Discontinuous Deformation Analysis Method for Wave Propagation Problems

Viscous boundaries are widely used in numerical simulations of wave propagation problems in rock mechanics and rock engineering. By using such boundaries, reflected waves from artificial boundaries can be eliminated; therefore, an infinite domain can be modeled as a finite domain more effectively and with a much greater accuracy. Little progress has been made, thus far, with the implementation and verification of a viscous boundary in the numerical, discrete element, discontinuous deformation analysis (DDA) method. We present in this paper a new viscous boundary condition for DDA with a higher absorbing efficiency in comparison to previously published solutions. The theoretical derivation of the new viscous boundary condition for DDA is presented in detail, starting from first principles. The accuracy of the new boundary condition is verified using a series of numerical benchmark tests. We show that the new viscous boundary condition works well with both P waves as well as S waves.     

A multiresolution strategy for solving landslides using the Particle Finite Element Method

We present an approach for the simulation of landslides using the Particle Finite Element Method of the second generation. In this work, the multiphase nature (granular phase and water) of the phenomenon is considered in a staggered fashion using a single, indeformable Finite Element mesh. A fractional step and a monolithic strategy are used for the water flow and granular phase, respectively. In this way, the maximum accuracy with minimal computational times is reached. The method is completed by adding the interaction terms due to drag and pressure forces, together with a moving mesh strategy to reduce the size of the computational domain.     

THE ‘NEW IMPLICIT METHOD’ FOR TUNNEL ANALYSIS

Tunnel excavation is a coupled three-dimensional problem dealing with two different structures: lining and rockmass. For a simple application it is useful to develop simplified methods by treating the problem as plane strain. If the problem of tunnel face advance presents an axisymmetric geometry, then we show that the major parameter governing the ground–interface–lining interaction is the convergence of the tunnel U₀ at the moment of the lining installation. The ‘New Implicit Method’ (NIM) presented in this paper makes use of principles similar to those of the ‘convergence–confinement’ method, but it provides a better appreciation of the coupled behaviour of rockmass and lining. For independent time constitutive laws (elasticity and plasticity), we point out that the convergence U₀ depends not only on the mechanical behaviour of the rockmass and on the distance from the tunnel face, as predicted by the ‘convergence–confinement’ method, but also on the stiffness of the lining previously set. We present the ‘NIM’ for elastic and perfect elastoplastic rockmasses without dilatancy for many criteria. The development of this new method is based on the results of tunnel calculations with an axisymmetric FEM numerical model that takes into account the three-dimensional aspect of the problem. Using this method is simple and its results agree well with the FEM numerical results. Its accuracy is highly satisfactory for a geotechnical study.