Performance of displacement finite elements for modelling incompressible materials

It is well established that severe numerical difficulties may arise when the displacement finite element method is used to analyse the behaviour of incompressible solids and this is particularly true for axisymmetric problems. These numerical difficulties are caused by excessive kinematic constraints and are reflected by strong oscillations in the calculated stress distribution and overestimations of collapse loads. The purpose of this paper is to present new displacement finite element formulations which are particularly suitable for axisymmetric analysis of incompressible materials. A direct comparison is made of the performance of various displacement finite elements in the analysis of elastic or plastic incompressible materials under axisymmetric loading conditions. Particular attention is focused on the performance of various axisymmetric displacement elements in predicting the stress field of incompressible or nearly incompressible materials. Copyright © 2000 John Wiley & Sons, Ltd.     

Numerical manifold method for dynamic nonlinear analysis of saturated porous media

It is well known that the Babuska–Brezzi stability criterion or the Zienkiewicz–Taylor patch test precludes the use of the finite elements with the same low order of interpolation for displacement and pore pressure in the nearly incompressible and undrained cases, unless some stabilization techniques are introduced for dynamic analysis of saturated porous medium where coupling occurs between the displacement of solid skeleton and pore pressure. The numerical manifold method (NMM), where the interpolation of displacement and pressure can be determined independently in an element for the solution of u–p formulation, is derived based on triangular mesh for the requirement of high accurate calculations from practical applications in the dynamic analysis of saturated porous materials. The matrices of equilibrium equations for the second‐order displacement and the first‐order pressure manifold method are given in detail for program coding. By close comparison with widely used finite element method, the NMM presents good stability for the coupling problems, particularly in the nearly incompressible and undrained cases. Numerical examples are given to illustrate the validity and stability of the manifold element developed. Copyright © 2006 John Wiley & Sons, Ltd.     

Softening,localization and stabilization: capture of discontinuous solutions in J2 plasticity

This paper exploits the concept of stabilization techniques to improve the behaviour of mixed linear/linear simplicial elements (triangles and tetrahedra) in incompressible or nearly incompressible situations. Elasto-J2-plastic constitutive behaviour has been considered with linear and exponential softening. Two different stabilization methods are used to attain global stability of the corresponding discrete finite element formulation. Implementation and computational aspects are also discussed, showing that a robust application of the proposed formulation is feasible. Numerical examples show that the formulation derived is free of volumetric locking and spurious oscillations of the pressure. The results obtained do not suffer from spurious mesh-size or mesh-bias dependence, comparing very favourably with those obtained with the standard, non-stabilized, approaches. Copyright © 2004 John Wiley & Sons, Ltd.     

Explanation of locking of four‐node plane element by considering it as elastic Dirichlet‐type boundary value problem

The aim of this study is to arrive at a better understanding of the phenomenon of locking of low‐order compatible displacement type of finite elements in particular for the hour‐glass mode of the plane four‐node element and dilative materials. To this end the properties of finite elements are investigated in an analytical way, where a finite element is considered as a plane boundary value problem with prescribed boundary displacement (Dirichlet problem). In this paper for the sake of simplicity the simplest possible linear comparison solid, namely isotropic linear elasticity, is applied, although recognizing fully that for a dilative material elasto‐plasticity would be more realistic. From the study described in this paper it is concluded that locking of the four‐node element is not due to any particular numerical formulation of this compatible finite element since, even the analytical solution suffers from this problem. The locking of this element is not related to incompressibility of the material either as the analytical solution shows locking to occur at a parameter set which differs significantly from the one in case of incompressibility. It is shown that locking is a consequence of the combination of the dilative material behaviour and the compatible displacement type of boundary conditions, which leads to infinite isotropic stresses in the element. These infinite isotropic stresses occur at the limit of uniqueness of the solution, which for this element is shown to occur outside the parameter range of the sufficiency of uniqueness. Copyright © 2000 John Wiley & Sons, Ltd.     

Fast iterative solution of large undrained soil‐structure interaction problems

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.     

Performance of Jacobi preconditioning in Krylov subspace solution of finite element equations

This paper examines the performance of the Jacobi preconditioner when used with two Krylov subspace iterative methods. The number of iterations needed for convergence was shown to be different for drained, undrained and consolidation problems, even for similar condition number. The differences were due to differences in the eigenvalue distribution, which cannot be completely described by the condition number alone. For drained problems involving large stiffness ratios between different material zones, ill‐conditioning is caused by these large stiffness ratios. Since Jacobi preconditioning operates on degrees‐of‐freedom, it effectively homogenizes the different spatial sub‐domains. The undrained problem, modelled as a nearly incompressible problem, is much more resistant to Jacobi preconditioning, because its ill‐conditioning arises from the large stiffness ratios between volumetric and distortional deformational modes, many of which involve the similar spatial domains or sub‐domains. The consolidation problem has two sets of degrees‐of‐freedom, namely displacement and pore pressure. Some of the eigenvalues are displacement dominated whereas others are excess pore pressure dominated. Jacobi preconditioning compresses the displacement‐dominated eigenvalues in a similar manner as the drained problem, but pore‐pressure‐dominated eigenvalues are often over‐scaled. Convergence can be accelerated if this over‐scaling is recognized and corrected for. Copyright © 2002 John Wiley & Sons, Ltd.     

Low-order stabilized finite element for the full Biot formulation in soil mechanics at finite strain

This article presents a novel finite element formulation for the Biot equation using low-order elements. Additionally, an extra degree of freedom is introduced to treat the volumetric locking steaming from the effective response of the medium; its balance equation is also stabilized. The accuracy of the proposed formulation is demonstrated by means of numerical analyses.     

Bearing capacity factors of circular foundations for a general c–ϕ soil using lower bound finite elements limit analysis

By using the lower bound limit analysis in conjunction with finite elements and linear programming, the bearing capacity factors due to cohesion, surcharge and unit weight, respectively, have been computed for a circular footing with different values of ?. The recent axisymmetric formulation proposed by the authors under ?=0 condition, which is based on the concept that the magnitude of the hoop stress (σ_θ) remains closer to the least compressive normal stress (σ₃), is extended for a general c–? soil. The computational results are found to compare quite well with the available numerical results from literature. It is expected that the study will be useful for solving various axisymmetric geotechnical stability problems. Copyright © 2010 John Wiley & Sons, Ltd.     

Numerical simulation of undrained insertion problems in geotechnical engineering with the Particle Finite Element Method (PFEM)

The paper presents total-stress numerical analyses of large-displacement soil-structure interaction problems in geomechanics using the Particle Finite Element Method (PFEM). This method is characterized by frequent remeshing and the use of low order finite elements to evaluate the solution. Several important features of the method are: (i) a mixed formulation (displacement-mean pressure) stabilized numerically to alleviate the volumetric locking effects that are characteristic of low order elements when the medium is incompressible, (ii) a penalty method to prescribe the contact constraints between a rigid body and a deformable media combined with an implicit scheme to solve the tangential contact constraint, (iii) an explicit algorithm with adaptive substepping and correction of the yield surface drift to integrate the finite-strain multiplicative elasto-plastic constitutive relationship, and (iv) the mapping schemes to transfer information between successive discretizations. The performance of the method is demonstrated by several numerical examples, of increasing complexity, ranging from the insertion of a rigid strip footing to a rough cone penetration test. It is shown that the proposed method requires fewer computational resources than other numerical approaches addressing the same type of problems.     

Two‐phase dynamic analysis by material point method

This paper extends the material point method to analyze coupled dynamic, two‐phase boundary‐valued problems via a velocity formulation, in which solid and fluid phase velocities are the variables. Key components of the proposed approach are the adoption of Verruijt's sequence of update steps when integrating over time and the enhancement of volumetric strains. The connection between fractional step method and the time‐stepping algorithm presented in this paper is addressed. Enhancement of volumetric strains allows lower order variations in pressure and mitigates spurious pressure fields and locking that plague low‐order finite‐element implementations. A stress averaging technique to smoothen stress variations is proposed, and the local damping procedure adopted by FLAC is extended to handle two‐phase problems. Special Kelvin‐Voigt boundaries are developed to suppress reflections at artificial boundaries. Idealized examples are presented to demonstrate the capability of the proposed framework to accurately capture the physics of wave propagation, consolidation and wave attack on a sea dike. Copyright © 2012 John Wiley & Sons, Ltd.     

u‐p semi‐Lagrangian reproducing kernel formulation for landslide modeling

This paper presents a u‐p (displacement‐pressure) semi‐Lagrangian reproducing kernel (RK) formulation to effectively analyze landslide processes. The semi‐Lagrangian RK approximation is constructed based on Lagrangian discretization points with fixed kernel supports in the current configuration. As a result, it tracks state variables at discretization points while allowing extreme deformation and material separation that is beyond the capability of Lagrangian formulations. The u‐p formulation following Biot theory is incorporated into the formulation to describe poromechanics of saturated geomaterials. In addition, a stabilized nodal integration method to ensure stability of the domain integration and kernel contact algorithms to model contact between bodies are introduced in the u‐p semi‐Lagrangian RK formulation. The proposed method is verified with several numerical examples and validated with an experimental result and the field data of an actual landslide.     

Relation between mean stress,thermodynamic, and lithostatic pressure

Pressure is one of the most important parameters to be quantified in geological problems. However, in metamorphic systems the pressure is usually calculated with two different approaches. One pressure calculation is based on petrological phase equilibria and this pressure is often termed thermodynamic pressure. The other calculation is based on continuum mechanics, which provides a mean stress that is commonly used to estimate the thermodynamic pressure. Both thermodynamic pressure calculations can be justified by the accuracy and applicability of the results. Here, we consider systems with low‐differential stress (<1 kbar) and no irreversible volumetric deformation, and refer to them as conventional systems. We investigate the relationship between mean stress and thermodynamic pressure. We discuss the meaning of thermodynamic pressure and its calculation for irreversible processes such as viscous deformation and heat conduction, which exhibit entropy production. Moreover, it is demonstrated that the mean stress for incompressible viscous deformation is essentially equal to the mean stress for the corresponding viscous deformation with elastic compressibility, if the characteristic time of deformation is five times longer than the Maxwell viscoelastic relaxation time that is equal to the ratio of shear viscosity to bulk modulus. For typical lithospheric rocks, this Maxwell time is smaller than c. 10,000 years. Therefore, numerical simulations of long‐term (>10 kyr) geodynamic processes, employing incompressible deformation, provide mean stress values that are close to the mean‐stress value associated with elastic compressibility. Finally, we show that for conventional systems the mean stress is essentially equal to the thermodynamic pressure. However, mean stress and, hence, thermodynamic pressure can be significantly different from the lithostatic pressure.     

A FIC‐based stabilized mixed finite element method with equal order interpolation for solid–pore fluid interaction problems

A new mixed displacement‐pressure element for solving solid–pore fluid interaction problems is presented. In the resulting coupled system of equations, the balance of momentum equation remains unaltered, while the mass balance equation for the pore fluid is stabilized with the inclusion of higher‐order terms multiplied by arbitrary dimensions in space, following the finite calculus (FIC) procedure. The stabilized FIC‐FEM formulation can be applied to any kind of interpolation for the displacements and the pressure, but in this work, we have used linear elements of equal order interpolation for both set of unknowns. Examples in 2D and 3D are presented to illustrate the accuracy of the stabilized formulation for solid–pore fluid interaction problems. Copyright © 2016 John Wiley & Sons, Ltd.     

Hybrid time integration and coupled solution methods for nonlinear finite element analysis of partially saturated deformable porous media at small strain

The goal of the paper is to determine the most efficient, yet accurate and stable, finite element nonlinear solution method for analysis of partially saturated deformable porous media at small strain. This involves a comparison between fully implicit, semi‐implicit, and explicit time integration schemes, with monolithically coupled and staggered‐coupled nonlinear solution methods and the hybrid combination thereof. The pore air pressure p_a is assumed atmospheric, that is, p_a=0 at reference pressure. The solid skeleton is assumed to be pressure‐sensitive nonlinear isotropic elastic. Coupled partially saturated ‘consolidation’ in the presence of surface infiltration and traction is simulated for a simple one‐dimensional uniaxial strain example and a more complicated plane strain slope example with gravity loading. Three mixed plane strain quadrilateral elements are considered: (i) Q4P4; (ii) stabilized Q4P4S; and (iii) Q9P4; “Q” refers to the number of solid skeleton displacement nodes, and “P” refers to the number of pore fluid pressure nodes. The verification of the implementation against an analytical solution for partially saturated pore water flow (no solid skeleton deformation) and comparison between the three time integration schemes (fully implicit, semi‐implicit, and explicit) are presented. It is observed that one of the staggered‐coupled semi‐implicit schemes (SIS(b)), combined with the fully implicit monolithically coupled scheme to resolve sharp transients, is the most efficient computationally. Copyright © 2015 John Wiley & Sons, Ltd.     

Propagation of a hydraulic fracture parallel to a free surface

This paper analyses the plane strain problem of a fracture, driven by injection of an incompressible viscous Newtonian fluid, which propagates parallel to the free surface of an elastic half‐plane. The problem is governed by a hyper‐singular integral equation, which relates crack opening to net pressure according to elasticity, and by the lubrication equations which describe the laminar fluid flow inside the fracture. The challenge in solving this problem results from the changing nature of the elasticity operator with growth of the fracture, and from the existence of a lag zone of a priori unknown length between the crack tip and the fluid front. Scaling of the governing equations indicates that the evolution problem depends in general on two numbers, one which can be interpreted as a dimensionless toughness and the other as a dimensionless confining stress. The numerical method adopted to solve this non‐linear evolution problem combines the displacement discontinuity method and a finite difference scheme on a fixed grid, together with a technique to track both crack and fluid fronts. It is shown that the solution evolves in time between two asymptotic similarity solutions. The small time asymptotic solution corresponding to the solution of a hydraulic fracture in an infinite medium under zero confining stress, and the large time to a solution where the aperture of the fracture is similar to the transverse deflection of a beam clamped at both ends and subjected to a uniformly distributed load. It is shown that the size of the lag decreases (to eventually vanish) with increasing toughness and compressive confining stress. Copyright © 2005 John Wiley & Sons, Ltd.     

A stabilized assumed deformation gradient finite element formulation for strongly coupled poromechanical simulations at finite strain

An adaptively stabilized finite element scheme is proposed for a strongly coupled hydro‐mechanical problem in fluid‐infiltrating porous solids at finite strain. We first present the derivation of the poromechanics model via mixture theory in large deformation. By exploiting assumed deformation gradient techniques, we develop a numerical procedure capable of simultaneously curing the multiple‐locking phenomena related to shear failure, incompressibility imposed by pore fluid, and/or incompressible solid skeleton and produce solutions that satisfy the inf‐sup condition. The template‐based generic programming and automatic differentiation (AD) techniques used to implement the stabilized model are also highlighted. Finally, numerical examples are given to show the versatility and efficiency of this model. Copyright © 2013 John Wiley & Sons, Ltd.     

Frictionless contact formulation for dynamic analysis of nonlinear saturated porous media based on the mortar method

A finite element algorithm for frictionless contact problems in a two‐phase saturated porous medium, considering finite deformation and inertia effects, has been formulated and implemented in a finite element programme. The mechanical behaviour of the saturated porous medium is predicted using mixture theory, which models the dynamic advection of fluids through a fully saturated porous solid matrix. The resulting mixed formulation predicts all field variables including the solid displacement, pore fluid pressure and Darcy velocity of the pore fluid. The contact constraints arising from the requirement for continuity of the contact traction, as well as the fluid flow across the contact interface, are enforced using a penalty approach that is regularised with an augmented Lagrangian method. The contact formulation is based on a mortar segment‐to‐segment scheme that allows the interpolation functions of the contact elements to be of order N. The main thrust of this paper is therefore how to deal with contact interfaces in problems that involve both dynamics and consolidation and possibly large deformations of porous media. The numerical algorithm is first verified using several illustrative examples. This algorithm is then employed to solve a pipe‐seabed interaction problem, involving large deformations and dynamic effects, and the results of the analysis are also compared with those obtained using a node‐to‐segment contact algorithm. The results of this study indicate that the proposed method is able to solve the highly nonlinear problem of dynamic soil–structure interaction when coupled with pore water pressures and Darcy velocity. Copyright © 2015 John Wiley & Sons, Ltd.     

A four‐node plane parametric element based on quadrilateral area coordinate and its application to coupled solid‐deformation/fluid‐flow simulation for porous geomaterials

A four‐node plane parametric element AQGβ6‐I is constructed on the basis of the quadrilateral area coordinate, the generalized conforming principle and the projection technique with a penalty factor β within an interval of 0–1. When β = 0, the element has excellent bending performance. When β = 1, the element can pass patch test strictly; its performance is as good as many famous elements. When β value is between 0 and 1, such as β = 0.5, the element can arrive at a compromise between (relatively) low sensitivity to mesh distortion and perfect convergence. The work provides an illuminating method to alleviate a difficult problem in finite element modelling using the four‐node quadrilateral element, which can pass the strict patch test, but has poor performance in bending dominated problem; on the contrary, it has excellent performance in bending dominated problem but cannot pass the strong patch test. The AQGβ6‐I with the convergence formulation (β = 1) is then applied to coupled solid‐deformation/fluid‐flow simulation for porous geomaterials. The computational examples are carried out to demonstrate that the AQGβ6‐I (β = 1) element is not only stable, reliable and efficient but also of high accuracy. The present study provides a good applicable element for finite element simulations of solid‐deformation/fluid‐flow for porous geomaterials. Copyright © 2014 John Wiley & Sons, Ltd.     

A method for frost jacking prediction of single pile in permafrost

In this paper, a model is proposed to simulate frost jacking performances of a pile foundation within an axisymmetric pile–soil system through a coupling strategy. We consider three diversified stages for frost heave of adjacent foundation soil below freezing point, where mathematical expressions for the volumetric strain are given in terms of volumetric ice content, negative temperature and porosity. A modified strain-softening model characterizing frozen soil–pile interactions is established based on experimental results, taking into account the effects of normal pressure, negative temperature and moisture content. The proposed computational approach is then illuminated and validated via the numerical example of a simplified bridge pile foundation under natural permafrost condition. Variation of temperature regime, volumetric ice content, displacement and stress over time is analyzed. This model can be further applied to evaluating effects of different countermeasures that mitigate frost jacking hazard of single pile subjected to cold climate.