Andrew J. Abbo

Refined explicit integration of elastoplastic models with automatic error control

Effective explicit algorithms for integrating complex elastoplastic constitutive models, such as those belonging to the Cam clay family, are described. These automatically divide the applied strain increment into subincrements using an estimate of the local error and attempt to control the global integration error in the stresses. For a given scheme, the number of substeps used is a function of the error tolerance specified, the magnitude of the imposed strain increment, and the non‐linearity of the constitutive relations. The algorithms build on the work of Sloan in 1987 but include a number of important enhancements. The steps required to implement the integration schemes are described in detail and results are presented for a rigid footing resting on a layer of Tresca, Mohr‐Coulomb, modified Cam clay and generalised Cam clay soil. Explicit methods with automatic substepping and error control are shown to be reliable and efficient for these models. Moreover, for a given load path, they are able to control the global integration error in the stresses to lie near a specified tolerance. The methods described can be used for exceedingly complex constitutive laws, including those with a non‐linear elastic response inside the yield surface. This is because most of the code required to program them is independent of the precise form of the stress‐strain relations. In contrast, most of the implicit methods, such as the backward Euler return scheme, are difficult to implement for all but the simplest soil models.

A smooth hyperbolic approximation to the Mohr-Coulomb yield criterion

Abstract The Mohr-Coulomb yield criterion is used widely in elastoplastic geotechnical analysis. There are computational difficulties with this model, however, due to the gradient discontinuities which occur at both the edges and the tip of the hexagonal yield surface pyramid. It is well known that these singularities often cause stress integration schemes to perform inefficiently or fail. This paper describes a simple hyperbolic yield surface that eliminates the singular tip from the Mohr-Coulomb surface. The hyperbolic surface can be generalized to a family of Mohr-Coulomb yield criteria which are also rounded in the octahedral plane, thus eliminating the singularities that occur at the edge intersections as well. This type of yield surface is both continuous and differentiable at all stress states, and can be made to approximate the Mohr-Coulomb yield function as closely as required by adjusting two parameters. The yield surface and its gradients are presented in a form which is suitable for finite element programming with either explicit or implicit stress integration schemes. Two efficient FORTRAN 77 subroutines are given to illustrate how the new yield surface can be implemented in practice.

AN AUTOMATIC LOAD STEPPING ALGORITHM WITH ERROR CONTROL

SUMMARY This paper presents an algorithm for controlling the error in non-linear finite element analysis which is caused by the use of finite load steps. In contrast to most recent schemes, the proposed technique is non-iterative and treats the governing load-deflection relations as a system of ordinary differential equations. This permits the governing equations to be integrated adaptively where the step size is controlled by monitoring the local truncation error. The latter is measured by computing the difference between two estimates of the displacement increments for each load step, with the initial estimate being found from the first-order Euler scheme and the improved estimate being found from the second-order modified Euler scheme. If the local truncation error exceeds a specified tolerance, then the load step is abandoned and the integration is repeated with a smaller load step whose size is found by local extrapolation. Local extrapolation is also used to predict the size of the next load step following a successful update. In order to control not only the local load path error, but also the global load path error, the proposed scheme incorporates a correction for the unbalanced forces. Overall, the cost of the automatic error control is modest since it requires only one additional equation solution for each successful load step. Because the solution scheme is non-iterative and founded on successful techniques for integrating systems of ordinary differential equations, it is particularly robust. To illustrate the ability of the scheme to constrain the load path error to lie near a desired tolerance, detailed results are presented for a variety of elastoplastic boundary value problems.

Efficiency of High-Order Elements in Large-Deformation Problems of Geomechanics

AbstractThis paper investigates the application of high-order elements within the framework of the arbitrary Lagrangian-Eulerian method for the analysis of elastoplastic problems involving large deformations. The governing equations of the method as well as its important aspects such as the nodal stress recovery and the remapping of state variables are discussed. The efficiency and accuracy of 6-, 10-, 15-, and 21-noded triangular elements are compared for the analysis of two geotechnical engineering problems, namely, the behavior of an undrained layer of soil under a strip footing subjected to large deformations and the soil behavior in a biaxial test. The use of high-order elements is shown to increase the accuracy of the numerical results and to significantly decrease the computational time required to achieve a specific level of accuracy. For problems considered in this study, the 21-noded elements outperform other triangular elements.

Undrained Stability of Dual Circular Tunnels

In this paper,numericallimit analysis and semianalytical rigid blocktechniques are used toinvestigate the effectof the tunnelspac- ingonthestabilityoftwocirculartunnelsexcavatedsidebyside.Thetunnelsaremodeledunderplane-strainconditions,whichimpliesthatthey are assumed to be infinitely long. Bounds on the stability of the tunnels are obtained using finite-element limit analysis, the numerical formu- lation of which is based on the upper and lower bounds theorems of classical plasticity. Solutions are obtained using advanced conic program- mingschemes tosolvethe resultingoptimization problems,and upperand lowerboundestimates on thestabilityof thetunnels areobtained for arangeofgeometries.Thesebounds,whichbracketthetruecollapseloadfromaboveandbelow,arefoundtodifferbyatmost5%forthecases where the solution does not approach zero. Results from this study are summarized in stability charts for use by practitioners. DOI: 10.1061/ (ASCE)GM.1943-5622.0000288.

Biot consolidation analysis with automatic time stepping and error control. Part 2 : Applications

The automatic time-stepping algorithms developed in a companion paper are used to study the behaviour of several problems involving the consolidation of porous media. The aim of these analyses is to demonstrate that the new procedures are robust, efficient, and can control the global temporal discretization (or time-stepping) error in the displacements to lie near a prescribed error tolerance. Copyright

Applications of adaptive time stepping in analysis of biot consolidation

The accuracy of finite element solutions for the consolidation of porous media is influenced by the number and size of the time increments used in the analysis. A solution algorithm for adaptively selecting time increments for the solution of elastic and elastoplastic coupled consolidation problems in finite element analysis has been developed by Sloan and Abbo (1999). By treating the governing consolidation relations as a system of 1st-order differential equations their algorithm utilized subincrementation to automatically adjust the size of time increments used in the analysis. Unlike other time stepping schemes, the procedure adjusts the time increments in order to control the error due to time stepping to lie near a specified tolerance. The algorithm was shown to be robust and to provide an efficient method for the solution of consolidation problems. In this paper the efficiency of the algorithm is further demonstrated through the analysis of the construction of an embankment on a deep layer of soft soil. The time increments required for the efficient and accurate analysis of the consolidation of porous media are shown to differ by orders of magnitude.