High performance interior point methods for three-dimensional finite element limit analysis

The ability to obtain rigorous upper and lower bounds on collapse loads of various structures makes finite element limit analysis an attractive design tool. The increasingly high cost of computing those bounds, however, has limited its application on problems in three dimensions. This work reports on a high-performance homogeneous self-dual primal-dual interior point method developed for three-dimensional finite element limit analysis. This implementation achieves convergence times over 4.5× faster than the leading commercial solver across a set of three-dimensional finite element limit analysis test problems, making investigation of three dimensional limit loads viable. A comparison between a range of iterative linear solvers and direct methods used to determine the search direction is also provided, demonstrating the superiority of direct methods for this application. The components of the interior point solver considered include the elimination of and options for handling remaining free variables, multifrontal and supernodal Cholesky comparison for computing the search direction, differences between approximate minimum degree [1] and nested dissection [13] orderings, dealing with dense columns and fixed variables, and accelerating the linear system solver through parallelization. Each of these areas resulted in an improvement on at least one of the problems in the test set, with many achieving gains across the whole set. The serial implementation achieved runtime performance 1.7× faster than the commercial solver Mosek [5]. Compared with the parallel version of Mosek, the use of parallel BLAS routines in the supernodal solver saw a 1.9× speedup, and with a modified version of the GPU-enabled CHOLMOD [11] and a single NVIDIA Tesla K20c this speedup increased to 4.65×

Three dimensional lower bound solutions for the stability of plate anchors in sand

Soil anchors are commonly used as foundation systems for structures that require uplift or lateral resistance. These types of structures include transmission towers, sheet pile walls and buried pipelines. Although anchors are typically complex in shape (e.g. drag or helical anchors), many previous analyses idealise the anchor as a continuous strip under plane strain conditions. This assumption provides numerical advantages and the problem can solved in two dimensions. In contrast to recent numerical studies, this paper applies three dimensional numerical limit analysis and axi-symetrical displacement finite element analysis to evaluate the effect of anchor shape on the pullout capacity of horizontal anchors in sand. The anchor is idealised as either square or circular in shape. Results are presented in the familiar form of breakout factors based on various anchor shapes and embedment depths, and are also compared with existing numerical and empirical solutions

Computational limit analysis for anchors and retaining walls

The computation of the bearing capacity of engineering structures commonly relays on results obtained for simple academic examples. Recent developments in computational limit analysis have allowed engineers to compute bounds of the bearing capacity of arbitrary geometries. We here extend these formulations to problems with practical interest such as retaining walls, anchors, or excavations with particular interface conditions. These situations require the special treatment of the contact conditions between different materials, or the modelling of joints and anchors. We demonstrate the potential of the resulting tool with some practical examples

The stability of inclined plate anchors in purely cohesive soil

Soil anchors are commonly used as foundation systems for structures requiring uplift resistance such as transmission towers, or for structures requiring lateral resistance, such as sheet pile walls. To date the design of these anchors has been largely based on empiricism. This paper applies numerical limit analysis and displacement finite element analysis to evaluate the stability of inclined strip anchors in undrained clay. Results are presented in the familiar form of break-out factors based on various anchor geometries. By obtaining both upper and lower bound limit analysis estimates of the pullout capacity, the true pullout resistance can be bracketed from above and below. In addition, the displacement finite element solutions provide an opportunity to validate these findings thus providing a rigorous evaluation of anchor capacity

Mesh generation for lower bound limit analysis

This paper describes a general strategy for generating lower bound meshes in D-dimensions. The procedure is based on a parametric mapping technique, coupled with midpoint splitting of subdomains, and permits the user to control the distribution of the discontinuities and elements precisely. Although it is not fully automatic, the algorithm is fast and automatically generates extension zones for problems with semi-infinite domains

A comparative study of preconditioning techniques for large sparse systems arising in finite element limit analysis

The efficiency of several preconditioned Conjugate Gradient (PCG) schemes for solving of large sparse linear systems arising from application of interior point methods to nonlinear Finite Element Limit Analysis (FELA) is studied. Direct solvers fail to solve these linear systems in large sizes, such as large 2D and 3D problems, due to their high storage and computational cost. This motivates using iterative methods. However, iterative solvers are not efficient for difficult problems without preconditioning techniques. In this paper, the effect of various preconditioning techniques on the convergence behavior of the preconditioned Conjugate Gradient (PCG) is investigated through a detailed comparative study. Furthermore, numerical results of applying PCG to several sample systems are presented and discussed thoroughly in a parametric study. Our results suggest that while incomplete Cholesky preconditioners are by far the most efficient techniques for sequential computations, significant gains may result from use of sparse approximate inverse methods in parallel environment in this field

Iterative solution of large sparse linear systems arising from application of interior point method in computational geomechanics

The efficiency of several preconditioned Conjugate Gradient (PCG) schemes for solving of large sparse linear systems arising from application of second order cone programming in computational plasticity problems is studied. Direct solvers fail to solve these linear systems in large sizes, such as three dimensional cases, due to their high storage and computational cost. This motivates using iterative methods. However, iterative solvers are not efficient without preconditioning techniques for difficult problems. In this paper, the effect of different incomplete factorization preconditioning techniques on the convergence behavior of the preconditioned Conjugate Gradient (PCG) method to solve these large sparse and usually ill-conditioned linear systems is investigated. Furthermore, numerical results of applying PCG to several sample systems are presented and discussed. Several suggestions are also made as potential research subjects in this field

Undrained limiting lateral soil pressure on a row of piles

The displacement finite element, lower and upper bound finite element limit analysis and analytical upper bound plasticity methods are employed to investigate the undrained limiting lateral resistance of piles in a pile row. Numerical analyses and analytical calculations are presented for various pile spacings and pile-soil adhesion factors. The numerical results are shown to be in excellent agreement with each other and also with the theoretical upper bounds produced by the analytical upper bound calculations. Based on the numerical and analytical results, an empirical equation is proposed for the calculation of the ultimate undrained lateral bearing capacity factor. This equation is subsequently used to calculate p-multipliers applicable to the lower part of piles in pile rows, which are compared to multipliers available in the literature (that are constant with depth). The comparison shows significant differences, indicating that the amount of reduction in lateral resistance due to group effects is not constant with depth as routinely assumed in practice

High performance interior point methods for three-dimensional finite element limit analysis

The ability to obtain rigorous upper and lower bounds on collapse loads of various structures makes finite element limit analysis an attractive design tool. The increasingly high cost of computing those bounds, however, has limited its application on problems in three dimensions. This work reports on a high-performance homogeneous self-dual primal-dual interior point method developed for three-dimensional finite element limit analysis. This implementation achieves convergence times over 4.5× faster than the leading commercial solver across a set of three-dimensional finite element limit analysis test problems, making investigation of three dimensional limit loads viable. A comparison between a range of iterative linear solvers and direct methods used to determine the search direction is also provided, demonstrating the superiority of direct methods for this application. The components of the interior point solver considered include the elimination of and options for handling remaining free variables, multifrontal and supernodal Cholesky comparison for computing the search direction, differences between approximate minimum degree [1] and nested dissection [13] orderings, dealing with dense columns and fixed variables, and accelerating the linear system solver through parallelization. Each of these areas resulted in an improvement on at least one of the problems in the test set, with many achieving gains across the whole set. The serial implementation achieved runtime performance 1.7× faster than the commercial solver Mosek [5]. Compared with the parallel version of Mosek, the use of parallel BLAS routines in the supernodal solver saw a 1.9× speedup, and with a modified version of the GPU-enabled CHOLMOD [11] and a single NVIDIA Tesla K20c this speedup increased to 4.65×