This paper describes a technique for computing lower bound limit loads in soil mechanics under conditions of plane strain. In order to invoke the lower bound theorem of classical plasticity theory, a perfectly plastic soil model is assumed, which may be either purely cohesive or cohesive-frictional, together with an associated flow rule. Using a suitable linear approximation of the yield surface, the procedure computes a statically admissible stress field via finite elements and linear programming. The stress field is modelled using linear 3-noded traingles and statically admissible stress discontinuities may occur at the edges of each triangle. Imposition of the stress-boundary, equilibrium and yield conditions leads to an expression for the collapse load which is maximized subject to a set of linear constraints on the nodal stresses. Since all of the requirements for a statically admissible solution are satisfied exactly (except for small round-off errors in the optimization computations), the solution obtained is a strict lower bound on the true collapse load and is therefore ‘safe’.

A major drawback of the technique, as first described by Lysmer,1 is the large amount of computer time required to solve the linear programming problem. This paper shows that this limitation may be avoided by using an active set algorithm, rather than the traditional simplex or revised simplex strategies, to solve the resulting optimization problem. This is due to the nature of the constraint matrix, which is always very sparse and typically has many more rows that columns. It also proved that the procedure can, without modification, be used to derive strict lower bounds for a purely cohesive soil which has increasing strength with depth. This important class of problem is difficult to tackle using conventional methods. A number of examples are given to illustrate the effectiveness of the procedure.

Lower bound limit analysis using finite elements and linear programming

This paper describes recent advances in stability analysis that combine the limit theorems of classical plasticity with finite elements to give rigorous upper and lower bounds on the failure load. These methods, known as finite-element limit analysis, do not require assumptions to be made about the mode of failure, and use only simple strength parameters that are familiar to geotechnical engineers. The bounding properties of the solutions are invaluable in practice, and enable accurate limit loads to be obtained through the use of an exact error estimate and automatic adaptive meshing procedures. The methods are very general, and can deal with heterogeneous soil profiles, anisotropic strength characteristics, fissured soils, discontinuities, complicated boundary conditions, and complex loading in both two and three dimensions. A new development, which incorporates pore water pressures in finite-element limit analysis, is also described. Following a brief outline of the new techniques, stability solutions ...

Geotechnical stability analysis

This paper describes a new formulation, based on linear finite elements and non-linear programming, for computing rigorous lower bounds in 1, 2 and 3 dimensions. The resulting optimization problem is typically very large and highly sparse and is solved using a fast quasi-Newton method whose iteration count is largely independent of the mesh refinement. For two-dimensional applications, the new formulation is shown to be vastly superior to an equivalent formulation that is based on a linearized yield surface and linear programming. Although it has been developed primarily for geotechnical applications, the method can be used for a wide range of plasticity problems including those with inhomogeneous materials, complex loading, and complicated geometry. Copyright © 2002 John Wiley & Sons, Ltd.

Lower bound limit analysis using non-linear programming

A new method for computing rigorous upper bounds on the limit loads for one-, two- and three-dimensional continua is described. The formulation is based on linear finite elements, permits kinematically admissible velocity discontinuities at all interelement boundaries, and furnishes a kinematically admissible velocity field by solving a non-linear programming problem. In the latter, the objective function corresponds to the dissipated power (which is minimized) and the unknowns are subject to linear equality constraints as well as linear and non-linear inequality constraints.

Provided the yield surface is convex, the optimization problem generated by the upper bound method is also convex and can be solved efficiently by applying a two-stage, quasi-Newton scheme to the corresponding Kuhn–Tucker optimality conditions. A key advantage of this strategy is that its iteration count is largely independent of the mesh size. Since the formulation permits non-linear constraints on the unknowns, no linearization of the yield surface is necessary and the modelling of three-dimensional geometries presents no special difficulties.

The utility of the proposed upper bound method is illustrated by applying it to a number of two- and three-dimensional boundary value problems. For a variety of two-dimensional cases, the new scheme is up to two orders of magnitude faster than an equivalent linear programming scheme which uses yield surface linearization. Copyright © 2001 John Wiley & Sons, Ltd.

Upper bound limit analysis using linear finite elements and non-linear programming

Upper bound limit analysis using discontinuous velocity fields

Finite element method and limit analysis theory for soil mechanics problems

We investigate the feasibility of using a known elastic target located near the seabed for the purpose of inferring characteristics of marine sediment. In the problem considered the object position and its burial depth are not known with precision. First, the admittance matrix of the elastic object is determined (numerically or experimentally) over a wide frequency range in the structural acoustic regime. Then, the equivalent source method (ESM) coupled with a spectral representation of the Green's functions in stratified domains is used to predict the object acoustic signature in various environments and experimental configurations. The resulting solver takes into account all multiple scattering between target (buried or not), sea floor, and sea surface and is not limited to short distances. After presenting the solution to the forward problem several synthetic inversions for sediment characteristics are shown. They are based upon a resonance-based misfit function we describe. The Bayesian procedure also infers object burial and source-object range, broadening its range of application.

Inferring the characteristics of marine sediments from the acoustic response of a known, partially buried object.

T-waves are generated by earthquakes and travel horizontally with the speed of sound. They are the most common earthquake sounds in the ocean and make strong contributions to the ambient sound field. However, the mechanisms of their generation remain poorly understood. Recent theoretical paper [O. A. Godin, J. Acoust. Soc. Am. 150, 3999–4017 (2021)] argues that scattering of ballistic waves from the earthquake focus by the ocean surface is a strong source of T-waves, which should be considered on par with previously identified generation mechanisms of wave diffraction on large bathymetric features and scattering by the rough seafloor. This paper compares the T-waves due to ocean surface roughness and a seamount in deep water. 3-D surface scattering and wave interaction with large-scale bathymetric features are modeled rigorously using time-domain spectral-element code SPECFEM3D, with the ocean surface being shaped by wind waves and sea swell and a large seamount located off the earthquake epicenter. These simulations are complemented by a normal-mode code to propagate the sound field from the generation region to long ranges. A mode projection technique is developed to efficiently couple SPECFEM3D to the normal mode code. The two mechanisms are found to generate T-waves of comparable amplitudes.

Numerical modeling of T-wave generation in a horizontally inhomogeneous ocean

This study presents simulation results on passive structural health monitoring of a vibrating elastic structure for defect localization. The object considered in this study consists of a shell equipped with vibration sensors surrounding a vibrating structure such that they are coupled at a finite number of points. An acoustic digital twin (ADT) is used to model the non-defective state of the external shell. A combination of the concept of an ADT together with an adjoint-based high-resolution array-processing approach was used to detect and locate a defect on the structure under inspection.

A. Bottero

Papers

Finite element method and limit analysis theory for soil mechanics problems

Inferring the characteristics of marine sediments from the acoustic response of a known, partially buried object.

Numerical modeling of T-wave generation in a horizontally inhomogeneous ocean

Acoustic digital twin for passive structural health monitoring.