The intersection or the overlap region of two n-dimensional ellipsoids plays an important role in statistical decision making in a number of applications. For instance, the intersection volume of two n-dimensional ellipsoids has been employed to define dissimilarity measures in time series clustering (see [M. Bakoben, T. Bellotti and N. M. Adams, Improving clustering performance by incorporating uncertainty, Pattern Recognit. Lett. 77 2016, 28–34]). Formulas for the intersection volumes of two n-dimensional ellipsoids are not known. In this article, we first derive exact formulas to determine the intersection volume of two hyper-ellipsoids satisfying a certain condition. Then we adapt and extend two geometric type Monte Carlo methods that in principle allow us to compute the intersection volume of any two generalized convex hyper-ellipsoids. Using the exact formulas, we evaluate the performance of the two Monte Carlo methods. Our numerical experiments show that sufficiently accurate estimates can be obtained for a reasonably wide range of n, and that the sample-mean method is more efficient. Finally, we develop an elementary fast Monte Carlo method to determine, with high probability, if two n-ellipsoids are separated or overlap.

Numerical techniques for the computation of strict bounds in limit analyses have been developed for more than thirty years. The efficiency of these techniques have been substantially improved in the last ten years, and have been successfully applied to academic problems, foundations and excavations. We here extend the theoretical background to problems with anchors, interface conditions, and joints. Those extensions are relevant for the analysis of retaining and anchored walls, which we study in this work. The analysis of three-dimensional domains remains as yet very scarce. From the computational standpoint, the memory requirements and CPU time are exceedingly prohibitive when mesh adaptivity is employed. For this reason, we also present here the application of decomposition techniques to the optimisation problem of limit analysis. We discuss the performance of different methodologies adopted in the literature for general optimisation problems, such as primal and dual decomposition, and suggest some strategies that are suitable for the parallelisation of large three-dimensional problems. The propo sed decomposition techniques are tested against representative problems.

Despite recent progress in the optimisation techniques, finite element stability analysis of realistic three-dimensional (3D) problems is still hampered by the size of the resulting optimisation problem. Current solvers may take a prohibitive computational time, if they give a solution at all. Possible remedies to this are the design of adaptive de-remeshing techniques, decomposition of the system of equations, or the decomposition of the optimisation problem. In this paper we concentrate on the last approach, and present an algorithm especially suited for limit analysis.The optimisation problems in limit analysis are convex but non-linear. This fact renders the design of decomposition techniques specially challenging. The efficiency of general approaches such as Benders or Dantzig-Wolfe is not always satisfactory, and strongly depends on the structure of the optimisation problem. We here present a new method that is based on rewriting the feasibility region of the global optimisation problem as the intersection of two subsets. By resorting to the Averaged Alternate Reflections (AAR) in order to find the distance between the sets, we achieve to solve the optimisation problem in a decomposed manner. We illustrate the method with some representative problems, and comment its efficiency with respect to other well-known decomposition algorithms.

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The analysis of the bearing capacity of structures with a rigid-plastic behaviour can be achieved resorting to computational limit analysis. Recent techniques [3], [4] have allowed scientists and engineers to determine upper and lower bounds of the load factor under which the structure will collapse. Despite the attractiveness of these results, their application to practical examples is still hampered by the size of the resulting optimisation process.We here propose a method for decomposing a class of convex nonlinear programmes which are encountered in limit analysis. These problems have second-order conic memberships constraints and a single complicating variable in the objective function. The method is based on finding the distance between the feasible sets of the decomposed problems, and updating the global optimal value according to the value of this distance. The latter is found by exploiting the method of Averaged Alternating Reflections (AAR), which is here adapted to the optimisation problem at hand. The method is specially suited for non-linear problems, and as our numerical results show, its convergence is independent of the number of variables of each sub-domain. We have tested the method with problems that have more than 10000 variables.