Application of discontinuity layout optimization to three-dimensional plasticity problems

Abstract

A new three-dimensional limit analysis formulation that uses the recently developed discontinuity layout optimization (DLO) procedure is described. With DLO, limit analysis problems are formulated purely in terms of discontinuities, which take the form of polygons when three-dimensional problems are involved. Efficient second-order cone programming techniques can be used to obtain solutions for problems involving Tresca and Mohr-Coulomb yield criteria. This allows traditional 'upper bound' translational collapse mechanisms to be identified automatically. A number of simple benchmark problems are considered, demonstrating that good results can be obtained even when coarse numerical discretizations are employed.

Multi-slicing strategy for the three-dimensional discontinuity layout optimization (3D DLO)

“…where X P is the relative displacement of node P , .B P;i d i / is the contribution of discontinuity i to the relative displacement of node P . ¶ B P;i is a coordinate transformation matrix (2 2 for 2D, see [7] for details and 3 3 for 3D, see Appendix A for details) transforming the local displacement jump d i (given in local coordinates) to the global coordinate system [7,8]. In 3D DLO, this compatibility relationship is formulated for every edge [8,9], such as edge PQ illustrated in Figure 3.…”

“…Belonging to the family of topology optimization, discontinuity layout optimization (DLO) is a recently presented, elegant and promising numerical method for determining (i) the critical layout of discontinuities in a body at failure and (ii) the associated upper bound limit load [5] for plasticity problems [6][7][8][9]. Introducing large numbers of potential discontinuities permitted to crossover one another in a given domain and specifying the loading and boundary conditions (live loads, dead loads and supports), DLO can automatically generate a set of activated discontinuities corresponding to the upper bound limit load.…”

“…In threedimensional (3D) DLO, on the other hand, the discontinuities are geometrically represented by polygons. Overlapping and crossing of different 3D discontinuities should be avoided to save computing resources [8,9], as an important pre-processing task. For the potential wide application of 3D DLO, a highly efficient pre-processing method is necessary.…”

Stability analysis of shotcrete supported crown of NATM tunnels with discontinuity layout optimization

“…Recently, an innovative type of topology optimization method coined “discontinuity layout optimization (DLO)” was presented in Smith and Gilbert, which determines the critical layout of discontinuities and associated upper‐bound limit load for the plane (2D) and 3‐dimensional (3D) plasticity problems. By introducing a great amount of potential discontinuities, the layout of activated discontinuity will be automatically obtained by solving the corresponding linear programming (LP) and second‐order cone programming problems.…”

Modelling rotational failure in confined geometries using DLO

“…Lysmer, 1970;Makrodimopoulos and Martin, 2006;Sloan, 1988) and the method of characteristics (Sokolovskii, 1965), through its ability to identify directly critical failure mechanisms in the form of velocity discontinuities for a prescribed numerical discretisation, and to handle singularities in a natural and fully general way. The application of DLO to three-dimensional problem and plane-strain problems involving purely translational failure mechanisms has been described by Hawksbee et al (2013) and Smith and Gilbert (2007), respectively, and to plane-strain rotational mechanisms in non-dilational materials by Smith and Gilbert (2013). Recently, a formulation suitable for the analysis of slabs has been also presented (Gilbert et al, 2014).…”