Upper bound finite element limit analysis method with discontinuous quadratic displacement fields and remeshing in non-h
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O R I G I NA L
Rui Sun
· Junsheng Yang · Yiding Zhao · Shouhua Liu
Upper bound finite element limit analysis method with discontinuous quadratic displacement fields and remeshing in non-homogeneous clays
Received: 3 January 2020 / Accepted: 2 October 2020 © Springer-Verlag GmbH Germany, part of Springer Nature 2020
Abstract As is known, limit analysis is a common way to study stability and failure mechanism of geotechnical engineering. This paper presents an upper bound finite element limit analysis (UB-FELA) formulation for nonhomogeneous clays using discontinuous quadratic displacement fields and second-order cone programming (SOCP). The details are provided about the application of SOCP to the UB-FELA formulation for nonhomogeneous clays. To reduce the calculation error, the analytic expressions are proposed concerning the rate of external work done by body forces and power dissipation caused by the plastic deformation following discontinuous quadratic displacement fields. With SOCP solver MOSEK, large-scale optimization problems of UB-FELA can be achieved in minutes via a desktop computer. The method is then applied to analyzing the stability of an undrained square tunnel and an undrained plane strain heading. The numerical results are compared with those reported in related studies. This method helps to obtain highly precise upper bound solution and clear failure mechanism of the computational domain. Keywords Non-homogeneous clays · Upper bound finite element · Second-order cone programming · Discontinuous quadratic displacement fields remeshing
1 Introduction An accurate calculation of the collapse load plays a greatly important role in evaluating the safety factors of the geotechnical engineering. The collapse load of a series of plane strain stability problems can be directly acquired based on finite element limit analysis (FELA). From a technical point of view, Sloan and Kleeman [1] firstly proposed a plane strain UB-FELA based on linear programming (LP) with 3-noded triangular element and velocity discontinuity at the interface between adjoining elements. Due to the need of LP, the nonlinear yield function should be linearized by means of many linear constraints, but this could generate significant calculation costs. Accordingly, some scholars made some useful attempts to apply nonlinear programming (NLP) in the FELA [2–5], hopefully to overcome the limitation that LP needs to linearize yield criterion. Lyamin and Sloan [2] developed a new UB-FELA based on NLP. Then, Krabbenhoft et al. [3] proposed a stress-based UB-FELA based on NLP. However, their efforts required smoothing of the yield function to avoid stress singularities. Other limit analysis formulations based on NLP were also adopted to investigate the lower bound (LB) [4] and upper bound (UB) problems [5], but their solutions did not involve lower or upper bounds in a strict sense. Recent trends in limit analysis demonstrate that it is of significant advantage to utilize cone optimization techniques, including second-order cone programming
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