Shakedown analysis plays a substantial role in safety assessment, especially in nuclear plant industry, chemical industry and civil engineering. This paper presents numerical investigations of shakedown problems with holes using the adaptive extended isogeometric analysis (XIGA). An adaptive strategy performed in the discretized framework of a kinematic approach and second-order cone programming (SOCP) is presented. The main idea is to integrate XIGA based on locally refined non-uniform rational B-splines (LR NURBS) into the SOCP to form an efficient approach for shakedown analysis. A SOCP form based on the von Mises yield criterion is established. By means of an effective SOCP solver, the arising optimization problem is then solved. In addition, the level set function is adopted to detect the position of holes. The L2-norm of plastic strain rates is used to guide local mesh refinement. The merits of the developed approach are its easy implementation and high computational efficiency. We consider several benchmark examples to illustrate the accuracy, reliability and efficiency of the developed method.

Limit analysis can be used to directly calculate the ultimate load of structures without considering cumbersome elastic–plastic analysis. We present an adaptive mesh refinement strategy by mean of extended isogeometric analysis (XIGA) in association with second-order cone programming (SOCP) for kinematic limit analysis of hole and inclusion problems. The location of hole and inclusion can be captured by level set function without regard for its interfaces. The local refined non-uniform rational B-splines (LR NURBS), allowing local refinement and modeling complex geometries, are selected as basis functions. The structured mesh refinement strategy is exploited to carry out local refinement on the basis of the indicator of L2-norm of plastic strain rates. The adaptive local refinement can detect the local plastic deformation regions, and further reveals the possible failure mechanisms. The discrete kinematic formulation of limit analysis can be rewritten as the form of SOCP, and further is solved efficiently with the help of the Mosek toolbox. All the desirable features of the present approach are illustrated through numerical experiments.

Limit analysis, without the complicated elasto-plastic computation, is an efficient method for estimating safety load of engineering structures. This paper develops a novel computational approach by integrating second-order cone programming (SOCP) into adaptive extended isogemetric elements (XIGA) for upper-bound limit analysis of cracked structures. The advantage of XIGA is to model cracks without considering the location of crack faces by introducing enrichment functions. The local refined (LR) B-splines, which have versatile and flexible local refinement ability, are adopted as basis functions in the XIGA. We use structured mesh refinement strategy to implement local refinement based on the indicator of 2-norm of plastic strain rates. Cracked structures are assumed under plane stress condition, and the von Mises yield criterion is used. Kinematic formulation of the limit analysis is translated into the form of SOCP, then is solved by the Mosek tool. The developed model is implemented and its accuracy and effectiveness are illustrated through several numerical examples. In addition, numerical results illustrate that the convergence rate of adaptive XIGA is faster than that of traditional XIGA.