Lower bound limit analysis of cohesive‐frictional materials using second‐order cone programming

The formulation of limit analysis by means of the finite element method leads to an optimization problem with a large number of variables and constraints. Here we present a method for obtaining strict lower bound solutions using second‐order cone programming (SOCP), for which efficient primal‐dual interior‐point algorithms have recently been developed. Following a review of previous work, we provide a brief introduction to SOCP and describe how lower bound limit analysis can be formulated in this way. Some methods for exploiting the data structure of the problem are also described, including an efficient strategy for detecting and removing linearly dependent constraints at the assembly stage. The benefits of employing SOCP are then illustrated with numerical examples. Through the use of an effective algorithm/software, very large optimization problems with up to 700 000 variables are solved in minutes on a desktop machine. The numerical examples concern plane strain conditions and the Mohr–Coulomb criterion, however we show that SOCP can also be applied to any other problem of lower bound limit analysis involving a yield function with a conic quadratic form (notable examples being the Drucker–Prager criterion in 2D or 3D, and Nielsen's criterion for plates). Copyright © 2005 John Wiley & Sons, Ltd.     

Second-order cone programming with warm start for elastoplastic analysis with von Mises yield criterion

The incremental problem for quasistatic elastoplastic analysis with the von?Mises yield criterion is discussed within the framework of the second-order cone programming (SOCP). We show that the associated flow rule under the von?Mises yield criterion with the linear isotropic/kinematic hardening is equivalently rewritten as a second-order cone complementarity problem. The minimization problems of the potential energy and the complementary energy for incremental analysis are then formulated as the primal-dual pair of SOCP problems, which can be solved with a primal-dual interior-point method. To enhance numerical performance of tracing an equilibrium path, we propose a warm-start strategy for a primal-dual interior-point method based on the primal-dual penalty method. In this warm-start strategy, we solve a penalized SOCP problem to find the equilibrium solution at the current loading step. An advanced initial point for solving this penalized SOCP problem is defined by using information of the solution at the previous loading step.     

Large‐deformation and friction analysis of non‐linear elastic cable networks by second‐order cone programming

A new formulation is presented for equilibrium shape analysis of cable networks considering geometrical and material non‐linearities. Friction between cables and joint devices is also considered. The second‐order cone programming (SOCP) problem which has the same solution as that of minimization of total potential energy is solved to obtain the equilibrium configuration. The optimality conditions are derived to verify that the solution satisfies equilibrium conditions and friction laws. Since no assumption on stress state is needed in the proposed method, no process of trial and error is involved. No effort is required to develop any analysis software because SOCP can be solved by using the well‐developed software based on the interior‐point method. Copyright © 2002 John Wiley & Sons, Ltd.     

Numerical shakedown analysis of axisymmetric sandwich shells: An upper bound formulation

Shakedown analysis of axisymmetric elastic–perfectly plastic sandwich shells is performed here using a new upper bound formulation based on a special form of Koiter's theorem concerning piecewise linearized yield surfaces. Starting from finite element techniques and the Tresca sandwich yield condition, shakedown analysis is reduced to a linear programming problem which is solved by a powerful simplex algorithm. Numerical results are given for a number of examples and a comparison is made with a previously computed lower bound formulation.     

Locking‐free discontinuous finite elements for the upper bound yield design of thick plates

This work investigates the formulation of finite elements dedicated to the upper bound kinematic approach of yield design or limit analysis of Reissner–Mindlin thick plates in shear‐bending interaction. The main novelty of this paper is to take full advantage of the fundamental difference between limit analysis and elasticity problems as regards the class of admissible virtual velocity fields. In particular, it has been demonstrated for 2D plane stress, plane strain or 3D limit analysis problems that the use of discontinuous velocity fields presents a lot of advantages when seeking for accurate upper bound estimates. For this reason, discontinuous interpolations of the transverse velocity and the rotation fields for Reissner–Mindlin plates are proposed. The subsequent discrete minimization problem is formulated as a second‐order cone programming problem and is solved using the industrial software package MOSEK . A comprehensive study of the shear‐locking phenomenon is also conducted, and it is shown that discontinuous elements avoid such a phenomenon quite naturally whereas continuous elements cannot without any specific treatment. This particular aspect is confirmed through numerical examples on classical benchmark problems and the so‐obtained upper bound estimates are confronted to recently developed lower bound equilibrium finite elements for thick plates. Copyright © 2015 John Wiley & Sons, Ltd.     

Internal blast loading in a buried lined tunnel

The paper presents a comprehensive approach to simulate an explosion occurring inside a buried axisymmetric lined cavity. The approach considers all the stages of the process: detonation of the internal charge; the shock wave propagation in the internal gas and its following interaction with the cavity's shell lining including multiple reflections; soil–structure dynamic interaction, including multiple gap opening/closure and wave propagation in the surrounding medium. The cavity's lining is modeled as a Timoshenko elastic plastic shell. The soil is modeled by the Grigoryan model that takes into account both bulk and shear elastic plastic behavior, including the effect of soil pressure on the yield strength for the stress tensor deviator. The gas-dynamics problem is solved by the modified Godunov method, based on a fixed Eulerian mesh with the so-called mixed cell. The variational difference method is applied to solve the problem in the soil and in the shell domains. The contact pressures acting on the lining due to both the detonation products on the internal side and the soil on its outer side are computed by solving the coupled system of finite differences equations of gas, shell and soil dynamics using a simple iteration method. The problems of a blast-hole charge and of a periodical system of spherical blast charges that are placed at equal distances from each other were solved. The explosion occurs within a cylindrical steel pipeline, surrounded by an elastic plastic loess soil. The effect of the soil elastic plastic pressure-density behavior and its shear properties on the soil–structure interaction (including the gap opening/closure process) was studied.     

3D混凝土打印进程中柱壳结构的力学性能研究

3D混凝土打印(3DCP)技术由于其快速制造的优势，在过去几十年里得到了迅速的发展。然而，在印刷过程中仍有许多问题需要解决，例如：目前的相关研究尚未建立能准确预测与分析3DCP圆柱壳的力学模型。该文利用Goldenveizer-Novozhilov壳体理论，加入打印进程参数，包括打印速率、打印材料固化特征、柱壳几何特征、以及自重作用影响，对3DCP圆柱壳的两种破坏机理：弹性屈曲和塑性破坏进行了分析，并在此基础上描述了柱壳形直立墙弹性屈曲与塑性坍塌间的竞争关系。参数模型、有限元模拟与已有相关试验的对比结果表明：该文提出的模型可以较好地预测3DCP圆柱壳的失效高度与失效形式，并为找寻最佳打印参数集给予理论指导。     

Active macro‐zone approach for incremental elastoplastic‐contact analysis

The symmetric boundary element method, based on the Galerkin hypotheses, has found an application in the nonlinear analysis of plasticity and in contact‐detachment problems, but both dealt with separately. In this paper, we want to treat these complex phenomena together as a linear complementarity problem. A mixed variable multidomain approach is utilized in which the substructures are distinguished into macroelements, where elastic behavior is assumed, and bem‐elements, where it is possible that plastic strains may occur. Elasticity equations are written for all the substructures, and regularity conditions in weighted (weak) form on the boundary sides and in the nodes (strong) between contiguous substructures have to be introduced, in order to attain the solving equation system governing the elastoplastic‐contact/detachment problem. The elastoplasticity is solved by incremental analysis, called for active macro‐zones, and uses the well‐known concept of self‐equilibrium stress field here shown in a discrete form through the introduction of the influence matrix (self‐stress matrix). The solution of the frictionless contact/detachment problem was performed using a strategy based on the consistent formulation of the classical Signorini equations rewritten in discrete form by utilizing boundary nodal quantities as check elements in the zones of potential contact or detachment. Copyright © 2013 John Wiley & Sons, Ltd.     

Optimization of clamped plastic shallow shells subjected to initial impulsive loading

An optimization technique is suggested for shallow spherical shells made of a ductile material and subjected to initial impact loading. The shell under consideration is pierced with a central hole and clamped at the outer edge. The optimal design of the shell of piece-wise constant thickness is established under the condition that the maximal residual deflection attains its minimal value for given total weight. The material of the shell is assumed to obey the Tresca yield condition and associated flow law. By the use of the method of mode form motions the problem is transformed into a particular problem of non-linear programming and solved numerically.     

Blast response of a lined cavity in a porous saturated soil

The paper proposes a comprehensive approach to simulate the blast response of a lined cavity in a porous soil. To calculate the soil–structure contact pressure, the coupled Godunov-variational-difference approach was developed. The lining is modeled by a Timoshenko elastic–plastic shell with kinematic linear hardening. To solve the problem in the lining domain, the variational-difference method is applied. The soil is modeled by the Lyakhov three-phase model that takes into account both bulk and shear elastic–plastic behavior, including the effect of soil pressure on the yield strength for the stress tensor deviator. The problem of blast wave propagation within the soil medium is solved by the Godunov method. The coupled approach to calculate the soil–lining contact pressure is based on the relationships on the shock and rarefaction waves with finite-difference equations of the shell motion using a simple iteration method. It allows the reduction of the contact problem to the self-similar symmetric Riemann problem. Solution of the problem of an explosion in a porous medium, and analysis of the soil–obstacle interaction under the blast action using the proposed method show good correspondence with available experimental results. Also, the plane problem of blast response of the circular cavity lined by a thin steel lining was solved. The effect of the gas volumetric content in the soil on the incident shock wave pressure as well as on the contact pressure and lining meridian strain was studied.     

Limit analysis of plates using the EFG method and second‐order cone programming

The meshless element‐free Galerkin (EFG) method is extended to allow computation of the limit load of plates. A kinematic formulation that involves approximating the displacement field using the moving least‐squares technique is developed. Only one displacement variable is required for each EFG node, ensuring that the total number of variables in the resulting optimization problem is kept to a minimum, with far fewer variables being required compared with finite element formulations using compatible elements. A stabilized conforming nodal integration scheme is extended to plastic plate bending problems. The evaluation of integrals at nodal points using curvature smoothing stabilization both keeps the size of the optimization problem small and also results in stable and accurate solutions. Difficulties imposing essential boundary conditions are overcome by enforcing displacements at the nodes directly. The formulation can be expressed as the problem of minimizing a sum of Euclidean norms subject to a set of equality constraints. This non‐smooth minimization problem can be transformed into a form suitable for solution using second‐order cone programming. The procedure is applied to several benchmark beam and plate problems and is found in practice to generate good upper‐bound solutions for benchmark problems. Copyright © 2009 John Wiley & Sons, Ltd.     

Nonlinear dynamic analysis with a 48 d.o.f. curved thin shell element

The developments of an existing 48 degrees-of-freedom (d.o.f.), curved, quadrilateral, thin shell element, for materially and geometrically nonlinear static analysis of shell structures, are extended for the study of dynamic responses of nonlinear shells. The variable-order polynomial representations of the shell surface and the non-axisymmetric definition of the shell boundaries allow the study of the dynamic behaviour of a class of shell structures more general than those treated by using flat plate elements and elements with assumptions of axisymmetry. The equations of motion are based on a Lagrangian frame of reference. A combination of step-by-step and iterative procedures is used for the solution of nonlinear equations. The incremental equations of motion are linearized for computation purposes, and an algorithm for numerical integration based on Newmark's generalized operator for dynamic analysis, using optional iteration, is adopted. The flow theory of plasticity is used in the inelastic range, and perfectly plastic or isotropic strain hardening materials are considered. The spread of plastic zones in the thickness direction is treated by using a layered model. Numerical examples presented include the dynamic analyses of a square plate, a circular annulus, a cylindrical panel and a spherical cap. Comparisons with existing solutions demonstrate the validity and accuracy of the present developments.     

A 3D upper bound limit analysis using radial point interpolation meshless method and second‐order cone programming

This paper presents a 3D formulation for quasi‐kinematic limit analysis, which is based on a radial point interpolation meshless method and numerical optimization. The velocity field is interpolated using radial point interpolation shape functions, and the resulting optimization problem is cast as a standard second‐order cone programming problem. Because the essential boundary conditions can be only guaranteed at the position of the nodes when using radial point interpolation, the results obtained with the proposed approach are not rigorous upper bound solutions. This paper aims to improve the computing efficiency of 3D upper bound limit analysis and large problems, with tens of thousands of nodes, can be solved efficiently. Five numerical examples are given to confirm the effectiveness of the proposed approach with the von Mises yield criterion: an internally pressurized cylinder; a cantilever beam; a double‐notched tensile specimen; and strip, square and rectangular footings. Copyright © 2016 John Wiley & Sons, Ltd.     

Non‐linear analysis of shells with arbitrary evolving cracks using XFEM

A new formulation and numerical procedures are developed for the analysis of arbitrary crack propagation in shells using the extended finite element method. The method is valid for completely non‐linear problems. Through‐the‐thickness cracks in sandwich shells are considered. An exact shell kinematics is presented, and a new enrichment of the rotation field is proposed which satisfies the director inextensibility condition. To avoid locking, an enhanced strain formulation is proposed for the 4‐node cracked shell element. A finite strain plane stress constitutive model based on the logarithmic corotational rate is employed. A cohesive zone model is introduced which embodies the special characteristics of the shell kinematics. Stress intensity factors are calculated for selected problems and crack propagation problems are solved. Copyright © 2004 John Wiley & Sons, Ltd.     

Mesh adaptive computation of upper and lower bounds in limit analysis

An efficient procedure to compute strict upper and lower bounds for the exact collapse multiplier in limit analysis is presented, with a formulation that explicitly considers the exact convex yield condition. The approach consists of two main steps. First, the continuous problem, under the form of the static principle of limit analysis, is discretized twice (one per bound) using particularly chosen finite element spaces for the stresses and velocities that guarantee the attainment of an upper or a lower bound. The second step consists of solving the resulting discrete non‐linear optimization problems. These are reformulated as second‐order cone programs, which allows for the use of primal–dual interior point methods that optimally exploit the convexity and duality properties of the limit analysis model. To benefit from the fact that collapse mechanisms are typically highly localized, a novel method for adaptive meshing is introduced. The method first decomposes the total bound gap as the sum of positive contributions from each element in the mesh and then refines those elements with higher contributions. The efficiency of the methodology is illustrated with applications in plane stress and plane strain problems. Copyright © 2008 John Wiley & Sons, Ltd.     

Upper bound shakedown analysis with the nodal natural element method

In this paper, a novel numerical solution procedure is developed for the upper bound shakedown analysis of elastic-perfectly plastic structures. The nodal natural element method (nodal-NEM) combines the advantages of the NEM and the stabilized conforming nodal integration scheme, and is used to discretize the established mathematical programming formulation of upper bound shakedown analysis based on Koiter’s theorem. In this formulation, the displacement field is approximated by using the Sibson interpolation and the difficulty caused by the time integration is solved by König’s technique. Meanwhile, the nonlinear and non-differentiable characteristic of objective function is overcome by distinguishing non-plastic areas from plastic areas and modifying associated constraint conditions and goal function at each iteration step. Finally, the objective function subjected to several equality constraints is linearized and the upper bound shakedown load multiplier is obtained. This direct iterative process can ensure the shakedown load to monotonically converge to the upper bound of true solution. Several typical numerical examples confirm the efficiency and accuracy of the proposed method.     

AN UPPER BOUND FINITE ELEMENT PROCEDURE FOR SOLVING LARGE PLANE STRAIN DEFORMATION

A unique and robust upper bound finite element procedure is developed for the analysis of large plastic deformation problems under plane strain condition. It can consistently treat problems with isotropic strain varying materials. It can also effectively solve problems with any initial ‘guessed’ velocity field, even from an random number generator. To explore and demonstrate the capability of this new approach, strip tension and plane strain compression problems are solved. For validation, the computed results are compared with existing analytical or experimental solutions in good agreement. The phenomenon of shear band formation can be simulated and, as expected, is found to develop more distinctly in strain softening materials than in perfectly plastic and strain hardening materials. © 1997 by John Wiley & Sons, Ltd.     

Lower‐bound limit analysis by using the EFG method and non‐linear programming

Intended to avoid the complicated computations of elasto‐plastic incremental analysis, limit analysis is an appealing direct method for determining the load‐carrying capacity of structures. On the basis of the static limit analysis theorem, a solution procedure for lower‐bound limit analysis is presented firstly, making use of the element‐free Galerkin (EFG) method rather than traditional numerical methods such as the finite element method and boundary element method. The numerical implementation is very simple and convenient because it is only necessary to construct an array of nodes in the domain under consideration. The reduced‐basis technique is adopted to solve the mathematical programming iteratively in a sequence of reduced self‐equilibrium stress subspaces with very low dimensions. The self‐equilibrium stress field is expressed by a linear combination of several self‐equilibrium stress basis vectors with parameters to be determined. These self‐equilibrium stress basis vectors are generated by performing an equilibrium iteration procedure during elasto‐plastic incremental analysis. The Complex method is used to solve these non‐linear programming sub‐problems and determine the maximal load amplifier. Numerical examples show that it is feasible and effective to solve the problems of limit analysis by using the EFG method and non‐linear programming. Copyright © 2007 John Wiley & Sons, Ltd.