A non‐linear programming approach to kinematic shakedown analysis of composite materials

Using a Representative volume element (RVE) to represent the microstructure of periodic composite materials, this paper develops a non‐linear numerical technique to calculate the macroscopic shakedown domains of composites subjected to cyclic loads. The shakedown analysis is performed using homogenization theory and the displacement‐based finite element method. With the aid of homogenization theory, the classical kinematic shakedown theorem is generalized to incorporate the microstructure of composites. Using an associated flow rule, the plastic dissipation power for an ellipsoid yield criterion is expressed in terms of the kinematically admissible velocity. By means of non‐linear mathematical programming techniques, a finite element formulation of kinematic shakedown analysis is then developed leading to a non‐linear mathematical programming problem subject to only a small number of equality constraints. The objective function corresponds to the plastic dissipation power which is to be minimized and an upper bound to the shakedown load of a composite is then obtained. An effective, direct iterative algorithm is proposed to solve the non‐linear programming problem. The effectiveness and efficiency of the proposed numerical method have been validated by several numerical examples. This can serve as a useful numerical tool for developing engineering design methods involving composite materials. Copyright © 2005 John Wiley & Sons, Ltd.     

用边界元方法和复合形法求解三维结构的下限安定载荷

基于安定分析的静力定理,建立了用常规边界元方法进行三维理想弹塑性结构安定分析的整套求解算法。下限安定分析所需的弹性应力场直接由边界元方法求出,所需的自平衡应力场由一组带有待定系数的自平衡应力场基矢量的线性组合进行模拟,这些自平衡应力场基矢量通过边界元弹塑性迭代计算获取。安定分析问题最终被归结为一系列未知变量较少的非线性数学规划子问题并通过复合形法直接求解。计算结果表明了算法的有效性。     

A unified mathematical programming formulation of strain driven and interior point algorithms for shakedown and limit analysis

A mathematical programming formulation of strain‐driven path‐following strategies to perform shakedown and limit analysis for perfectly elastoplastic materials in an FEM context is presented. From the optimization point of view, standard arc‐length strain‐driven elastoplastic analyses, recently extended to shakedown, are identified as particular decomposition strategies used to solve a proximal point algorithm applied to the static shakedown theorem that is then solved by means of a convergent sequence of safe states. The mathematical programming approach allows: a direct comparison with other non‐linear programming methods, simpler convergence proofs and duality to be exploited. Owing to the unified approach in terms of total stresses, the strain‐driven algorithms become more effective and less non‐linear with respect to a self‐equilibrated stress formulation and easier to implement in the existing codes performing elastoplastic analysis. The elastic domain is represented avoiding any linearization of the yield function so improving both the accuracy and the performance. Better results are obtained using two different finite elements, one with a good behavior in the elastic range and the other suitable for performing elastoplastic analysis. The proposed formulation is compared with a specialized implementation of the primal–dual interior point method suitable to solve the problems at hand. Copyright © 2011 John Wiley & Sons, Ltd.     

Kinematical shakedown analysis with temperature‐dependent yield stress

This paper describes a direct shakedown analysis of structures subjected to variable thermal and mechanical loading. The classical kinematical shakedown theorem is modified to be implemented with any displacement‐based finite elements. The plastic incompressibility condition is imposed by the penalty function method. The shakedown limit is found via a non‐linear mathematical programming procedure. Two numerical shakedown methods are developed and implemented to provide alternative numerical means. The temperature‐dependent material model is included in theoretical and numerical calculation in a simple way. Its effect on shakedown limit is investigated. The numerical examination for some pressure vessel structures subjected to thermal and mechanical loading shows a satisfying precision and efficiency of the methods presented. Copyright © 2001 John Wiley & Sons, Ltd.     

A microscopic nonlinear programming approach to shakedown analysis of cohesive–frictional composites

A microscopic approach together with nonlinear programming technique and finite element method is developed for shakedown analysis of a composite which has cohesive–frictional constituents. The macroscopic shakedown limit of a composite subject to cyclic loading is calculated in a direct way and the macro–micro relation is quantitatively evaluated. First, by means of the homogenization theory, the classical kinematic theorem of shakedown analysis is generalized to incorporate the microstructure – Representative Volume Element (RVE) chosen from a periodic heterogeneous material. Pressure-dependence and non-associated plastic flow of cohesive–frictional constituent materials are formulated into shakedown analysis. Based on the mathematical programming technique and the finite element method, the numerical micro-shakedown model is finally formulated as a nonlinear programming problem subject to only a few equality constraints, which is solved by a generalized Lagrangian-penalty iterative algorithm. The proposed approach provides a direct approach for determining the reduced macroscopic strength domain of heterogeneous or composite materials due to cyclic loading. Meanwhile, it can capture different plastic behaviors of materials and therefore the developed method has a wide applicability.     

A selective strategy for shakedown analysis of engineering structures

Determining the load‐bearing capacity of engineering structures is essential for their design. In the case of varying thermo‐mechanical loading beyond the elastic limit, the statical shakedown analysis constitutes a particularly suitable tool for this. The application of the statical shakedown theorem, however, leads to a nonlinear convex optimization problem, which is typically characterized by large numbers of variables and constraints. In the present work, this optimization problem is solved by a primal–dual interior‐point algorithm, which shows a remarkable performance due to its problem‐tailored formulation. Nevertheless, the solution procedure remains still a demanding task from computational point of view. Thus, the aim of this paper is to tackle the task of solving large‐scale problems by use of a new so‐called selective algorithm. This algorithm detects automatically the plastically most affected zones within the structure, which have the highest influence on the solution. The entire system is then reduced to a substructure consisting of these zones, based upon which a new optimization problem can be set up, which is solved with significantly less effort. Consequently, the running time decreases drastically, as is shown by application to numerical examples from the field of power plant engineering. Copyright © 2013 John Wiley & Sons, Ltd.     

Optimal shakedown analysis of plane reinforced concrete frames according to Eurocodes

We present an updated mathematical model of shakedown optimization for reinforced concrete plane frames subjected to variable and repeated uncertain loading within a known domain. In such structures, plastic redistribution of forces is known to occur, and various mechanisms of system collapse at shakedown have been identified, such as plastic yielding and sign-changing. We develop a general nonlinear mixed-integer optimization problem that reduces to a linear programming problem, and we demonstrate the duality of the linear programming problem for the static and kinematic formulations. We derive strength conditions according to Eurocode 2 and an iterative process of optimization, where stiffness properties of frame elements are allowed to vary. The frame cross-sections are rectangular and made from doubly reinforced concrete; the material is considered composite. We successfully demonstrate the numerical optimization procedure on a two-storey reinforced concrete plane frame. We present variations of interaction loci of each optimized section in graphical form.     

A mixed integer programming for robust truss topology optimization with stress constraints

This paper presents a mixed integer programming (MIP) formulation for robust topology optimization of trusses subjected to the stress constraints under the uncertain load. A design‐dependent uncertainty model of the external load is proposed for dealing with the variation of truss topology in the course of optimization. For a truss with the discrete member cross‐sectional areas, it is shown that the robust topology optimization problem can be reduced to an MIP problem, which is solved globally. Numerical examples illustrate that the robust optimal topology of a truss depends on the magnitude of uncertainty. Copyright © 2010 John Wiley & Sons, Ltd.     

Shakedown of a cracked body consisting of kinematic hardening material

In this paper, the shakedown behaviour of a cracked body is studied. The main idea is to consider the crack as a notch. Then no singular stresses appear at crack tip. Due to the local character of the problem, Melan's shakedown theorem is used. By solving shakedown as an optimization problem, the limited stress intensity factor (SIF) for shakedown K_sh is obtained. It is found that the shakedown limit SIF of a cracked body is proportional to the initial yield stress σ_y of the material times the square root of the effective crack tip radius π, i.e. K_sh ∝ σ_y√ϱ. Comparison of shakedown limit SIFs with fatigue thresholds for certain materials, so far as can be found in literature, shows that these two quantities agree well with each other. This agreement indicates that shakedown of the cracked body is one of the reasons for arrest of the crack under cyclic loads. Shakedown investigation is then a new method for predicting the fatigue threshold of a cracked body. Thus, a transition from shakedown to cyclic fracture mechanics has been achieved.     

Confidence structural robust optimization by non‐linear semidefinite programming‐based single‐level formulation

Structural robust optimization problems are often solved via the so‐called Bi‐level approach. This solution procedure often involves large computational efforts and sometimes its convergence properties are not so good because of the non‐smooth nature of the Bi‐level formulation. Another problem associated with the traditional Bi‐level approach is that the confidence of the robustness of the obtained solutions cannot be fully assured at least theoretically. In the present paper, confidence single‐level non‐linear semidefinite programming (NLSDP) formulations for structural robust optimization problems under stiffness uncertainties are proposed. This is achieved by using some tools such as S‐procedure and quadratic embedding for convex analysis. The resulted NLSDP problems are solved using the modified augmented Lagrange multiplier method which has sound mathematical properties. Numerical examples show that confidence robust optimal solutions can be obtained with the proposed approach effectively. Copyright © 2010 John Wiley & Sons, Ltd.     

Bounds to Shakedown Loads for a Class of Deviatoric Plasticity Models

The problem of estimating bounds to shakedown loads for problems governed by a class of deviatoric plasticity models including those of Hill, von Mises, and Tresca is addressed. Assuming that an exact elastic solution is available, an upper bound to the elastic shakedown multiplier can be obtained relatively easily using the plastic shakedown theorem. A procedure for computing this upper bound for arbitrary load domains is presented. A number of problems are then examined and it is found that the elastic shakedown factor is given as the minimum of the plastic shakedown factor and the classical limit load factor. Finally, some exact solutions to a number of two dimensional problems are given.     

Alternative approach to shakedown as a solution of a min-max problem

Summary Melan's classical shakedown theorem for continuous media considered as a problem of methematical programming with constraints is reformulated and reduced to a solution of a certain min-max problem. A similar approach is presented for the structural theory described in terms of generalized variables. A distinction is made between alternating plasticity and incremental collapse modes in the analysis of structures with nonsandwich cross-sections.     

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.     

弹塑性结构安定性上限分析的数值方法及应用

建立了复杂变化载荷作用下理想弹塑性结构安定上限分析的有限元数学规划格式。利用研究结构在基准载荷域各个角点处安定的办法,避开了机动定理中对时间积分的困难,提出了一种直接迭代算法求解,以克服目标函数非线性非光滑所导致的困难。该格式同时考虑了温度对屈服极限的影响。     

Robust compliance topology optimization based on the topological derivative concept

In this paper, we present an approach for robust compliance topology optimization under volume constraint. The compliance is evaluated considering a point‐wise worst‐case scenario. Analogously to sequential optimization and reliability assessment, the resulting robust optimization problem can be decoupled into a deterministic topology optimization step and a reliability analysis step. This procedure allows us to use topology optimization algorithms already developed with only small modifications. Here, the deterministic topology optimization problem is addressed with an efficient algorithm based on the topological derivative concept and a level‐set domain representation method. The reliability analysis step is handled as in the performance measure approach. Several numerical examples are presented showing the effectiveness of the proposed approach. Copyright © 2015 John Wiley & Sons, Ltd.     

A robust optimization methodology for preliminary aircraft design

This article focuses on a robust optimization of an aircraft preliminary design under operational constraints. According to engineers' know-how, the aircraft preliminary design problem can be modelled as an uncertain optimization problem whose objective (the cost or the fuel consumption) is almost affine, and whose constraints are convex. It is shown that this uncertain optimization problem can be approximated in a conservative manner by an uncertain linear optimization program, which enables the use of the techniques of robust linear programming of Ben-Tal, El Ghaoui, and Nemirovski [Robust Optimization, Princeton University Press, 2009]. This methodology is then applied to two real cases of aircraft design and numerical results are presented.     

Difference of convex functions algorithms (DCA) for image restoration via a Markov random field model

In this paper, we introduce a novel approach in the nonconvex optimization framework for image restoration via a Markov random field (MRF) model. While image restoration is elegantly expressed in the language of MRF’s, the resulting energy minimization problem was widely viewed as intractable: it exhibits a highly nonsmooth nonconvex energy function with many local minima, and is known to be NP-hard. The main goal of this paper is to develop fast and scalable approximation optimization approaches to a nonsmooth nonconvex MRF model which corresponds to an MRF with a truncated quadratic (also known as half-quadratic) prior. For this aim, we use the difference of convex functions (DC) programming and DC algorithm (DCA), a fast and robust approach in smooth/nonsmooth nonconvex programming, which have been successfully applied in various fields in recent years. We propose two DC formulations and investigate the two corresponding versions of DCA. Numerical simulations show the efficiency, reliability and robustness of our customized DCAs with respect to the standard GNC algorithm and the Graph-Cut based method—a more recent and efficient approach to image analysis.     

A Nonlinear Optimization Algorithm for Lower Bound Limit and Shakedown Analysis

Limit and shakedown analysis theorems are the theories of classical plasticity for the direct computation of the load-carrying capacity under proportional and varying loads. Based on Melan's theorem, a solution procedure for lower bound limit and shakedown analysis of three-dimensional (3D) structures is established making use of the finite element method (FEM). The self-equilibrium stress fields are expressed by linear combination of several basic self-equilibrium stress fields with parameters to be determined. These basic self-equilibrium stress fields are elastic responses of the body to imposed permanent strains obtained through elastic-plastic incremental analysis by the three-dimensional finite element method (3D-FEM). The Complex method is used to solve the resulting nonlinear programming directly and determine the maximal load amplifier. The numerical results show that it is efficient and accurate to solve three-dimensional limit and shakedown analysis problems by using the 3D-FEM and the Complex method. The limit analysis is treated here as a special case of shakedown analysis in which only proportional loading is considered.