تجزیه و تحلیل با استفاده از روش المان محدود PP-تطبیقی ​​و برنامه ریزی خطی

In this paper, we investigate the use of an adaptive pp-version of the finite element method to perform shakedown analyses of 2D plane strain problems within the static approach. Moreover, efficient piecewise linearizations of the yield surfaces are carried out in a semi-adaptive fashion so that we need to solve the more tractable linear, rather than nonlinear, programming problems. State-of-the-art linear programming solvers, based on the very efficient interior point methodology, are used for solving the optimization problems. Various numerical examples are provided to compare the efficiency of the proposed approach with those of uniform and nonuniform pp-mesh models and nonlinear programming.

The use of simplified methods is an appealing approach to obtain some essential information, such as the load-carrying capacity of structural systems, for use in preliminary design or safety evaluation. Such, so-called “direct”, schemes avoid computationally expensive step-by-step analyses that trace inelastic structural responses to a given loading history. One such method is limit analysis which is used to estimate the collapse load of perfectly-plastic media under a monotonically increasing load regime. A shakedown analysis, a generalization of limit analysis, is also commonly used when the external loading, as is often the case, is repeated in nature with its precise history being unknown except for its upper and lower limits. Performing efficient and accurate shakedown analyses constitutes the focus of the present paper. The occurrence of shakedown implies that dissipative, yielding processes eventually cease, while its obverse leads to failure by either incremental collapse (characterized by unbounded deformation growth for each cycle of loading) or alternating plasticity (eventually leading to fractures by a low cycle fatigue type phenomenon). For an overview, the interested reader is referred, for instance, to the still useful proceedings of the 1977 NATO conference held at Waterloo [1], to the monograph by König [2], to such classical papers as Belytschko [3] and Maier [4], and to the key survey articles by König and Maier [5], Maier et al. [6] and Maier [7]. Worthy of mention are some more recent works, and the numerous references contained therein, such as those by Chen and Ponter [8], Staat and Heitzer [9], and Vu et al. [10]. As discussed in [11], two main difficulties, which are in fact typical of computational plasticity problems in general, are encountered when computing limit and shakedown loads. Firstly, in plane strain and 3D problems with certain yield conditions such as von Mises, volumetric or isochoric locking may occur. Secondly, practically motivated structures often lead to large numerical models that can become computationally infeasible or even intractable. Tin-Loi and Ngo [11] explored the use of the pp-version of the finite element method (FEM) for overcoming locking when carrying out limit analyses. This method was found to be robust and accurate, and was subsequently extended to solve shakedown problems as well [12]. However, as pointed out in these papers, whilst increasingly more accurate estimates of both collapse and shakedown limits are obtained as the degree of polynomial pp is raised, simply increasing pp uniformly does incur a computing cost for high order elements, primarily in the effort required to solve the large size nonlinear programming (NLP) problems that arise. An attempt to reduce computational costs was explored recently by Ngo and Tin-Loi [13] who used a pp-adaptive scheme. Despite the fact that plasticity was involved, the adaptive process was carried out in the preoptimization (elastic) stage only. The uniform pp-version FEM approach was modified, as is conventional [14], [15] and [16], by using some error estimators to evolve the original FE mesh into a nonuniform one for which the number of degrees of freedom (DOFs) is much less than in the similar (i.e. of the same geometrical discretization) pp-uniform mesh. Whilst reduction in the size of the problem showed a significant gain in computational time in the nonlinear optimization solution–the most time consuming phase of solving limit and shakedown problems–it was concluded that further computational efficiency was desirable. In the present paper, we investigate use of piecewise linear (PWL) yield surfaces in concert with the pp-adaptive FEM. The primary reason for this approach is to take advantage of the fact that linear programming (LP) problems are far easier to solve than NLP problems (see e.g. [17] and [18]), especially with the ready availability of state-of-the-art interior point LP solvers, as provided, for instance, via the GAMS modeling system [19] which is available freely through the NEOS server over the internet [20]. The advantages of PWL yield surfaces, leading to vastly improved computational efficiency and a mathematical structure more susceptible for theoretical developments, have been widely recognized since the 1970s (see e.g. [1] and [4]). Ad hoc [17], [21] and [22] as well as automatic methods [23], [24] and [25] for piecewise linearizing yield surfaces have been proposed in the literature. It should be noted that the latter schemes are not necessarily more efficient. For instance, Cannarozzi’s [24] nontraditional linearizing procedure appears to be attractive at first sight. It iteratively approaches the plastic admissibility domain through a sequence of circumscribing PWL yield polyhedra, with each polyhedron representing a better approximation to the actual yield domain than the preceding one. The procedure is theoretically capable of evaluating the limit load as accurately as required, if it is implemented with a complete search for points in the structure where the yield condition is violated. However, such a search can become very expensive, and may also require too many associated optimization runs (i.e. one run per linearization). In this paper, the adaptive scheme of Tin-Loi [25] is used. Basically, starting from a coarse discretization, the PWL surface is subsequently refined locally from information provided by the initial run. This local refinement has the effect of not only improving the solution accuracy but also keeping the LP problem size as small as possible. The organization of this paper is as follows. In Section 2, we present some fundamental notions related to the shakedown problem within our adopted static approach, including its appropriate discretization by the pp-version FEM. The adaptive scheme we use to evolve a uniform mesh into a nonuniform one at the elastic stage is also briefly presented. Section 3 deals with the construction of PWL yield surfaces using a heuristic, semi-adaptive approach. Some 2D numerical examples involving the limit and shakedown analyses of plane strain structures obeying von Mises yield criterion are presented in Section 4. This class of structures has been chosen in view of its propensity to lock when modeled using conventional displacement hh-elements. We conclude in Section 5.

Motivated by the need to increase the performance of our developed pp-FEM approach [12] for shakedown analysis, we propose, in this paper, the incorporation of adaptivity at two levels. Firstly, at the elastic computation phase, the pp-uniform mesh is evolved into a nonuniform one for which the associated nonlinear optimization problem is easier to solve in view of its reduced size. More importantly, however, a heuristic semi-adaptive piecewise linearization of the yield surface is also introduced to enable a far more tractable LP problem to be solved instead of an NLP problem. A key feature of this linearization is the description of the PWL yield surface in efficient vertex, rather than hyperplane, formulation so that the number of constraints involved in the LP problem is minimized. In effect, increased linearization does not result in an increase in constraints for the vertex description; it only increases the number of variables. Our extensive computational testing, four examples of which are given in this paper, attest to the effectiveness of our proposed approach. First, we need to note that the addition of these adaptive facilities does not impair in any way the locking-free performance of the pp-FEM approach, as clearly evidenced by the fact that all four examples concern plane strain von Mises instances, for which the traditional low order hh-approach would fail to solve if no special provisions, such as reduced integration, were employed. Moreover, the reductions in computation times achieved by the LP approach (using modern state-of-the-art and freely available interior point solvers) at the typical optimization “bottleneck” stage are remarkable in all instances. This is especially so when the associated reduction in accuracy is minimal. We have considered only 2D problems but extension to cater for 3D should not entail any conceptual difficulties.