Topology optimization based on finite strain plasticity

In this paper infinitesimal elasto-plastic based topology optimization is extended to finite strains. The employed model is based on rate-independent isotropic hardening plasticity and to separate the elastic deformation from the plastic deformation, use is made of the multiplicative split of the deformation gradient. The mechanical balance laws are solved using an implicit total Lagrangian formulation. The optimization problem is solved using the method of moving asymptotes and the sensitivity required to form convex separable approximations is derived using a path-dependent adjoint strategy. The optimization problem is regularized using a PDE-type filter. A simple boundary value problem where the plastic work is maximized is used to demonstrate the capability of the presented model. The numerical examples reveal that finite strain plasticity successfully can be combined with topology optimization.     

On two-scale adaptive FE analysis of micro-heterogeneous media with seamless scale-bridging

In principle, two approaches are possible for resolving strong material micro-heterogeneity: one approach is to adopt homogenization with the underlying assumption of scale separation, whereas the other approach is to completely resolve the fine scale(s) in a single-scale computation. The point of departure for this paper is a recently proposed algorithm for scale-transition such that the two extreme approaches are bridged in a “seamless” fashion. Numerical homogenization is carried out locally, where needed, based on the relation of the macro-scale mesh diameter to the typical length scale of the subscale structure. Moreover, the macroscale mesh adaptivity is driven by an estimation of discretization errors. In the present paper, we generalize this procedure by introducing two-scale adaptivity, whereby subscale discretization errors are viewed as model errors from the macroscale perspective. Numerical examples, adopting elastic–plastic subscale material properties, illustrate the principle and the effectiveness of the adaptive procedure.     

Comparison of models for finite plasticity: A numerical study

We introduce a general framework for the numerical approximation of finite multiplicative plasticity. The method is based on a fully implicit discretization in time which results in an iteratively evaluated stress response; the arising nonlinear problem is then solved by a Newton method where the linear subproblems are solved with a parallel multigrid method. The procedure is applied to models with different elastic free energy functionals and a plastic flow rule of von Mises type. In addition these models are compared to a recently derived frame indifferent approximation of finite multiplicative plasticity valid for small elastic strains which leads to linear balance equations. Rate independent and rate dependent realizations of the former models are considered. We demonstrate by various 3D simulations that the choice of the elastic free energy is not essential (for material parameters representative for metals) and that the new model gives the same response quantitatively and qualitatively as the standard models.     

Fast evolution of image manifolds and application to filtering and segmentation in 3D medical images

In many instances, numerical integration of space-scale PDEs is the most time consuming operation of image processing. This is because the scale step is limited by conditional stability of explicit schemes. We introduce the unconditionally stable semiimplicit linearized difference scheme that is fashioned after additive operator split (AOS) [Weickert, J. et al. (1998)], [Goldenberg, R et al., (2001)] for Beltrami and the subjective surface computation. The Beltrami flow [Kimmel, R. (1997) (1999)], [Sochen, N. et al. (1998)], is one of the most effective denoising algorithms in image processing. For gray-level images, we show that the flow equation can be arranged in an advection-diffusion form, revealing the edge-enhancing properties of this flow. This also suggests the application of AOS method for faster convergence. The subjective surface [Sarti, A. et al. (2002)] deals with constructing a perceptually meaningful interpretation from partial image data by mimicking the human visual system. However, initialization of the surface is critical for the final result and its main drawbacks are very slow convergence and the huge number of iterations required. We first show that the governing equation for the subjective surface flow can be rearranged in an AOS implementation, providing a near real-time solution to the shape completion problem in 2D and 3D. Then, we devise a new initialization paradigm where we first "condition" the viewpoint surface using the fast-marching algorithm. We compare the original method with our new algorithm on several examples of real 3D medical images, thus revealing the improvement achieved.     

Improving angular error via systematically designed near-circular Gaussian-based feature extraction operators

In image filtering, the ‘circularity’ of an operator is an important factor affecting its accuracy. For example, circular differential edge operators are effective in minimising the angular error in the estimation of image gradient direction. We present a general approach to the computation of scalable circular low-level image processing operators that is based on the finite element method. We show that the use of Gaussian basis functions within the finite element method provides a framework for a systematic and efficient design procedure for operators that are scalable to near-circular neighbourhoods through the use of an explicit scale parameter. The general design technique may be applied to a range of operators. Here we evaluate the approach for the design of the image gradient operator. We illustrate that this design procedure significantly reduces angular error in comparison to other well-known gradient approximation methods.     

Cyclic elastic-plastic dynamic analysis by the finite element method

A finite element approach for cyclic elastic-plastic dynamic analysis is presented. A hardening model suited for cyclic plasticity behavior is incorporated. It is composed of several yield surfaces, and nonlinear stress-strain curves can be included. The central difference timewise operator is employed to solve the equations of motion. Comparison is made with the Newmark operator. Numerical examples illustrate the effect of cyclic plastic deformations on the dynamic response of simple problems. Comparison is presented for the structural behavior as predicted by the present hardening model and by the isotropic hardening model.     

The partitioning of QSDF computation graphs

In a previous paper the Quasi Synchronous Data Flow (QSDF) model of computation was presented. A data flow graph is partitioned into disjoint directed paths, called computation paths, with the object of permitting a processor to execute along such a path in a sequential manner as far as possible. Such sequential execution will be facilitated to the extent that each operator along the path, when encountered by the executing processor, will have been supplied with any operand(s) it may require from nodes on other computation paths. If such is found not to be the case, execution of the path must be suspended and will be resumed at such time as the operand arrives.

In this paper, the question of how best to effect such a partition is addressed. A definition of optimality under the QSDF execution discipline is obtained and a closed form for the value of the cardinality of the partition is derived. An algorithm for obtaining an optimal partition is presented and some properties of the resulting partition are examined. Only static program behavior is considered in the analysis.     

Adaptive Allocation of Independent Tasks to Maximize Throughput

In this paper, we consider the task allocation problem for computing a large set of equal-sized independent tasks on a heterogeneous computing system where the tasks initially reside on a single computer (the root) in the system. This problem represents the computation paradigm for a wide range of applications such as SETI@home and Monte Carlo simulations. We consider the scenario where the systems have a general graph-structured topology and the computers are capable of concurrent communications and overlapping communications with computation. We show that the maximization of system throughput reduces to a standard network flow problem. We then develop a decentralized adaptive algorithm that solves a relaxed form of the standard network flow problem and maximizes the system throughput. This algorithm is then approximated by a simple decentralized protocol to coordinate the resources adaptively. Simulations are conducted to verify the effectiveness of the proposed approach. For both uniformly distributed and power law distributed systems, a close-to-optimal throughput is achieved, and improved performance over a bandwidth-centric heuristic is observed. The adaptivity of the proposed approach is also verified through simulations.     

On anisotropic flow rules in multiplicative elastoplasticity at finite strains

Hill’s anisotropic formulation of the flow rule is extended so as to fit in the realm of multiplicative finite strain plasticity. The anisotropic Hill-type yield criterion is formulated in terms of purely material quantities. The list of arguments of the flow function includes a non-symmetric material Eshelby-like stress tensor, as well as structural tensors that describe the anisotropy at hand. The formulation is exemplified on orthotropy, where three structural tensors are employed. The fact that the stress tensor is not symmetric necessitates a special treatment of the flow function, where representation theorems of tensor valued function with non-symmetric arguments are invoked. The consequences of such a definition on the resulting inelastic rate are discussed in full. It is shown that the corresponding resulting rate, as defined at the actual configuration, is not symmetric any more. Accordingly, the rate naturally includes a plastic material spin. Moreover, we deal with the theoretically interesting question of how to define spin-free rates. It is also demonstrated that the flow function must depend not only on the stress tensor and on adequate structural tensors, but also on the deformation itself in form of the right Cauchy-Green tensor C. However, this surprising dependency, which must obey a specific form, can be justified as physically meaningful. Various numerical examples of large plastic deformations of structural components are presented, that underpin the capabilities of the formulation.     

Nonlocal large deformation and localization behavior of metals

The present paper deals with a nonlocal continuum plasticity model which includes the dependence of the yield function on a nonlocal equivalent plastic strain measure. Particular attention is focused on the formulation of a generalized I₁–J₂ yield criterion to describe the effect of hydrostatic stress on the plastic flow properties of metals, and the nonlocal equivalent plastic strain is defined as a weighted average of the corresponding local measure taken over the neighboring material points of the body. The nonlocal yield condition leads to a partial differential equation which is solved using the finite difference method at each iteration of a loading step. Since this requires no additional boundary conditions, the displacement-based finite element procedure is governed by the standard principle of virtual work, and the associated linearized variational equations are obtained in the usual manner from a consistent linearization algorithm. Numerical simulations of the elastic–plastic deformation behavior of ductile metal specimens show the influence of the various model parameters on the deformation and localization prediction. The proposed nonlocal theory preserves well-posedness of the governing equations in the post-localization regime and prevents pathological mesh sensitivity of the numerical results. The internal length scale incorporated in the model determines the size of the localized shear bands.     

需求点随机的分批配送VRP模型与算法研究

针对城市配送中需求点不确定的现象, 在分批配送车辆路径问题中引入随机需求点进行研究.建立带修正的随机规划模型, 采用先验优化策略, 根据分批配送的特点, 在自适应大邻域搜索算法中引入改进的分割插入算子进行求解.在调整的Solomon算例上进行的测试表明, 允许分批配送在大部分算例中的费用低于不允许分批配送的情形.通过分析计算过程中各个算子权重变化, 确定性最差删除算子和随机删除算子在求解此类问题时表现较好; 贪婪插入算子、后悔插入算子表现较好; 而分割插入算子虽然权重较低, 但能对解产生质的影响.     

Stabilization Methods for Spectral Element Computations of Incompressible Flows

A stabilization method for the spectral element computation of incompressible flow problems is investigated. It is based on a filtering procedure which consists in filtering the velocity field by a spectral vanishing Helmholtz-type operator at each time step. Relationship between this filtering procedure and SVV-stabilization method, introduced recently in [JCP, 2004, 196(2), p680], is established. A number of numerical examples are presented to show the accuracy and stabilization capability of the method.     

A posteriori error estimator and adaptive procedures for computation of shakedown and limit loads on pressure vessels

Adaptive finite element procedures are presented for the computation of upper bounds estimates of limit and shakedown loads for pressure vessels. The method consists of an h-type adaptive mesh refinement strategy based upon an a-posteriori error estimator measured by the energy norm. The problem is formulated in a kinematic approach using Koiter's shakedown theorem. A constitutive model, for elastic-perfectly plastic materials, relates the plastic strains increments and curvatures to plastic multipliers through the flow law associated with a shell piecewise-linear yield surface (hexagonal prism). A consistent relationship between nodal displacements and nodal plastic multipliers is enforced by minimizing the strain residual between the total strain and the plastic strain increments, which is measured with respect to the energy norm. Discretization of the shell into finite elements allows the reduction of the problem to a minimization problem which is solved by linear programming.     

Unconditionally stable element-by-element algorithms for dynamic problems

A collection of results is presented regarding the consistency, stability and accuracy of operator split methods and product formula algorithms for general nonlinear equations of evolution. These results are then applied to the structural dynamics problem. The basic idea is to exploit an element-by-element additive decomposition of a particular form of the discrete dynamic equations resulting from a finite element discretization. It is shown that such a particular form of the discrete dynamic equations is obtained when velocity and stress are taken as unknowns. By applying the general product formula technique to the element-by-element decomposition, unconditionally stable algorithms are obtained that involve only element coefficient matrices. The storage requirements and operation counts are comparable to those of explicit methods. The method places no restriction on the topology of the finite element mesh.     

Cap models for clay strata to footing loads

Plasticity cap models are applied to obtain progressive failure solutions of flexible and smooth strip footings as well as rigid and rough strip footings on an overconsolidated stratum of clay. The comparative study of the elastic-plastic small deformation response of clay to footing loads is made within the framework of finite element analysis with different plasticity models based on a non-associated and the associated flow rule. More specifically, the analyses include the following features: (i) Drucker-Prager perfect plastic model with different methods in determining the material constants using the associated flow rule, (ii) Drucker-Prager perfect plastic model with the associated flow rule and a non-associated flow rule, and (iii) Drucker-Prager perfect plastic yield surface with a work-hardening plane cap and a work-hardening elliptic cap and their associated flow rules.     

A 3D finite element with planar symmetry for limit analysis computations

A formulation for finite element limit analysis of a certain class of 3D perfectly plastic solids governed by von Mises’ plasticity condition is presented. A planar symmetry constraint for both geometry and displacement field is assumed to analyze plane problems where the variable nature of transverse dissipation must be considered. A mixed locking free and low distortion sensitive element is formulated on the basis of the natural approach. The solution procedure exploits the kinematic theorem of limit analysis, cast in the form of a minimum problem for a convex but non-smooth dissipation functional. Applications to a notched specimen and to a bolted joint are presented to stress the importance of transverse effects in some problems commonly modeled as purely 2D.     

Multispike interactions in a stochastic model of spike-timing-dependent plasticity

Recently we presented a stochastic, ensemble-based model of spike-timing-dependent plasticity. In this model, single synapses do not exhibit plasticity depending on the exact timing of pre- and postsynaptic spikes, but spike-timing-dependent plasticity emerges only at the temporal or synaptic ensemble level. We showed that such a model reproduces a variety of experimental results in a natural way, without the introduction of various, ad hoc nonlinearities characteristic of some alternative models. Our previous study was restricted to an examination, analytically, of two-spike interactions, while higher-order, multispike interactions were only briefly examined numerically. Here we derive exact, analytical results for the general n-spike interaction functions in our model. Our results form the basis for a detailed examination, performed elsewhere, of the significant differences between these functions and the implications these differences have for the presence, or otherwise, of stable, competitive dynamics in our model.     

Total-FETI domain decomposition method for solution of elasto-plastic problems

In this paper we present an algorithm for the parallel solution of the rate-independent elasto-plastic problems with kinematic hardening. We assume the von Mises plastic criterion and the associated plastic flow rule. The time discretization is based on the implicit Euler method. The corresponding one-time-step problem is formulated in the incremental form with respect to the unknown displacement and discretized spatially by the finite element method. We use an ‘external’ algorithm based on a linearization of the elasto-plastic stress–strain relation by the corresponding tangential operator and we parallelize the arising linearized problem by the Total-FETI method. The numerical experiments were carried out using our novel C/C++ library FLLOP (FETI Light Layer On top of PETSc) at HECToR supercomputer located at EPCC, UK.