求解高维复合体函数的智能优化算法

由于粒子群算法在处理高维复杂函数时存在容易陷入局部最优的问题,提出了多种群子空间学习粒子群算法(SLPSO),采用多种群进化模式,在粒子更新公式中加入了全局最优粒子,加快了粒子收敛速度,同时在种群之间采用了交叉学习的方法,大大提高了算法的全局搜索能力.另外,还增加了一种子空间学习方法,充分地利用粒子的历史经验,有效地避免了陷入局部最优的问题.通过在高维基准测试函数的仿真实验表明,SLPSO算法的测试结果都明显优于其他两种算法,随着函数维数增加,SLPSO算法测试结果的下降幅度明显低于其他两种算法.在6个极其复杂的复合函数的测试中,SLPSO算法有2个测试函数结果非常接近理论值,其他4个也明显优于其他三种算法.     

基于模糊c均值算法和人工蜂群算法的无监督波段选择

波段选择是高光谱影像处理中一种重要的降维方法.在类标签不可获得的情况下,如何选择出一个具有代表性的波段子集是一个挑战性的问题.为了解决高光谱数据维数灾难以及光谱空间冗余的问题,基于模糊C均值算法(Fuzzy c-means,FCM),人工蜂群算法(Artificial Bee Colony, ABC)与极大熵准则(Maximum Entropy, ME),文章提出了一种新的无监督波段选择方法.该方法首先通过FCM算法将相似的波段划分到一个波段子集中,然后以ME为ABC算法中的适应度函数,寻找优化的波段子集.为验证该算法的有效性,在三个典型的高光谱数据集上,将所提出的方法和其它一些有效的波段选择算法进行了分类精度和计算时间对比.实验结果表明,所提出的算法不但可以得到高的分类精度,同时在计算时间上也具有明显的优势.     

带有摩擦的单边接触大变形问题的研究：（Ⅱ）非线性有限元解及应用

在文[1]所建立的增量变分力程的基础上,本文建立了单边接触弹塑性大变形问题的非线性有限元增量方程;进一步阐述了基于拖带坐标系的大变形有限元方法的特点,建立了实用的大变形接触模型.作为应用实例,计算了悬臂梁、厚圆板的弹性接触大变形和金属圆环弹塑性接触大变形问题,得到了令人满意的计算结果.     

采用复合规则约束的多种群自适应进化算法

针对种群固定的进化算法容易使个体集中分布在局部区域,不利于处理大尺度空间和多峰类型的优化问题,提出了一种多种群分布并且动态变化的种群自适应进化算法.采用Logistic模型模拟多个种群在有限资源下的竞争关系,设计了稳定性规则、熵规则和精英规则以确定不同种群的Logistic模型参数,从而控制种群数量的变化.同时,算法引入了算术内插和外插两种交叉算子,使得各个种群依据自身类型来缩小或扩展搜索空间.此外,算法还通过周期性的调整规则重新构建种群和分配资源.通过5组大尺度和多峰优化问题的测试结果表明,所提的种群自适应方法能够有效改善算法的寻优性能,在达到同等优化水平时所提算法消耗的函数调用次数为对比算法的61.08%～91.55%.     

一种基于支持向量域描述的区间数分类

研究了基于数据的区间数智能决策分析,提出了一种基于区间数的支持向量域多分类软计算方法,该方法可以直接处理特征空间为区间数的多分类问题,拓展了支持向量域多分类算法应用的范围.     

基于改进遗传算法的集合覆盖问题

集合覆盖问题是组合优化中的典型问题,在日常生活中有着广泛的应用.提出了一种改进遗传算法来解决集合覆盖问题.算法对标准遗传算法的改进主要表现在:1)结合启发式算法和随机生成,设计了新的产生初始种群的方法;2)引入修补操作处理不可行解使其转换成可行解;3)对重复个体进行处理再利用;4)对多点交叉进行推广,提出了新的交叉算子;5)针对可行解和不可行解,采取两种自适应多位变异操作.数值实验结果表明该算法对于解决规模较大的集合覆盖问题是有效的.     

弹性接触问题有限元分析的“广义子结构法”

本文提出了一种解决弹性接触问题的快速增量算法,措施是改革了通用的子结构法.     

带有摩擦的单边接触大变形问题的研究(Ⅰ)——增量变分方程

本文以非线性连续体几何场论为基本理论和方法,建立了拖带坐标下弹塑性大变形增量变分方程的更一般表示式.给出了二维、三维连续体接触边界变化率公式,得到了变边界接触大变形增量变分公式和速率型变分不等式,为有限元计算求解带有摩擦弹塑性大变形接触问题提供了理论基础.     

一种新的求解变分不等式的惯性双次梯度外梯度算法（英文）

当可行集为一光滑凸函数的下水平集时,文献[Optimization,2020,69(6):1237-1253]提出了一种惯性双次梯度外梯度算法来求解Hilbert空间中的单调且Lipschitz连续的变分不等式问题.该算法在每次迭代中仅需向一个半空间计算两次投影,并得到了算法的弱收敛结果.本文通过使用黏性方法以及在惯性步采用新的步长来修正该算法.在适当的假设条件下证明了新算法所生成的序列能强收敛到变分不等式的一个解.此外,新算法在每次迭代中也仅需向半空间计算两次投影.     

大规模多设施Weber问题的改进Cooper算法

 多设施Weber问题（multi-source Weber problem,MWP）是设施选址中的重要模型之一,而Cooper算法是求解MWP最为常用的数值方法.Cooper算法包含选址步和分配步,两步交替进行直至达到局部最优解.本文对Cooper算法的选址步和分配步分别引入改进策略,提出改进Cooper算法：选址步中将Weiszfeld算法和adaptive Barzilai-Borwein （ABB）算法结合,提出收敛速度更快的ABB-Weiszfeld算法求解选址子问题;分配步中提出贪婪簇分割策略来处理退化设施,由此进一步提出具有更好性质的贪婪混合策略.数值实验表明本文提出的改进策略有效地提高了Cooper算法的计算效率,改进算法有着更好的数值表现.     

3D dynamic modeling of powder forming processes via a simple and efficient node-to-surface contact algorithm

In this paper, a simple and efficient contact algorithm is presented for the evaluation of density distribution in three-dimensional dynamic modeling of powder compaction processes. The contact node-to-surface algorithm is employed to impose the contact constraints in large deformation frictional contact, and the contact frictional slip is modified by the Coulomb friction law to simulate the frictional behavior between the rigid punch and the work-piece. The 3D nonlinear contact friction algorithm is employed together with a double-surface cap plasticity model within the framework of large finite element deformation in order to predict the non-uniform relative density distribution during the dynamic simulation of powder die-pressing. The accuracy and robustness of contact algorithm is verified by the impact analysis of two elastic rods, which is compared with the analytical solution. Finally, the performance of computational schemes is illustrated in dynamic modeling of a set of powder components.     

Path-following and augmented Lagrangian methods for contact problems in linear elasticity

A certain regularization technique for contact problems leads to a family of problems that can be solved efficiently using infinite-dimensional semismooth Newton methods, or in this case equivalently, primal–dual active set strategies. We present two procedures that use a sequence of regularized problems to obtain the solution of the original contact problem: first-order augmented Lagrangian, and path-following methods. The first strategy is based on a multiplier-update, while path-following with respect to the regularization parameter uses theoretical results about the path-value function to increase the regularization parameter appropriately. Comprehensive numerical tests investigate the performance of the proposed strategies for both a 2D as well as a 3D contact problem.     

A solution approach for contact problems based on the dual interpolation boundary face method

The recently proposed dual interpolation boundary face method (DiBFM) has been shown to have a much higher accuracy and improved convergence rates compared with the traditional boundary element method. In addition, the DiBFM has the ability to approximate both continuous and discontinuous fields, and this provides a way to approximate the discontinuous pressure at a contact boundary. This paper presents a solution approach for two dimensional frictionless and frictional contact problems based on the DiBFM. The solution approach is divided into outer and inner iterations. In the outer iteration, the size of the contact zone is determined. Then the elements near the contact boundary are updated to approximate the discontinuous pressure. The inner iteration is used to determine the contact state (sticking or sliding), and is only performed for frictional contact problems. To make the system of equations solvable, the contact constraints and some supplementary equations are also given. Several numerical examples demonstrate the validity and high accuracy of the proposed approach. Furthermore, due to the continuity of elements in DiBFM and the detection of the contact boundary, the pressure oscillations near the contact boundary can be treated.     

One- and two-level Schwarz methods for variational inequalities of the second kind and their application to frictional contact

We present and analyze subspace correction methods for the solution of variational inequalities of the second kind and apply these theoretical results to non smooth contact problems in linear elasticity with Tresca and non-local Coulomb friction. We introduce these methods in a reflexive Banach space, prove that they are globally convergent and give error estimates. In the context of finite element discretizations, where our methods turn out to be one- and two-level Schwarz methods, we specify their convergence rate and its dependence on the discretization parameters and conclude that our methods converge optimally. Transferring this results to frictional contact problems, we thus can overcome the mesh dependence of some fixed-point schemes which are commonly employed for contact problems with Coulomb friction.     

Quasistatic Frictional Problems for Elastic and Viscoelastic Materials

We consider two quasistatic problems which describe the frictional contact between a deformable body and an obstacle, the so-called foundation. In the first problem the body is assumed to have a viscoelastic behavior, while in the other it is assumed to be elastic. The frictional contact is modeled by a general velocity dependent dissipation functional. We derive weak formulations for the models and prove existence and uniqueness results. The proofs are based on the theory of evolution variational inequalities and fixed-point arguments. We also prove that the solution of the viscoelastic problem converges to the solution of the corresponding elastic problem, as the viscosity tensor converges to zero. Finally, we describe a number of concrete contact and friction conditions to which our results apply.     

A dynamic viscoelastic contact problem with normal compliance,finite penetration and nonmonotone slip rate dependent friction

We consider a mathematical model which describes the dynamic evolution of a viscoelastic body in frictional contact with an obstacle. The contact is modelled with normal compliance and unilateral constraint, associated to a rate slip-dependent version of Coulomb’s law of dry friction. In order to approximate the contact conditions, we consider a regularized problem wherein the contact is modelled by a standard normal compliance condition without finite penetrations. For each problem, we derive a variational formulation and an existence result of the weak solution of the regularized problem is obtained. Next, we prove the convergence of the weak solution of the regularized problem to the weak solution of the initial nonregularized problem. Then, we introduce a fully discrete approximation of the variational problem based on a finite element method and on a second order time integration scheme. The solution of the resulting nonsmooth and nonconvex frictional contact problems is presented, based on approximation by a sequence of nonsmooth convex programming problems. Finally, some numerical simulations are provided in order to illustrate both the behaviour of the solution related to the frictional contact conditions and the convergence result.     

A nonsmooth optimization approach for time-dependent hemivariational inequalities

This paper is devoted to the study of time-dependent hemivariational inequality. We prove the existence and uniqueness of its solution, provide a fully discrete scheme, and reformulate this scheme as a series of nonsmooth optimization problems. The introduced theory is later applied to a sample quasistatic contact problem that describes a viscoelastic body in frictional contact with a foundation. This contact is governed by a nonmonotone friction law with dependence on the normal component of displacement and the tangential component of velocity. Finally, computational simulations are performed to illustrate the obtained results.     

速多极边界元法解的存在唯一性

本文对求解3维弹性摩擦接触问题的快速多极边界元法(FM- BEM)在数学理论上作了深入探讨.首先,利用向量和子空间理论找出快速优化广义极小残余算法(GMRES(m) )求解边界元方程组所满足的代数条件,使对工程用FM- BEM解的研究转化为对代数问题的讨论,然后,分三步证明了FM- BEM解的存在唯一性,为FM- BEM求解弹性摩擦接触工程问题提供强有力的数学支撑.     

A class of optimal control problems for piezoelectric frictional contact models

We consider control problems for a mathematical model describing the frictional bilateral contact between a piezoelectric body and a foundation. The material’s behavior is modeled with a linear electro–elastic constitutive law, the process is static and the foundation is assumed to be electrically conductive. Both the friction and the electrical conductivity conditions on the contact surface are described with the Clarke subdifferential boundary conditions. The weak formulation of the problem consists of a system of two hemivariational inequalities. We provide the results on existence and uniqueness of a weak solution to the model and, under additional assumptions, the continuous dependence of a solution on the data. Finally, for a class of optimal control problems and inverse problems, we prove the existence of optimal solutions.     

Coupled FEM-BEM for elastoplastic contact problems using Lagrange multipliers

The coupling of the elastoplastic finite element and elastic boundary element methods for two-dimensional frictionless contact stress analysis is presented. Interface traction matching (boundary element approach), which involves the force terms in the finite element analysis being transformed to tractions, is chosen for the coupling method. The analysis at the contact region is performed by the finite element method, and the Lagrange multiplier approach is used to apply the contact constraints. Since the analyses of elastoplastic problems are non-linear and involve iterative solution, the reduced size of the final system of equations introduced by combining the two methods is very advantageous, especially for contact problems where the nature of the problem also involves an iterative scheme.