一类约束不可微优化问题的区间极大熵方法

本文研究求解不等式约束离散ｍｉｎｉｍａｘ问题的区间算法,其中目标函数和约束函数是 Ｃ~１类函数．利用罚函数法和极大熵函数思想将问题转化为无约束可微优化问题,讨论了极大熵函数的区间扩张,证明了收敛性等性质,提出了无解区域删除原则,建立了区间极大熵算法,并给出了数值算例．该算法是收敛、可靠和有效的．     

非线性整数规划的一个新的无参数填充函数算法

离散填充函数是一种用于求解多极值优化问题最优解的一种行之有效的方法.已被证明对于求解大规模离散优化问题是有效的.本文基于改进的离散填充函数定义,构造了一个新的无参数填充函数,并在理论上给出了证明,提出了一个新的填充函数算法.该填充函数无需调节参数,而且只需极小化一次目标函数.数值结果表明,该算法是高效的、可行的.     

基于磨光反演映射的拓扑优化ICM方法

对结构拓扑优化ICM(independent continuous mapping)方法中的磨光映射和过滤映射加以拓广,利用反演映射极限形式的磨光特性构造其与过滤映射相协调的复合映射.由于该复合映射的叠加离散效应,首先引入幂函数和正弦函数的复合形式过滤函数,用ICM方法建立位移约束下重量最小为目标的连续体结构拓扑优化模型,并采用二次规划精确对偶算法进行求解.再将求得的离散解为主的连续最优解依照动态反演策略,用最佳阈值和理性反演函数求出最严格的0-1离散解,给出了拓扑优化"离散→连续"和"连续→离散"先后相反的二阶段解法.基于MATLAB软件平台开发了相应的拓扑优化计算程序,给出的数值算例对该文提出的方法进行验证,结果表明:该方法计算效率高,最优解灰度单元少,反演后结构重量更小,并且能够计算出更合理的结构拓扑.     

一种单位化的增量梯度算法

研究目标函数是若干光滑函数和的可分离优化问题,提出了一种单位化增量梯度算法。该算法每次子迭代只需要计算一个（或几个）分量函数的单位负梯度方向作为迭代方向。在一定条件下,证明了采用发散步长的单位化增量梯度算法的收敛性。作为应用,新算法和Bertsekas D P,Tsitsikils J N提出的（没有单位化）增量梯度算法分别用来求解稳健估计问题和源定位问题。数值例子表明,新算法优于（没有单位化）增量梯度算法。     

框架结构多目标优化方法

提出了一种框架结构多目标优化方法。基于力法给出了框架结构的等效刚度、最大弯矩随结构材料、几何等设计参数的变化,利用动能等效给出了框架结构的等效质量,进而得到框架结构一阶固有频率随结构材料、几何等参数的变化,用有限元方法验证了所得频率、弯矩公式的正确性。以固有频率、最大弯矩为优化目标函数构造框架结构的多目标优化模型,通过单目标优化得到频率、弯矩最优值,以单目标优化值为基础构造全局目标优化函数,将多目标优化转化为单目标优化。     

非线性约束优化的光滑化平方根罚函数

介绍一种非线性约束优化的不可微平方根罚函数,为这种非光滑罚函数提出了一个新的光滑化函数和对应的罚优化问题,获得了原问题与光滑化罚优化问题目标之间的误差估计. 基于这种罚函数,提出了一个算法和收敛性证明,数值例子表明算法对解决非线性约束优化具有有效性.     

连续时间变界截尾的两步算法及它的一个应用

一、引言自从五十年代 Robbins-Monro 提出随机逼近算法以来,不仅离散时间量测,而且连续时间量测的算法也引起人们很大的关注。对连续时间的量测误差也从最简单的独立增量过程研究到相关增量及其他一般的过程。 近年来对离散时间的量测的算法出现了一些新的想法。文献[5]中提出了随机变界截尾算法,去掉了对算法事先要求有界的条件。研究了两步算法,去掉了对某一 Liapunov 函数存在性的要求。[8]进一步把[5]和[6]的方法结合起来,在较弱的条件下求解了优化问题。本文把这些思想结合起来用到连续时间量测,给出一个连续时间的二步算法,用随机变界截尾的方法,既不事先假定算法有界,也不额外假定某一 Liapunov 函数的存在性,在较弱条件下证明了算法向函数的极值几乎处处收敛。本文还以组合债券问题为例,说明在实际中确实出现连续量测下求极值的问题。我们对组合债券这一实例给出了数学模型。它正好可以用本文给出的理论方法解决。     

关于九参数拟协调板元

1980年以来,唐立民等提出一种拟协调元法,用来构造椭圆型方程的离散格式.粗略地讲,该法将每个单元上的能量表达式所含导数项的面积分(假设问题二维的),用格林公式转化为单元边界上的线积分,然后采用某种数值积分,将线积分进行离散.对只含函数项的面积分,也用相应的数值积分进行离散.用此法计算单元刚度阵,比较简单、灵活.     

基于CUDA-GPU架构的超二次曲面离散单元并行算法

大规模离散元的并行计算通常基于理想的球体单元，然而自然界或工业生产中普遍存在的是由非球形颗粒组成的复杂体系，其在不同空间尺度下的动力学行为及力学性质与球形颗粒具有显著差异.基于连续函数包络的超二次曲面单元能有效地构造非球形颗粒的几何形态，并通过非线性Newton迭代算法准确计算单元间的作用力.针对非球形颗粒间接触判断的复杂性及其大规模离散元计算的需求，该文发展了基于CUDA-GPU构架下超二次曲面单元并行算法.该方法在球形颗粒并行计算的基础上，通过核函数建立单元包围盒的粗判断列表及Newton迭代的细判断列表，并优化并行算法和内存访问模式以提高算法的计算效率.为检验超二次曲面并行算法的可靠性，对非球形颗粒的流动过程进行离散元模拟, 并与试验结果进行对比验证.在此基础上，进一步分析了颗粒单元不同长宽比和表面尖锐度对颗粒材料流动特性的影响，为非球形颗粒材料的大规模离散元模拟提供一种有效的数值方法.     

强Wolfe线搜索下一个新的优化算法

本文主要研究了一个新的优化算法.首先,利用给出的新的公式和强Wolfe线搜索,证明了该算法在不要求搜索方向满足共轭性条件下具有充分下降性和全局收敛性;其次,利用目标函数为一致凸函数的假设,证明了该算法具有线性收敛速率;最后,利用数值试验,验证了新算法是有效的、可行的.     

Multiple-Load Truss Topology and Sizing Optimization: Some Properties of Minimax Compliance

This paper considers the mathematical properties of discrete or discretized mechanical structures under multiple loadings which are optimal w.r.t. maximal stiffness. We state a topology and/or sizing problem of maximum stiffness design in terms of element volumes and displacements. Multiple loads are handled by minimizing the maximum of compliance of all load cases, i.e., minimizing the maximal sum of displacements along an applied force. Generally, the problem considered may contain constraints on the design variables. This optimization problem is first reformulated in terms of only design variables. Elastic equilibrium is hidden in potential energy terms. It is shown that this transformed objective function is convex and continuous, including infinite values. We deduce that maximum stiffness structures are dependent continuously on the bounds of the element volumes as parameters. Consequently, solutions to sizing problems with small positive lower bounds on the design variables can be considered as good approximations of solutions to topology problems with zero lower bounds. This justifies heuristic approaches such as the well-known stress-rationing method for solving truss topology problems.     

An efficient sensitivity computation strategy for the evolutionary structural optimization (ESO) of continuum structures subjected to self-weight loads

This work presents a modified version of the evolutionary structural optimization procedure for topology optimization of continuum structures subjected to self-weight forces. Here we present an extension of this procedure to deal with maximum stiffness topology optimization of structures when different combinations of body forces and fixed loads are applied. Body forces depend on the density distribution over the design domain. Therefore, the value and direction of the loading are coupled to the shape of the structure and they change as the material layout of the structure is modified in the course of the optimization process. It will be shown that the traditional calculation of the sensitivity number used in the ESO procedure does not lead to the optimum solution. Therefore, it is necessary to correct the computation of the element sensitivity numbers in order to achieve the optimum design. This paper proposes an original correction factor to compute the sensitivities and enhance the convergence of the algorithm. The procedure has been implemented into a general optimization software and tested in several numerical applications and benchmark examples to illustrate and validate the approach, and satisfactorily applied to the solution of 2D, 3D and shell structures, considering self-weight load conditions. Solutions obtained with this method compare favourably with the results derived using the SIMP interpolation scheme.     

Evolutionary structural optimization for problems with stiffness constraints

This paper presents a simple evolutionary procedure based on finite element analysis to minimize the weight of structures while satisfying stiffness requirements. At the end of each finite element analysis, a sensitivity number, indicating the change in the stiffness due to removal of each element, is calculated and elements which make the least change in the stiffness; of a structure are subsequently removed from the structure. The final design of a structure may have its weight significantly reduced while the displacements at prescribed locations are kept within the given limits. The proposed method is capable of performing simultaneous shape and topology optimization. A wide range of problems including those with multiple displacement constraints, multiple load cases and moving loads are considered. It is shown that existing solutions of structural optimization with stiffness constraints can easily be reproduced by this proposed simple method. In addition some original shape and layout optimization results are presented.     

Bi-directional evolutionary topology optimization of geometrically nonlinear continuum structures with stress constraints

This paper proposes a design method to maximize the stiffness of geometrically nonlinear continuum structures subject to volume fraction and maximum von Mises stress constraints. An extended bi-directional evolutionary structural optimization (BESO) method is adopted in this paper. BESO method based on discrete variables can effectively avoid the well-known singularity problem in density-based methods with low density elements. The maximum von Mises stress is approximated by the p-norm global stress. By introducing one Lagrange multiplier, the objective of the traditional stiffness design is augmented with p-norm stress. The stiffness and p-norm stress are considered simultaneously by the Lagrange multiplier method. A heuristic method for determining the Lagrange multiplier is proposed in order to effectively constrain the structural maximum von Mises stress. The sensitivity information for designing variable updates is derived in detail by adjoint method. As for the highly nonlinear stress behavior, the updated scheme takes advantages from two filters respectively of the sensitivity and topology variables to improve convergence. Moreover, the filtered sensitivity numbers are combined with their historical sensitivity information to further stabilize the optimization process. The effectiveness of the proposed method is demonstrated by several benchmark design problems.     

Variational sensitivity analysis and changes in the material configuration

This contribution is concerned with some aspects of variational design sensitivity analysis in the physical and material configuration. Sensitivity analysis is a branch of structural optimization, e.g. shape or topology optimization. In these disciplines we consider variations of the material configuration and we are interested in the change of the state variables and the objective functional due to these variations. In the context of structural optimization this is termed as design sensitivity analysis. The sensitivities are required in order to solve the corresponding Lagrangian equation within standard nonlinear programming algorithms. In many engineering applications, the energy functional of the problem is used as the objective functional. In this paper, we consider variations of the energy with respect to the state and the design and we investigate sensitivity relations for the physical and material problem. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

An evolution equation based approach to topology optimization

The objective of topology optimization is to find a mechanical structure with maximum stiffness and minimal amount of used material for given boundary conditions [2]. There are different approaches. Either the structure mass is held constant and the structure stiffness is increased or the amount of used material is constantly reduced while specific conditions are fulfilled. In contrast, we focus on the growth of a optimal structure from a void model space and solve this problem by introducing a variational problem considering the spatial distribution of structure mass (or density field) as variable [3]. By minimizing the Gibbs free energy according to Hamilton's principle in dynamics for dissipative processes, we are able to find an evolution equation for the internal variable describing the density field. Hence, our approach belongs to the growth strategies used for topology optimization. We introduce a Lagrange multiplier to control the total mass within the model space [1]. Thus, the numerical solution can be provided in a single finite element environment as known from material modeling. A regularization with a discontinuous Galerkin approach for the density field enables us to suppress the well-known checkerboarding phenomena while evaluating the evolution equation within each finite element separately [4]. Therefore, the density field is no additional field unknown but a Gauß-point quantity and the calculation effort is strongly reduced. Finally, we present solutions of optimized structures for different boundary problems. (© 2016 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Topology Optimization of Flexible Multibody Systems Using Equivalent Static Loads and Displacement Fields

Topology optimization techniques are applied in most cases for static applications. However, recently topology optimization procedures for structures under dynamic loads have been the focus of several studies. In this work, a topology optimization scheme for flexible multibody systems using equivalent static loads and displacement fields is investigated. The optimization problem is formulated using a homogenization method, more precisely, the solid isotropic material with penalization (SIMP) approach. The objective function in the optimization problem is the compliance and the method of moving asymptotes is used as optimizer. The objective function and the sensitivities are computed directly from the displacement field computed in the dynamic simulation. The examples of a 2-arm manipulator and a slider-crank mechanism are presented and the results are discussed to verify the improved dynamical behavior through this optimization method. (© 2014 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Combination of fuzzy ranking and simulated annealing to improve discrete fracture inversion

Mathematical and computational modelling of discrete fracture networks is critical for the exploration and development of natural resource reservoirs. Utilizing the concept of fuzzy memberships, this paper advances the fundamental understanding in fracture network inversion and presents a systematic procedure to solve the most important problem in global optimization (simulated annealing): objective function formulation. First, a comprehensive field study identifies all potential components of an objective function. The components are statistical, geostatistical, mathematical and spatial measurements of fracture properties (location, orientation and size). The characteristic measurements can be input in parametric or non-parametric, discrete or continuum forms. Next, sensitivity analysis and fuzzy logic are combined to rank the candidate components based on their effects on the final objective function value and optimization convergence. The process negates guess works in objective function formulation by automatic selection of highly ranked components and their corresponding weighting factors. A case study is applied to a surface DFN in New York. The derived discrete fracture network is representative of the field data.     

Una formulación de mínimo peso con restricciones en tensión en optimización topológica de estructuras

Optimum design of structures has been traditionally focused on the analysis of shape and dimensions optimization problems. However, more recently a new discipline has emerged: the topology optimization of the structures. This discipline states innovative models that allow to obtain optimal solutions without a previous definition of the type of structure being considered. These formulations obtain the optimal topology and the optimal shape and size of the resulting elements. The most usual formulations of the topology optimization problem try to obtain the structure of maximum stiffness. These approaches maximize the stiffness for a given amount of material to be used. These formulations have been widely analyzed and applied in engineering but they present considerable drawbacks from a numerical and from a practical point of view. In this paper the author propose a different formulation, as an alternative to maximum stiffness approaches, that minimizes the weight and includes stress constraints. The advantages of this kind of formulations are crucial since the cost of the structure is minimized, which is the most frequent objective in engineering, and they guarantee the structural feasibility since stresses are constrained. In addition, this approach allows to avoid some of the drawbacks and numerical instabilities related to maximum stiffness approaches. Finally, some practical examples have been solved in order to verify the validity of the results obtained and the advantages of the proposed formulation.     

应力和位移约束下连续体结构拓扑优化

同时考滤应力和位移约束的连续体结构拓扑优化问题,很难用现有的均匀方法或变密度方法等求解。主要困难在于难以建立应力和位移约束与拓扑设计变量间显式关系式;即使建立了这种关系,也由于优化问题规模过大,利用常规的数学规划方法难以求解。隋允康、杨德庆曾提出了基于独立连续拓扑变量及映射变换（ＩＣＭ）的桁架结构拓扑优化模型。本文在此基础上,建立了以重量为目标,考虑应力和位移约束的连续体结构拓扑优化模型,并推导出