On ε-Constraint Based Methods for the Generation of Pareto Frontiers

primal-dual-based method头部航程规划问题（Primal-Dual based method）引言：头部航程规划问题是一种常见的优化问题，它的目标是在给定航线网络和航线容量的情况下，找到一种合理的飞行计划，以最小化总飞行成本并满足航线容量限制。

这个问题在航空运输管理中具有重要的实际意义。

本文将介绍一种基于Primal-Dual方法的解决方案，解释其原理和应用。

第一部分：问题描述在头部航程规划问题中，我们考虑一个航线网络G=(V, E)，其中V是航线节点的集合，E是连接这些节点的航线的集合。

每条航线e属于E都有一个容量限制Ce，表示该航线能够承载的最大飞行量。

同时，每条航线e 可以有一个飞行成本Ce，表示单位距离的运营成本。

我们的目标是找到一种最优的头部航程规划方案，以最小化总飞行成本，并且满足航线容量限制。

第二部分：Primal-Dual算法简介Primal-Dual算法是一种常用的解决线性规划问题的方法，它将原问题转化为对偶问题，并通过不断调整原问题和对偶问题之间的关系，逐步找到最优解。

在头部航程规划问题中，我们可以将其转化为一个线性规划问题，然后应用Primal-Dual算法求解。

第三部分：Primal-Dual算法的步骤1. 初始化：- 将所有航线的飞行量设置为0，即x_e=0，这意味着初始航程规划方案中没有航线被使用。

- 创建对偶变量p_v，表示每个节点的流平衡。

初始化为0。

2. 进入迭代过程：- 根据当前航线规划方案（即飞行量x_e）计算对偶变量p_v。

这可以通过使用Bellman-Ford算法来计算最短路径和对应的费用，将费用作为对偶变量p_v。

- 根据对偶变量p_v，计算每条航线的费用约束，即c_e = Ce - p_u + p_v，其中u和v是航线e的起始节点和终止节点。

- 利用线性规划方法求解最小费用流问题，即最小化目标函数Σ_c_e*x_e，同时满足容量约束Σ_x_e≤Ce，其中x_e是航线e的飞行量。

primal-dual-based method -回复什么是原始对偶法方法（primal-dual-based method）？原始对偶法方法是一种优化算法，用于求解具有原始对偶结构的优化问题。

在这种方法中，优化问题被分解成原始问题和对偶问题，并通过迭代地求解它们之间的对偶性关系来获得最优解。

原始对偶法方法的一般框架如下：1.确定原始问题：将给定的优化问题表达为其原始形式，并定义目标函数和约束条件。

原始问题通常是一个最小化问题，其目标函数为要最小化的目标值。

2.确定对偶问题：通过定义对偶变量和对偶约束来构造对偶问题。

对偶问题通常是一个最大化问题，其目标函数是原始问题的拉格朗日对偶函数。

3.建立对偶性关系：根据原始问题和对偶问题之间的对偶性关系，构建一个线性方程系统。

4.迭代求解：通过迭代地更新原始变量和对偶变量来求解原始问题和对偶问题。

在每次迭代中，通过将原始变量和对偶变量向梯度下降方向更新，逐渐向最优解逼近。

5.停止准则：定义一个停止准则，用于判断迭代是否收敛。

常用的停止准则包括目标函数值的变化或对偶变量的变化小于某个阈值。

下面以一个具体的优化问题来说明原始对偶法方法的应用。

假设我们有一个线性规划问题，即最小化目标函数f(x) = c^T x，其中x 是一个n 维向量，满足线性约束Ax = b，其中A 是一个m×n 的矩阵，b 是一个m 维向量。

我们可以将该线性规划问题分解为原始问题和对偶问题。

原始问题：最小化f(x) = c^T x约束条件Ax = b对偶问题：最大化g(y) = b^T y约束条件A^T y + s = c，其中s 是一个n 维向量，称为松弛变量。

我们可以根据原始问题和对偶问题之间的对偶性关系，建立一个线性方程系统。

对于原始问题，我们可以使用梯度下降法进行迭代求解。

在每次迭代中，根据梯度下降方向更新原始变量x，直到满足停止准则。

对于对偶问题，我们可以使用梯度上升法进行迭代求解。

Simulate While You Design with Inventor and Simulation DFM Robert Savage – Advanced Solutions Inc., Education Specialist - Primary SpeakerKevin J. Smedley – FS-Elliott Co., LLC – Co-SpeakerMD5511 Simulation DFM or “Design for Manufacturing” software was created by Autodesk after they purchased Moldflow. It was inspired by a Moldflow software called “Moldflow Cad Doctor”, but DFM is not a Moldflow product… It is, however, powered by Moldflow. It was created to be used by designers and engineers to help verify the part design and cut down on costly mistakes at the part level. DFM loads inside your 3D design tool to give you near-real-time feedback on the quality of your plastic part, as you design it. Simulation DFM software reviews things, such as nominal wall thickness, draft angle, material cost, part recyclability, sink marks, fill patterns, and more, to grade your part in the categories of manufacturability, cost, and environmental impact. In this class, we will look at how we can analyze the part files, during the design process. We will also investigate the effect changes to the configurations and the parts have.Learning ObjectivesAt the end of this class, you will be able to:1. Learn how to navigate the Simulation DFM interface2. Learn how to analyze the model for possible defects3. Learn how to make changes to the model's material4. Review cost effects of part changesAbout the SpeakersRobert Savage is an Education Specialist at Advanced Solutions, Inc. He is a 15-year design veteran who has designed everything from molds and molded parts to robots. At Remotec, a division of Northrop Grumman Corporation, he spent 5 years as a designer in the development group, as well as CAD and Vault Administrator. He is an Autodesk Certified Instructor and a Certified Inventor Professional. He has used Inventor software since its inception, as well as being well-versed in a variety of other design software. He has 10 years of experience teaching 3D design software, including Product Design Suite Ultimate software, Factory Design Suite Ultimate software, Simulation Moldflow software, and Vault Professional software. Email: *****************************Kevin Smedley is currently Cad Manager for Product Engineering at FS-Elliott Company, a Global Centrifugal Compressor Manufacturer with facilities in Asia, Europe, South America and headquartered in Export, Pennsylvania. He has 26 years of Autodesk applications experience in design/drafting and as an Autodesk VAR channel Technical/Mechanical Applications Engineer. He spent 16 years, in the Autodesk channel, implementing 3D processes and creating Vault database strategies, along with training users. Kevin has created best practices and workflows, as a Certified Inventor and Vault Specialist. He’s presented lectures and hands-on labs at Autodesk University for 7 years, as wel l as attending AU for 10 years. Kevin’s philosophy is summed up in three words… Communication, Consistency, and Standards. Kevin continues pursuing best practices and processes for the Product Design Suite and Vault Pro data management. Email: ***********************ContentsLearning Objectives (1)About the Speakers (1)Chapter 1 (3)Navigate the Interfaces (3)Widget (4)Standalone (7)Chapter 2 (8)Analyzing the Model (8)Indicators (8)Pull Direction (11)Injection Locations (14)Animation Tools (11)Finished Part View (15)Saving and Exporting (17)Chapter 3 (18)Making Changes (18)Configure Rules (18)Materials (24)Chapter 4 (26)Cost Effect of Part Changes (26)Mold Cost (26)Material Cost (27)Production Cost (27)Chapter 1Navigate the Interfaces∙Widget∙StandaloneSimulation DFM (Simulation Design for Manufacturing) is part of the Autodesk Digital Prototyping solutions. Simulation DFM also runs as a standalone tool and as a widget inside your Autodesk Inventor software. In addition, it will work in PTC Creo, and Solidworks. Simulation DFM is designed to show real-time evaluation of your plastic components, focusing on Manufacturability, Cost Efficiency, and Plastic Material Impact. It is intended to be used during the 3D part design process by both designers and engineers to minimize errors and improve design efficiencies. The widget usesgreen/yellow/red indicators for visual feedback in areas,such as wall thickness, draft angles, sink marks, fillpatterns, and part recyclability. As the design processbuilds, the DFM evaluates and updates the output.Autodesk has a portfolio of Simulation products which fits into a number of different situations. The portfolio make up includes the following:Mechanical SimulationMoldflow SimulationComposite SimulationComputational Fluid Dynamics SimulationStructural AnalysisWidgetSimulation DFM will run as a widget inside Autodesk Inventor, as well as PTC Creo, and Solidworks, giving the same analysis for each of the products. The widget is similar to a toolbar, but instead of icons, it has indicators that change colors from Green to Yellow to Red, showing the severity of the issue. In this class, we are going to be focusing on using DFM, while running Inventor software. It can be set to load automatically in the software by going to the Tools Tab then selecting the Add-Ins Button. Inside the Add-In Manage, you can set Autodesk Simulation DFM to Load/Unload or Load Automatically.The Widget has 3 main parts: the Indicators, the Menu, and the Refresh Button. The Widget Indicator displays the results of the analysis in the 3 different categories: Manufacturability, Cost Efficiency, and Plastic Material Impact.The Widget Menu is a dropdown menu in the lower right hand corner of the page. In the drop down, you will find the options and toolbars for defining injector locations and Fill Animation, as well as A Finish Part Previewer. The Refresh tool, which can be set to automatically refresh, will show as a button in the upper left hand corner of the tools. The Refresh tool is what you would select to tell it to refresh your calculations.If an indicator has a yellow triangle with an exclamation mark on it, you have received an Alert. This means you have one or more subcategories with an issue, which you need to address. If you select on the indicator, it will display the subcategory with the issue. If you select the subcategory, it will then display the issue on the model.IndicatorsPlastic Material ImpactCost EfficiencyManufacturabilityMenuRefreshThe Refresh can be used as an Automatic Refresh by checking the option in the dropdown list on the widget, or used as a manual refresh to be selected as desired.ToolbarsThere are 2 toolbars available in the dropdown menu on the widget. They are named, “Injection location” and “Animation”. In the software, they display as toolbars. However, in the standalone they show up as tabs in the ribbon.Injection locationAnimation (Fill Animation)StandaloneSimulation DFM will not only run inside of a variety of design software, but it also runs as a standalone where it will read and analyze 11 different file formats. The standalone will run the same tools as the add-on software. In the standalone, you have the same tools as the widget, but you have an increase in file formats that you can work with.It also contains 2 different ways of using the tools. You can choose to use the widget or the ribbon to access the different tools.Chapter 2Analyzing the Model∙Indicatorso Manufacturabilityo Cost Efficiencyo Plastic Material Impact∙Pull Direction∙Fill Tools∙Injector Locations∙Finished Part Preview∙Exporting ResultsIn this Chapter, we are going to focus on how the tools are used to analyze your design. The tools are used the same, whether you are using the Widget or the Standalone tools. The Indicator tools will give you visual feedback on the thresholds that are configured in the system. The toolbars will allow you to set injector locations and view the fill results based on the injector locations.Indicatorso Manufacturabilityo Cost Efficiencyo Plastic Material ImpactManufacturability, Cost Efficiency, and Plastic Material Impact are indicators on the widget, but they are rollups of different subcategories. They indicate the overall effect of the different subcategories by using different colors. By selecting the indicator, it will list the subcategories that are not at optimal levels in the alerts section and all levels will be listed in the information section.There are 3 buttons in the Indicators. They each have different areas they are evaluating. The button changes colors from green to yellow to red, and if you put your cursor over the indicator it will give you an overall percentage of the cumulative subcategories. If you select the indicator, you will get a drop down that has two tabs: Alerts and Information. In the Alerts area, you will see any subcategory that has a higher chance of causing an issue in manufacturing. Selecting a subcategory will also highlight the areas on the screen where the issue exists.The Information tab will let you see the results from all of the sub categories. Selecting a subcategory will either giving you an on-the-screen representation of the issues, or a dialog box indicating why the rating is not high.In the design software, the Indicators and subcategories recalculate, as you make modifications to the model or when you tell it to refresh. Unless, you have it set to Auto-refresh.Animation ToolsThe Animation tool will play, pause, and replay the fill animation, based on the injector locations. This can be used with the Injector tools to see how the part will fill. These tools are used together to determine results, like, Weld Lines and Sink Marks.Pull DirectionThe Pull Direction is a tool that is only available in the standalone. The icon for the tool launches a tab on the ribbon, allowing you to define or move the pull direction of the part. It launches a triad that you can place and adjust, by making adjustments to the Red, Green or Blue circular rings.Injection LocationsThe Injection Locations tool will allow you to Move, Add and Delete Injector Locations. There is, by default, 1 injector placed by Simulation DFM that is used in several of the calculations. By moving or adding locations, it will prompt the software to redo the calculations and change the results.Finished Part ViewThe Finished Part View tool is located in the dropdown list on the Widget and the Ribbon on the standalone. It generates a rendered view of the part to help show any defects that may be caused by fill or weld line issues. Inside the rendered window, you can make adjustments to the material under the Edit drop down menu and highlight the defects under the view dropdown. You can also save the file out as an image file under the File dropdown.Saving and ExportingWhen you save the file in Inventor, any adjustments you make to the DFM portion of the file is saved, so that the next time you open the file the changes are still there. If you have the software, you can also export the information to the Autodesk MoldFlow.My Analysis ProcessIn this section, I want to go through the DFM Analysis process I use on plastic parts. Every part is different, as is every user. That is why this is intended as a general process guideline.1. Open or create the file to the point that you are ready to start analyzing it.2. Update or turn on automatic updates.3. Review manufacturability alerts and information.4. Check your fill location or locations.5. Make necessary changes to the part.6. Repeat 3 and 4 till you have it the way you want it.7. Review cost and material alerts and information to see if changes are needed.8. Run finish part preview.9. Save and export if needed.Chapter 3Making Changes∙Configure Rules∙MaterialsIn this section, we are going to look at the different areas you can configure to changes the results that DFM displays in the indicators. We are also going to see how to change materials and what affect the material changes have to the results.Configure RulesThe Configuration Rules can be accessed from the drop down menu on the widget or the ribbon in the standalone. In the rules configuration area, you can select what rules affect the indicators. In the sub categories, you can adjust the percentages displayed in the indicator.Note:∙Modifications to the configuration will apply to all parts∙Not all parameters can be configured∙Because an indicator result is a combination of several parameters, the effect of altering a parameter may not be obvious∙Realistic parameter values need to be appliedIn the rule configuration dialog box, you can adjust what calculations are included in the indicators by checking or unchecking the rule in the main indicator headers.Manufacturability: Wall Thickness calculates the percentage above and below the median wall thickness.Manufacturability: Undercuts calculate the percentage of undercuts to total surface.Manufacturability: Draft Angle calculates a percentage of drafted faces, based on a defined minimum draft angle.Manufacturability: Weld Lines will check weakness and visual flaws caused by joining flow.Manufacturability: Sink Mark checks for surface depressions.Manufacturability: Filling calculates fill ability, based on pressure location and wall thickness.Manufacturability: Knife Edge (New in the 2015 software version) allows you to define a minimum thin edge.Cost Efficiency: Mold Cost is based on size and complexity of the mold tooling.Cost Efficiency: Material Cost calculates raw material cost and volume.Cost Efficiency: Production Cost is based on cycle times.Plastic Material Impact: Carbon Footprint calculates the carbon dioxide required to produce each.Plastic Material Impact: Embodied Energy calculates the energy in BTU’s required to make each part.Plastic Material Impact: Recyclability calculates material recycle rate.Plastic Material Impact: Embodied Water calculates the water required to produce the part.If you have a threshold you want to meet, most of these allow you to add or append the percentages.You also have a button at the bottom of the page that will “Restore All Defaults”.MaterialsMaterial is changed differently in the standalone verses the widget in Inventor. In the widget (Inside Autodesk Inventor), the material is changed in the same way you would for any other part. You can change it using the physical tab, in the part iProperties, or the material drop down menu on the ribbon. Both of these options use the Autodesk Materials Library.In the standalone, you would use the Select Material tool on the Home tab of the ribbon. In the Material selection box, materials are listed by name, as well as by common uses. It also displays a cost indicator and plastic material impact indicator. You can sort the materials by Application and Attributes, using the drop down list at the top of the box.Chapter 4Cost Effect of Part Changes∙Mold Cost∙Material Cost∙Production CostIn this section, we are going to look at how cost effect is determined based on three main categories: Mold Cost, Material Cost, and Production Cost. There is no true way for the software to calculate the actual cost of the mold, the material, or the production of the part. What it does is display the Cost Efficiency as a percentage of the three categories. These calculations adjust with every effected change made to the part, both physical and material.Mold CostMold Cost checks the size and complexity of the mold tooling. As you make changes to the parts to reduce size and complexity, the system will automatically recalculate and adjust.Material CostThe Material Cost is based on part size and raw material cost. As you adjust the model size or material, whether it is in the CAD software or in the standalone application, it will adjust the calculations.Production CostProduction Cost is based on Part Cycle Time, which is based on material properties, injector quantities and fill pattern.ConclusionSimulate While You Design with Inventor and Simulation DFM ∙Learn how to navigate the Simulation DFM interface∙Learn how to analyze the model for possible defects∙Learn how to make changes to the model's material∙Review cost effects of part changesIn this class, we looked at using Simulation DFM inside of Inventor and as a standalone, to improve the design process of creating plastic parts. Simulation DFM is an analysis tool that allows you to see the effect different variables have on the overall manufacturability of a part. You can also see real-time updates on how changes to the file and material effect the overall manufacturability of the part.In Chapter 1, we went through the interface of both the widget that runs inside of Inventor, as well as the standalone interface. We also looked at the different areas in the interface and what they mean.In Chapter 2, we explored the different analysis tools that exist within the software and how to apply them to the model.In Chapter 3, we talked about the effect changes to the rules, as well as the material, have on the calculations.In Chapter 4, we looked at how the cost effect was calculated and how changes effect the results of the calculations.I want to thank you for attending this class. I know you have a lot of class choices at AU2015. If you have any questions on the information, please feel free to email me at*****************************.I would also like to thank Kevin Smedley for his assistance in this class. He not only kept me on track, but on point.。

Introduction to CAD & CAE计算机辅助设计与仿真_东南大学中国大学mooc课后章节答案期末考试题库2023年1.Physical-based simulation is a key function in _____software参考答案:Computer-aided Engineer2.Images have infinite resolution.参考答案:错误3.Rendering is an operation that converts _________参考答案:graphics to images4.The basic elements in graphics include _________参考答案:Points_Lines_Colors5.The following optimization problem is a/an ___.【图片】参考答案:Constraint nonlinear optimization problem6.Particle systems can be used to preview the effects like fire and crowds.参考答案:正确7.If a soft object is rotated and translated to a new pose, its distortion energy is____.参考答案:8.We may use ____ as proxies for fast query of collision detection betweenmeshes.参考答案:Bounding boxes_Bounding spheres9.Octree is more memory-efficient than uniform occupancy grid.参考答案:正确10.In rigid body simulation, we need to compute ____ for updating thetranslation of the rigid object.参考答案:Force11.In rigid body simulation, we need to compute ____ for updating the rotation ofthe rigid object.参考答案:Torque12.The iso-levels of a function 【图片】 is a/an ____.参考答案:Implicit surface13.Rendering is an operation that converts ______.参考答案:graphics to images14.As the simulated annealing algorithm proceeds, negative values of 【图片】are increasingly easy to accept.参考答案:错误15. A point has __ degrees of freedom to move in a solid cylinder, and has __degrees of freedom to move on a cylindrical surface.参考答案:3; 216.Physical-based simulation is a key function in ____ software.参考答案:Computer-aided Engineer17.Here we use 【图片】 to represent a point. Now we have an implicit curvewhose function is 【图片】. When we feed the point 【图片】 into thefunction 【图片】, the result is ___, so we know that the point 【图片】 is outside the curve. When we feed the point 【图片】 into the function 【图片】, the result is ___, so we know that the point \beta is inside the curve.参考答案:18; -218.Typically, a hinged joint has ___ translation DOF and ___ rotational DOF.参考答案:0; 119.If we want to detect collision between two balls and enumerate each pairfaces on the two balls for intersection detection. Both two balls are lowresolution with 32 triangular surfaces. So that we should operate ___detections.参考答案:102420.An ellipse can be represented as【图片】. This curve representation is a/an____.参考答案:Parametric curve21.In an inside-outside test, we shoot a ray from a point and count theintersections. If the point is inside the polygon, the number of theintersections is most likely to be ___.参考答案:Odd22.____ is used to simulate the motion of the end-effector of an articulated robotmanipulator (robot arm), if we are given the rotation angles of each joint.参考答案:Forward kinematics23. A rotation matrix is an orthogonal matrix.参考答案:正确24.Shear is a nonlinear transform.参考答案:错误25. A ball-socket joint has _____.参考答案:Three rotational degrees of freedom.26.Finite element method is usually used to simulate a/an___.参考答案:soft object27.If in a task of car-body design, we hope the shape is as smooth as possible.This is the ____ of the design problem.参考答案:Design goal28.____ can be used to find the local minimum of an optimization problem.参考答案:Gradient descent29.Which of the following statements regarding simulated annealing is false?____参考答案:The temperature is typically set to increase over the iterations of the optimization.30.Boolean operation on solids includes ___.参考答案:Union_Intersect_Subtraction31.NURBS curve is a representation of ____.参考答案:Parametric curve32.Here we use【图片】to represent a point. Now we have an implicit curvewhose function is【图片】. When we feed the point【图片】into thefunction【图片】, the result is ___, so we know that the point【图片】is outside the curve. When we feed the point【图片】into the function【图片】, the result is ___, so we know that the point \beta is inside the curve.参考答案:18; -233.三维物体的点云表示缺少连接拓扑信息。

primal-dual-based method -回复什么是基于原始-对偶方法（primal-dual based method）？基于原始-对偶方法是一种数值优化技术，用于解决约束优化问题。

与传统的优化方法相比，基于原始-对偶方法具有很多优势，特别适用于求解大规模问题。

它结合了原始问题和对偶问题之间的关系，通过不断调整原始问题和对偶问题的变量来求得最优解。

原始问题是指包含有约束条件的优化问题，而对偶问题是根据原始问题的约束条件所构建的一个辅助问题。

原始问题和对偶问题之间的关系通过拉格朗日乘子法来建立。

在基于原始-对偶方法中，我们首先将原始问题转化为对偶问题。

这可以通过建立拉格朗日函数来实现。

拉格朗日函数是原始问题的目标函数和约束函数的线性组合，其中约束通过乘以拉格朗日乘子来表示。

然后，对拉格朗日函数求极小值，就可以得到对偶问题。

接下来，我们通过交替迭代的方式，不断调整原始问题和对偶问题的变量，来逐步逼近最优解。

在每一次迭代中，我们首先通过求解对偶问题的最优解来更新原始问题的变量。

我们使用对偶问题的最优解作为原始问题的界限（dual-feasible bound），来约束原始问题的搜索空间。

然后，我们更新对偶问题的变量，以找到更好的原始问题界限。

这样，我们反复进行迭代，直到原始问题和对偶问题的解逐渐趋于最优解。

基于原始-对偶方法有很多应用领域，其中最重要的领域之一是凸优化。

凸优化是一类具有凸约束条件的优化问题，凸优化问题在实际应用中非常常见。

基于原始-对偶方法能够高效地求解凸优化问题，且具有良好的收敛性能。

此外，基于原始-对偶方法还被广泛应用于机器学习和计算机视觉等领域。

在这些领域中，我们常常需要处理大规模数据和高维特征，传统的优化方法往往难以处理这些问题。

基于原始-对偶方法通过对偶问题的更新来加速收敛速度，并提高求解效率。

总之，基于原始-对偶方法是一种非常强大的优化技术，用于解决约束优化问题。

Toward the Next Generation of Recommender Systems:A Survey of the State-of-the-Art andPossible ExtensionsGediminas Adomavicius,Member,IEEE,and Alexander Tuzhilin,Member,IEEE Abstract—This paper presents an overview of the field of recommender systems and describes the current generation ofrecommendation methods that are usually classified into the following three main categories:content-based,collaborative,and hybrid recommendation approaches.This paper also describes various limitations of current recommendation methods and discussespossible extensions that can improve recommendation capabilities and make recommender systems applicable to an even broader range of applications.These extensions include,among others,an improvement of understanding of users and items,incorporation of the contextual information into the recommendation process,support for multcriteria ratings,and a provision of more flexible and less intrusive types of recommendations.Index Terms—Recommender systems,collaborative filtering,rating estimation methods,extensions to recommender systems.æ1I NTRODUCTIONR ECOMMENDER systems have become an important research area since the appearance of the first papers on collaborative filtering in the mid-1990s[45],[86],[97]. There has been much work done both in the industry and academia on developing new approaches to recommender systems over the last decade.The interest in this area still remains high because it constitutes a problem-rich research area and because of the abundance of practical applications that help users to deal with information overload and provide personalized recommendations, content,and services to them.Examples of such applica-tions include recommending books,CDs,and other products at [61],movies by MovieLens [67],and news at VERSIFI Technologies(formerly )[14].Moreover,some of the vendors have incorporated recommendation capabilities into their commerce servers[78].However,despite all of these advances,the current generation of recommender systems still requires further improvements to make recommendation methods more effective and applicable to an even broader range of real-life applications,including recommending vacations,certain types of financial services to investors,and products to purchase in a store made by a“smart”shopping cart[106]. These improvements include better methods for represent-ing user behavior and the information about the items to be recommended,more advanced recommendation modeling methods,incorporation of various contextual information into the recommendation process,utilization of multcriteria ratings,development of less intrusive and more flexible recommendation methods that also rely on the measures that more effectively determine performance of recommen-der systems.In this paper,we describe various ways to extend the capabilities of recommender systems.However,before doing this,we first present a comprehensive survey of the state-of-the-art in recommender systems in Section2.Then, we identify various limitations of the current generation of recommendation methods and discuss some initial ap-proaches to extending their capabilities in Section3.2T HE S URVEY OF R ECOMMENDER S YSTEMS Although the roots of recommender systems can be traced back to the extensive work in cognitive science[87], approximation theory[81],information retrieval[89], forecasting theories[6],and also have links to management science[71]and to consumer choice modeling in marketing [60],recommender systems emerged as an independent research area in the mid-1990s when researchers started focusing on recommendation problems that explicitly rely on the ratings structure.In its most common formulation, the recommendation problem is reduced to the problem of estimating ratings for the items that have not been seen by a user.Intuitively,this estimation is usually based on the ratings given by this user to other items and on some other information that will be formally described below.Once we can estimate ratings for the yet unrated items,we can recommend to the user the item(s)with the highest estimated rating(s).More formally,the recommendation problem can be formulated as follows:Let C be the set of all users and let S be the set of all possible items that can be recommended, such as books,movies,or restaurants.The space S of.G.Adomavicius is with the Carlson School of Management,University ofMinnesota,32119th Avenue South,Minneapolis,MN55455.E-mail:gedas@.. A.Tuzhilin is with the Stern School of Business,New York University,44West4th Street,New York,NY10012.E-mail:atuzhili@.Manuscript received8Mar.2004;revised14Oct.2004;accepted10Dec.2004;published online20Apr.2005.For information on obtaining reprints of this article,please send e-mail to:tkde@,and reference IEEECS Log Number TKDE-0071-0304.1041-4347/05/$20.00ß2005IEEE Published by the IEEE Computer Societypossible items can be very large,ranging in hundreds of thousands or even millions of items in some applications,such as recommending books or CDs.Similarly,the user space can also be very large—millions in some cases.Let u be a utility function that measures the usefulness of item s to user c ,i.e.,u :C ÂS !R ,where R is a totally ordered set (e.g.,nonnegative integers or real numbers within a certain range).Then,for each user c 2C ,we want to choose such item s 02S that maximizes the user’s utility.More formally:8c 2C;s 0c ¼arg max s 2Su ðc;s Þ:ð1ÞIn recommender systems,the utility of an item is usually represented by a rating ,which indicates how a particular user liked a particular item,e.g.,John Doe gave the movie “Harry Potter”the rating of 7(out of 10).However,as indicated earlier,in general,utility can be an arbitrary function,including a profit function.Depending on the application,utility u can either be specified by the user,as is often done for the user-defined ratings,or is computed by the application,as can be the case for a profit-based utility function.Each element of the user space C can be defined with a profile that includes various user characteristics,such as age,gender,income,marital status,etc.In the simplest case,the profile can contain only a single (unique)element,such as User ID.Similarly,each element of the item space S is defined with a set of characteristics.For example,in a movie recommendation application,where S is a collection of movies,each movie can be represented not only by its ID,but also by its title,genre,director,year of release,leading actors,etc.The central problem of recommender systems lies in that utility u is usually not defined on the whole C ÂS space,but only on some subset of it.This means u needs to be extrapolated to the whole space C ÂS .In recommender systems,utility is typically represented by ratings and is initially defined only on the items previously rated by the users.For example,in a movie recommendation application (such as the one at ),users initially rate some subset of movies that they have already seen.An example of a user-item rating matrix for a movie recommendation application is presented in Table 1,where ratings are specified on the scale of 1to 5.The “ ”symbol for some of the ratings in Table 1means that the users have not rated the corresponding movies.Therefore,the recommendation engine should be able to estimate (predict)the ratings of the nonrated movie/user combinations and issue appropriate recommendations based on these predictions.Extrapolations from known to unknown ratings are usually done by 1)specifying heuristics that define the utility function and empirically validating its performanceand 2)estimating the utility function that optimizes certain performance criterion,such as the mean square error.Once the unknown ratings are estimated,actual recommendations of an item to a user are made by selecting the highest rating among all the estimated ratings for that user,according to (1).Alternatively,we can recommend the N best items to a user or a set of users to an item.The new ratings of the not-yet-rated items can be estimated in many different ways using methods from machine learning,approximation theory,and various heuristics.Recommender systems are usually classified according to their approach to rating estimation and,in the next section,we will present such a classification that was proposed in the literature and will provide a survey of different types of recommender systems.The commonly accepted formulation of the recommendation problem was first stated in [45],[86],[97]and this problem has been studied extensively since then.Moreover,recommender systems are usually classified into the following categories,based on how recommendations are made [8]:.Content-based recommendations :The user will be recommended items similar to the ones the user preferred in the past;.Collaborative recommendations :The user will berecommended items that people with similar tastes and preferences liked in the past;.Hybrid approaches :These methods combine colla-borative and content-based methods.In addition to recommender systems that predict the absolute values of ratings that individual users would give to the yet unseen items (as discussed above),there has been work done on preference-based filtering ,i.e.,predicting the relative preferences of users [22],[35],[51],[52].For example,in a movie recommendation application,prefer-ence-based filtering techniques would focus on predicting the correct relative order of the movies,rather than their individual ratings.However,this paper focuses primarily on rating-based recommenders since it constitutes the most popular approach to recommender systems.2.1Content-Based MethodsIn content-based recommendation methods,the utility u ðc;s Þof item s for user c is estimated based on the utilities u ðc;s i Þassigned by user c to items s i 2S that are “similar”to item s .For example,in a movie recommendation application,in order to recommend movies to user c ,the content-based recommender system tries to understand the commonalities among the movies user c has rated highly in the past (specific actors,directors,genres,subject matter,TABLE 1A Fragment of a Rating Matrix for a Movie Recommender Systemetc.).Then,only the movies that have a high degree of similarity to whatever the user’s preferences are would be recommended.The content-based approach to recommendation has its roots in information retrieval[7],[89]and information filtering[10]research.Because of the significant and early advancements made by the information retrieval and filtering communities and because of the importance of several text-based applications,many current content-based systems focus on recommending items containing textual information,such as documents,Web sites(URLs),and Usenet news messages.The improvement over the tradi-tional information retrieval approaches comes from the use of user profiles that contain information about users’tastes, preferences,and needs.The profiling information can be elicited from users explicitly,e.g.,through questionnaires, or implicitly—learned from their transactional behavior over time.More formally,let ContentðsÞbe an item profile,i.e.,a set of attributes characterizing item s.It is usually computed by extracting a set of features from item s(its content)and is used to determine the appropriateness of the item for recommendation purposes.Since,as mentioned earlier, content-based systems are designed mostly to recommend text-based items,the content in these systems is usually described with keywords.For example,a content-based component of the Fab system[8],which recommends Web pages to users,represents Web page content with the 100most important words.Similarly,the Syskill&Webert system[77]represents documents with the128most informative words.The“importance”(or“informative-ness”)of word k j in document d j is determined with some weighting measure w ij that can be defined in several different ways.One of the best-known measures for specifying keyword weights in Information Retrieval is the term frequency/inverse document frequency(TF-IDF)measure[89]that is defined as follows:Assume that N is the total number of documents that can be recommended to users and that keyword k j appears in n i of them.Moreover,assume that f i;j is the number of times keyword k i appears in document d j.Then, T F i;j,the term frequency(or normalized frequency)of keyword k i in document d j,is defined asT F i;j¼f i;jmax z f z;j;ð2Þwhere the maximum is computed over the frequencies f z;j of all keywords k z that appear in the document d j. However,keywords that appear in many documents are not useful in distinguishing between a relevant document and a nonrelevant one.Therefore,the measure of inverse document frequencyðIDF iÞis often used in combination with simple term frequencyðT F i;jÞ.The inverse document frequency for keyword k i is usually defined asIDF i¼log Nn i:ð3ÞThen,the TF-IDF weight for keyword k i in document d j is defined asw i;j¼T F i;jÂIDF ið4Þand the content of document d j is defined asContentðd jÞ¼ðw1j;...w kjÞ:As stated earlier,content-based systems recommend items similar to those that a user liked in the past[56],[69], [77].In particular,various candidate items are compared with items previously rated by the user and the best-matching item(s)are recommended.More formally,let ContentBasedP rofileðcÞbe the profile of user c containing tastes and preferences of this user.These profiles are obtained by analyzing the content of the items previously seen and rated by the user and are usually constructed using keyword analysis techniques from information retrieval.For example,ContentBasedP rofileðcÞcan be defined as a vector of weightsðw c1;...;w ckÞ,where each weight w ci denotes the importance of keyword k i to user c and can be computed from individually rated content vectors using a variety of techniques.For example,some averaging approach,such as Rocchio algorithm[85],can be used to compute ContentBasedP rofileðcÞas an“average”vector from an individual content vectors[8],[56].On the other hand,[77]uses a Bayesian classifier in order to estimate the probability that a document is liked.The Winnow algorithm[62]has also been shown to work well for this purpose,especially in the situations where there are many possible features[76].In content-based systems,the utility function uðc;sÞis usually defined as:uðc;sÞ¼scoreðContentBasedP rofileðcÞ;ContentðsÞÞ:ð5ÞUsing the above-mentioned information retrieval-based paradigm of recommending Web pages,Web site URLs, or Usenet news messages,both ContentBasedP rofileðcÞof user c and ContentðsÞof document s can be represented as TF-IDF vectors~w c and~w s of keyword weights.Moreover, utility function uðc;sÞis usually represented in the information retrieval literature by some scoring heuristic defined in terms of vectors~w c and~w s,such as the cosine similarity measure[7],[89]:uðc;sÞ¼cosð~w c;~w sÞ¼~w cÁ~w sjj~w c jj2Âjj~w s jj2¼P Ki¼1w i;c w i;sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiP Ki¼1w2i;cqffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiP Ki¼1w2i;sq;ð6Þwhere K is the total number of keywords in the system.For example,if user c reads many online articles on the topic of bioinformatics,then content-based recommenda-tion techniques will be able to recommend other bioinfor-matics articles to user c.This is the case because these articles will have more bioinformatics-related terms(e.g.,“genome,”“sequencing,”“proteomics”)than articles on other topics and,therefore,ContentBasedP rofileðcÞ,as defined by vector~w c,will represent such terms k i with high weights w ic.Consequently,a recommender system using the cosine or a related similarity measure will assign higher utility uðc;sÞto those articles s that have high-weighted bioinformatics terms in~w s and lower utility to the ones where bioinformatics terms are weighted less.Besides the traditional heuristics that are based mostly on information retrieval methods,other techniques for content-based recommendation have also been used,such as Bayesian classifiers[70],[77]and various machine learning techniques,including clustering,decision trees, and artificial neural networks[77].These techniques differ from information retrieval-based approaches in that they calculate utility predictions based not on a heuristic formula,such as a cosine similarity measure,but rather are based on a model learned from the underlying data using statistical learning and machine learning techni-ques.For example,based on a set of Web pages that were rated as“relevant”or“irrelevant”by the user,[77]uses the naive Bayesian classifier[31]to classify unrated Web pages.More specifically,the naive Bayesian classifier is used to estimate the following probability that page p j belongs to a certain class C i(e.g.,relevant or irrelevant) given the set of keywords k1;j;...;k n;j on that page:PðC i j k1;j&...&k n;jÞ:ð7ÞMoreover,[77]uses the assumption that keywords are independent and,therefore,the above probability is proportional toPðC iÞYxPðk x;j j C iÞ:ð8ÞWhile the keyword independence assumption does not necessarily apply in many applications,experimental results demonstrate that naı¨ve Bayesian classifiers still produce high classification accuracy[77].Furthermore,both Pðk x;j j C iÞand PðC iÞcan be estimated from the underlying training data.Therefore,for each page p j,the probability PðC i j k1;j&...&k n;jÞis computed for each class C i and page p j is assigned to class C i having the highest probability[77].While not explicitly dealing with providing recommen-dations,the text retrieval community has contributed several techniques that are being used in content-based recommen-der systems.One example of such a technique would be the research on adaptive filtering[101],[112],which focuses on becoming more accurate at identifying relevant documents incrementally by observing the documents one-by-one in a continuous document stream.Another example would be the work on threshold setting[84],[111],which focuses on determining the extent to which documents should match a given query in order to be relevant to the user.Other text retrieval methods are described in[50]and can also be found in the proceedings of the Text Retrieval Conference (TREC)().As was observed in[8],[97],content-based recommender systems have several limitations that are described in the rest of this section.2.1.1Limited Content AnalysisContent-based techniques are limited by the features that are explicitly associated with the objects that these systems recommend.Therefore,in order to have a sufficient set of features,the content must either be in a form that can be parsed automatically by a computer(e.g.,text)or the features should be assigned to items manually.While information retrieval techniques work well in extracting features from text documents,some other domains have an inherent problem with automatic feature extraction.For example,automatic feature extraction methods are much harder to apply to multimedia data,e.g.,graphical images, audio streams,and video streams.Moreover,it is often not practical to assign attributes by hand due to limitations of resources[97].Another problem with limited content analysis is that,if two different items are represented by the same set of features,they are indistinguishable.Therefore,since text-based documents are usually represented by their most important keywords,content-based systems cannot distin-guish between a well-written article and a badly written one,if they happen to use the same terms[97].2.1.2OverspecializationWhen the system can only recommend items that score highly against a user’s profile,the user is limited to being recommended items that are similar to those already rated. For example,a person with no experience with Greek cuisine would never receive a recommendation for even the greatest Greek restaurant in town.This problem,which has also been studied in other domains,is often addressed by introducing some randomness.For example,the use of genetic algorithms has been proposed as a possible solution in the context of information filtering[98].In addition,the problem with overspecialization is not only that the content-based systems cannot recommend items that are different from anything the user has seen before.In certain cases,items should not be recommended if they are too similar to something the user has already seen,such as a different news article describing the same event.Therefore, some content-based recommender systems,such as Daily-Learner[13],filter out items not only if they are too different from the user’s preferences,but also if they are too similar to something the user has seen before.Furthermore,Zhang et al.[112]provide a set of five redundancy measures to evaluate whether a document that is deemed to be relevant contains some novel information as well.In summary,the diversity of recommendations is often a desirable feature in recommender systems.Ideally,the user should be pre-sented with a range of options and not with a homogeneous set of alternatives.For example,it is not necessarily a good idea to recommend all movies by Woody Allen to a user who liked one of them.2.1.3New User ProblemThe user has to rate a sufficient number of items before a content-based recommender system can really understand the user’s preferences and present the user with reliable recommendations.Therefore,a new user,having very few ratings,would not be able to get accurate recommendations.2.2Collaborative MethodsUnlike content-based recommendation methods,collabora-tive recommender systems(or collaborative filtering systems) try to predict the utility of items for a particular user based on the items previously rated by other users.More formally, the utility uðc;sÞof item s for user c is estimated based on the utilities uðc j;sÞassigned to item s by those users c j2C who are“similar”to user c.For example,in a movierecommendation application,in order to recommend movies to user c ,the collaborative recommender system tries to find the “peers”of user c ,i.e.,other users that have similar tastes in movies (rate the same movies similarly).Then,only the movies that are most liked by the “peers”of user c would be recommended.There have been many collaborative systems developed in the academia and the industry.It can be argued that the Grundy system [87]was the first recommender system,which proposed using stereotypes as a mechanism for building models of users based on a limited amount of information on each individual ing stereotypes,the Grundy system would build individual user models and use them to recommend relevant books to each ter on,the Tapestry system relied on each user to identify like-minded users manually [38].GroupLens [53],[86],Video Recommender [45],and Ringo [97]were the first systems to use collaborative filtering algorithms to automate prediction.Other examples of collaborative recommender systems include the book recommendation system from ,the PHOAKS system that helps people find relevant information on the WWW [103],and the Jester system that recommends jokes [39].According to [15],algorithms for collaborative recom-mendations can be grouped into two general classes:memory-based (or heuristic-based )and model-based .Memory-based algorithms [15],[27],[72],[86],[97]essentially are heuristics that make rating predictions based on the entire collection of previously rated items by the users.That is,the value of the unknown rating r c;s for user c and item s is usually computed as an aggregate of the ratings of some other (usually,the N most similar)users for the same item s :r c;s ¼aggr c 02^Cr c 0;s ;ð9Þwhere ^Cdenotes the set of N users that are the most similar to user c and who have rated item s (N can range anywhere from 1to the number of all users).Some examples of the aggregation function are:ða Þr c;s ¼1N Xc 02^C r c 0;s ;ðb Þr c;s¼k X c 02^Csim ðc;c 0ÞÂr c 0;s ;ðc Þr c;s ¼"rc þk Xc 02^Csim ðc;c 0ÞÂðr c 0;s À"rc 0Þ;ð10Þwhere multiplier k serves as a normalizing factor and is usually selected as k ¼1 P c 02^Cj sim ðc;c 0Þj ,and where the average rating of user c ,"rc ,in (10c)is defined as 1"r c ¼À1 j S c j ÁX s 2S cr c;s;where S c ¼f s 2S j r c;s ¼ g :ð11ÞIn the simplest case,the aggregation can be a simple average,as defined by (10a).However,the most common aggregation approach is to use the weighted sum,shown in (10b).The similarity measure between users c and c 0,sim ðc;c 0Þ,is essentially a distance measure and is used as aweight,i.e.,the more similar users c and c 0are,the more weight rating r c 0;s will carry in the prediction of r c;s .Note that sim ðx;y Þis a heuristic artifact that is introduced in order to be able to differentiate between levels of user similarity (i.e.,to be able to find a set of “closest peers”or “nearest neighbors”for each user)and,at the same time,simplify the rating estimation procedure.As shown in (10b),different recommendation applications can use their own user similarity measure as long as the calculations are normalized using the normalizing factor k ,as shown above.The two most commonly used similarity measures will be described below.One problem with using the weighted sum,as in (10b),is that it does not take into account the fact that different users may use the rating scale differently.The adjusted weighted sum,shown in (10c),has been widely used to address this limitation.In this approach,instead of using the absolute values of ratings,the weighted sum uses their deviations from the average rating of the correspond-ing user.Another way to overcome the differing uses of the rating scale is to deploy preference-based filtering [22],[35],[51],[52],which focuses on predicting the relative prefer-ences of users instead of absolute rating values,as was pointed out earlier in Section 2.Various approaches have been used to compute the similarity sim ðc;c 0Þbetween users in collaborative recom-mender systems.In most of these approaches,the similarity between two users is based on their ratings of items that both users have rated.The two most popular approaches are correlation and cosine-based .To present them,let S xy be the set of all items corated by both users x and y ,i.e.,S xy ¼f s 2S j r x;s ¼ &r y;s ¼ g .In collaborative recom-mender systems,S xy is used mainly as an intermediate result for calculating the “nearest neighbors”of user x and is often computed in a straightforward manner,i.e.,by computing the intersection of sets S x and S y .However,some methods,such as the graph-theoretic approach to collaborative filtering [4],can determine the nearest neighbors of x without computing S xy for all users y .In the correlation-based approach,the Pearson correlation coefficient is used to measure the similarity [86],[97]:sim ðx;y Þ¼Ps 2S xyðr x;s À"rx Þðr y;s À"r y ÞffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiP s 2S xyðr x;s À"r x Þ2P s 2S xyðr y;s À"ry Þ2r :ð12ÞIn the cosine-based approach [15],[91],the two users x and y are treated as two vectors in m -dimensional space,where m ¼j S xy j .Then,the similarity between two vectors can be measured by computing the cosine of the angle between them:sim ðx;y Þ¼cos ð~x ;~y Þ¼~x Á~yjj ~x jj 2Âjj ~y jj 2¼Ps 2S xy r x;s r y;s ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiP s 2S xyr 2x;sr ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiP s 2S xyr2y;sr ;ð13Þwhere ~x Á~y denotes the dot-product between the vectors ~xand ~y .Still another approach to measuring similarity between users uses the mean squared difference measure1.We use the r c;s ¼ notation to indicate that item s has not been rated by user c .。

全国计算机四级考试选择题及答案全国计算机四级考试选择题及答案全国计算机等级考试采用全国统一命题，统一考试的形式。

今天，店铺特意为大家推荐全国计算机四级考试选择题及答案，希望大家喜欢！选择题(1) 8位二进制原码表示整数的范围是____。

A) 0~+128 B) -128~+128 C) 0~+127 D)-127~+127(2) 在计算机运行时，建立各寄存器之间的“数据通路”并完成取指令和执行指令全过程的部件是____。

A) 时序产生器 B) 程序计数器 C) 操作控制器 D) 指令寄存器(3) 在数据传送过程中，为发现误码甚至纠正误码，通常在源数据数据上附加“校验码”。

其中功能较强的是____。

A）奇偶校验码 B）循环冗余码 C）交叉校验码 D) 横向校验码(4) 设有下三角距阵A[0..10,0..10],按行优先顺序存放其非零元素，则元素A[5,5]的存放地址为____。

A) 110 B) 120 C) 130 D) 140(5) 若一棵二叉树中，度为2的节点数为9，则该二叉树的叶结点数为____。

A) 10 B) 11 C) 12 D) 不确定(6) 设根结点的层次为0，则高度为k的二叉树的最大结点数为____。

A)2k-1 B) 2k C) 2k+1-1 D) 2k+1(7) 设待排序关键码序列为 (25,18,9,33,67,82,53,95,12,70)，要按关键码值递增的顺序排序，采取以第一个关键码为分界元素的快速排序法，第一趟排序完成后关键码为33被放到了第几个位置？____。

A) 3 B) 5 C) 7 D) 9(8) 如下所示是一个带权连通无向图，其最小生成树各边权的总和为____。

A) 24 B) 25 C) 26 D) 27(9) 下列命题中为简单命题的是____。

A)张葆丽和张葆华是亲姐妹 B) 张明和张红都是大学生C) 张晖或张旺是河北省人 D) 王际广不是工人(10) 设p:天下大雨，q:我骑自行车上班。

excelbaseentity method 使用-回复ExcelBaseEntity是一种在Excel中使用的方法，用于简化和优化数据管理和分析任务。

它为用户提供了一些有用的功能和方法，从而使数据处理更加高效和方便。

在本文中，我们将一步一步地回答关于ExcelBaseEntity 的问题，并探讨其在实际应用中的重要性和优势。

第一步：什么是ExcelBaseEntity方法？ExcelBaseEntity是一种基于Excel的数据处理方法，它提供了一系列用于处理和分析数据的功能和方法。

它是在Excel中编写的宏，可以通过VBA （Visual Basic for Applications）来访问和使用。

ExcelBaseEntity使用一种名为"实体"（Entities）的概念来管理数据，并提供了许多工具和函数来帮助用户执行各种数据处理任务。

第二步：为什么要使用ExcelBaseEntity方法？使用ExcelBaseEntity方法可以带来许多好处和优势。

首先，它提供了一种更高效和可重用的方式来处理数据。

通过使用实体的概念，用户可以将数据分组和组织起来，并在需要时轻松地对其进行操作和分析。

此外，ExcelBaseEntity还提供了许多内置的函数和算法，可以节省用户大量的时间和精力。

其次，ExcelBaseEntity可以提高数据处理的准确性和一致性。

用户可以定义各种规则和约束，以确保数据的合法性和正确性。

这些规则可以应用于不同的实体和属性，并在用户进行数据输入和编辑时进行验证。

这样可以避免常见的错误和问题，并确保数据的质量和准确性。

第三步：ExcelBaseEntity的主要功能是什么？ExcelBaseEntity提供了一系列有用的功能和方法，用于处理和分析数据。

以下是其中一些主要功能：1. 数据导入和导出：用户可以使用ExcelBaseEntity将数据从各种来源导入到Excel中，并将结果导出到其他格式。

蔚 constraint method
蔚（Wèi）constraint method是一种常用的约束方法，也称为蔚式约束法。

它是一种用于求解非线性规划问题的数学算法。

该方法最初由美国数学家Stephen Boyd和Lieven Vandenberghe 在他们的书籍《Convex Optimization》中提出。

该方法基于凸优化理论，适用于具有凸目标函数和凸约束条件的问题。

蔚constraint method的基本思想是将非线性规划问题转化为一系列等价的线性规划问题，并通过不断迭代来逼近最优解。

该方法通过引入松弛变量和对偶变量，将原始问题转化为一系列线性规划子问题，并通过求解这些子问题来逐步逼近最优解。

蔚constraint method具有以下优点：
1. 能够处理大规模复杂的非线性规划问题；
2. 可以保证收敛到全局最优解；
3. 可以处理不等式约束、等式约束和无约束情况。

但是，该方法也存在以下缺点：
1. 迭代次数较多，计算时间较长；
2. 对于某些特殊情况下可能会出现数值不稳定或发散现象。

总之，蔚constraint method是一种重要的求解非线性规划问题的方法，具有广泛的应用前景。

第50 卷第 3 期2023年3 月Vol.50，No.3Mar. 2023湖南大学学报（自然科学版）Journal of Hunan University（Natural Sciences）基于结点作用力和拉普拉斯平滑的单层球面网格处理方法郭小农，李政宁，计丽艳†，欧阳辉（同济大学土木工程学院，上海 200092）摘要：通过基于结点间作用力的方法和拉普拉斯平滑方法对传统凯威特型单层球面网壳进行了网格优化，得到两种新型的网格形式. 首先介绍了两种算法的计算原理及计算过程：基于结点间作用力的方法将杆件假设为弹簧，将结点向不平衡力的方向移动，从而得到一种杆件长度更加均匀的网格（SF网格）；采用拉普拉斯平滑方法也可得到一种更加流畅的网格（LS网格）. 然后，对凯威特网格、SF网格、LS网格进行了对比：在几何指标方面，对比了三种网格的杆件平均长度、杆件长度变化率和网格三角形形状系数；在力学性能方面，对比了三种网格在不同荷载模式下的弹性应变能和非线性稳定承载力. 数值分析结果表明，SF网格比LS网格具有更好的几何指标；在大部分情况下，SF网格与LS网格具有更低的应变能；三种网格的稳定极限承载力较为接近. 最后，从整体上看，SF网格具有最佳的整体力学性能；LS网格具有更好的网格流畅性；SF网格和LS网格的力学性能和网格流畅性均优于传统凯威特型单层球面网壳.关键词：网壳；结构优化；网格生成；拉普拉斯平滑中图分类号：TU391； TU317.1 文献标志码：ASingle-layer Spherical Mesh Processing Method Based on ForceBetween Nodes and Laplacian SmoothingGUO Xiaonong，LI Zhengning，JI Liyan†，OUYANG Hui（College of Civil Engineering， Tongji University， Shanghai 200092， China）Abstract：The traditional Kiewitt reticulated single layer shell mesh was optimized by the method based on the Spring Force between nodes and the Laplacian Smoothing method， and two new spherical mesh forms are obtained. Firstly， the calculation principle and process of the two algorithms are introduced： the method based on the force be‐tween the nodes assumes the member as a spring， and moves the node in the direction of the unbalanced force， so as to obtain a grid （SF mesh） with a more uniform length of the member. At the same time， a smoother mesh （LS mesh）can be obtained by using the Laplacian smoothing method. Then， the Kiewitt mesh， SF mesh， and LS mesh were compared. In terms of geometric indices， the average length of the member， the change rate of member length and the shape coefficient of the mesh triangle of the three kinds of reticulated shells were compared. In terms of mechani‐∗收稿日期：2022-03-10基金项目：国家自然科学基金资助项目（51878473），National Natural Science Foundation of China（51878473）作者简介：郭小农（1977—），男，四川金堂人，同济大学副教授，博士生导师，工学博士† 通信联系人，E-mail：******************.cn文章编号：1674-2974（2023）03-0071-09DOI：10.16339/ki.hdxbzkb.2023032湖南大学学报（自然科学版）2023 年cal properties， the elastic strain energy and nonlinear stability bearing capacity of three kinds of reticulated shells under different load modes are compared. The numerical analysis results show that the SF mesh has better geometric indices than the LS mesh. In most cases， the SF mesh and LS have lower strain energy； and the stability ultimate bearing capacity of the three meshes is relatively close. On the whole， the SF mesh has the best overall mechanical properties； the LS mesh has the best mesh fluency； SF mesh and LS mesh are both better than the traditional Kiewitt reticulated shell.Key words：reticulated shell；structural optimization；mesh generation；Laplacian smoothing大跨度空间结构因其可以提供较大的建筑活动空间，且具有受力合理、重量轻、杆件规格统一、安装方便等特点［1］，目前已经广泛地应用在体育场、航站楼等大型标志性建筑中.球面网壳由于形状规则、构造简单，应用尤为广泛［2-4］. 根据网格划分方式，球面网壳可以分为凯威特网壳、三向网格球面网壳、肋环形球面网壳、施威德勒型球面网壳等［2］. 其中凯威特网壳和三向网格球面网壳均为三角形网格且没有杆件交叉，在工程中应用更多. 凯威特网格与三向网格具有相同的拓扑形式. 凯威特网壳先将球面的圆心角按环数等分，形成多个不同大小的圆环，并将圆环继续等分得到最终的网格. 三向网格是先在平面中通过平行的直线进行划分，然后将交点垂直投影至球面得到的. 三向网格会在矢跨比较大或环数较大时出现畸变，适用性相对有限. 从生成方式来看，两种网格都具有较为严格的几何约束，因此可能存在更优的球面网格划分方式.空间网格结构的出现是复杂建筑外形发展的重要一步. 传统的网格生成方法已经不再适用于复杂的异形建筑，随着有限元分析技术（FEA）的不断发展，不少学者开始基于计算机进行网格划分的研究. 常见的非结构化网格划分算法有Delaunay三角划分、波前推进算法和映射方法等. Su等［5］在波前法的基础上，用主应力线来调整网格的走向，提出了改进的波前法，实现建筑曲面网格划分的自动化. 危大结等［6］和潘炜等［7］利用将曲面展开的方法实现了曲面与平面的双向映射，并结合平面网格的生成方法，实现对自由曲面的网格划分. Persson等［8］提出了平面不规则图形的网格生成方法，该方法将网格类比为桁架结构. Shimada等［9］、Zhou等［10］提出，将网格点类比为相互作用的弹性气泡，通过求解系统的平衡状态，得到结点在平面内的最优分布，最后通过将平面网格映射到曲面上得到空间网格. 李铁瑞等［11］提出了一种以网格均匀性为目标的划分方法，首先在曲面上初始布点，对点云进行均匀化，之后求得点云的Voronoi图，进而得到相应网格. Liu等［12］提出了一种基于库仑定律的自由曲面三角形网格划分方法，利用该方法可以得到一个具有特定拓扑的自由形状网格结构，结构中所有结点之间的距离都很近. 以上不同方法皆可对异形曲面进行网格划分，不同的方法各有优劣，但得到的网格常会存在杆件长度差异较大或杆件不够流畅的现象，因此往往需要对网格做进一步处理.本文介绍了两种针对球面网壳的网格处理方法：基于结点间相互作用力的网格处理方法（Spring Force，简称SF方法）和拉普拉斯平滑的方法（Lapla‐cian Smoothing，简称LS方法）. 将两种网格处理方法应用于球面网壳中，基于数值分析结果对处理后网壳的几何指标和力学指标进行对比，尝试得到一些更优的球面网格划分方式. 此外，本研究也可为自由曲面网格处理提供参考.1 网格生成方法1.1 基于结点间相互作用力的方法基于结点间相互作用力的方法将网壳中的杆件视为一种特殊的线性弹簧，弹簧的原长为网壳中杆件的平均长度. 结点在弹簧恢复力的作用下发生移动，即杆件长度大于杆件平均长度时，会产生缩短的趋势；当杆件长度小于杆件平均长度时，会产生伸长的趋势. 在每一步迭代过程中，根据网壳中当前的结点位置，计算各结点受到的弹簧作用，对结点进行移72第 3 期郭小农等：基于结点作用力和拉普拉斯平滑的单层球面网格处理方法动. 由于杆件产生的合力会使结点偏离球面，因此在每一步移动结点结束后，将结点位置重新投影回球面. 不断迭代，直至达到收敛条件. 计算方法的具体步骤如图 1所示.算法的具体实现过程如下：1） 生成初始凯威特网壳通过网壳结构的跨度L 、矢高f 、环数m 等基本参数，计算得到球面半径R ，然后以球心为原点，通过凯威特网壳的空间几何关系生成网壳的结点坐标及杆件连接关系.2） 对结点进行分类对于球面网壳，通常选取最外环结点作为支座结点. 支座结点在整个计算过程中的坐标保持不变；除支座外的其余结点为内部结点，每个内部结点周围均连接有6根杆件.3） 根据当前位置计算结点合力首先根据网格当前形状计算杆件的平均长度-l .将所有杆件视为原长为-l 的线性弹簧，按式（1）计算第i 根杆件在当前状态下的恢复力f i .f i =k (l i --l ).（1）式中：l i 为杆件的当前长度；k 为弹簧的刚度.在本算法的计算过程中，假设单位力会使结点在单个迭代步中移动单位距离. 因此k 的取值会影响整体迭代过程中结点的移动速度，为防止迭代不收敛，k 的取值宜大于0、小于等于1，本文计算中k =1.将结点坐标向量记为P i ={x i ，y i ，z i }，i 表示结点编号，x i 、y i 、z i 为结点坐标. 对于网壳内部杆件，用e i ， j 表示与第i 个结点相连的第j 根杆件的杆件编号，用n i ， j 表示与第i 个结点相连的第j 根杆件的另一端结点的编号. 对于网壳内部结点，按式（2）可计算其受到的合力F i . 式中f e i ，j表示第i 个结点所连第j 根杆件的弹簧力大小，如图2所示.F i =∑j =16éëêêùûúúf e i ，j×P n i ，j-P i l e i ，j.（2）4） 计算合力的切向分力、移动结点，并将结点投影回球面网壳中与同一结点相连的各个杆件并不在同一平面内，因此结点P i 处的合力F i 可能为空间中的任意方向，如图 3（a ）所示. 为消除合力F i 的径向分力的影响，计算合力F i 在结点P i 处切平面的分力F τ i ，使结点沿着球面切向分力的方向移动. 结点的移动向量M i =F τ i .将结点P i 按向量M i 进行移动，得到移动后结点坐标P i '. 如图 3（b ）所示，结点在移动后会产生微小的球面外位移，为保证结点严格位于球面上，将结点P i '投影至球面，得到移动后的坐标P i "，完成一次迭代.5） 重复步骤3）、4），直至满足收敛条件程序需要判断迭代是否满足收敛条件，满足时迭代停止，得到最终结果；若不满足，则将上一步得到的结果作为当前网壳形状，不断重复过程3）、4）的计算，直至达到收敛条件. 随着迭代次数的增加，杆图 1 基于结点间相互作用力的计算方法Fig.1 Calculation method based on the force between nodes（a ）切向分力 （b ）结点坐标修正图3 结点移动的计算方法Fig.3 Calculation method of node movement图2 杆件作用力示意Fig.2 Indication of the rod force73湖南大学学报（自然科学版）2023 年件的长度逐渐均匀，网壳中结点受到的合力也逐渐变小. 当网壳中结点坐标的移动变得非常小时，认为达到了收敛条件. 定义各结点移动长度向量V ={V 1，V 2，⋯，V i ，⋯，V n }，V i 表示第i 个结点在当前迭代步的移动距离，即V i =|P i "-P i |. 当向量V 的模|V |<δ时停止迭代，其中δ为迭代收敛精度. δ可根据结点坐标需要的精度取值，δ越小，迭代次数越多，最终得到的坐标精度越高. 本文示例中，结点坐标的单位为米，δ取0.001，即对应坐标的精度小于1 mm.1.2 拉普拉斯平滑方法拉普拉斯平滑是一种常用于网格处理和图像去噪的算法［13］. 拉普拉斯平滑的本质是用相邻结点坐标的加权平均值来代替原结点坐标. 拉普拉斯平滑通过不断地调整结点的位置，在不改变网格拓扑的情况下，将原先杂乱的网格变得更加流畅. 在进行一次拉普拉斯平滑时，对于网格上的原结点，它的新坐标为与它相邻结点坐标的加权平均［14］. 以凯威特网壳为基础网格进行拉普拉斯平滑的具体步骤如下：1） 生成初始凯威特网壳2） 计算迭代后结点坐标首先固定边缘结点，外圈结点在拉普拉斯平滑过程中保持不变. d i 表示第i 个结点的度数，即该结点连接的杆件数量；w i 表示第i 个结点在进行拉普拉斯平滑计算时的权重；n i ，j 表示与第i 个结点相邻的第j 个结点的编号. 对于网壳内部结点，迭代后的坐标按式（3）计算. 计算过程如图 4所示.P new i =∑j =1d i(w n i ，jP n i ，j)/∑j =1d iw n i ，j.（3）结点权重w i 的取值一般根据结点的度数确定. 权重w i 会影响结点在拉普拉斯平滑过程中的重要性，w i 越大表示第i 个结点周围的点在拉普拉斯平滑时会更靠近该结点. 不同的w i 的取值会得到不同的计算结果，其具体取值可根据网格形状进行调整以寻找最佳参数. 本文计算过程中，适当增大度数较小的结点的权重以增大其在拉普拉斯平滑过程中的重要性，当d i =3时，w i 取1.5；当d i =4或6时，w i 取1.0.3） 将结点投影回球面常规的拉普拉斯平滑虽然可提升曲面质量，但过度的拉普拉斯平滑会导致曲面的坍缩. 为保证在平滑过程中结点不偏离球面，每进行一次平滑后，将结点投影回球面.4） 重复过程3）、4）至满足收敛条件迭代过程的收敛条件与1.1节中相同，当网壳中结点坐标的移动变得非常小时，认为达到收敛条件.1.3 运算结果使用Python 语言按以上算法编写参数化程序，生成不同参数下的凯威特网壳，并分别使用以上两种方法处理. K6表示未处理的凯威特网壳；SF （Spring force ）表示采用基于结点间相互作用力方法处理得到的网壳；LS （Laplacian Smoothing ）表示采用拉普拉斯平滑方法处理得到的网壳.图 5汇总了矢跨比f /L 为1/6，环数m =6、8、10、12的 K6、SF 、LS 三种网壳的俯视图.由图5可发现，由于K6网壳是通过几何方法生成的，因此其俯视图中有明显的环肋；SF 网壳和LS 网壳在中心区域有着三向网格的特点，两者在整体外观上较为接近，但也存在一定差异. 图 6展示了SF 网格与LS 网格叠加对比的效果. 可发现，SF 网格与LS 网格在中心区域几乎重合，两者差异主要在网格边缘：LS 网格中的杆件在网壳边缘处更加流畅.2 几何指标对比为分析两种处理方法对原网壳在几何方面的处理效果，本节对跨度为40 m ，矢跨比f /L 取1/3、1/4、 1/5、1/6、1/8、1/10，环数m 取6、8、10、12的K6、SF 、LS 三种网壳的几何指标进行了对比.2.1 杆件长度分布本文中的两种处理方法均不改变网格的拓扑关系，因此三种网壳有着相同的杆件数量. 杆件的平均长度-l 可表征使用相同截面时的材料用量. 网壳结构的构件一般同时承受压力和弯矩作用，杆件稳定为主要控制因素［15］，因此网壳中的杆件不宜过长. lmax图 4 拉普拉斯平滑计算过程Fig.4 Laplacian smoothing calculation process74第 3 期郭小农等：基于结点作用力和拉普拉斯平滑的单层球面网格处理方法表示网壳的最大杆件长度. 杆长分布均匀有利于充分发挥截面的利用率，并可提高加工下料时的材料利用率. 杆件长度的分布情况可通过长度变化系数来衡量［16］. 长度变化系数r 为杆件长度的标准差s 与杆件平均长度-l 的比值，其计算方法如式（4）和式（5）.s 2=∑i =1n (l i --l )2n -1，（4）r =s /-l .（5）图 7（a ）（b ）（c ）分别给出了环数m =12时三种网壳的-l 、r 、l max 随矢跨比变化曲线.从图 7（a ）可发现，两种处理方法均可降低原K6网格的平均杆件长度，处理效果较为接近；从图 7（b ）可发现，两种处理方法均可使网格杆件长度更加均匀，LS 方法处理的效果更加明显；从图 7 （c ）可发现，两种处理方法都明显降低了杆件的最大长度.2.2 三角形的形状系数平均值三角形的形状系数可用来表征其接近正三角形的程度［16］6）计算.q =（6）式中：A t 表示三角形的面积；l a 、 l b 和l c 表示三角形的三条边的长度. q 的计算结果为0~1，对于等边三角形，q =1；当三角形三点共线时，q =0. 网壳中所有三角形的形状系数平均值-q 可以表征网壳中三角形网格的均匀程度.图7（d ）中给出了m =12时三种网壳的-q 与矢跨比的关系. 由图7（d ）可知，采用两种处理方法均可使网格中的三角形形状更接近正三角形. 当矢跨比较大时，采用LS 方法处理后的网格会在形状系数方面优于SF 网格.3 力学性能对比为探究采用SF 和LS 方法处理后网壳的力学性能，本节对三种网壳在不同参数下的应变能及弹塑性非线性稳定承载力进行了对比.3.1 弹性应变能荷载作用下结构的弹性应变能可用来表征其整体刚度，应变能E 的计算公式如式（7）.E =12U T ΚU .（7）式中：U 表示结点的位移向量；K 表示结构的整体刚度矩阵. 应变能越小代表结构整体刚度越大［17］.本节对比计算了上述三种网壳在不同参数下的弹性应变能. 计算参数如下：网壳跨度L =40 m ；矢跨比f /L 取1/3、1/4、1/5、1/6、1/8、1/10；环数m 取6、8、10、12；恒荷载取1.0 kN/m 2；活荷载取0.5 kN/m 2 （按水平图 5 不同环数下三种网格的俯视图Fig.5 Top view of three grids with different number of rings图 6 SF 网壳与LS 网壳对比Fig.6 Comparison of SF reticulated shelland LS reticulated shell75湖南大学学报（自然科学版）2023 年面积投影计算，并分别考虑满跨和半跨的情况）；杆件材料为铝合金6061-T6，弹性模量取70 GPa ，按理想弹性材料计算；杆件采用梁单元，截面为工字形截面H300×200×10×12，弱轴方向指向球心；网壳结点均刚接，约束条件为周边固定铰支座.表 1和表 2汇总了三种网壳在各参数下的应变能计算结果.从表中数据可知，在同一跨度下，网壳的应变能会随着矢跨比和环数的增大而减小. 随着矢跨比增大，网壳的拱效应增强，整体刚度更大；随着环数增加，网格密度变大，材料用量增大，因此整体刚度也会提高.为分析经过SF 处理和LS 处理后网壳的应变能变化情况，对比了三种网壳应变能与K6网壳应变能的相对比值. 图 8给出了环数m =12时三种网壳的应变能相对比值随矢跨比的变化曲线. 图中E f 代表“恒载+满跨活载”作用下三种网壳的应变能与K6网壳应变能的相对比值，E h 代表在“恒载+半跨活载”作用下的应变能相对比值. 从图 8可发现，在矢跨比较小表1 “恒载+满跨活载”作用下结构的应变能Tab.1 Strain energy of structure under dead loadand full span live loadkN·mm681012网壳类型K6SF LS K6SFLS K6SFLS K6SFLS矢跨比1/107.2566.9737.0775.5685.2715.3224.5074.2324.2583.7813.5333.5461/84.8704.6894.7783.7373.5453.5893.0262.8462.8692.5382.3762.3881/63.0272.9353.0232.3232.2192.2651.8801.7811.8071.5781.4871.5011/52.3392.2882.3881.7961.7321.7861.4541.3911.4211.2201.1611.1791/41.8651.8561.9901.4361.4111.4861.1651.1361.1790.9780.9500.9751/31.8061.8582.0991.4111.4361.5741.1531.1661.2470.9720.9791.030表2 “恒载+半跨活载”作用下结构的应变能Tab.2 Strain energy of structure under dead loadand half span live loadkN·mm681012网壳类型K6SF LS K6SFLS K6SFLS K6SFLS矢跨比1/105.4265.2035.2734.1693.9403.9733.3773.1663.1822.8342.6442.6521/83.6553.5093.5692.8092.6582.6872.2762.1362.1511.9101.7841.7921/62.2822.2042.2641.7541.6701.7011.4211.3421.3591.1931.1211.1301/51.7671.7201.7881.3591.3041.3411.1011.0481.0690.9240.8760.8871/41.4081.3931.4871.0861.0611.1130.8810.8540.8840.7400.7140.7321/31.3561.3871.5601.0601.0721.1710.8660.8700.9290.7310.7310.767（a ）杆件平均长度（b ）杆长变化系数（c ）最大杆件长度（d ）三角形形状系数平均值图7 网壳的几何参数对比Fig.7 Comparison of geometric parameters of reticulated shells76第 3 期郭小农等：基于结点作用力和拉普拉斯平滑的单层球面网格处理方法时，SF 网壳与LS 网壳相比K6网壳具有更低的应变能，且SF 网壳要优于LS 网壳；随着矢跨比增大，SF 网壳与LS 网壳的应变能会大于K6网壳. 当矢跨比大于1/4后，LS 的力学性能不优于K6.3.2 非线性稳定承载力网壳结构的非线性稳定承载力是网壳结构设计时的关键指标［17］. 对比三种网壳在不同参数下的非线性稳定承载力，计算模型在3.1节模型的基础上有一定调整：每根杆件等分为4段；考虑材料塑性，材料简化为理想弹塑性，屈服强度为f 0.2=240 MPa ；铝合金网壳会由于螺栓滑移或施工误差产生初始缺陷［18］，球面网壳是典型的缺陷敏感型结构，因此计算非线性稳定承载力时应考虑网壳的初始缺陷［19］；结构初始缺陷采用一致缺陷模态法施加，缺陷的形状取第1阶屈曲模态，按照《空间网格结构技术规程》（JGJ 7—2010）［20］，缺陷幅值取L /300. 在ANSYS 软件中采用弧长法计算非线性稳定承载力.表 3和表 4分别列举了计算得到的稳定承载力系数.表 3为“恒载+满跨活载”作用下的计算结果，表 4为“恒载+半跨活载”作用下的计算结果. 从表中稳定系数结果可发现：在相同跨度下，网壳的稳定承载系数会随着矢跨比和环数的增大而增大. 对比表 3和表 4可发现，当网壳矢跨比较小时，半跨活荷载的稳定承载力系数更低；当矢跨比逐渐增大时，满跨活荷载的稳定承载力系数更低.定义C f 为网壳在恒载加满跨活载作用下的稳定承载力系数与K6网壳稳定承载力系数的相对比值，C h 为恒载加半跨活载作用下的相对比值.图 9（a ）和（b ）分别表示SF 网格和LS 网格在不同参数下的C f 值的散点图；图 9（c ）和（d ）分别表示SF 网格和LS 网格在不同参数下的C h 值的散点图.从图中可知，C f 和C h 的值随矢跨比的变化没有呈现明显规律， 除了个别参数下SF 网壳与LS 网壳的稳定承载力系数相比K6网壳有较大提升外，多数情况下三种网壳的稳定承载力系数较为接近. 计算各个参数的相对比值的平表4 “恒载+半跨活载”稳定承载力系数Tab.4 The stable bearing capacity coefficient under deadload and half span live loadm681012网壳类型K6SF LS K6SFLS K6SFLS K6SFLS矢跨比1/105.6425.9375.7148.7419.5799.55810.91111.98011.99013.49813.48411.2441/87.1027.7177.56511.03212.16111.94517.40917.65517.44621.18021.30421.0771/69.1869.6659.32814.62415.88115.50123.63222.23822.10926.10926.92428.5811/510.27010.85710.35216.62217.81817.12522.99925.03124.52531.69231.82228.8041/410.93111.79210.61518.33119.12718.25225.90527.84626.85634.98635.34538.4101/311.23011.1839.24718.27819.51517.55026.54927.62126.19535.41335.84634.972表3 “恒载+满跨活载”稳定承载力系数Tab.3 The stable bearing capacity coefficient under deadload and full span live loadm681012网壳类型K6SF LS K6SFLS K6SFLS K6SFLS矢跨比1/105.2865.2915.2389.33210.40910.09911.99912.75412.76914.24316.04515.4271/86.5416.4506.58711.17510.24110.14316.66327.08427.29220.03132.67732.5601/68.1638.0378.28413.72613.42213.17922.91722.97322.71827.61827.69027.5551/59.0199.6179.07915.05015.30114.83521.95420.88720.53230.25747.07748.2121/49.73610.4079.48115.94616.95816.06923.64123.43022.74736.51429.44329.2651/39.56810.2018.62215.90017.16115.48222.90424.31522.97631.43031.26030.281（a ）“恒载+满跨活载”（b ）“恒载+半跨活载”图8 应变能对比结果Fig.8 Comparison results of strain energy77湖南大学学报（自然科学版）2023 年均值可得到：在“恒载+满跨活载”的情况下，SF 网壳整体提升约6.7%，LS 网壳整体提升约4.7%；在“恒载+半跨活载”的情况下，SF 网壳整体提升约4.5%，LS 网壳整体提升约1.6%. 总体而言，SF 网壳整体上具有更好的稳定承载性能.4 结论1） 本文分别介绍了基于结点间作用力（SF ）和拉普拉斯平滑（LS ）的网格处理方法. 采用两种方法对凯威特型球面网壳进行了网格优化，得到两种新的球面网格划分方式.2） 将三种网壳在不同尺寸下的几何参数与力学性能对比发现：几何参数方面，SF 方法和LS 方法均可显著提升原网格的几何指标，SF 网格的杆件分布更均匀，LS 网格中的三角形在整体上更均匀；力学性能方面，SF 网格具有更低的应变能，且两种方法均不会降低网壳的非线性承载力.3） 通过两种方法得到的新型网格形式比传统凯威特网壳更具优势，可应用于实际工程.4） 两种网格处理方法在原理上具有普适性，可进一步扩展至自由曲面网格的处理，使杆件长度均匀，进而提升网格的整体质量. 难点在于如何控制自由曲面网格在处理过程中保持原有形状.参考文献［1］董石麟．我国大跨度空间钢结构的发展与展望［J ］．空间结构，2000，6（2）：3-13．DONG S L ．The development and prospect of long-span steel space structures in China ［J ］．Spatial Structures ，2000，6（2）：3-13．（in Chinese ）［2］董石麟，姚谏．钢网壳结构在我国的发展与应用［J ］．钢结构，1994，9（1）：21-31．DONG S L ，YAO J ．Development and application of steel latticed shell structures in China ［J ］．Steel Construction ，1994，9（1）：21-31．（in Chinese ）［3］杨联萍，韦申，张其林．铝合金空间网格结构研究现状及关键问题［J ］．建筑结构学报，2013，34（2）：1-19．YANG L P ，WEI S ，ZHANG Q L ．Aluminum reticulated spatial structures ：state of the art and key issues ［J ］．Journal of BuildingStructures ，2013，34（2）：1-19．（in Chinese ）［4］吴金志，宋子魁，孙国军，等．铝合金单层网壳结构的工程应用与研究进展［J ］．建筑结构，2021，51（17）：129-140．WU J Z ，SONG Z K ，SUN G J ，et al ．Engineering application and research progress of aluminum alloy single-layer latticed shell structure ［J ］．Building Structure ，2021，51（17）：129-140．（inChinese ）［5］SU L ，ZHU S L ，XIAO N ，et al ．An automatic gridgeneration（a ）SF 网格，“恒载+满跨活载”（b ）LS 网格，“恒载+满跨活载”（c ）SF 网格，“恒载+半跨活载”（d ）LS 网格，“恒载+半跨活载”图9 稳定承载力系数提升效果Fig.9 Improved effect of stable bearing capacity coefficient78第 3 期郭小农等：基于结点作用力和拉普拉斯平滑的单层球面网格处理方法approach over free-form surface for architectural design［J］．Journal of Central South University，2014，21（6）：2444-2453．［6］危大结，舒赣平．自由曲面网格的划分与优化方法［J］．建筑结构，2013，43（19）：48-53．WEI D J，SHU G P．Mesh generation and optimization method forfree-form surface grid［J］．Building Structure，2013，43（19）：48-53．（in Chinese）［7］潘炜，吴慧，李铁瑞，等．基于曲面展开的自由曲面网格划分［J］．浙江大学学报（工学版），2016，50（10）：1973-1979．PAN W，WU H，LI T R，et al．Grid generation on free-formsurface based on surface flattening［J］．Journal of ZhejiangUniversity （Engineering Science），2016，50（10）：1973-1979．（inChinese）［8］PERSSON P O，STRANG G．A simple mesh generator in MATLAB［J］．SIAM Review，2004，46（2）：329-345．［9］SHIMADA K，GOSSARD D C．Automatic triangular mesh generation of trimmed parametric surfaces for finite elementanalysis［J］．Computer Aided Geometric Design，1998，15（3）：199-222．［10］ZHOU Y Q，NIE Y F，ZHANG W W．A modified bubble placement method and its application in solving elliptic problemwith discontinuous coefficients adaptively［J］．InternationalJournal of Computer Mathematics，2017，94（6）：1268-1289．［11］李铁瑞，吴慧，王奇胜，等．基于离散化的复杂曲面建筑网格划分方法［J］．湖南大学学报（自然科学版），2018，45（7）：48-53．LI T R，WU H，WANG Q S，et al．Grid generation method forcomplex surface buildings by discretization［J］．Journal ofHunan University （Natural Sciences），2018，45（7）：48-53．（inChinese）［12］LIU F C，FENG R Q，TSAVDARIDIS K D．A novel progressive grid generation method for free-form grid structure design andcase studies［J］．Journal of Building Engineering，2021，34：101866．［13］XIAO L，YANG G X，ZHAO K Y，et al．Efficient parallel algorithms for 3D Laplacian smoothing on the GPU［J］．AppliedSciences，2019，9（24）：5437．［14］PAN W，LU X Q，GONG Y H，et al．HLO：half-kernel Laplacian operator for surface smoothing［J］．Computer-Aided Design，2020，121：102807．［15］ZHU S J，GUO X N，LIU X，et al．The in-plane effective length of members in aluminum alloy reticulated shell with gusset joints［J］．Thin-Walled Structures，2018，123：483-491．［16］WANG Q S，YE J，WU H，et al．A triangular grid generation and optimization framework for the design of free-form grid shells［J］．Computer-Aided Design，2019，113：96-113．［17］ZHU S J，OHSAKI M，GUO X N，et al．Shape optimization for non-linear buckling load of aluminum alloy reticulated shells withgusset joints［J］．Thin-Walled Structures，2020，154：106830．［18］刘俊，罗永峰，郭小农，等. 螺栓滑移引起的铝合金板式节点网壳变形研究［J］. 湖南大学学报（自然科学版），2020，47（9）：40-47．LIU J，LUO Y F，GUO X N，et al．Study on structural deformationof reticulated shells with aluminum alloy gusset joints caused bybolt slippage［J］．Journal of Hunan University （NaturalSciences），2020，47（9）：40-47．（in Chinese）［19］郭小农，朱劭骏，熊哲，等．K6型铝合金板式节点网壳稳定承载力设计方法［J］．建筑结构学报，2017，38（7）：16-24．GUO X N，ZHU S J，XIONG Z，et al．Design method for bucklingcapacity of K6 single-layer reticulated shells with aluminum alloygusset joints［J］．Journal of Building Structures，2017，38（7）：16-24．（in Chinese）［20］空间网格结构技术规程：JGJ 7—2010［S］．北京：中国建筑工业出版社，2011．Technical specification for space frame structures：JGJ 7—2010［S］．Beijing：China Architecture & Building Press，2011．（inChinese）79。

基于GWLN方法的冗余机械臂关节力矩约束控制陈鹏;项基;韦巍【摘要】针对冗余机械臂受到的关节驱动力矩有限的约束,提出基于广义加权最小范数法(GWLN)的算法.通过引入辅助变量,考虑重力和科里奥利力的影响,排除现有方法对机械臂低速运行的假定.在逆运动学求解时,对扩展变量的加权范数优化,使得规划关节加速度所需的力矩指令保持在驱动器输出范围之内.该算法的有效性通过数学证明得到验证.在MATLAB ROBOTIC TOOLBOX中对PUMA560机械臂的仿真结果证实,基于GWLN方法控制的机械臂在遵循关节力矩约束的同时,准确地完成操作任务.同零空间力矩优化方法及现有加权方法的仿真结果对比显示,基于GWLN方法的控制算法能够更加有效地保证力矩约束,具有更好的稳定性.%A general weighted least norm (GWLN) based algorithm was proposed to solve the torque limit constraint for a redundant manipulator.An auxiliary variable was introduced to consider the Coriolis force and the gravity's effect in order to eliminate the assumptions made by other algorithms in existence that the manipulator is running slow.The torque command for the planned joint acceleration was kept in the actuators' output ranges by optimizing the extended variable's weighted norm during inverse kinematic solution.The validity of the algorithm was demonstrated by a mathematical proof.A simulation of PUMA 560 manipulator in the MATLAB ROBOTIC TOOLBOX shows that the manipulator controlled by the GWLN based method can comply with the joint torque constraint and accomplish the manipulation parison with the results generated by the nullspace torque optimization method and the existing weighted norm method shows that the GWLN based method is more effective and stable.【期刊名称】《浙江大学学报（工学版）》【年(卷),期】2017(051)001【总页数】8页(P68-74,105)【关键词】冗余机械臂;关节力矩约束;广义加权最小范数法;驱动器输出饱和【作者】陈鹏;项基;韦巍【作者单位】浙江大学电气工程学院,浙江杭州310027;浙江大学电气工程学院,浙江杭州310027;浙江大学电气工程学院,浙江杭州310027【正文语种】中文【中图分类】TP241冗余机械臂可以利用系统的冗余特性完成更复杂的操作任务,例如在不影响操作任务的情况下避免同环境发生碰撞,保持可观的可操作度以避免位形奇异等.然而,冗余机械臂的运动受到驱动元件的约束限制.例如,关节坐标应位于物理结构所限定的范围内；关节转速应位于驱动器最大输出转速及减速器所允许的最大输出转速范围内；关节驱动力矩应在驱动器最大输出力或力矩范围内.以上约束限制了机械臂关节的运动能力,通过正向运动学映射影响了机械臂在操作空间的运动性能.如果缺乏恰当的考虑,那么机械臂的控制算法将产生无法实现的逆运动学解,导致操作任务的执行出现偏差[1],甚至为系统带来损害.关节力矩约束对于关节力矩控制和速度控制都起着重要的制约作用,通过动力学模型制约关节加速度范围,从而限定关节速度以及关节坐标的变化.针对关节力矩约束下机械臂的控制问题,众多学者采用不同的方法来求解.根据采用的求解思路,这些方法大致可分为两类：力矩优化法和约束求解法.前者通过对与关节力矩相关的性能指标的优化,以减小关节力矩指令,使其在驱动器的输出范围之内.例如,Hollerbach等[2]通过设计零空间关节加速度使关节力矩向量与代数中心距离的2范数最小.Liu等[3-4]求解了操作任务运动学约束下关节力矩2范数最优问题.Zhao等[5-7]分别通过对关节加速度加权范数优化来减小接近输出饱和的关节的指令力矩.郭宪等[8]通过对关节力矩的无穷范数优化实现最小力矩幅值.闫彩霞等[9]通过对关节力矩进行加权优化来平衡各关节力矩负荷.金波等[10]通过对系统能耗进行优化来完成力矩分配.这些优化指标将各个关节力矩综合为单一指标,没有直接考虑各个关节力矩的输出范围,不能严格保证关节力矩指令在物理允许范围内. 约束求解法将最大关节输出力矩作为关节力矩指令的边界,对操作任务求解以确保遵循关节力矩约束.例如,Bianco等[11]通过对操作任务轨迹切向速率及高阶导数规划的办法来求解关节力矩有限的约束问题.该方法在逆运动学求解过程中没有考虑系统冗余特性,不适合冗余机械臂.陈伟海等[12]对关节力矩再分配,以保证各关节的实际输出力矩在其驱动能力范围内.该方法需要进行二次计算,不仅增加了运算量,并且再分配的结果可能导致新的关节发生力矩饱和.Zhang等[13]将关节力矩约束和操作任务作为二次型优化问题求解的可行域.该方法未考虑到操作任务与关节驱动力矩约束相矛盾的情况.针对上述方法的缺点,本文提出基于GWLN的冗余机械臂有限关节力矩约束求解方法：将有限关节力矩的非线性约束局部线性化为线性约束,引入辅助变量将非齐次约束转化为扩展系统变量的齐次约束,采用GWLN方法进行求解,保证关节力矩约束.由于该方法直接对力矩约束进行求解,避免了力矩再分配方法需要二次计算运算量大的缺点.该算法的有效性通过理论分析得到证明,对比仿真结果表明,采用该方法更有效地保证了关节力矩约束.假定机械臂m维运动学操作任务可以通过二阶运动学模型表示为式中：q∈Rn为关节坐标,x∈Rm为操作任务坐标, J(q)=∂x/∂q∈Rm×n为操作任务坐标对关节坐标的雅可比矩阵.为了实现期望的操作空间的加速度要求关节空间加速度指令为式中为雅可比矩阵零空间内加速度项,取z=0n×1可得最小2范数的关节加速度指令.当机械臂具有冗余自由度时,操作空间的加速度可以对应不同的关节加速度指令为在不影响操作任务的前提下进行性能优化和约束任务实现提供了可能.由动力学模型可知,机械臂在q和关节速度下产生关节加速度为的运动所需的驱动力矩为式中：M(q)∈Rn×n为系统惯性矩阵为科里奥利——向心力矩阵,g(q)∈Rn为作用于关节的重力向量.由于驱动元件的输出能力受到电源电流的限制,关节力矩在有限范围内：式中：τi,max、τi,min分别为第i关节驱动力矩τi允许取值范围的上、下限.当关节加速度指令过大时,所需关节力矩指令τc会超出驱动器的输出能力范围.由于驱动力不足,实际关节加速度与规划关节加速度出现偏差,从而导致操作任务出现执行误差.为了避免这一问题,需要在机械臂的控制过程中考虑驱动器输出能力有限的约束.综合式(1)、(4),可得关节力矩约束下的机械臂控制问题：式(5)中的约束为关节加速度指令的非线性约束,且约束边界随关节坐标及关节速度的变化而变化,增加了求解难度.考虑到关节力矩约束仅在指令关节力矩超过驱动元件的输出范围时产生作用,仿照Flacco等[14]采用的方法,可将力矩饱和关节的输出力矩限定在饱和输出极限τi=τi,max或τi=τi,min,以充分利用关节驱动能力,从而将不等式约束转化为等式约束,即式中：τi>0.5(τi,max+τi,min)时,力矩约束边界τi,lim=τi,max,否则τi,lim=τi,min；Mi和Ci分别为惯性矩阵M和科里奥利力矩阵C的第i行.式(6)为线性约束作用下的线性系统的求解问题,可以采用梯度投影法或基于任务优先级的方法求解[15-16].基于优先级的方法的分层算法结构较复杂,并且在任务优先级变化时可能导致控制量的不连续,不利于系统的平稳运行.此外,由于约束条件具有高优先级,相应关节的力矩将始终处于饱和输出的状态,不会随操作任务的变化而退出饱和状态,约束将持续作用.采用基于广义加权最小范数的方法来求解冗余机械臂关节力矩约束控制问题.广义加权最小范数法[17-18]被应用于线性齐次约束下线性系统的最小二乘问题的求解,成功地解决了冗余机械臂关节限位和操作空间避障问题.注意到,式(6)中的约束为非齐次约束,不能直接应用广义加权最小范数法进行求解.对广义加权最小范数法进行扩展：引入辅助变量s并且令扩展后的系统变量为扩展后的系统方程为.式中为需要关节加速运动产生的操作任务空间加速度.式(6)中的关节力矩约束可以通过表示为记由式(7)和(8),可得力矩约束下控制问题模型：式(9)为线性齐次约束下的线性系统求解问题,可以应用广义加权最小范数法求解.构建所有力矩饱和关节的约束任务矩阵及变换矩阵T=[KT,K⊥]T,其中K⊥∈R(n+1)×(n+1-p)为K的正交补.式(9)可以通过虚拟关节变量重写为式中：虚拟关节变量任务对虚拟关节变量的雅可比矩阵构建加权矩阵W：式中：子对角加权矩阵W1=diag(w1,…,wp)∈Rp×p为约束方向K的权值,子对角加权矩阵W2=diag (wp+1,…,w(n+1))∈R(n-p+1)×(n-p+1)为正交补方向(K⊥)T 的权值.采用加权最小范数法求解式(10),可得具有最小加权范数的解T为对式(12)两边左乘T-1,可得扩展系统变量：.根据广义加权最小范数法可知,当W1→∞时,满足力矩约束式(8),可以通过下面的定理证明.定理：当约束对应的权值wi趋于无穷时,式(13)满足证明：设约束任务矩阵K的奇异值分解为K=USVT,其中U∈Rp×p、V∈R(n+1)×(n+1)为酉矩阵,S=[Σp,0p×(n-p+1)]∈Rp×(n+1)的奇异值位于对角子矩阵Σp∈Rp×p中.将V分块为其中V1∈R(n+1)×p,V2∈R(n+1)×(n+1-p),则K⊥=V2.此时,变换矩阵T为.将式(11)、(14)代入式(13),可得.当W1的对角元素趋于∞时有.此时在K方向的投影,即约束任务值为.由与V2的正交性可知于是解(13)满足约束条件(8),力矩约束得到保证.注：当加权后的扩展雅可比矩阵奇异时,需要引入阻尼项来保证加权后的扩展雅可比矩阵的伪逆有界.然而,引入阻尼后的扩展后系统方程存在残差,即扩展的辅助变量值可能不严格等于1.这一问题可以通过采用变权重最小二乘法[19]来解决,在此不作详述.基于GWLN方法的约束求解与基于动力学灵活性指标优化算法[5]均采用加权范数法进行求解,然而两者略有不同,分析比较如下.1) 基于动力学灵活性指标优化算法忽略了科里奥利力及重力项,仅对关节加速运动所需力矩进行优化,当关节重力矩和科里奥利力之和超过力矩输出极限时,该方法将失效；基于GWLN的方法通过辅助变量的引入,将科里奥利力及重力的影响考虑在内,因而能够严格保证关节力矩约束.2) 基于动力学灵活性指标优化算法的优化效果使得关节加速运动所需力矩在输出饱和关节的投影分量趋于零,未充分利用临近饱和关节的驱动能力且易造成指令关节力矩跳变及振荡；基于GWLN的方法借助辅助变量,使相应关节的指令力矩在饱和输出边界上,控制的结果更平滑.基于GWLN的方法与基于动力学灵活性指标的优化算法相比,更加具有优点,可以看作对后者的补充和完善.为了验证该算法的有效性,在MATLAB Robotic Toolbox工具箱[20]中对PUMA 560机械臂在关节力矩约束下的操作任务进行仿真.仿真的设定如下.PUMA 560机械臂的连杆Denavit-Hartenberg参数如表1所示.在机械臂腕关节末端添加了执行器.执行器的端点在关节6坐标系下的坐标执行器的惯性模型被简化为质量为2.5 kg的质点,质心位置在关节6坐标系中的坐标m.机械臂的初始关节坐标q(0)=[0,π/4,π,0,π/4,0] rad.操作任务轨迹为沿Ox方向的线段,任务轨迹方程为式中：t为时间坐标.任务持续的时间为0.5s,仿真步长设定为5 ms.各个关节的关节力矩输出范围均为[-60,60]N·m.由于末端执行机构的轨迹逐渐远离机械臂的基座,整个操作过程机械臂作伸展动作,水平方向2、3关节的重力矩增加.为了避免离散化误差,在此引入闭环误差反馈：式中：e(t)为操作任务误差，e(t)=x(t)-xr(t)；kp=kv=20为误差反馈系数.在不考虑关节力矩约束控制的情况下,最小2范数关节加速度逆运动学解所需的关节力矩峰值可以达到120 N·m,大大超出了关节驱动器的能力范围,如图1所示.由于关节驱动器无法满足关节力矩输出要求,实际输出的关节力矩仅能保持在饱和极限值.因系统驱动力不足,实际的关节加速度将偏离最小范数关节加速度指令,如图2所示.图中，e为加速度偏差.尽管只有第2关节的关节力矩达到输出饱和,由于惯量矩阵M(q)的耦合,其他关节的加速度偏离了关节加速度指令.通过正向运动学映射,实际的操作任务加速度将偏离操作任务加速度指令.采用本文方法来求解关节力矩约束.注意到式(8)将使得关节力矩保持在饱和极限值.为了避免在关节力矩约束未被激活时使关节力矩幅度增长的副作用,仅当第i关节的力矩τi超过设定阈值τthres时,关节力矩约束任务Ki被激活.将约束任务矩阵修改为Ki(k)=.式中：k表示当前控制周期,k-1表示上一控制周期.当时,式(20)将使得τi(k)=τi(k-1),即τi保持不变,因而不会超过输出极限.相应地,将约束任务权值设置为由式(21)可知,当τi≤τi,thres时,wi=1,式(13)退化为关节加速度最小2范数解.当τi=τi,lim时,wi=∞,关节力矩指令τi,c将保持在输出极限τi,lim.在式(13)的作用下,整个过程的指令关节力矩曲线如图3所示.从图3可以看出,各关节的指令力矩平滑变化,且均保持在设定的输出范围内,因而关节实际输出力矩与关节力矩指令一致.其中,关节2力矩的最大值为59.994 N·m,几乎在其饱和输出边界上,满足关节力矩约束.算法的有效性可以从相邻控制周期关节力矩的增量变化中看出,如图4所示.在t=0.3 s前后,随着关节2的力矩值趋向输出极限,指令力矩的增量Δτc2逐渐减小到0,即满足约束在式(13)的控制下,整个过程的关节加速度曲线及关节速度曲线分别如图5、6所示.由于关节驱动力矩未超出极限,实际关节加速度与规划关节加速度指令一致,从而保证了操作任务的准确性.当操作任务结束时,关节速度及关节加速度不为零,该现象是由关节力矩约束导致的.随着操作任务的进行,各个关节的重力负荷逐渐增加,如图7所示.在t=0.3 s之后,关节2的重力负荷已经超过了关节力矩极限.此时,机械臂系统不能在该位形下保持零关节速度和零关节加速度的静止状态.在式(13)的作用下,关节力矩约束使得关节2加速力矩子项减小为负值,保证关节2的指令力矩在力矩极限范围内,如图8所示.采用Hollerbach提出的零空间力矩优化算法以及赵占芳提出的动力学灵活性指标优化算法对同一力矩约束下的操作任务进行求解,得到的指令关节力矩结果如图9、10所示.零空间力矩优化算法在t=0.38 s后,关节3的指令关节力矩超出了实际关节输出极限,如图9所示.此时,实际关节加速度和指令关节加速度之间将出现偏差,导致操作任务产生执行偏差.这一结果验证了关节力矩指标优化不保证关节力矩约束的结论.基于动力学灵活性指标优化算法的结果如图10所示.当某一关节指令力矩趋近输出饱和极限时,对动力学灵活性指标的优化作用使得相应关节所需要的加速力矩项趋于0,如图11所示.由于忽略了科里奥利力以及重力项的作用,该方法不能保证指令关节力矩在关节输出极限内.此外,该方法将输出饱和关节的加速力矩子项减小到零,既没有充分利用关节力矩,也容易导致关节力矩指令突变并引起振荡,文献[9]中对关节力矩加权优化的方法同样存在稳定性的问题.基于GWLN的方法通过辅助变量的引入,将重力项以及科里奥利力——离心力项包含在力矩约束中,避免了上述问题,具有更准确、平滑的效果.针对有限关节力矩约束下冗余机械臂的控制问题,本文提出基于广义加权最小范数法的求解算法.该方法在保证关节力矩约束的同时,准确地完成了操作任务,充分利用了系统的冗余特性.算法的有效性既通过理论分析得到了证明,也由在MATLAB Robotic Toolbox 中的仿真结果得到了验证.与现有加权算法的结果对比可知,基于GWLN方法的控制算法更加有效地保证关节力矩约束.当操作任务的要求完全超出关节力矩范围时,关节力矩约束不能通过基于GWLN的运动学规划求解方法得以实现,任务轨迹的执行将产生偏差.此时,须对操作任务规划调整来保证对任务轨迹的跟踪,具体方法还有待研究.此外,本文提出的算法未考虑机械臂模型偏差及外界扰动存在的情况.如何在模型偏差以及扰动存在的情况下,实现冗余机械臂的力矩约束控制是今后的研究方向之一.【相关文献】[1] KIRCANSKI M, KIRCANSKI N. Resolved rate and acceleration control in the presence of actuator constraints [J]. IEEE Control Systems, 1998, 18(1): 42-47.[2] HOLLERBACH J M, SUH K C. Redundancy resolution of manipulators through torque optimization [J]. IEEE Journal of Robotics and Automation, 1987, 3(4): 308-316.[3] LIU S, WANG J. Bi-criteria torque optimization ofredundant manipulators based on a simplified dual neural network [C]∥2005 IEEE International Join t Conference on Neural Networks. Montreal: IEEE, 2005:2796-2801.[4] 熊有伦.机器人学[M].北京：机械工业出版社,1993: 263-266.[5] ZHAO Z, WU Z, LU J, et al. Dynamic dexterity of redundant manipulators [C]∥ IEEE International Conference on Systems, Man and Cybernetics. Vancouver: IEEE, 1995: 928-933.[6] 陈伟海.冗余自由度机器人优化控制研究[D].北京：北京航空航天大学, 1996. CHEN Wei-hai. Research on control of robot with redundant degree of freedom [D]. Beijing: Beihang University, 1996.[7] 郭大忠,柳洪义,张威,等.冗余度机器人运动学和动力学同时优化[J].东北大学学报:自然科学版,2008,29(7): 1008-1011. GUO Da-zhong, LI Hong-yi, ZHANG Wei, et al. Simultaneously kinematical and dynamical optimizations of redundant robots [J]. Journal of Northeastern University: Natural Science, 2008, 29(7): 1008-1011.[8] 郭宪,王明辉,李斌,等.基于最小无穷范数的蛇形机器人最优力矩控制[J].机器人, 2014, 36(01): 8-13. GUO Xian, WANG Ming-hui, LI Bin, et al. Optimal torque control of a snake-like robot based on the minimum infinity norm [J]. Robot, 2014, 36(01): 8-13.[9] 闫彩霞,闫楚良,陆震.基于加权矩阵的过驱动并联机构驱动力矩调节法[J].吉林大学学报：工学版,2008,38(5): 1215-1219. YAN Cai-xia, YAN Chu-liang, LU Zhen. Approach to coordinate driving torque of redundant actuated parallel manipulator based on weighted matrix [J]. Journal of Jilin University: Engineering and Technology Edition, 2008, 38(5): 1215-1219.[10] 金波,陈诚,李伟.基于能耗优化的六足步行机器人力矩分配[J].浙江大学学报:工学版, 2012,46(07): 1168-1174. JIN Bo, CHEN Cheng, LI Wei. Optimization of energy-efficient torque distribution for hexapod walking robot [J]. Journal of Zhejiang University: Engineering Science, 2012, 46(07): 1168-1174.[11] BIANCO C G L, GERELLI O. Trajectory scaling for a manipulator inverse dynamicscontrol subject to generalized force derivative constraints [C]∥International Conference on Intelligent Robots and Systems. St. Louis: IEEE, 2009: 5749-5754.[12] 陈伟海,武桢,丁希伦,等.冗余度机器人动力学容错控制技术研究[J].北京航空航天大学学报,2000, 26(6): 726-730. CHEN Wei-hai, WU Zhen, DING Xi-lun,et al.Research on fault tolerant control of dynamically redundant manipulators [J]. Journal of Beijing University of Aeronautics And Astronautics, 2000, 26(6): 726-730.[13] ZHANG Yu-nong, WANG Jun. A dual neural network for constrained joint torque optimization of kinematically redundant manipulators [J]. IEEE Transactions on Systems Man and Cybernetics Part B, 2002, 32(5):654-662.[14] FLACCO F, LUCA A D, KHATIB O. Motion control of redundant robots under joint constraints: saturation in the null space [C]∥2012 IEEE International Conference on Robotics and Automation (ICRA). St. Paul: IEEE, 2012: 285-292.[15] LIEGEOIS A. Automatic supervisory control of the configuration and behavior of multibody mechanisms [J]. IEEE Transactions on Systems Man and Cybernetics, 1977,7(12): 868-871.[16] NAKAMURA Y, HANAFUSA H, YOSHIKAWA T. Task-priority based redundancy control of robot manipulators [J]. International Journal of Robotics Research, 1987, 6(2): 3-15. [17] XIANG J, ZHONG C W, WEI W. General-weighted least-norm control for redundant manipulators [J]. IEEE Transactions on Robotics, 2010, 26(4):660-669.[18] 钟琮玮.仿人型乒乓球机械手运动学及动力学控制方法研究[D].杭州：浙江大学, 2011. ZHONG Cong-wei. Kinematic and dynamic control of a humanoid pingpong manipulator[D]. Hangzhou: Zhejiang University, 2011.[19] XIANG J, ZHONG C W, WEI W. A varied weights method for the kinematic control of redundant manipulators with multiple constraints [J]. IEEE Transactions on Robotics, 2012, 28(2): 330-340.[20] CORKE P. Robotics, vision and control [M]. Berlin: Springer, 2011.。

基于约束矩阵和遗传算法的装配线平衡优化方法兰世海,李蓓智,杨建国,周亚勤(东华大学机械工程学院,上海201620)摘要:针对装配线平衡问题(AL B),归纳装配作业的三种基本约束关系,并通过约束矩阵描述装配作业的约束与优先权关系。

利用遗传算法对装配线平衡问题(ALB)进行分析和求解。

算法中,提出基于约束矩阵的作业序列编码策略,设计遗传操作(选择、交叉、变异),并通过基于模拟退火机制的精英策略加速了算法收敛。

最后,用实例证明此算法的有效性。

关键词:装配线平衡;约束矩阵;遗传算法;精英策略中图分类号:T H39119文献标识码:A文章编号:1671)3133(2007)03)0084)04A sse m b l y li ne ba l anci ng op ti m ization ba sed on restr ictedm a tr ix and genetic a lgor ith mLan Sh i2ha,i L i Be i2zh,i Yang Jian2guo,Zhou Ya2qi n(College ofM echan ica lEngineeri n g,Donghua University,Shangha i201620,C HN)Abstr ac t:Three bas i c restricted re l a tio nsh i ps of asse mb l y tas ks a re presented for Assemb l y L i ne Balanc i ng(ALB).The restr i ctio n and pr i or ity of tasks a re described accordi ng restricted m atri x.Asse m bl y Li ne Ba lanc i ng is analyzed and stud i ed usi ng genetic al2 gor ith m.In this research,t he tacti cs of tasks sequence cod i ng based restr i cted m atri x is presen ted.Geneti c opera tions(se lecti on, crossover and mu tati on)are deve l oped.A lgorith m beco m es more astri ngent by e lite tactic based on si m ulated anneali ng.F i na ll y an ill ustra ti ve exa m ple i s gi ven to testify the vali d ity of t h is a l gorith m.K ey word s:Asse m bly li ne balanc i ng;R estr i c ted m a trix;Geneti c a l gorith m;E lite tacti c自Salverson1955年提出装配线平衡(A sse mb l y L i n e Ba lanc i n g,ALB)问题[1]以来,ALB一直是国内外学者的研究热点。

收稿日期:2000-10-22基金项目:国家高技术863计划资助项目(863-511-820-020) 作者简介:欧阳兴(1968-),男,江西都昌人,博士生,100083,北京.有限元网格结点编号欧阳兴 陈中奎 施法中(北京航空航天大学机械工程及自动化学院)摘 要:在有限元分析中,求解高阶线性代数方程组时整体刚度矩阵所需存储与由网格结点编号决定的顺序有关.在基于等带宽存储的求解法与基于变带宽存储的求解法的基础上推导出它们的关系.据此,提出了有限元网格结点编号的前沿法与矩形法,并给出了这两种编号法的内存消耗与结点数量的关系.理论分析和实例表明这两种编号法能有效地减少计算机内存消耗.关 键 词:有限法;刚度矩阵;线性方程;结点编号;排序中图分类号:TB 115文献标识码:A 文章编号:1001-5965(2002)03-0339-041 问题的提出在有限元分析中,由于求解如下形式的高阶线性代数方程组,需要耗费计算机大量的内存和计算时间:K x =f(1)其中 K 是整体刚度矩阵;x 是未知向量;f 是已知向量.K 具有大型、对称、带状、稀疏、正定和主元占优等特点.有限元分析的求解效率很大程度上取决于方程组(1)的求解方法,包括K 的存储方法.因为(1)的求解时间在有限元分析的求解时间中占了很大的比重.当单元增多、结点增多、网格加密和未知数大量增加时,尤为如此.为了提高有限元分析效率,必须结合K 的特点,研究出方程组(1)的高效求解方法.方程组(1)的求解方法主要有两大类:直接求解法和迭代求解法.直接求解法以高斯消元法为基础,求解效率高.迭代求解法有赛德尔迭代法和超松弛迭代法.高斯消元法是目前求解线性方程组最普遍的方法,有一般高斯消元法和基于带状存储的高斯消元法.当K 的阶数非常高时,为了减小内存消耗,考虑到K 的对称性和带状性,带状存储法只存储以主对角线为中心的斜带形区域的半边.带状存储法有等带宽存储法和变带宽存储法.基于变带宽存储的求解法与基于等带宽存储的求解法相比算法更复杂,但内存消耗更少,因而有限元分析中越来越多地采用基于变带宽存储的求解法,如活动列求解器(active column solv -ers)[1]和前沿求解法(frontal solution methods)[2].带状存储法中,(1)所需要的存储空间不仅与(1)中x 的个数有关,而且与x 的顺序有关.而(1)中x 的分量次序与有限元网格结点的编号有关,不同的有限元网格结点编号很可能使(1)所需要的存储空间差别很大.文献[3],[4]提出了有限元网格结点的编号算法.这些算法是通过减小K 的带宽来减少方程组(1)所需要的存储空间,因而只适合于等带宽存储法.本文提出了一种新的有限元网格结点编号算法,这种算法仅与网格模型中结点空间位置及结点的连接关系有关,与网格模型中结点初始编号无关,既适合于等带宽存储法,又适合于变带宽存储法,并且使(1)需要的内存空间非常少.2 结点编号问题的数学描述设一个有限元网格模型总共n 个结点和e 个单元,结点编号和单元编号都从0开始,对于i =0,1,,,n -1,具有编号为i 的结点记为v i ,对于i =0,1,,,e -1,具有编号为i 的单元记为u i .设A =(a ij )n @n 为表示有限元网格模型的结点相连矩阵,a ij 它的第i 行第j 列元素,因而:1)a ij =1,如果i =j 或者v i ,v j 在同一单元内;2002年6月第28卷第3期北京航空航天大学学报Journal of Beijing University of Aeronautics and Astronautics June 2002Vol.28 No 132)a ij=0,其它.A是n阶对称方阵.如果每个结点都仅有一个自由度,则不仅K与A阶数相同(都是n),而且它们的零元素与非零元素对应位置完全相同,即它们具有相同的稀疏结构.设每个结点的自由度为f,在A=(a ij)n@n中,将每个1换成f@f个1组成的方阵,将每个0换成f@f个0组成的方阵,这样得到f@n阶方阵记B,则B与K具有相同的稀疏结构,因此有限元网格结点编号算法只需考虑A.以下元素都是针对A,行标和列标都从0开始.对于i=0,1,,,n-1,记第i行第1个元素1的列标为s i,显然s i=min{j|v j与v i在同一单元内,j=0,1,,,n-1}(2)又记l i=i-s i+1(3) s i称为结点v i的最小相关结点号,l i称为第i行半行带宽或称为结点v i的半行带宽.对于j=0,1,,,n-1,记第j列第1个元素1的行标为t j,称c j=j-t j+1为第j列半列带宽.A的半带宽B可以从以下两式得到:B=max{l i|i=0,1,,,n-1}(4)B=max{c j|j=0,1,,,n-1}(5)对应于等带宽存储法,A的半带形区域(包括主对角线)元素个数为A0=nB-B(B-1)P2(6)此时,K的半带形区域(包括主对角线)元素个数为K0=[nB-B(B-1)P2]f2(7)对应于变带宽存储法,A的半带形区域(包括主对角线)元素个数为A1=E n-1i=0l i=E n-1i=0(i-s i+1)(8)此时,K的半带形区域(包括主对角线)元素个数为K1=f2E n-1i=0(i-s i+1)(9)显然A0[A1,K0[K1,因此,变带宽存储法比等带宽存储法更节省内存.由n个结点组成的有限元网格有n!个不同的结点编号,对于这n!个不同的结点编号中任意一个结点编号(记为h,h=0,1,,,n!-1),由有限元网格模型中单元与结点的相互关系,可以立即计算其对应的B,A0,A1,K0和K1.可将B, A0,A1,K0和K1看成以h为自变量的函数,记为B(h),A0(h),A1(h),K0(h)和K1(h).因此,对应于等带宽存储法和变带宽存储法,有限元网格模型的结点最佳编号是分别求解g,使B(g)=min{B(h),h=0,1,,,n!-1}(10)和A1(g)=min{A1(h),h=0,1,,,n!-1}(11)虽然n!是有限数,但对于一般的有限元网格模型来说,n!非常大,所以不可能通过直接求出n!个B(h)及A1(h),再比较n!个B(h)或A1(h)的大小,来求出h.本文根据结点与单元的相互关系导出以下算法.3算法原理算法用到的数据结构包括:整个网格模型的数据包括结点数组和单元数组,每个结点的数据包括结点坐标、结点编号和结点所属的单元数组,每个单元的数据包括单元编号和依序构成单元的结点数组.由(2)、(3)、(4)、(6)、(8)、(10)、(11)等式可归纳出两种结点编号方法.311前沿法先寻找一个边角结点.因边角结点所在的单元数少,可以通过比较结点所属的单元数组的大小,使结点所属的单元数组最小的结点编号为0,依次给0,1,,,n-2号结点所在单元的未编号结点进行编号.图1采用4@5个四边形单元组成的网格模型的结点编号作为前沿法结点编号的示意.24232221202915141312192887611182732510172601491625图1前沿法结点编号312矩形法对矩形区域来说,采用前沿法的图1不是最佳结点编号,图2才是最佳结点编号.一般地,若整个网格模型是一矩形区域,由(a-1)(b-1)个340北京航空航天大学学报2002年四边形单元组成,共有ab个结点,排列成a行b 列.若按图2所示进行结点编号,可得如下结论.49141924293813182328271217222716111621260510152025图2矩形区域的最佳结点编号由(4)式得:B=a+2(12)由(6)式得:A0=(a+2)ab-(a+2)(a+1)P2(13)由(8)式得:A1=1+2+,+2+(a+1)+(a+2)+,+ (a+2)+,+(a+1)+(a+2)+,+(a+2)=ba2+2ba-a2-b(14)当b=a时,由(14)式得:A1=a3+a2-a(15)在(14)式中,交换a与b可得:A1=ab2+2ab-b2-a(16)因为当a<b时,ba2+2ba-a2-b<ab2+ 2ab-b2-a.因此,图3中先按行对结点进行编号比图2中先按列对结点进行编号得到的A1和B要大.在图2中,将从左到右作为x轴正向,从下到上作为y轴正向,定义关系/<0为p(x0,y0)< q(x1,y1),当且仅当x0<x1或x0=x1且y0<y1.则图2所示的结点编号可看作结点从小到大的排序.对于一般的网格模型,虽然单元和结点不如图1~图3所示的矩形区域的单元和结点那样排列得非常规则,但由于同一单元的所有结点的空间位置接近,因而可以采用图2所示的方法进行结点编号.若一般的网格模型中结点、单元的连接关系不与图2所示的矩形区域的单元和结点连接关系相似,则需计算多个排序方法所得编号对应的B或A1,再比较求出最好的结点编号.以下叙述求一般的网格模型的最佳的结点编号方法.24252627282918192021222312131415161767891011012345图3矩形区域的行优先编号如果待编号网格模型是平面网格模型,即所有结点在同一平面内,则可认为网格模型中所有结点有一个坐标分量是同一常数.否则,经平移、旋转坐标系变换可将所有结点的一个坐标分量变换成同一常数.不妨设所有结点z坐标相同,因而z坐标与结点编号无关,下面只有两个坐标的点可认为省写了z坐标.分别定义关系/<0为1)p(x0,y0)<q(x1,y1)当且仅当x0<x1或x0=x1且y0<y1.2)p(x0,y0)<q(x1,y1)当且仅当x0<x1或x0=x1且y0>y1.3)p(x0,y0)<q(x1,y1)当且仅当y0<y1或y0=y1且x0<x1.4)p(x0,y0)<q(x1,y1)当且仅当y0<y1或y0=y1且x0>x1.按这4种结点/<0关系的定义,分别排序得到对应的4种结点编号及对应的B或A1,再比较求出最好的结点编号.如果待编号网格模型不是平面网格模型,则必须把结点按x,y,z3个坐标排序编号.和上述方法一样,因为x,y,z3个坐标分量的排列次序可以任意,而且每个坐标分量可以从大到小也可以从小到大.因而有48种3坐标字典排序方法来定义结点/<0关系.同样分别排序得到对应的48种结点编号及对应的B或A1,再比较求出最好的结点编号.313前沿与矩形结合法大部分情况下矩形法比前沿法更节省内存,只有在一些特殊情况下前沿法比矩形法更节省内存.无论如何,都可以将两种方法结合起来,即通过计算和比较两种方法对应的B或A1,求出最好的结点编号.4实例与总结本文处理的一个实例中,整个网格模型有6684个结点和7321个单元,其中三角形单元1452个,四边形单元5869个.采用变带宽存储法并且实数用双精度浮点数(double)存贮.每个结点自由度为5个.未经本文结点编号而采用初始结点编号需要5801511926MB内存;应用本文前沿法,结果只需要1271346289MB内存;应用本文矩形法,结果只需要861866638MB内存.341第3期欧阳兴等:有限元网格结点编号由(15)式可知,当网格模型基本呈正方形时应用矩形法,K所需要存贮元素个数为O(n3P2).同样可推出,当网格模型基本呈正方形时,应用本文前沿法,K所需要存贮元素个数为O(n3P2).由(14)式可知,当网格模型为长条矩形,K所需要存贮元素个数比网格模型基本呈正方形时还要少.应用矩形法,当网格模型形状非常复杂时与网格模型形状为正方形时基本接近.而大多数结点编号算法K所需要存贮元素个数为O(n2).因而本文算法明显优于这些算法.一般地,设每个实数需要r个字节内存(采用双精度浮点实数r=8,采用单精度浮点实数r= 4),每个结点的自由度为f,采用矩形法K所需要内存大约为r f2n3P2,采用前沿法K所需要内存大约为4P3r f2n3P2.参考文献[1]Bathe K J.Fi nite elenent procedures in engineeri ng analysis[M].Prentice-Hall,Inc,Enge wood Cli ffs,NJ,1982.441~449.[2]Irons B M.A frontal solution program for finite element analysis[J].Int J Numer M eth Engng,1986,23:239~256.[3]Cuthill E,M ckee J.Reducing the bandwidth of s parse symme tricma xtrices[A].Proc24th Nat Conf of the ACM[C].1969.157~ 172.[4]Gibbs N E,Poole W G,Stockmeyer P K.An al gori thm for reducethe bandwidth and profile of a sparse matrix[J].SIMA J Num Anal, 1976,13(2):236~250.Numbering of Fin ite Element Mesh NodesOUYANG Xing C HEN Zhong-kui SHI Fa-zhong(Beijing University of Aeronautics and As tronautics,School of Mechanical Engineering and Automati on) Abstract:In finite element analysis,the storage needed by a total stiffness matrix for solving a large-scale sys-tem of linear equations is related to the sequenace determined by numbering of mesh nodes.Based on the solutions for both constant and varible bandwidth storages,their relationship is derived.On the above basis,the frontal meth-od and rectangle method are proposed,the relationships between memory spending for both methods and a mount of nodes are given.Theoretical analyzing and practical examples have proved that the two methods can efficiently de-crease the memory spending of computer.Key words:finite element methods;stiffness matrix;linear equations;numbering of nodes;sorting 342北京航空航天大学学报2002年。

无约束极值问题的拟合方法王薇;徐以凡【期刊名称】《运筹学学报》【年(卷),期】2003(007)001【摘要】我们在本文中从一个完全不同的观点提出了一个用于求解无约束最优化问题的拟合算法.算法中的迭代方向是从函数拟合中得到,而不是由传统的拟牛顿方程得到.此方法有许多好的性质,并且在较弱的假设下证明算法是线性收敛的.%In this paper, we proposes a modified fitting algorithm from a total different point of view for unconstrained optimization. In algorithm the iterative direction is from the function fitting other than the traditional quasi-Newton equation. Our method has many good properties and is linearly convergent under some mild assumptions.【总页数】7页(P46-52)【作者】王薇;徐以凡【作者单位】上海大数学系,上海,200436;复旦大学管理学院,上海,200433【正文语种】中文【中图分类】O22【相关文献】1.分别用耿贝尔(Gumbel)极值分布和广义(GEV)极值分布拟合五华县降水极值 [J], 赵立2.用无约束最优化方法拟合布里渊散射数据中存在的问题的讨论 [J], 王越;刘玉龙3.确定物理实验的拟合曲线的极值的方法 [J], 雷桂林;薛怀庆4.在无约束极值问题中运用正交法改进模矢法初探 [J], 姜效先5.基于g-h分布的极值分布拟合新方法 [J], 吴建刚因版权原因，仅展示原文概要，查看原文内容请购买。

无约束最优化问题的BFGS松弛异步并行算法李文敬;刘之家;蓝贞雄【摘要】In order to solve the sequential solution speed slow problem of the large-scale non-linearity optimization problems, for solving unconstrained optimization problems, chaotic relaxation asynchronous iterative parallel algorithm is proposed. BFGS sequential algorithm of unconstrained optimization problems are parallelized in the PC cluster environment. It utilizes CHOLESKY method to decompose coefficient to make the linear equation group of matrix for the symmetry. To solve the solution vector and non-linear search step of Wolfe-Powell by chaotic relaxation asynchronous parallel method, it parallels to the revised formula BFGS. The BFGS relaxation asynchronous parallel algorithm is build. Time complexity and speedup of the algorithm are analyzed. The experimental results of PC cluster show that this algorithm better solves the system load unbalanced problem, improves the speed of problem solving, the algorithm has the linear speedup ratio.%为解决大规模非线性最优化问题的串行求解速度慢的问题,提出应用松弛异步并行算法求解无约束最优化问题.根据无约束最优化问题的BFGS串行算法,在PC机群环境下将其并行化.利用CHOLESKY方法分解系数为对称正定矩阵的线性方程组,运用无序松弛异步并行方法求解解向量和Wolfe-Powell非线性搜索步长,并行求解BFGS修正公式,构建BFGS松弛异步并行算法,并对算法的时间复杂性、加速比进行分析.在PC机群的实验结果表明,该算法提高了无约束最优化问题的求解速度且负载均衡,算法具有线性加速比.【期刊名称】《计算机工程与应用》【年(卷),期】2012(048)017【总页数】4页(P44-47)【关键词】最优化问题;BFGS公式;松弛异步;并行算法;加速比【作者】李文敬;刘之家;蓝贞雄【作者单位】广西师范学院计算机与信息工程学院,南宁530023;广西师范学院计算机与信息工程学院,南宁530023;广西师范学院计算机与信息工程学院,南宁530023【正文语种】中文【中图分类】TP393在工农业生产、工程技术、交通运输、生产管理、经济计划、国防、金融等很多实际问题都可以抽象转化成最优化问题。