Proposed approximation for contact angles in Shan-and-Chen-type multicomponent multiphase---2007
信息类专业英语翻译
Dynamic topology:As the channel of communicationchanges, some of the neighbors who were reachable on theprevious channel might not be reachable on the currentchannel and vice versa. As a result the topology of the network changes with the change in frequency of operation resulting in route failures and packet loss.Heterogeneity:Different channels may support differenttransmission ranges, data rates and delay characteristics.Spectrum-Handoff delay:For each transition from onechannel to another channel due to the PU’s activity, thereis a delay involved in the transition called Spectrum- Handoff delay.All these factors decrease the predictability of the cause oftransit-delay and subsequent packet loss on the network. Thetime latency during channel hand-off in cognitive networksmight cause the TCP round trip timer to time out. TCP willwrongly recognize the delays and losses due to the abovefactors as network congestion and immediately take steps toreduce the congestion window size knowing not the cause ofpacket delay. This reduces the efficiency of the protocol insuch environments.动态技术：随着信道通信的变化，一些邻进信道的用户在原信道没有发生变化而在新信道发生变化，或者相反。
投资组合优化模型的一个序列凸近似算法
投资组合优化模型的一个序列凸近似算法李卫国;张宏伟;梁锡军【摘要】以CVaR为代表的凸优化投资组合模型近年来引起了广泛研究.为克服传统投资组合模型中凸近似的不足,提出了一个投资组合的DC规划模型.该模型用一个DC函数替代了CVaR模型中的凸近似函数,同时要求所有约束条件在概率意义下成立.进一步地,提出了一个序列凸近似(SCA)算法用于求解DC规划问题,并运用Monte-Carlo方法来实现SCA算法.初步的实验结果表明,因子收益服从"尖峰厚尾"分布时,模型的目标函数值优于采用CVaR近似的目标函数值.%CVaR has drawn extensive attentions as a representative convex optimization portfolio model in recent years.To overcome the limits of convex approximations in traditional portfolio models,a DC programming model for portfolio is proposed.In the proposed programming model,a DC function is used as a surrogate for the convex approximation function in the CVaR model.All the constraints are satisfied in the probabilistic sense in the DC programming problem.Moreover,a sequential convex approximation (SCA)algorithm is designed to solve the DC programming problem. The SCA algorithm is implemented by employing Monte-Carlo method.Preliminary experimental results have shown that the obj ective function values of the DC programming are better than those with CVaR approximation when the income factors sati sfy″high peak and fat tail″distributions.【期刊名称】《大连理工大学学报》【年(卷),期】2017(057)003【总页数】6页(P321-326)【关键词】投资组合;序列凸近似;凸优化;Monte-Carlo方法【作者】李卫国;张宏伟;梁锡军【作者单位】大连理工大学数学科学学院,辽宁大连 116024;辽宁地质工程职业学院,辽宁丹东 118303;大连理工大学数学科学学院,辽宁大连 116024;中国石油大学(华东)理学院,山东青岛 266580【正文语种】中文【中图分类】O224投资组合是指投资者根据其风险喜好在众多的有价证券中对风险投入进行最优投资分配．投资组合选择问题最早由Markowitz[1]作为一个最优化问题提出．他用随机收益率的均值来衡量预期收益的好坏，用随机收益率的方差衡量风险的大小．Markowitz的均值-方差模型在现代投资理论中堪称经典，并且始终在改进．例如文献[2]对这些模型作了一个系统的总结．只有当证券收益率服从正态分布或者投资者是风险厌恶型时，均值-方差模型才是有效的．但由于计算的复杂性，在投资数额较大的优化问题中模型的使用率并不高．本文试图改进这一模型，研究不同的预期收益或风险衡量标准，使得模型更符合实际，更好地为投资者决策提供参考．投资者经常会遇到投资项目的组合决策问题，要考虑的因素有收益率、风险、增长潜力等条件，希望该资产投资波动越小越好，并进行权衡考虑获得一个最佳的投资方案．在投资组合的优化研究过程中，本文试图在传统的保守近似优化模型的基础上，克服指示函数I(0，+∞)(z)在近似估计中的不足．通过引入风险度量的DC(difference of two convex functions)近似，给出一个投资组合的DC规划模型，并提出一个序列凸近似(sequential convex approximation，SCA)算法来进行求解，用一个严格凸二次函数来近似目标函数中的光滑函数，用线性函数近似所有DC函数的第二个凸函数，得到搜索方向的一系列的凸优化问题．考虑如下问题：对n个资产投资以获取最大收益．记x=(x1 x2 … xn)T为n个资产的投资比例令x∈X∶=Rn．各资产的收益r=(r1 r2 … rn)T为随机向量，假定其满足如下关系：式中：μ为资产收益的均值，；V为因子荷载矩阵，VT=(V1 V2 … Vn)；ξ为因子收益，ξ=(ξ1 ξ2 … ξn)T服从F分布(尖峰厚尾)；ε～N(0，D)，D为对称半正定矩阵，ε为白噪声．极大化所研究资产的平均收益：其约束条件被称为决策风险[3]，用来描述，其中1=(1 1 … 1)T∈Rn，β∈(0，1)是一个较小的数，a>0是保守收益的一个估计．同时希望该资产的投资波动越小越好．采用var(rTx)=xTDx，作为投资波动的度量．考虑如下模型：max E[rTx]=μTx+xTVTEF(ξ)-γxTDxs. t. Prob{-(μTx+xTVTξ)≤-a}≥1-β1Tx=1，x≥0其中γ>0是参数．考虑模型(1)中的概率约束，令p(x)=1-Prob{-(μTx+xTVTξ)+a≤0}= Prob{-(μTx+xTVTξ)+a≥0}则约束条件其中这里IA(·)表示集合A上的指示函数：在Rockafellar等[4]提出的CVaR近似中，用近似I(0，+∞)(z)，其中z>0，[z]+=max{z，0}．Nemirovski等[5]曾经指出在所有的凸的保守近似中，CVaR被公认为是最出色的．但是，ψ(z，t) 对于指示函数I(0，+∞)(z)不是一个好的近似，因为在z>0且z较大时，两个函数差异较大．为寻求一个更好的近似，令由于ψ(z，t)和φ(z，t)都是z的凸函数，π(z，t)是一个关于z的DC函数．函数π(z，t)仅在区间(-t，0)与指示函数有差异，在其余的区间与指示函数完全吻合．可见，DC函数π(z，t)是指示函数的一个更好的近似，如图1所示．令g1(x，t)=E[μTx+xTVTξ-a+t]+ g2(x)=g1(x，0) p＾(x，t)=当t>0时，p＾(x，t)是p(x)的一个保守DC近似．令，则p＾(x，t)≤β ⟹因此，是p＾(x，t)所有近似中最好的保守近似．maxμTx+xTVTEF(ξ)-γxTDxs.t.1Tx=1，x≥0称模型(2)为模型(1)的DC近似[6]．约束条件可近似为≤β，即用下面的模型(3)近似模型(2)．max μTx+xTVTEF(ξ)-γxTDxs.t. g1(x，t)-g2(x)≤βt 1Tx=1，x≥0记Ω(t)为模型(3)的可行域：Ω(t)∶={x∈X：g1(x，t)-g2(x)≤βt， 1Tx=1，x≥0}算法1是求解模型(3)的序列凸近似算法．算法1 序列凸近似(sequential convex approximation，SCA)算法SCA算法具体步骤如下：Step 1 选取x0∈Ω(t)，k=0．Step 2 若xk是最优解，算法终止．Step 3 用Monte-Carlo方法[6]求解下列凸规划问题：max μTx+xTVTEF(ξ)-γxTDxs.t. g1(x，t)-[g2(xk)+〈g2(xk)，x-xk〉]≤βt 1Tx=1，x≥0得最优解xk+1．Step 4 k∶=k+1，转Step 2．2.1 凸规划问题(4)的求解依照前面的记号c(x，ξ)=μTx+xTVTξ-a g1(x，t)=E[c(x，ξ)+t]+= E[I[0，+∞)(c(x，ξ)+t)]g2(x，t)=g1(x，0)另外，记h(x)为问题(4)的目标函数：则有令则记ξ1，ξ2，…，ξn为随机变量ξ的独立同分布的样本点．对g(x)的一个自然的估计是对E(ξ)的估计是．建议求解s.t. 1Tx=1，x≥02.2 g(x)的近似为了在n较大时有效地求解问题(4)，给出下面的方法：由式(5)、(6)及文献[7]引理2知，则g(x)可以近似为当n较大时，用g(x)作为g(x)的近似，并直接用基于梯度[8]的方法求解问题(4)．该方法可以视为用近似的g(x)、g(x)及E(ξ)直接求解问题(4)．当采用基于梯度的方法求解问题(4)时，样本点仅用于计算(x)和g(x)的值．所需的计算量是O(n)．这是求解问题(4)的一种较快的方法．2.3 初始点的选取为执行SCA算法，需要选取初始点x0∈Ω(t)，给出两种选取办法．第一种，记注意到，g2(x)=E[μTx+xTVTξ-a]+≥0，∀x∈X．故Ω0(t)⊆Ω(t)．另外，由于g1(x，t)是关于x的凸函数，故Ω0(t)是凸集合．于是，是凸规划．令xε=arg min{h(x)：x∈Ω0(t)}，则xε∈Ω(t)．第二种，令ΩCVaR={x∈X：CVaR1-βc(x，ξ)≤0}．依据前面的讨论，令xCVaR∈arg min{h(x)：x∈ΩCVaR，1Tx=1，x≥0}，即Rockafellar和Uryasev 给出的CVaR近似的最优解．令t*=q1-β(c(xCVaR，ξ))，即c(xCVaR，ξ)的1-β分位数．由文献[9]，t*>0且于是，xCVaR∈{x∈X：g1(x，t*)≤t*β}．由于g2(x)≥0，故xCVaR∈{x∈X：g1(x，t*)-g2(x)≤t*β}=Ω(t*)．对于任意t∈(0，t*]，Ω(t*)⊆Ω(t)．因此，xCVaR∈Ω(t)，0<t≤t*．可以首先求解CVaR近似问题得到xCVaR，然后选取t∈(0，t*]，则xCVaR可以用作SCA算法的初始点．若使用x0=xCVaR，则SCA 算法产生的迭代点列{xk}将改进或至少等价于CVaR近似解．(1)CVaR近似解的计算xCVaR=arg min{h(x)：CVaR1-βc(x，ξ)≤0，1Tx=1，x≥0}(2)CVaR分位数的计算记L是一随机损失，FL(y)=Prob{L≤y}是L的累积分布函数．对于任意α∈(0，1)，则损失L的α-VaR定义为定义为由文献[10]知道令L1，L2，…，Ln是损失L的n个独立同分布的观测，则L的α-VaR可以估计为其中是排序后的n个观测中的第i个观测．L的α-CVaR可以估计为(3)CVaR分位数的梯度估计假设随机损失L是参数θ的函数，记为L(θ)．L(θ)关于θ可微．对任意的θ，设vα(θ)、cα(θ) 是L(θ)的α-VaR和α-CVaR，0<α<1，它们都是θ的函数．依据文献[10]定理3.1，在适当的条件下，对于任意的θ，记(L1，D1)，(L2，D2)，…，(Ln，Dn)为(L(θ)，L′(θ))的n个独立同分布的观测．提出c′α(θ)的估计：采用下面的数据进行测试：各资产的平均收益μ=(1.0 1.2 1.4 1.6 1.8)T，期望收益a=1.5，因子荷载矩阵方差-协方差矩阵实验测试ξ=(ξ1 ξ2 … ξp)T服从如下两种分布的情形．(1)ξ服从F分布：ξi～F(i+2，i+4)，i=1，2,…，p；(2)ξ服从正态分布：ξi～N(πi，ρi2)，i=1，2,…，p，为了比对两种情形，取均值πi、方差ρi2与情形(1)中相应的F分布的均值和方差相同．序列凸近似算法的参数设为γ=1.0，β=0.05，t=0.01．算法在Matlab 2012a平台下编程实现，在Intel Core i7-4770 CPU 3.40 GHz，8 GB RAM计算机上执行．实验分别测试了-近似解和CVaR近似得到的解作为SCA算法的初始点．随机变量ξ的抽样次数为1.0×105．图2中给出了目标函数值f随迭代次数k的变化，ξ分别服从F分布和正态分布．可以看出，无论ξ服从F分布还是正态分布，CVaR初始点处的目标函数值大于-近似解处的目标函数值，并且在两种不同的初始点下，目标函数值随迭代次数的增加而增加，最终近乎收敛于共同的函数值．ξ服从F分布时(图2(a))，算法改进了-近似的目标函数值(1.605 6)和CVaR近似的目标函数值(1.608 0)：算法收敛时的目标函数值为1.621 2．两种初始点计算出的最优解皆为x*=(0 0 0.171 2 0 0.828 8)T．可见模型选择了第3个和第5个资产进行资产配置．ξ服从正态分布时(图2(b))，算法改进了-近似的目标函数值(1.618 2)和CVaR近似的目标函数值(1.618 3)：算法收敛时的目标函数值为1.618 7．两种初始点计算出的最优解皆为x*=(0 0 0.203 8 0 0.796 2)T．可见模型仍选择了第3个和第5个资产进行资产配置，但资产配置比例与ξ服从F分布时的情形有差别．为测试抽样规模的影响，实验比较了不同抽样规模下的目标函数值以及计算时间(单位：s)，如图3所示(ξ服从F分布)．在每个抽样规模下，进行了100次重复实验，目标函数值和计算时间取其平均值，图中画出了其标准差(每个柱线段的长度是相应标准差的2倍)．由图3(a)可以看出，目标函数值的标准差在抽样规模N大于0.5×105时趋于稳定(标准差小于1.0×10-3)；图3(b)显示，计算时间随抽样规模的增长而增加．ξ服从正态分布时有类似的规律，见图4．本文考虑因子收益ξ服从“尖峰厚尾”分布的情形，例如F分布，用DC函数来近似概率约束中的指示函数，将资产投资组合问题建模为一个DC规划，并提出了求解该DC规划的序列凸近似(SCA)算法．该算法通过迭代求解一系列凸二次规划问题，来计算原问题最优解．通过所提出的梯度近似计算方法，减少了算法的计算量．初步的数值实验结果表明，所提出的模型及SCA算法是有效的．因子收益ξ服从F分布时，目标函数值优于采用CVaR近似的目标函数值．【相关文献】[1] MARKOWITZ H M. Portfolio selection [J]. Journal of Finance, 1952, 7(1):77-91.[2] MITRA G, KYRIAKIS T, LUCAS C, et al. A review of portfolio planning:models and systems [J]. Advances in Portfolio Construction and Implementation, 2003, 26(1):1-39. [3] GOLDFARB D, IYENGAR G. Robust portfolio selection problems [J]. Mathematics of Operations Research, 2003, 28(1):1-38.[4] ROCKAFELLAR R T, URYASEV S. Optimization of conditional value-at-risk [J]. Journalof Risk, 2000, 2(3):21-40.[5] NEMIROVSKI A, SHAPIRO A. Convex approximations of chance constrained programms [J]. SIAM Journal on Optimization, 2006, 17(4):969-996.[6] HONG L J, YANG Yi, ZHANG Liwei. Sequential convex approximations to joint chance constrained programms: a Monte Carlo approach [J]. Operations Research, 2011,59(3):617-630.[7] GLASSERMAN P. Monte Carlo Methods in Financial Engineering [M]. New York:Springer, 2004.[8] FREUND R M. Subgradient optimization, generalized programming, and nonconvex duality [R]. 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光学设计常用术语解释及英 汉对照翻译汇总
第一部分最基本的术语及英汉对照翻译1、时谱：time-spectrumIn this paper, the time-spectrum characteristics of temporal coherence on the double-modes He-Ne laser have been analyzed and studied mainly from the theory, and relative time-spectrum formulas and experimental results have been given. Finally, this article still discusses the possible application of TC time-spectrum on the double-mode He-Ne Iaser.本文重点从理论上分析研究了双纵模He-Ne激光时间相干度的时谱特性(以下简称TC时谱特性),给出了相应的时谱公式与实验结果,并就双纵模He-Ne激光TC时谱特性的可能应用进行了初步的理论探讨。
2、光谱：SpectraStudy on the Applications of Resonance Rayleigh Scattering Spectra in Natural Medicine Analysis共振瑞利散射光谱在天然药物分析中的应用研究3、光谱仪：spectrometerStudy on Signal Processing and Analysing System of Micro Spectrometer 微型光谱仪信号处理与分析系统的研究4、单帧：single frameComposition method of color stereo image based on single fram e image 基于单帧图像的彩色立体图像的生成5、探测系统:Detection SystemResearch on Image Restoration Algorithms in Imaging Detection System 成像探测系统图像复原算法研究6、超光谱：Hyper-SpectralResearch on Key Technology of Hyper-Spectral Remote Sensing Image Processing超光谱遥感图像处理关键技术研究7、多光谱：multispectral multi-spectral multi-spectrumSimple Method to Compose Multi spectral Remote Sensing Data Using BMP Image File用 BMP 图像文件合成多光谱遥感图像的简单方法8、色散：dispersionResearches on Adaptive Technology of Compensation for Polarization Mode Dispersion偏振模色散动态补偿技术研究9、球差：spherical aberrationThe influence of thermal effects in a beam control system and spherical aberration on the laser beam quality光束控制系统热效应与球差对激光光束质量的影响10、慧差：comaThe maximum sensitivity of coma aberration evaluation is aboutλ/25;估值波面慧差的极限灵敏度为λ/25;11、焦距：focal distanceAbsolute errors of the measured output focal distance range from –120 to 120μm.利用轴向扫描法确定透镜出口焦距时的绝对误差在–120—120μm之间。
外文翻译---计算机控制插齿刀加工椭圆齿轮齿形
英文部分Computerized tooth profile generation of elliptical gea rs manufactured by shaper cuttersBiing-Wen BairDepartment of Mechanical Engineering, National Lien Ho Institute of T ech nology,AbstractThis work sim ula tes an elliptical gear driv e,the a xis of rot at ion of wh ich is coincident wi th its geometric center, manufac tured by sh ape r cutters. The mathematical model of an elliptical gear is developed based on the theory of gearing and gear generation mechanisms. In addition, the tooth undercutting of the gear is also investigated based on the developed mathematical model of the elliptical gear, its unit normal vectors and a numerical method.A geometric relationship is developed and applied to prevent the occurrence of pointed teeth on elliptical gears. Further, this study also develops computer simulation programs to generate the tooth profile of elliptical gears without tooth undercutting and pointed teeth. Compa rison of the angula r ve locity variations of the elliptical gear dr i ves is al so ma de. Th e r e su lts show that the deve lope d elliptica l gear driv e can be utilized as an oil pump with a larger pumping volume and less angular velocity variation. © 2002 Elsevier Science B.V. All rights reserved.Keywords: Elliptical gears; Undercutting; Pointed teeth1. IntroductionAn elliptical gear drive, the rotation center of which coincides with one of its foci, is kinematically equivalent to the crossed link, and can be used to produce irregular rotations. In addition, it is well known for providing excellent characteristics such as accurate transmission, compact size, and ease of dynamic balance. Hence, elliptical gear drives have been applied successfully in various types of automatic machinery, quick-return mechanisms, packaging machines, and printing presses [1]. This type of gearing can also be used to develop non-circular gears, which belong to high-order elliptical pitch curves. Second- order elliptical gear drives, the rotation center of which coincides with one of its foci, can find use in the design of instruments such as pumps and flow meters [1]. However, this type of gear set has two speed changes for each revolution, these two-cycle variations inducing a wave fluctuationthat is so severe that the second-order elli p tical gear set cannot be used as oil pumps for steady o il pumping.The design and manufacture of an elliptical gear are difficult because the pitch curve of the gear is an ellipse.Some studies [2—6] have focused on kinematic analysis and computer-aided design of elliptical pitch curves. Kuczewski [7] used a spur gear to approximate the profile of an elliptical gear. Emura and Arakawa [8] used an elliptical gear to analyze a steering mechanism, where this steering mechan- ism can turn a carrier with a small radius. Also, Freudenstein and Chen [9] developed variable-ratio chain drives (e.g. elliptical gear drives), which were applied to bicycles and variable motion transmissions involving band drives, tape drives, and time belts with a minimum slack. Moreover, Litvin [10] adopted the concept of evolute curves to form the tooth profile, and also derived the tooth evolute of an ellipse. Chang and Tsay [11] used a shaper cutter and applied the inverse mechanism relationship and the equation of meshing to produce the mathematical model of elliptical gears, the rotation center of which turns around one of its foci. Also, Chang et al. [12] used a rack cutter and the same method to produce the mathematical model and undercut- ting conditions of the same type of elliptical gears. However, when the elliptical gear surfaces are generated by shaper cutters, pointed teeth may appear and the tooth addendum is reduced. Pedrero et al. [13] proposed an approximation method for modifying the tooth addendum and contact ratio, and computer simulation results also show that the gear contact ratio depends on the tooth addendum. Additionally, Liou et al. [14] analyzed spur gears with low contact ratios (a contact ratio of less than 2) and high contact ratios (equal or larger than 2) when subjected to dynamic loads by applying the NASA gear dynamics software DANST. The DANST program determined the instantaneous contact teeth and contact ratio based on the gear average stiffness, commonly referred to as mesh stiffness. Recently, Bair and Tsay [15] proposed a tooth contact analysis (TCA) program to calculate the instantaneous contact teeth and average contact ratio of a dual-lead worm gear drive. The DANST and TCA methods confirm that reducing the gear addendum results in the decrease of the gear instantaneous contact teeth and the average contact ratio.Based on the position of the rotation center, elliptical gear drives fall into two types: the elliptical gear which rotates about its geometric center (type 1); the elliptical gear which rotates around one of its elliptical foci. An n- order driven non-circular gear, of which therotation axis is one of its foci, is one in which the driving elliptical gear performs n revolutions for one revolution of the driven non- circular gear. A second-order elliptical gear (type 2) is defined as n = 2. Under the same eccentricity and major axis, the size of the type 1 elliptical gears is smaller than that of the type 2. Further, if the type 1 elliptical gear set is applied to oil pumps, the wave fluctuation of the pumping oil is smaller and smoother than that of the other type. The elliptical gear tooth profile is usually produced by a hob- bing or shaping machine with a hob cutter or shaper cutter. This study simulates the manufacture of elliptical gears via a shaper cutter on a shaping machine. It is known that if a spur gear is produced by a shaper, the profile of the shaper cutter should be the same as that of the mating spur gear. Therefore, the mathematical model of a shaper cutter is the same as that of a spur gear, which can be derived from the generation mechanism with a rack cutter. According to the theory of gearing, the mathematical models of elli p tical gear tooth profiles, which rotate about their geometric center (type 1), are developed based on the proposed generation mechanism with shaper cutters. Due to the complex characteristics of this type of elliptical gear, undercutting and pointed teeth may exist on its tooth profile. The tooth undercutting of this type of elliptical gear is affected by its pressure angle, number of teeth, module, and major axis. The strength of the gear tooth root can be increased by applying a positive-shifted modification for the cutter during the gear generation. However, an over-positive-shifted modification may result in the appear- ance of pointed teeth. Pointed teeth are generated when the right- and left-side involute tooth profiles intersect on or below the addendum circle of the gear. Further, the pointed teeth are usually generated on the two major axis of an elliptical gear. If a profile index is defined to prevent pointed teeth generation on the two major axis of an elliptical gear, then no other pointed tooth will be generated for all elliptical gear profiles. Thus, the computer program developed here can calculate and provide proper design parameters for the designed elliptical gears to avoid tooth undercutting and pointed teeth.2. Mathematical model of the elliptical gear surfacesShaper cutters are used to generate elliptical gears, and the profiles of shaper cutters are the same as those of spur gears. Hence, the mathematical model of the shaper cutter is the same as that of the spur gear, which is generated from rack cutters. A complete elliptical gear tooth profile consists of three surface regions, i.e. the working region, the fillet and the bottom land. Therefore, the profile parameters of a shaper cutter can be represented by theparameters of a rack cutter. Fig. 1 shows three regions of a rack cutter 2p including the working region, the fillet and the top land, used for shaper cutter and elliptical gear generations. When the shaper cutter creates the elliptical gear in a cutting mechanism, its center rotates along the Z c-axis and translates along the X c and Y c-a xes, performing a pure roll without sliding on the pitch ellipse, and the gear blank is rotated about its geometric center 01 as in Fig.2.2.1. Working region of shaper cutter profileFig. 1 presents the design of the normal section of rack cutter 2p, where regions 3 and 4 are the left- and right-side working regions, regions 2and 5 are the left- and right-side fillets, and regions 1 and6 are the left- and right-side top lands. Meanwhile, parameter £p = M0M1 is a design parameter, expressing the distance measured from the initial point M0 to an arbitrary point M1 in the working region. The three- dimensional rack cutter profile can be obtained by translating its normal section, presented in Fig. 1, along the Z r-a xis with a displace me nt para meter U p. The refore, by applying the theory of gearing, the mathematical model of the working region of the shaper cutter can be represented in the coordinate system S c (X c , Y c , Z c ) by the following equation (Litvin, 1989):Fig. 1. Normal section of a rack cutter 2p for generating the driving shaper cutter.Fig. 2. Kinematic relationship between the shaper cutter and the generated gear.where A0 is the design parameter used to determine the addendum of the shaper, B0 the tooth width of the shaper, r s the pitch radius of the shaper, c c the generated angle of the shaper and i/i n the pressure angle as shown in Fig. 1. In Eq. (1), the upper sign indicates the right-side shaper surface while the lower sign represents the left-side shaper surface. The normal vector of the working region of the shaper cutter surface can be obtained as follows:2.2. Locus of the shaper cutterFig. 2displays the kinematic relationship between the shaper cutter and the generated elliptical gear. During the elliptical gear generation, the shaper cutter rotates about the Z c-axis and translates along a curve that keeps the shaper centrode and elliptical pitch of the generated gear in tangency at their instantaneous pitch point, I. The coordinate systems displayed in Fig. 2are the Cartesian coordinate system with right-handed three mutual perpendicular axes. It is noted that the Z-axis is not shown for simplicity. Coordinate system S c (X c , Y c , Z c ) is attached to the shaper cutter, and coordinate system S1(X1, Y1, Z1) is attached to the generated elliptical gear of which the rotation center is coincident with the gear geometric center. Parameter y1 is the angle formed by the X1-axis and the tangent line which corresponds to the tangency of the shaper centrode and elliptical pitch at their instantaneous pitch point, I. The rotation angle of the elliptical gear is 7t/2—y1, and anglec1 is a function of y1. Parameter c s, measured from the line0c I to the Y c-axi s of the shaper cutter, represents the rota- tional angle of the shaper. Let R1 denotes the position vector of the generated elliptical gear profile and R c represents the position vector of the shaper cutter surface. By applying thefollowing homogeneous coordinate transformation matrix equation, the locus (family) of the shaper cutter represented in coordinate system S1 can be obtained as follows:Substituting Eq. (1) into Eq. (3) provides the locus of shaper surfaces represented in coordinate system S1 as follows:Fig. 3. Tangent line of the ellipse.and the corresponding normal vector can also be obtained by:WhereIn Eq. (6), parameter 21 is the eccentricity of the ellipse, a1 the major semi-axis and b1 the minor semi-axis. Expressing the pitch curve of the elliptical gear, r1(c1), using the Cartesian coordinate system, the x1 and y1 compone nts along the coordinate axes are: AndReferring to Fig. 3, the unit tangent vector to the pitch curve at point I is positive in the fourth quadrant. The tangent vector of the pitch curve can be obtained by differentiating Eqs.(7) and (8) with respect to parameter c1 and then normalizing the results. This process results in the unit tangent vector of the pitch curve as follows:As Fig. 3 indicates, the unit tangent vector t1 of the pitch curve can also be represented in terms of angle y1 by the following equation:According to Eqs. (9) and (10), angle y1 can be expressed in terms of c1 as follows：AndAs shown in Fig. 2, the arc length of the shaper cutter, measured from the starting point N to the instantaneous pitch point I along the circular pitch, is equal to the arc length measured from the starting point M to the instantaneous pitch point I along the pitch ellipse. According to integral operation, the arc length can be expressed as follows:中文翻译计算机控制插齿刀加工椭圆齿轮齿形白炳文国立联合大学机械工程系摘要这个工作是模拟椭圆齿轮传动,轴围绕其几何中心旋转,用插齿刀加工。
Empirical processes of dependent random variables
2
Preliminaries
n i=1
from R to R. The centered G -indexed empirical process is given by (P n − P )g = 1 n
n
the marginal and empirical distribution functions. Let G be a class of measurabrocesses that have been discussed include linear processes and Gaussian processes; see Dehling and Taqqu (1989) and Cs¨ org˝ o and Mielniczuk (1996) for long and short-range dependent subordinated Gaussian processes and Ho and Hsing (1996) and Wu (2003a) for long-range dependent linear processes. A collection of recent results is presented in Dehling, Mikosch and Sorensen (2002). In that collection Dedecker and Louhichi (2002) made an important generalization of Ossiander’s (1987) result. Here we investigate the empirical central limit problem for dependent random variables from another angle that avoids strong mixing conditions. In particular, we apply a martingale method and establish a weak convergence theory for stationary, causal processes. Our results are comparable with the theory for independent random variables in that the imposed moment conditions are optimal or almost optimal. We show that, if the process is short-range dependent in a certain sense, then the limiting behavior is similar to that of iid random variables in that the limiting distribution is a Gaussian process and the norming √ sequence is n. For long-range dependent linear processes, one needs to apply asymptotic √ expansions to obtain n-norming limit theorems (Section 6.2.2). The paper is structured as follows. In Section 2 we introduce some mathematical preliminaries necessary for the weak convergence theory and illustrate the essence of our approach. Two types of empirical central limit theorems are established. Empirical processes indexed by indicators of left half lines, absolutely continuous functions, and piecewise differentiable functions are discussed in Sections 3, 4 and 5 respectively. Applications to linear processes and iterated random functions are made in Section 6. Section 7 presents some integral and maximal inequalities that may be of independent interest. Some proofs are given in Sections 8 and 9.
材料成型及控制工程专业外语阅读
3. 1. 1 Dynamic Structural Change
During the deformation of austenite at hot-working temperatures and constant strain rate, the characteristic form of stress-strain curve observed is illustrated in Fig. 3. 1. These curves are for low-alloy steels,tested in torsion,but are similar to those obtained for other steels in the austenitic condition tested in torsion,tension,or compression. After initial rapid work-hardening the curves go through a maximum associated with the occurrence of dynamic recrystallization. The peak in flow stress occurs after some low fraction of recrystal lization has taken place so the strain to the peak (εp) is always greater than the critical strain for dynamic recrystallization (εc).
It can be seen from Fig.3.1 that εp increases systematically with Zener-Hollomon parameter (Z),independent of the particular combination of strain rate (ε) and temperature (T/K) in the relationship Z = εexp Qdef/RT
强力手段
研究资料：并无有效的研究显示对肌力训练者有提升运 动表现的效能。
以肌力训练者为行销对象的营养补品
营养补充：育亨宾
广告宣传：提升血清中雄性激素量；а-2受体阻断剂； 降低体脂肪。
研究资料：并无有效的研究显示对肌力训练者有提升运 动表现的效能，需进一步研究以证明对肌力 训练者降体重有效。
以肌力训练者为行销对象的营养补品
Proposed mechanism through which ergogenic aids might work－ Counteract CNS inhibition （对抗中枢神经抑制）
Anabolic steroids（合成类固醇） Amphetamines （安非他明）
Proposed mechanism through which ergogenic aids might work－ Counteract or delay onset or perception of fatigue（对抗或延迟疲劳感觉）
研究资料：研究显示不影响肌力训练者血清中男性激 素量，肌肉或肌力。
以肌力训练者为行销对象的营养补品
营养补充：镁
广告宣传：促进蛋白质合成或肌肉收缩；增长肌肉及力量
研究资料：研究资料模棱两可，但大都无法提升肌力 训练者运动表现。
以肌力训练者为行销对象的营养补品
营养补充：中链三酸甘油脂
广告宣传：增加热能，有助降低脂肪
Alcohol（酒精） Beta-blocking drugs（ β 阻断剂） Hypnosis（催眠术） Stress management（压力应对）
Proposed mechanism through which ergogenic aids might work－ Weight loss or weight gain（增减体重）
一种改进的高斯频率域压缩感知稀疏反演方法(英文)
AbstractCompressive sensing and sparse inversion methods have gained a significant amount of attention in recent years due to their capability to accurately reconstruct signals from measurements with significantly less data than previously possible. In this paper, a modified Gaussian frequency domain compressive sensing and sparse inversion method is proposed, which leverages the proven strengths of the traditional method to enhance its accuracy and performance. Simulation results demonstrate that the proposed method can achieve a higher signal-to- noise ratio and a better reconstruction quality than its traditional counterpart, while also reducing the computational complexity of the inversion procedure.IntroductionCompressive sensing (CS) is an emerging field that has garnered significant interest in recent years because it leverages the sparsity of signals to reduce the number of measurements required to accurately reconstruct the signal. This has many advantages over traditional signal processing methods, including faster data acquisition times, reduced power consumption, and lower data storage requirements. CS has been successfully applied to a wide range of fields, including medical imaging, wireless communications, and surveillance.One of the most commonly used methods in compressive sensing is the Gaussian frequency domain compressive sensing and sparse inversion (GFD-CS) method. In this method, compressive measurements are acquired by multiplying the original signal with a randomly generated sensing matrix. The measurements are then transformed into the frequency domain using the Fourier transform, and the sparse signal is reconstructed using a sparsity promoting algorithm.In recent years, researchers have made numerous improvementsto the GFD-CS method, with the goal of improving its reconstruction accuracy, reducing its computational complexity, and enhancing its robustness to noise. In this paper, we propose a modified GFD-CS method that combines several techniques to achieve these objectives.Proposed MethodThe proposed method builds upon the well-established GFD-CS method, with several key modifications. The first modification is the use of a hierarchical sparsity-promoting algorithm, which promotes sparsity at both the signal level and the transform level. This is achieved by applying the hierarchical thresholding technique to the coefficients corresponding to the higher frequency components of the transformed signal.The second modification is the use of a novel error feedback mechanism, which reduces the impact of measurement noise on the reconstructed signal. Specifically, the proposed method utilizes an iterative algorithm that updates the measurement error based on the difference between the reconstructed signal and the measured signal. This feedback mechanism effectively increases the signal-to-noise ratio of the reconstructed signal, improving its accuracy and robustness to noise.The third modification is the use of a low-rank approximation method, which reduces the computational complexity of the inversion algorithm while maintaining reconstruction accuracy. This is achieved by decomposing the sensing matrix into a product of two lower dimensional matrices, which can be subsequently inverted using a more efficient algorithm.Simulation ResultsTo evaluate the effectiveness of the proposed method, we conducted simulations using synthetic data sets. Three different signal types were considered: a sinusoidal signal, a pulse signal, and an image signal. The results of the simulations were compared to those obtained using the traditional GFD-CS method.The simulation results demonstrate that the proposed method outperforms the traditional GFD-CS method in terms of signal-to-noise ratio and reconstruction quality. Specifically, the proposed method achieves a higher signal-to-noise ratio and lower mean squared error for all three types of signals considered. Furthermore, the proposed method achieves these results with a reduced computational complexity compared to the traditional method.ConclusionThe results of our simulations demonstrate the effectiveness of the proposed method in enhancing the accuracy and performance of the GFD-CS method. The combination of sparsity promotion, error feedback, and low-rank approximation techniques significantly improves the signal-to-noise ratio and reconstruction quality, while reducing thecomputational complexity of the inversion procedure. Our proposed method has potential applications in a wide range of fields, including medical imaging, wireless communications, and surveillance.。
A Robust Optimization Approach to Inventory theory
A Robust Oபைடு நூலகம்timization Approach to Inventory Theory
Sloan School of Management and Operations Research Center, Massachusetts Institute of Technology, E53-363, Cambridge, Massachusetts 02139, dbertsim@ Department of Industrial and Systems Engineering, Lehigh University, Mohler Building, Bethlehem, Pennsylvania 18015, aurelie.thiele@
1. Introduction
Optimal supply chain management has been extensively studied in the past with much theoretical success. Dynamic programming has long emerged as the standard tool for this purpose, and has led to significant breakthroughs as early as 1960, when Clark and Scarf (1960) proved the optimality of base-stock policies for series systems in their landmark paper. Although dynamic programming is a powerful technique as to the theoretical characterization of the optimal policy for simple systems, the complexity of the underlying recursive equations over a growing number of state variables makes it ill suited for the computation of the actual policy parameters, which is crucial for real-life applications. Approximation algorithms have been developed to address those issues. These include stochastic approximation (see Koshner and Clark 1978) and infinitesimal perturbation analysis (IPA) (see Glasserman 1991, Ho and Cao 1991), where a class of policies, e.g., base-stock, characterized by a set of parameters, is optimized using simulation-based methods (see Fu 1994, Glasserman and Tayur 1995, Kapuscinski and Tayur 1999). IPA-based methods assume knowledge of the underlying probability distributions and restrict their attention to cer150
土力学 英译汉
Soil mechanics is concerned with the use of the laws of mechanics and hydraulics in engineering problems related to soils . soil is natural aggregate of mineral grains , with or without organic constituents formed by the chemical and mechanical weathering of rock . it consists of three phases : solid mineral matter , water , and air or other gas . soil are extremely variable in composition , and it was this heterogeneity that long discouraged scientific studies of these deposits . gradually , the investigation of failures of retaining walls , foundations , embankments , pavements, and other structures resulted in a body of knowledge concerning the nature of soils and their behaviour sufficient to give rise to soil mechanics as a branch of engineering science .土力学是一门在工程上与土有关的科学,它使用了力学和水力学的定律。
土是含或不含有机质的矿物颗粒自然的堆积,是由岩石受到化学和力学机械风化而成。
Transactions of the Institute of Measurement and Control-2013-Juan-1008-15
Adaptive fuzzy approach for non-linearity compensation in MEMS gyroscope
Wanru Juan and Juntao Fei
Transactions of the Institute of Measurement and Control 35(8) 1008–1015 Ó The Author(s) 2013 Reprints and permissions: /journalsPermissions.nav DOI: 10.1177/0142331212472224
Corresponding author: Juntao Fei, Hohai University, 200 North Jinling Road, Changzhou, China. Email: jtfei@
Downloaded from at Southeast University on May 10, 2015
Fei and Juan
1009
motor and an XY-Positioning Table using fuzzy logic control. Park and Han (2011) derived backstepping and recurrent fuzzy neural controller for a robot manipulator with deadzone and friction.
Jiangsu Key Laboratory of Power Transmission and Distribution Equipment Technology, College of Computer and Information, Hohai University, Changzhou, China
英语国际音标讲解(详细版)
Frictional: [tr], [dr], [ts]
01
[tr] as in "tree"
02
[dr] as in "dress"
The tip of the tongue is placed on the alveolar ridge, and air is passed through with a trimming sound
01
[ ɑː] Is the sound of "car" in "card"
02
[ ɒ] Is the sound of "dog" in "dog"
[u] ː] Is the sound of "goose" in "goose"
03
03 Consistent phone
Explores: [p], [t], [k]
03
[ ŋ]: This symbol represents the nasal consistent found at the end of the word "sing" It is produced by closing the lips and allowing air to pass through the nacal activity
Methods and Techniques for Phonetic Alphabet Learning
• Practice: Learning the Phonetic Alphabet requires regular practice and repetition to familiarize yourself with the symbols and their corresponding sounds
变频驱动电机轴承电腐蚀的预防和测试方法研究
变频驱动电机轴承电腐蚀的预防和测试方法研究薛利晨（天津机电职业技术学院，天津300350）[摘要] PWM变频器所产生的共模电压会在电机寄生电容的耦合作用下引起轴承电流，进而造成轴承电腐蚀，使得轴承过早失效。
本文针对变频器控制电机中共模电压引起轴承电应力这一固有现象，分析了变频驱动电机中的电容耦合电流，并在此基础上提出了基于共模电流和轴承电流传输路径的轴承电腐蚀预防措施，以为降低轴承电应力提供解决方法。
为了验证预防措施的有效性，又碍于现场直接测量轴承电流的局限性，本文还描述了相关的间接测量方法及其作用，为解决轴承电腐蚀问题提供了测试依据。
[关键词] 变频驱动电机；轴承电腐蚀；轴承电流；轴电压；PWM[中图分类号] TM303.5 [文献标志码] A [文章编号] 1000-3983(2020)04-0045-07The Research and Analysis on the Prevention and Test Method of Electric Corrosion ofConverter Driven Motor BearingXUE Lichen(Tianjin Vocational College of Mechanics and Electricity, Tianjin 300350, China) Abstract: The common mode voltage produced by PWM inverter causes the bearing current under the coupling action of parasitic capacitance of the motor, which will lead to the premature failure of the bearing due to the electric corrosion of the bearing. In this paper, the capacitance coupling current in converter driven motor is analyzed, aiming at the inherent phenomenon of bearing electric stress caused by the common mode voltage of converter driven motor. On the basis of this, the preventive measures of bearing electric corrosion based on common mode current and bearing current transmission path are proposed to provide solutions for reducing bearing electric stress. In order to verify the effectiveness of preventive measures and the limitation of the direct measurement of bearing current in the field, the related indirect measurement methods and their functions are described.Key words: converter driven motor; electric corrosion of bearing; bearing current; shaft voltage; PWM0前言由于变频器驱动交流电机具有很好的控制和运行特性，促使这种驱动方式占据了当前驱动技术的核心地位。
关于手机控的英语作文
The prevalence of smartphones has dramatically changed the way we communicate, access information,and entertain ourselves.This phenomenon has given rise to a new group of individuals known as cell phone addicts or phone zombies.Here are some points to consider when discussing the topic of phone addiction in an English essay:1.Ubiquity of Smartphones:The essay could start by highlighting how smartphones have become an integral part of our daily lives.They are no longer just communication devices but have evolved into multifunctional tools that cater to various needs such as social networking,gaming,photography,and more.2.Convenience and Dependency:Discuss the convenience that smartphones offer,which leads to a high level of dependency.People rely on their phones for everything from checking the weather to making payments,which can make it difficult to disconnect.3.Social Media Influence:The impact of social media on phone addiction is significant. Platforms like Facebook,Instagram,and Twitter keep users constantly engaged,checking for updates and notifications,which can lead to compulsive phone checking.4.Health Implications:The essay should address the potential health risks associated with excessive phone use,including issues like eye strain,poor posture,and even mental health concerns such as anxiety and depression.5.Impact on Relationships:Discuss how being a phone addict can affect interpersonal relationships.The constant presence of a phone can lead to less facetoface interaction, which can weaken social bonds and communication skills.6.Distraction and Productivity:The essay could explore how the constant use of smartphones can be a distraction,affecting productivity at work or school.It can also lead to a decrease in attention span and an inability to focus on tasks for extended periods.7.Technological Solutions:Propose potential solutions to the problem of phone addiction. This could include the use of apps that limit screen time,setting boundaries for phone use, or even digital detox programs.8.Balancing Act:Conclude the essay by emphasizing the importance of finding a balance between the benefits of smartphones and the potential pitfalls of overuse.Encourage readers to be mindful of their phone usage and to prioritize reallife interactions and experiences.Remember to structure your essay with a clear introduction,body paragraphs that exploreeach point in detail,and a conclusion that summarizes the main ideas and offers a final thought on the topic.。
下垂控制下柔性配电网有功功率调制技术
《电气自动化》2021年第43卷第1期电力系统及其自动化Power System & Automation下垂控制下柔性配电网有功功率调制技术罗剑#,林虎##黄志清2,刘建成##夏树贵2,艾小清3(1.贵州电网物资有限公司，贵州贵阳550005 ;2.贵州电网公司铜仁供电局，贵州铜仁554300;3.贵州大学电气工程学院，贵州贵阳550025 )摘要：为解决传统配电网馈线负载不均衡情况，利用多端S O P构成柔性配电网，结合下垂控制提出一种基于均衡控制的功率调制技 $通采集馈线送出功率实时调节下制换流器的有功功率给 衡接入多端S O P的各回馈线负载均衡。
利用P S C A D/E M T D C搭建柔性配电网模型验证所提方法的有效性，对基于多端S O P的柔性配电网推广具有一定指导意义。
关键词&多端;柔性;馈$负载均衡;下 制;设备利用率D O I：10.3969/j.issn.1000 -3886.2021.01.026[中图分类号]T M46 [文献标志码]A[文章编号]1000 -3886(2021)01 -0092 -03ActivePower Modulation Technology for FlexibleDistribution Networks under Droop ControlLuo Jian1，Lin H u1，Huang Zhiqing2 , Liu Jiancheng1，Xia Shugui2，Ai Xiaoqing3(1.Guizhou Power Grid Materials Co. ，Ltd. ，Guiyang Guizhou 550005，China；2.Tongren Power Supply Bureau，Guizhou Power Grid Co. ，Tongren Guizhou554300，China;3. College of Electrical E nginering，Guizhou University，Guiyang Guizhou 550025，China)A b s tr a c t:In order to solve the problem 〇0feeder load unbalance in the tra d itio n a l d is trib u tio n n e tw o rk,a m u lti-te rm in a l S O P w a s used toconstruct a fle x ib le d is trib u tio n ne tw ork.A power m odulation technology based on balanced con tr co n tro l.By co lle ctin g feeder ou tput p o w e r,the active power o f the droop con trol converter was adjusted in the re a l-tim e w a y,andcommands were given fo r balanced access to feedback lin e s o f the m u lti-te rm in a l SOP to achieve load balance.The v a lid ity o f theproposed approach was ve rifie d through a m odel o f the fle x ib le d is trib u tio n netw ork established by use o f PSCAD/E M T D C.Theproposed approach had a certain g u id in g significa ncc fo r the popularization o f the fle x ib le d is trib u tio n netw ork based on m u lti-te rm in a lSOP.K e y w o r d s:m u lti-te rm in a l;fle x ib le;fe e d e r;load b a la n c in g;droop c o n tro l;equipm ent u tiliz a tio n rate0引言城市配电网各馈线之间通过联络开关连接形成环网接线模 式，但在实际上联络 通 断状态，配网各馈线之间无功率 ，导致设备利用率较低。
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西 安电 子 科 技 大 学 学 报 (自 然 科 学 版 )
基于最小生成树的图数据库索引算法
ISSN 1000-9825, CODEN RUXUEW E-mail: jos@Journal of Software, Vol.20, Supplement, December 2009, pp.144−153 © by Institute of Software, the Chinese Academy of Sciences. All rights reserved. Tel/Fax: +86-10-62562563∗基于最小生成树的图数据库索引算法李楠, 高宏+, 李建中(哈尔滨工业大学计算机科学与技术学院,黑龙江哈尔滨 150001)Minimal Spanning Tree Based Graph Indexing AlgorithmLI Nan, GAO Hong+, LI Jian-Zhong(Department of Computer Science and Technology, Harbin Institute of Technology, 150001, China)+ Corresponding author: E-mail: honggao@Li N, Gao H, Li JZ. Minimal spanning tree based graph indexing algorithm. Journal of Software, 2009,20(Suppl.):144−153. /1000-9825/09018.htmAbstract: Graphs have become popular for modeling structured data. As a result, graph indexing technique hascome to play an essential role in query processing. This paper investigates the issues of indexing graphs andpropose an approximation solution. The proposed approach, called MSTA, makes use of minimal spanning tree asbasic indexing feature. By containment relation of edge lists and maximal common subgraph based graph distance,those minimal spanning trees are organized into an indexing structure named MST tree. MST tree can support manykinds of queries efficiently, such as subgraph queries. The performance study shows that index size andconstructing time of traditional methods are tens or even a hundred times larger than MSTA.Key words: graph database; graph index; approximation algorithm; subgraph query摘 要: 对复杂数据进行图模式建模近几年越来越流行,因此,在查询执行的优化过程中图索引技术变得至关重要.研究了图模式的索引问题,并且提出了一种近似的索引方法,称为MSTA方法.MSTA方法利用最小生成树结构作为索引特征,依据最小生成树边序列的包含关系和基于最大公共子图的图距离度量,将最小生成树组织到一个称为MST树的索引结构中.MST树索引结构可以高效地支持多种查询,例如子图查询.MSTA方法具备高效的索引性能.在索引大小和索引建立时间方面,传统方法是MSTA方法的数十倍,甚至上百倍.MSTA方法虽然不能返回完整结果,但是可以返回经图距离度量排序最好的部分结果.关键词: 图数据库;图索引;近似算法;子图查询近年来,用图对复杂数据进行建模的应用渐渐兴起.图广泛应用于生物领域[1,2]、化学领域[3]、社会科学领域[4]、WEB领域[5]、XML领域[6],甚至是基于内容的图片检索和视频检索领域[7,8].在上述领域中,一般用无向图将结构化的数据建模成图数据,用图数据库管理及查询图数据.在用图数据库管理及查询图数据时,随着数据量的增大,返回的结果数目也随之增加.但是在实际应用中,用户可能并不需要得到全部的结果,只需要依据某种度量返回最好的部分结果.这与搜索引擎的原理十分相似.∗Supported by the National Natural Science Foundation of China under Grant Nos.60933001, 60773063 (国家自然科学基金); theNational Basic Research Program of China under Grant No.2006CB303000 (国家重点基础研究发展计划(973))Received 2009-05-01; Accepted 2009-07-20李楠等:基于最小生成树的图数据库索引算法145例如在图片检索时,用户可能只需要更贴近查询要求的几个图片;在蛋白质研究中,研究人员可能只需要返回与结构基因组同源性最相近的部分化合物;又如在WEB领域中,用户可能只需要系统返回最贴近查询要求的1个或几个网页.在上述这些应用情景中,用户更希望数据管理系统能够快速的返回满足要求最好的部分结果,而不是等待很长时间返回全部结果.数据库中的数据量越大,这种趋势就越明显.例如,我们的实验表明,我们的方法可以用8秒对一个拥有32 000个图的数据库建立索引,查询时返回了最好的绝大部分结果,而GraphGrep[9]则需要28分钟建立索引,查询时返回全部结果.若假设此时我们的图数据库是应用在图片检索领域中的,在信息高速发展的当代,我们有理由相信用户更可能会花费8秒获得最满足要求的绝大部分图片,而不是等待28分钟获得全部满足要求的图片.图距离度量广泛应用于图的相似性查询领域.研究中我们发现,在子图查询等多种查询应用中,距离更小的图是更加满足要求的图.例如,在药物化合物检测中,如果检测出的化合物与查询化合物的距离更小,那么检测出的化合物具备该功能的可能性就更强.本文提出了一种索引方法:MSTA(minimal spanning tree based approximation)方法.MSTA方法采用最小生成树作为索引特征,根据边序列的包含关系和基于最大公共子图的图距离度量,将所有的最小生成树组织到MST树索引结构中,并在该索引结构上支持了子图查询,查询返回最好的部分结果.正是由于这种近似的查询算法,使得索引性能和查询性能大为提高.传统的索引方法在索引大小和索引建立时间上是MSTA方法的几倍甚至几十倍.MSTA方法的查询执行时间也要比传统方法小得多.在建立MST树索引结构时,MSTA方法的主要原理是将图之间的子图关系先映射为最小生成树之间的子树关系,再映射为MST树中的祖先后代节点关系.本文的主要贡献如下:(1) 提出了MSTA索引方法,子图查询时高效地返回最好的部分结果.MSTA方法的索引性能大大的优于传统方法.(2) 利用最小生成树作为图数据库的索引特征,并对这种索引方式的可行性进行了理论分析和大量的实验辅证.本文第1节介绍问题定义、MSTA索引原理.第2节介绍MSTA索引方法.第3节介绍MST树上的子图查询算法.实验结果在第4节介绍.第5节给出本文的结论.1 MSTA索引原理1.1 问题定义定义1(图). 图G是一个四元组(V,E,∑,l),其中非空集合V代表顶点集合,E⊆V×V代表边的集合,∑代表标号的集合,l:V∪E→∑代表为顶点和边分配标号的映射函数.定义2(子图). 给定图G=(V,E,∑,l)和图G′=(V′,E′,∑′,l′),图G′是图G的子图,如果满足以下条件,则有l(v)= l′(v),l(e)=l′(e):(1) V′⊆V;(2) E′⊆E;(3) ∑=∑′,∀v∈V′,e∈E′.定义3(同构). 对于图G=(V,E,∑,l)与图G′=(V′,E′,∑′,l′),若存在一个双射f:V↔V′,且f满足下列条件时,则称G 与G′是同构的:(1) ∀u∈V,l(u)=l′(f(u));(2) ∀u,v∈V,((u,v)∈E)⇔((f(u),f(v))∈E′);(3) ∀(u,v)∈E,l(u,v)=l′(f(u),f(v)).定义4(子图同构). 给定两个图G=(V,E,∑,l)和图G′=(V′,E′,∑′,l′),一个从G到G′的子图同构是一个单射函数f:V→V′,其满足:(1) ∀u∈V,l(u)=l′(f(u));(2) ∀(u,v)∈E,(f(u),f(v))∈E′且l((u,v))=l′((f(u),f(v))).如果存在一个从G到G′的子图同构,则称G为G′的子图,记为G⊆G′.146 Journal of Software软件学报 V ol.20, Supplement, December 2009 定义5(子图查询). 给定一个图数据集合D={G1,G2,…,G n}和查询图Q,一个子图查询是要从D中查找图集合S,使得S={G|Qi ⊆G,Gi i∈D}.子图查询返回D中所有以查询图为子图的图.1.2 原理分析MSTA的索引原理可以用图1形象地表示出来,首先由图G计算最小生成树T G,然后根据图Q的顶点由T G导出一个子图T Q,那么T Q一定是Q的最小生成树.所以,如果我们知道了G和Q的最小生成树T G和T Q,且T G包含T Q,那么G就很可能包含Q.所以,MSTA索引方法是为T G构造MST索引树.当需要进行子图查询时,首先计算查询图Q的最小生成树T Q.然后搜索MST索引树,若索引中存在某一T G,使得T Q是T G的子树,那么T G所对应的G可能包含Q,将G加入到候选集中.生成导出生成T G1T qG1QFig.1 MSTA graph indexing theory图1 MSTA 图索引原理为了更加快速地判断两棵树之间的子树关系,我们在MST树中用一种特殊的方法来组织最小生成树.在MST树中,若T Q是T G的子树,那么T Q一定是T G的祖先节点.但是出于效率方面的考虑,我们不将所有的子树关系都映射为这种祖先后代关系,而是只映射图距离最小的部分.总的来说,MSTA将图之间的子图关系映射为最小生成树之间的子树关系.查询时,将所有最小生成树与查询图的最小生成树具有子树关系的图加入到候选图集中.为了高效地判断生成树间的这种子树关系,我们在组织MST树索引结构时,尽可能地将部分距离最小的树组织到一起.1.3 可信性分析对于大小为n的图数据库D,查询图为q,D q={g|g∈D,g⊇q},且不存在图g1,g1∈D,g1∉D q,g1⊇q,则D q为最优化解.假设MSTA算法对于一个查询图为q的查询返回的结果为D q1={g|g∈D,g⊇q},且D q1⊆D q,那么D q1为近似解.根据前面的定义,子图查询时一个求最大解的问题,即0≤D q1≤D q,其近似比ρ(n)=|D q|/|D q1|.D q={g1,g2,…,g n},D q中哪些图会被包含到D q1中,这取决于两点.首先,这与插入的顺序有关.若图g i⊆g j,对应的最小生成树T i⊆T j,对应的最小生成树的边集合E i⊆E j.此时,若g i在g j之前插入,则g i会以较大的概率被包含到D q1中,否则,很有可能不被包含到D q1中.这与MST树索引的建立算法有关.其次,这与图数据集合中图的标号的分布有关.图数据集合标号通常具有不可预测性.虽然大多数文献生成模拟数据时标号的分布符合泊松分布,但是我们也很难计算准确的近似比的量值.1.4 补充方法MSTA方法是近似方法,所以不能返回全部结果.为了使子图查询能够返回更多的结果,我们可以在建立索引时对所有的图数据库中的图,按照图的大小排序,按照非递减的顺序将图插入到MST树结构中建立索引.这样会加大MSTA方法的近似度.2 MSTA的索引方法在本节,我们首先介绍建立MST树索引结构所需要的最小生成树产生算法;然后介绍最小生成树的子树测试算法和最大公共子图距离度量的相关知识;最后详细介绍MST树索引结构.李楠等:基于最小生成树的图数据库索引算法1472.1 最小生成树产生算法如图2所示的图数据库,假设我们要以最小生成树为特征对该数据库建立索引,那么我们首先需要对每一个图求得最小生成树.由于我们所处理的图都是标号图,所以我们对Kruskal算法进行了如下改进:(1) 图G=(V,E,∑,l),∀e=(v1,v2)∈E,我们将边e的权重定义为一个三元组(l(e),l(v1),l(v2)),其中l(e)是边e的标号,l(v1)和l(v2)分别是e关联的两个顶点v1和v2的标号;(2) 两个边e i和e j权重的大小是通过三元组(l(e i),l(v1i),l(v2i))和(l(e j), l(v1j),l(v2j))按照顺序依次比较来决定;(3) 用二元对(v,n)记录节点所在集合,并将其组织成平衡二叉树的形式.A A A AB CG1B CG2DB D B DCG3G4G5B DCFig.2 A sample database图2 示例数据库由于我们用边及相关的两个顶点共3个标号构成的三元组来表示这条边,这种表示与边或顶点的ID无关.若一个图有多个相同标号的边,则可能会得到多个最小生成树.但这些最小生成树所对应的边的三元组序列是一样的,所以从三元组边序列角度,每个图都有唯一的最小生成树.算法1. GetMinimalSpanningTree.输入:图G=(V,E,∑,l).输出:最小生成树的边集合A.1. A←∅.2. for ∀v∈V do:3. 初始化其点集合.4. for ∀e∈E do:5. 计算其权重.6. 建立包含所有的点集合的平衡二叉树.7. 按照权重排序所有边,得到非递减的边集合E sort.8. for ∀e=(v1,v2)∈E sort do:9. 查找v1和v2分别对应的二元组(v1,n1),(v2,n2).10. if n1≠n211. n2←n1.12. A←A∪{e}.13. return A.2.2 最小生成树子树测试算法由算法1得到的最小生成树是一棵无根树,实际上就是一个边的子集.每条边e通过三元组(l(e),l(v1),l(v2))表示,与边及顶点的ID无关.对于图G1,G2的最小生成树T1,T2,这两棵树所对应的边集合为E1和E2,若T1是T2的子树,则E1⊆E2.所以,如果我们想测试T1和T2之间的子树关系,只要测试E1和E2之间的子集关系即可.我们可以通过构造一个二分图,计算二分图的最大匹配来测试这种子集关系.首先,我们将每个用三元组表示的边看作一个二分图的顶点.若两个顶点的三元组相同,则在这两个顶点之间建立一条边.最后,在这个二分图上求最大匹配.最大匹配的大小即为两个边集合交集的大小.这个方法可以进一步通过定义边的三元组的匹配程度得到两个边集合的匹配程度,将布尔的交集程度转换为带权重的交集程度.计算最大匹配的复杂度为148Journal of Software 软件学报 V ol.20, Supplement, December 2009O (n 5/2).由于边的三元组是有序的,所以可以用一种简单的方法求得交集的大小.首先将两个边集合排序成有序的序列,然后顺序匹配求出交集大小.这个简单方法的复杂度为O (n 2).算法过程如下:算法2. GetEdgeIntersect. 输入:边集合E 1,E 2,假设|E 1|≤|E 2|. 输出:交集大小ω. 1. 列表L 1←sort (E 1). 2. 列表L 2←sort (E 2). 3. 交集大小ω←0.4. 下一个匹配开始位置loc ←0.5. for i : 0→L 1.size do:6. forj :loc →L 2.size do: 7. if 边L 1[i ]与边L 2[j ]匹配, then 8. loc ←i +1; ω←ω+1; break. 9. 返回ω.2.3 最大公共子图的图距离度量在许多研究工作中阐述了最大公共子图的图距离度量,两个图之间的距离用下面的公式计算:121212|(,)|(,)1|max(||,||)|mcs G G d G G G G =− (1)其中,|mcs (G 1,G 2)|是G 1和G 2的最大公共子图大小.假设G 1和G 2的最小生成树是T 1和T 2.由于图的大小是由图的顶点数目度量的,则有|G 1|=|T 1|,|G 2|=|T 2|,可得|mcs (G 1,G 2)|≤|mcs (T 1,T 2)|=|E 1∩E 2|.故图距离度量可以用如下公式计算:12121212||(,)(,)1max(||,||)E E d G G d T T E E ∩≥=−(2)2.4 MST 树结构若要对一个图数据库建立MST 树索引结构,则首先需要求出所有图的最小生成树,即用算法1求得一个边集合E t .将边集合E t 排序得到有序序列L t ,L t 即为三元组序列{(α1,β1,γ1),(α2,β2,γ2),…}.我们将这些边序列组织到MST 树中.在MST 树中,子节点对应的最小生成树比父节点所对应的最小生成树多一条边,且子节点所对应的最小生成树包含父节点所对应的最小生成树的所有边.若父节点与最小生成树T p 相关联,子节点与最小生成树T q 相关联,T p ,T q 分别对应边集合E p ,E q ,那么T p 是T q 的子树,即E p ⊆E q .在MST 树中,有些节点与最小生成树相关联,并且与导出这棵最小生成树的图相关联;而其他一些节点是在MST 树的插入中加入的,不与具体的图相关联.这些为了保证MST 树每层增加一条边而填充的节点,称为哑元节点.处于MST 树的第n 层的节点所关联的最小生成树具有n 条边.若需要向MST 树中插入某个图的索引,则首先需要对该图最小生成树的边序列排序,然后确定其插入位置.若在预计的插入位置上已经存在一个非哑元节点,则我们关联该图;若预计的插入位置上已经存在一个哑元节点,则我们同时关联该图和该图的最小生成树;若预计的插入位置上无节点存在,则建立新的节点以及到达该节点的路径上的所有节点,关联该图和该图的最小生成树.在确定插入位置的过程中,我们从根节点出发,不断地选择可以作为该最小生成树的子树的子节点前进,若有多个子节点满足要求,则利用公式(2)选择图距离最小的一个子节点.若该最小生成树有n 条边,那么这个选择过程到达第n 层为止.所到达的位置即为预计的插入位置.例如一个节点T p ,T p 的两个子节点为T c 1,T c 2,待插入的新节点为T i .若T i 包含T c 1和T c 2,那么选择d (T i ,T j )最小的子节李楠 等:基于最小生成树的图数据库索引算法149点前进;若两个节点都不被包含,那么T i 作为T p 的第3个子节点插入.对于图2中的图数据库,我们用算法1得到如图3所示的最小生成树边序列.G 1,G 2,G 3,G 4,G 5所对应的边序列分别为T 1,T 2,T 3,T 4,T 5.由于所有顶点的标号都相同,我们用0表示.于是,示例图数据库的索引结构如图4所示.T 1(G 1)T 2(G 2)T 3(G 3)T 4(G 4)T 5(G 5)AB 0AC 0AB 0 AC 0 BD 0 AB 0AD 0AB 0AD 0BC 0BC 0BD 0Fig.3 Edge sets of minimal spanning trees 图3 示例图数据库的最小生成树边集合T 2(G 2)T 4(G 4)AB 0 AC 0AB 0AD 0BC 0BD 0BC 0AB 0AB 0 AC 0 BD 0 AB 0AD 0BC 0T 3(G 3)T 1(G 1)T 5(G 5)Fig.4 Structure of MST tree图4 MST 树结构3 MSTA 的子图查询算法MSTA 索引方法利用最小生成树作为索引结构;利用子图、子树和子树、父子节点两个映射关系来近似查询结果;利用插入时选择图距离最小的节点来保证返回的查询结果是距离最小的部分结果.假设子图查询的查询图为Q ,那么查询需要返回所有包含Q 的图.我们可以先计算Q 的最小生成树T ,然后利用T 的有序边序列在MST 树中定位到一个节点.在以该节点为根的MST 子树中,将所包含的所有图加入到候选图集合中.最后,我们利用精确的子图同构算法进行验证.算法具体描述如下:算法3. SubGraphQuery.输入:查询图Q ,MST 树索引结构IndexTree . 输出:结果图集合Ans .1. E q ←GetMinimalSpanningTree (Q ).2. L q ←sort (E q ).3. loc ←locate (L q ).4. 候选图集合CS ←IndexTree 以loc 为根的子树中所包含的所有图.5. for each g ∈CS do:6. if q 是g 的子图, Ans ←g .7. 返回Ans .150Journal of Software 软件学报 V ol.20, Supplement, December 20094 实验结果本节我们将分别使用真实数据集合和模拟数据集合来评估MSTA 索引方法的索引性能和查询性能.真实数据集采用NCI-HIV 化合物的CI 类别真实数据集作为实验用数据集.NCI-HIV 化合物可在/下载,CI 类别大约有42 000个化合物.模拟数据集用GIndex [10]中所叙述的方法生成.所有实验都在内存为512MB,CPU 为Pentium 4,2.93GHz,操作系统为Windows XP 的PC 机上进行.所有算法都在STL 库支持下用C++实现,用DEV 内嵌的G++编译.我们主要与GraphGrep 算法[9]、GIndex 算法[10]在索引大小、索引建立时间和子图查询执行时间这3个方面进行了对比.同时,我们计算了召回率以说明近似算法的可信性. 4.1 真实数据实验结果对于索引性能的可扩展性分析,我们从CI 类中随机抽取大小为2 000,4 000,8 000,16 000,32 000共5个数据集,并在这5个数据集上分别建立索引,然后测量索引的大小、索引的建立时间.对于子图查询性能,我们从CI 类别随机抽取大小为10 000的数据集,查询Query 大小分别为5,10,15,20,25(根据查询图中的节点个数计算).对于每种大小的查询,我们从CI 类别里各随机抽取1 000个符合大小要求的图.对这5组各1 000个查询测量查询时间等,然后取平均值.对于GraphGrep [9]算法,我们采用路径长度lp =4,对于GIndex [10]算法,我们采用GSpan [11]挖掘频率为10%的频繁子图作为索引特征. 4.1.1 MSTA 索引性能分析图5展示了GraphGrep,MSTA 和GIndex 的索引大小.MSTA 的索引大小随着数据库大小呈线性增长; GIndex 的索引大小约是MSTA 的索引大小的2倍左右;GraphGrep 的索引大小约是MSTA 的索引大小的10倍左右.图6展示了GraphGrep,MSTA 和GIndex 的索引建立时间.MSTA 的索引建立时间随着数据库大小呈线性增长;而GraphGrep 的索引建立时间的增长较为显著,约是MSTA 的40倍~200倍;GIndex 约是MSTA 的40 倍~60倍.200400600800100012001400160018002K4K 8K 16K 32K数据库中图的个数索引大小 (M B )GraphGrepMSTAGIndex2004006008001000120014001600180020002K4K 8K 16K 32K 数据库中图的个数索引建立时间 (s )GraphGrepMSTAGIndexFig.5 Comparison of index size图5 索引大小比较Fig.6 Comparison of construction time图6 索引建立时间比较4.1.2 MSTA 查询性能分析图7展示了GraphGrep,MSTA 和GIndex 索引下的子图查询执行时间.我们可以看到,GraphGrep 与MSTA 的查询执行时间随着查询的增大而减小;而GIndex 的查询执行时间随着查询图的增大而增加.GraphGrep 的查询执行时间约是MSTA 的2倍~3倍,GIndex 相对于MSTA 的查询执行时间的倍数在3倍~21倍之间增长.图8是GraphGrep,MSTA 与GIndex 索引下的子图查询的候选集与结果集的比率.MSTA 的比率在GraphGrep 和GIndex 之间,约在130~230之间变动.图9是MSTA 索引结构子图查询执行的访问率.我们可以看出,随着查询图的增大,访问率基本成降低趋势,李楠 等:基于最小生成树的图数据库索引算法151都在15%以下.图10展示了子图查询的召回率约在80%~90%的结果,也就是说,MSTA 平均可以返回全部结果的80%~90%.2468101214161820 5 10152025查询图大小执行时间 (s )GraphGrep MSTAGIndex200400600800100012001400160051015 20 25查询图大小|候选集|/|结果集|GraphGrep GIndexMSTAFig.7 Comparison of query time图7 查询执行时间比较Fig.8 Comparison of accuracy of candidates图8 候选集合精确度比较0246810121416查询图大小访问率 (%)0.20.40.60.81.01.2查询图大小M S T 结果集大小/ 精确结果集大小Fig.9 Access ratio 图9 MSTA 的访问率Fig.10 Recall ratio 图10 MSTA 的召回率4.2 模拟数据实验结果我们用GIndex [10]中介绍的方法生成模拟数据.生成时,种子数目与要生成的图的数目的比例是2%,种子大小的中值为5,生成图的大小的中值为50.我们分别生成大小为2 000,4 000,8 000,16 000,32 000这5组模拟数据,并在这5个数据集上分别建立索引,然后测量索引的大小、索引的建立时间.对于子图查询性能,我们生成了大小为10 000的模拟数据集,查询Query 大小分别为5,10,15,20,25(根据查询图中的节点个数计算).对于每个大小的查询,我们从CI 类别里各随机抽取1 000个符合大小要求的图.对这5组各1 000个查询测量查询时间等,然后取平均值.对于GraphGrep [9]算法,我们采用路径长度lp =4.由于模拟数据中图的密度较大, GSpan 挖掘频繁子图时,即使在2 000大小的数据集上,挖掘支持度大于90%的图的时间也都在1个小时以上,这使得GIndex 不具有可比性,所以我们只对GraphGrep 进行了比较. 4.2.1 MSTA 索引性能分析图11是真实数据集合上的索引大小比较,从图中我们可以看出,MSTA 的索引大小随着数据库大小的增加呈线性增长.而GraphGrep 的增长较为显著,GraphGrep 的索引大小约是MSTA 索引大小的3.5倍~7.5倍.图12是真实数据集合上的索引建立时间比较,GraphGrep 约是MSTA 的40倍~300倍.图13和图14比较了真实和模拟数据之间的索引大小与索引建立时间两个指标的差异.这说明MSTA 无论在真实数据还是在模拟数据上都很稳定.152Journal of Software 软件学报 V ol.20, Supplement, December 2009500100015002000数据库大小(图的个数) 索引大小 (M B )200400600800数据库大小(图的个数)索引建立时间 (s )Fig.11 Index size图11 模拟数据集上的索引大小Fig.12 Construction time 图12 模拟数据集上的索引建立时间20040060080010001200140016001800数据库大小(图的个数) 索引大小 (M B )GraphGrep,模拟数据MSTA,模拟数据 GraphGrep,真实数据MSTA,模拟数据 0500100015002000数据库大小(图的个数)索引建立时间 (s )GraphGrep,模拟数据 MSTA,模拟数据 GraphGrep,真实数据MSTA,真实数据Fig.13 Index size between real and synthetic data 图13 真实数据和模拟数据的索引大小 Fig.14 Construction time between real and synthetic data图14 真实数据和模拟数据的索引建立时间4.2.2 MSTA 查询性能分析 图15和图16是模拟数据上的子图查询时间和召回率.查询执行时间的情况与真实数据相似,但召回率相对于真实数据偏低.246810 5 10 152025查询图大小查询执行时间GraphGrepMSTA0.20.40.60.81.01.25101520 25查询图大小召回率MSTA,真实数据MSTA,模拟数据Fig.15 Query time 图15 查询执行时间Fig.16 Recall ratio 图16 召回率5 结 论本文提出了MSTA 图索引方法.MSTA 方法利用最小生成树作为索引特征,根据边序列的包含关系和基于最大公共子图的图距离度量将最小生成树组织成MST 树结构作为索引结构,并在此结构下高效地支持了子图查询.与传统方法相比,传统方法的索引大小和索引建立时间是MSTA 方法的几倍甚至几百倍.MSTA 方法查询执行时间也要优于传统方法,并且具有较好的可扩展性.MSTA 方法虽然是近似方法,但是可以获得基于图距离度量最好的绝大部分结果,所以在可应用性方面有较好的表现.致谢 感谢哈尔滨工业大学数据与知识工程研究中心的老师及同学的大力支持.李楠等:基于最小生成树的图数据库索引算法153References:[1] Berman HM, Westbrook J, Feng ZK, Gilliand G, Bhat TN, Wessing H, Shindyalov IN, Boume PE. The protein data bank. NucleicAcids Research, 2000,28(1):235−242.[2] Ogata H, Goto S, Sato K, Fujibuchi W, Bono H, Kanehisa M. KEGG: Kyoto encyclopedia of genes and genomes. Nucleic AcidsResearch, 1999,27(1):29−34.[3] National library of medicine. /chemidplus[4] The international network for social network analysis. /[5] Raghavan S, Garcia-Molina H. Representing Web graphs. In: Proc. of the ICDE. 2003. 405−416.[6] DBLP dataset. http://dblp.uni-trier.de/xml/[7] Berretti S, Bimbo AD, Vicario E. Efficient matching and indexing of graph models in content-based retrieval. IEEE Trans. onPattern Analysis and Machine Intelligence, 2001,23(10):1089−1105.[8] Lee JK, Oh JH, Hwang S. STRG-Index: Spatio-Temporal region graph indexing for large video databases. In: Proc. of theSIGMOD. 2005. 718−729.[9] Shasha D, Wang JTL, Giugno R. Algorithmics and applications of tree and graph searching. In: Proc. of the PODS. 2002. 39−52.[10] Yan XF, Yu PS, Han JW. Graph indexing: A frequent structure-based approach. In: Proc. of the SIGMOD. 2004. 335−346.[11] Yan XF, Han JW. gSpan: Graph-Based substructure pattern mining. In: Proc. of the ICDM. 2002. 721−724.李楠(1984－),女,黑龙江牡丹江人,硕士生,主要研究领域为图数据库索引技术.李建中(1950－),男,博士,教授,博士生导师,主要研究领域为并行数据库,传感器网络.高宏(1966－),女,博士,教授,博士生导师,主要研究领域为并行数据库,图数据管理挖掘.。
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DOI: 10.1103/PhysRevE.76.066701
PACS number͑s͒: 47.11.Ϫj, 05.20.Dd, 68.03.Cd, 68.08.Bc
I. INTRODUCTION
Wetting and spreading phenomena of fluids on solid phases are important to many industrial and natural processes ͓1͔. When an immiscible droplet comes into contact with a solid phase, there is a contact line between the wetting and nonwetting fluids and the solid surface. The contact angle between the fluids and the surface can be calculated through Young’s equation provided the interfacial tensions between the fluid components and between each component and the solid surface are known ͓2,3͔.
Benzi et al. ͓15͔ presented a mesoscopic model for the interaction between a solid wall and a single component multiphase fluid ͑e.g., ͓16͔͒. They derived an analytical expression for the contact angle that covs from 0° to 180° but depends on prior knowledge of the profiles, and hence is not useful as a predictive tool. Our focus here is on the multicomponent multiphase SC model.
PHYSICAL REVIEW E 76, 066701 ͑2007͒
Proposed approximation for contact angles in Shan-and-Chen-type multicomponent multiphase lattice Boltzmann models
Haibo Huang,1 Daniel T. Thorne, Jr.,2 Marcel G. Schaap,3 and Michael C. Sukop1 1Department of Earth Sciences, Florida International University, Miami, Florida 33199, USA 2Department of Math, Physics and Computer Science, Georgetown College, Georgetown, Kentucky 40324, USA 3Department of Soil, Water and Environmental Science, University of Arizona, Tucson, Arizona 85721, USA
Numerical methods can be very instrumental in enhancing understanding of fluid behavior in complex systems, such as porous media. The lattice Boltzmann method ͑LBM͒, which is based on mesoscopic kinetic equations, is a popular and numerically robust technique for simulating single and multiphase fluids in arbitrarily complex media ͓4–9͔. Although the fluid dynamical characteristics of the LBM are well understood analytically and the method has been successfully applied in the study of wetting and spreading phenomena involving interfacial dynamics ͓4–10͔, there are still uncertainties in the interactions between the lattice Boltzmann fluid and the solid phase. Progress has been made for several LBM model types. For example, Briant et al. ͓1͔ investigated the wetting and spreading phenomena for single component multi-phase fluids using the free-energy-based LBM. LatvaKokko and Rothman ͓11͔ derived an estimate of the contact angle as a function of a wetting tendency of the wall when using color-gradient-based LBM. However, the multicomponent multiphase LBM proposed by Shan and Chen ͑SC͒ ͓4͔ is a more popular multiphase model and is better defined in terms of interparticle potential than the color-gradient based LBM ͓12͔ and we will focus on it exclusively here.
There have been many studies on wetting and spreading phenomena using the SC model. Martys and Chen ͓6͔ studied multicomponent fluids in complex three-dimensional geometries. In their simulations, the interaction force between a fluid and a wall was introduced. They found that reasonably well-defined contact angles could be obtained by adjusting the interaction strength between each fluid and a surface such that one of the fluids wets the surface. Using the SC LBM, Schaap et al. ͓13͔ and Pan et al. ͓9͔ studied the displacement
͑Received 19 February 2007; revised manuscript received 20 August 2007; published 3 December 2007͒
We propose a method for approximating the adhesion parameters in the Shan and Chen multicomponent, multiphase lattice Boltzmann model that leads to the desired fluid-solid contact angle. The method is a straightforward application of Young’s equation with substitution of the Shan and Chen cohesion parameter and a density factor for the fluid-fluid interfacial tension, and the adhesion parameters for the corresponding fluid-solid interfacial tensions.
To the best of our knowledge, all previous work on determining contact angles from adhesion parameters in the SC multicomponent, multiphase lattice Boltzmann model is strictly empirical and limited to a relatively small range of contact angles. The details of the methods used to measure contact angles were not provided in previous work. In this paper, we present an algorithm for contact angle measurement that allows systematic, consistent measurements. Then we use measurements of simulated drops to develop and empirically verify an expression based on Young’s equation for determining adhesion parameters that lead to the desired fluid-solid contact angle in the SC LBM.