Stability of infinite dimensional stochastic evolution

一类随机SEIQR传染病模型的动力学行为分析李雪;胡良剑【摘要】考虑疾病传播过程中随机因素的影响,研究一类具有潜伏期和隔离仓室的SIR传染病模型(SEIQR).通过构造Lyapunov函数并利用It^o公式,证明了随机SEIQR传染病模型存在唯一的全局正解,得到其关于相应的确定性模型的无病平衡点和地方病平衡点渐近稳定的充分性条件.【期刊名称】《纺织高校基础科学学报》【年(卷),期】2019(032)001【总页数】7页(P37-43)【关键词】随机传染病模型;It(^o)公式;Lyapunov函数;平衡点;渐近稳定【作者】李雪;胡良剑【作者单位】东华大学理学院,上海 201620;东华大学理学院,上海 201620【正文语种】中文【中图分类】O211.630 引言传染病一直以来是医学界普遍关注的重要问题,为了更好地预防和治疗传染病,学者们通过建立数学模型对传染病模型进行定性和定量研究。

最经典的传染病模型就是由Kermack和Mckendrick于1927年创立的SIR(susceptible-infective-removed)仓室模型[1]。

自此,很多传染病模型在此基础之上建立起来[2-3]。

然而,传染病的传播会受到外部环境的随机干扰。

研究在随机因素影响下的传染病模型所具有的性质对疾病的控制有重要意义。

近些年来,国内外学者对于随机传染病模型在生物,医药等领域的应用和研究取得了很多进展[4-6]。

考虑疾病潜伏期的SEIR模型加入随机扰动,已有大量研究[7-10]。

YANG等[11]考虑了随机SIR及SEIR传染病模型解的全局正性,并得到了求解关于相应的确定性系统的无病平衡点及地方病平衡点的渐近行为。

魏凤英等[12]进一步研究了一类具有非线性发病率的随机SEIR传染病模型的灭绝性及平稳分布等问题。

另一方面,也有部分学者考虑将具有隔离项的传染病模型加入随机扰动,ZHANG等[13]对一类非线性发病率的确定性和随机SIQS模型进行研究,给出了模型解是否稳定的阀值R0。

一类具有随机扰动的传染病SEIR模型稳定性分析徐敏;胡良剑;丁永生【摘要】考虑疾病传播过程中的随机干扰,运用随机人口建模中参数扰动的标准化技术,建立了一类具有随机扰动的传染病SEIR(susceptible-exposed but not infectious-infectious-removed)模型,证明了模型解的存在唯一性及非负性,并研究了无病平衡点满足p阶矩指数稳定的条件.研究结果为传染病预防与控制提供一定的理论依据与决策支持.%Stochasticity is introduced into an epidemics SEIR (susceptible-exposed but not infectiousinfectious-removed) model via the technique of parameter perturbation which is standard in stochastic population modeling.The existence and uniqueness are also proved for the solution.The model possessing non-negative solutions established are also proved as desired in any population dynamics.And the condition is obtained for the trivial solution of the model to be p-th moment exponential stable.In this way,theoretical foundation and decision support is provided for epidemics' prevention and control.【期刊名称】《东华大学学报（自然科学版）》【年(卷),期】2017(043)002【总页数】4页(P305-308)【关键词】随机扰动;SEIR模型;无病平衡点;p阶矩指数稳定性【作者】徐敏;胡良剑;丁永生【作者单位】上海对外经贸大学商务信息学院,上海201620;东华大学理学院,上海201620;东华大学信息科学与技术学院,上海201620【正文语种】中文【中图分类】X43;O141.41927年, 文献[1]报道了著名的SIRS(susceptible-infective-removed-susceptible)模型, 用于传染病传播过程的建模, 并将人群分成3类, 即易感者(susceptible)、感染者(infective)和移出者(removed).这类模型适用于疾病无潜伏期的情况, 然而, 对大部分传染病而言,从感染到发病需要一段时间, 因此, 具有潜伏期的传染病的传播可以用SEIR(susceptible-exposed but not infectious-infectious-removed)模型建模, 近年来, 有大量学者对传染病SEIR模型进行研究.文献[2]研究了一类具有年龄依赖潜伏期和复发的SEIR模型的全局渐近稳定性; 文献[3]研究了一类具有非线性发病率的SIR(susceptible-infectious-removed)和SEIR模型的动力学行为; 文献[4]研究了一类具有一般发病率的非自主性SEIRS(susceptible-exposed but not infectious-infectious-removed-susceptible))模型, 得出了疾病灭绝的条件.文献[5]得到了一类SEIR模型的解析解, 通过仿真对该模型的精确解和解析解进行对比.疾病的传播一般会受到随机因素的干扰, 基于此, 文献[6]建立了一类带有无限时滞的两群组的随机SEIR模型, 得到了其渐近稳定性的条件并进行了仿真.文献[7]建立了一类带跳的随机SEIR模型, 研究了解的渐近稳定性.文献[8]建立了一类具有独立随机扰动的SEIR模型, 证明了其无病平衡点的几乎必然指数稳定性.文献[9]研究了一类具有随机扰动的多群组的SEIR 模型.文献[10]建立了一类带有随机扰动的多群组SEIR模型和SIR模型, 并研究了其随机渐近稳定性.在考虑到疾病的传播率受到环境因素的干扰下, 文献[11]建立了一类带有环境噪声干扰的随机SEIR模型, 并证明了其全局渐近稳定性.本文对文献[11]中的随机SEIR模型补充证明了其解的存在唯一性以及非负性, 进一步得到了该模型的平凡解p阶矩稳定性的条件, 为具有潜伏期的传染病预防及控制提供一定的理论依据与决策支持.不失一般性, 设t时刻总人口数为N(t), 假设不考虑人口的迁移, 令N(t)=N, S(t)表示t时刻未染病但有可能被疾病传染的人数占总人口数的比例, I(t)表示t时刻已被传染成病人而且具有传染力的人数占总人口数的比例.病人在被感染后成为患病者之前有一段病菌潜伏期, 假定在潜伏期内的感染者没有传染力, 记t时刻潜伏者人数占总人口数的比例为E(t), R(t)表示t时刻已从染病者康复的人数占总人口数的比例, 并假定病人康复后具有永久免疫力.考虑一类具有常数移民的SEIR传染病模型:假设总人口恒定, 即在疾病流行期间, 考虑人口的出生与自然死亡等变化, 假定出生率(单位时间内出生的人数在总人数中的比例)与死亡率相等, 用b表示, λ表示疾病传播率, 疾病的平均潜伏期为1/ω, γ表示病人康复后退出系统的移出率.对该确定性的SEIR模型而言, 文献[12]得出了疾病再生数R0为当R0≤1时, 无病平衡点P0(N, 0, 0, 0)全局渐近稳定; 当R0>1时, 无病平衡点P0(N, 0, 0, 0)不稳定.考虑疾病传播过程中环境因素的干扰, 将环境噪声引入模型, 考察其解的特性, 利用随机人口建模中的标准技术, 假设疾病传播率λ受到环境因素的干扰, 假设噪声的干扰强度为σ, 用λ+σdωt代替λ, 得到一类随机SEIR模型如下:ωt是一维纳过程(布朗运动), 故满足在证明解的非负性之前, 有必要做一些说明, 令同时令x(t)=(S(t), E(t), I(t), R(t)),y(t)=(S(t), E(t)).在证明主要定理之前先给出引理1.引理1 如下不等式成立:证明：对∀u>0定义函数由可知f(u)在u=2处取得最小值, 于是f(u)≥f(2), 得证.定理1 假定b, λ, σ, ε, γ均为正实数, 则对任何的初始值方程(3)～(6)在t≥0上有唯一解, 且解满足对∀a.s.证明：根据已知E(t)≥I(t), 则由式(5)得:于是同理, 由于I(t)≥R(t), 由式(6)得:下证对∀t≥0, S(t)>0, E(t)>0 a.s.由于方程(3)～(4)的系数是局部李普希兹连续的, 故对任何的初始值, 在t∈[0, τe)上方程(3)～(6)有唯一的局部解y(t), 为了证明解是全局的, 需要证明τe=∞ a.s., 取充分大正数k0使得y0的每一分量都属于区间[1/k0, k0], 对每一个整数k≥k0, 定义停时τk为显然当k→∞时, τk单调递增, 令则τ∞≤τe a.s., 下证τ∞=∞反证, 假设τ∞≠∞, 则存在一对常数T>0和ε∈(0, 1)使得P{τ∞≤T}>ε, 故∃k1≥k0使得对∀k≥k1, 有P{τk≤T}≥ε, 定义函数由于对∀u>0, u+1-log(u)≥0, 故V(x)≥0, 由公式得由引理1和V的定义可知, E(t)≤2V(x)-(4-2log2), 于是有这里c2=max(c1, 2λ), 故当t1≤T时, 有于是根据Gronwall不等式, EV(y(τk∧T))≤c3, c3=(V(y0)+c2T)ec2T, 对k≥k1, 令P(Ωk≥ε), 则P(Ωk≥ε), 于是由Ωk的定义可知c3≥E[1Ωk(ω)V(y(τk, ω))]≥ε([k+1-log(k)]∧[(1/k)+1+log(k)])令k→∞, 得∞>c3=∞, 矛盾, 于是τ∞=∞ a.s.笔者已得出模型(3)～(6)的解全局渐近稳定性的条件[11], 更进一步, 这里讨论方程(3)～(6)的平凡解的p阶矩指数稳定性.不失一般性, 在[0, ∞)上具有初始值x(0)=x0的d维随机微分方程为:假定对∀t≥t0, f(0, t)=0, g(0, t)=0, 则对应于初始条件x(t0)=0, 方程(7)有解x(t)≡0, 该解称为平凡解或平衡位置.定义1[13] 若存在一对正数λ和C使得对∀t≥0, E|x(t; x0)|p≤C|x0|pe-λt, 则称式(7)的平凡解是p阶矩指数稳定的.定理2 若对每一个固定的正整数p, 若<0, 且γ>b成立, 则方程(3)～(6)的平凡解是p阶矩指数稳定的.证明: 令V(E)=lnE, 由公式得两边从0到t积分, 得于是E(E(t))≤E(0)·eκ t·E(eσ ω(t)), 由于ω(t)～N(0, t), 得于是则对任何正整数p, 有其中令V(S(t))=ln(1-S), 由公式得,同上可得,其中κ2=λ-b, 根据定义1及已知条件可知, 定理2成立.本文在建立一类具有随机扰动的传染病SEIR模型的基础上, 利用公式证明了解的存在唯一性及非负性, 更进一步得出了模型的平凡解p阶矩稳定的条件, 为传染病的预防与控制提供了一定条件的决策支持与理论依据.【相关文献】[1] KERMACK W O, MCKENDRICK A G. Contributions to the mathematical theory of epidemics[C]//Proceedings of the Royal Society of London. 1927: 700-721.[2] LIU L L, WANG J L, LIU X N.Global stability of an SEIR epidemic model with age-dependent latency and relapse[J]. Nonlinear Analysis(Real World Applications), 2015, 24: 18-35.[3] LIU J L, PENG B Y, ZHANG T L.Effect of discretization on dynamical behavior of SEIR and SIR models with nonlinear incidence[J]. Applied Mathematics Letters, 2015, 39: 60-66.[4] MATEUSA J P, SILVA C M.A non-autonomous SEIRS model with general incidence rate[J]. Applied Mathematics and Computation, 2014, 247: 169-189.[5] WANG X Y, WEI L J, ZHANG J. Dynamical analysis and perturbation solution of an SEIR epidemic model[J]. Applied Mathematics and Computation, 2014, 232: 479-486.[6] LIU M, BAIA C Z, WANG K. Asymptotic stability of a two-group stochastic SEIR model with infinite delays[J]. Communications in Nonlinear Science and Numerical Simulation, 2014, 19(10): 3444-3453.[7] ZHANG X H, WANG K.Stochastic SEIR model with jumps[J]. Applied Mathematics and Computation, 2014, 239(15) : 133-143.[8] WITBOOI P J. Stability of an SEIR epidemic model with independent stochastic perturbations [J]. Physica A: Statistical Mechanics and its Applications, 2013, 392(20): 4928-4936.[9] YANG Q S, MAO X R. Extinction and recurrence of multi-group SEIR epidemic models with stochastic perturbations [J]. Nonlinear Analysis: Real World Applications, 2013, 14(3): 1434-1456.[10] YUAN C J, JIANG D Q, O’REGAN D, et al. Stochastically asymptotically stability of the multi-group SEIR and SIR models with random perturbation [J]. Communications in Nonlinear Science and Numerical Simulation, 2012, 17(6): 2501-2516.[11] XU M, HU L J.Global asymptotic stability of an SEIR model with random perturbation [J]. Journal of Donghua University (English Edition), 2014, 31(2): 152-154.[12] LI M Y, GRAEF J R, WANG L, et al. Global dynamics of a SEIR model with varying total population size[J], Mathematical Biosciences, 1999, 160(2): 191-213.[13] MAO X R.Stochastic differential equations and applications[M]. Chichester: Ellis Horwood, 1997: 206-212.。

Transient dynamic analysis of a floating beam–water interactionsystem excited by the impact of a landing beam Original ResearchArticleJournal of Sound and Vibration, Volume 303, Issues 1-2, 5 June2007, Pages 371-390J.Z. Jin, J.T. XingClose preview | Related articles | Related reference w ork articlesAbstract | Figures/Tables | ReferencesAbstractAn aircraft, idealised as an elastic beam with supporting system, lands with speed on anaircraft carrier or floating airport. The floating platform is idealised as a flexible beam floatingin an infinite water domain. The water is assumed incompressible and inviscid. The surface disturbance satisfies a linear free surface wave condition and an undisturbed condition atinfinity. A mathematical model is developed to describe this complex fluid–structureinteractive dynamical system, which exhibits strong coupling between the system'scomponents. An effective numerical scheme is established assuming that the two beam-likestructures are two solid substructures with motions represented by their respective modefunctions and the infinite fluid domain is modelled by a boundary element scheme. A mixedmode function-boundary element method is developed to solve the transient dynamics of thesystem in terms of the time histories of the beam displacement responses and waterpressure. The presented data demonstrates the applicability of the developed mathematicalmodel and numerical approach and some insights of the complex interactive processexhibited by the system.A rticle Outline1. Introduction2. Governing equations2.1. Fluid domain2.2. Landing beam2.3. Floating beam2.4. Fluid–structure interaction interface conditionPurchase$ 31.502.5. Initial conditions3. Mode equations of solid substructures3.1. Mode functions of a free –free beam3.2. Mode equation of landing beam3.3. Mode equation of floating beam4. Boundary element equation of the fluid domain5.Mixed mode function-boundary element equations6. Numerical solution7. Numerical results7.1. A simulation for a car running test7.2. An example simulating aircraft landing impacts8. ConclusionsAcknowledgements Appendix A. AppendixReferences427 Influence of modelling and solution methods on the FE-simulation of the post-buckling behaviour ofstiffened aircraft fuselage panels Original ResearchArticleComposite Structures , Volume 73, Issue 2, May 2006,Pages 229-236P. Linde, A. Schulz, W. RustClose preview | Related articles | Related reference w ork articles Abstract | Figures/Tables | ReferencesAbstractStiffened fuselage panels with laminated constructions play an increasing rolein aircraft design. The static behaviour through the buckling- and post-bucklingregime until failure has to be established. Apart from analytical calculations,experimental tests for different load combinations are indespensible, both ofwhich are expensive and time consuming.Purchase $ 31.50The virtual testing described here is based on a development project aiming at reducing the amount of experimental tests, and narrowing the numerical predictions to experimental results.A tool developed for parametric modelling and simulation of test shells is discussed.The numerical model is based on layered shell elements in ANSYS and for special purposes in LS-DYNA. It is outlined how far the behaviour of laminates (interaction of different and anisotropic materials, delamination and splices) can be simulated in this context.Results are given for welded panels and for fibre metal laminate panels. Comparison with experimental data is made. Recommendations for future research is provided.A rticle Outline1. Introduction1.1. General1.2. Objective1.3. Scope2. Model2.1. Geometry model2.2. Model for implicit analysis2.3. Model for explicit analysis2.4. FE-model for LS-DYNA2.5. Splices2.6. Delaminations2.7. Boundary conditions3. Computation 3.1. General 3.2. Static solution3.3. Implicit dynamic solution3.4. Explicit dynamic solution4. Results5. Summary, conclusions5.1.Summary5.2. Conclusions5.3. RecommendationsAcknowledgementsReferences428 Extratropical case study of stratosphere –troposphere exchange using multivariate analyses from mozaicaircraft data Original Research ArticleAtmospheric Environment , Volume 39, Issue 35,November 2005, Pages 6537-6549François Borchi, Emmanouil Oikonomou, Alain Marenco Close preview | Related articles | Related reference w ork articlesAbstract | Figures/Tables | ReferencesAbstractA multivariate analysis methodology, applied to ozone, water vapour andpotential temperature data collected from MOZAIC aircraft allowed to identifyand to quantify three types of air masses directly linked tostratosphere-troposphere exchanges (STE). These air masses occurred inFebruary 1997 over the North Atlantic during the development of a Rossbywave, which is manifested in the form of four different structures, namelytrough, ridge, streamer and cut-off low (COL). Here a study is conducted on 20isobaric (i.e. at 230 hPa) flights crossing all these upper-level structures. It isshown that the first type of air mass corresponds to a mixed zone between thePurchase $ 35.95stratosphere and the troposphere when the tropopause is poorly defined and when there exist medium values and weak gradients of ozone, potential temperature and humidity. The second type of air mass reveals an irreversible transport from the troposphere to the stratosphere for all these structuresl eading to a ―wet‖ stratosphere with high values of water vapour. The third type of air mass corresponds to the classical stratosphere with high values of ozone and low values of water vapour. The comparison with the model results of Kowol-Santen et al. (2000) shows that these first two air masses are directly linked to either diabatic processes or turbulent diffusion depending on the orientation and the type of the structure. The western side of the trough (with no mixing zone on the eastern side) and the ridge are dominated by clear air turbulence (CAT). On the other hand, the decay of the COL seems to be the result of strong convection on its northern and eastern flanks. Concerning the decay of the streamer, it is found to be due to both diabatic and turbulent processes. The above results suggest that the existence of a mixing zone during STE seems to be a more realistic concept instead of that where the tropopause is considered as a surface.A rticle Outline1. Introduction2. The MOZAIC database3. Methodology of MA4. Evolution of a trough structure in February 19975. Air masses in the trough, ridge, COL and streamer structures5.1. The trough and ridge structures5.2. The COL structure5.3. The streamer structure6. Discussion and comparison with model result of KS007. Conclusions Acknowledgements References429 Adaptive actuator failure compensation for nonlinearMIMO systems with an aircraft controlapplication Original Research ArticleAutomatica, Volume 43, Issue 11, November 2007,Pages1869-1883Xidong Tang, Gang Tao, Suresh M. JoshiClose preview | Related articles | Related reference w ork articlesAbstract | Figures/Tables | ReferencesAbstractA direct adaptive approach is developed for control of a class of multi-inputmulti-output (MIMO) nonlinear systems in the presence of uncertain failures ofredundant actuators. An adaptive failure compensation controller is designedwhich is capable of accommodating uncertainties in actuator failure timeinstants, values and patterns. A realistic situation is studied with fixed groupingof actuators and proportional actuation within actuator groups. The adaptivecontrol system is analyzed, to show its desired stability and asymptotictracking properties in the presence of actuator failure uncertainties. As anapplication, such an adaptive controller is used for actuator failurecompensation of a twin otter aircraft longitudinal model, with design conditionsverified and control structure and adaptive laws developed for a nonlinearaircraft dynamic model. The effectiveness of adaptive failure compensation is demonstrated by simulation results.A rticle Outline1. Introduction2. Problem statementPurchase$ 35.953. Adaptive failure compensation design4. Application to aircraft flight control5. ConclusionsReferencesVitae430 An aircraft acquisition decision model under stochastic demand Original Research ArticleJournal of King Saud University - Science, In Press, Corrected Proof, Available online 28 December 2010Hooi Ling Khoo, Lay Eng TeohClose preview | PDF (445 K) | Related artic les | Related reference w orkarticlesAbstract | Figures/Tables | ReferencesAbstractThis study proposes the probabilistic dynamic programming model to address the stochastic demand issue in aircraft acquisition problem. A probable phenomena is defined to comprehend the uncertain state variables so that the targeted level of service could be achieved profitably by the airline company. The objective function and the constraints have a linear expression with respect to the decision variables, and hence the proposed model is then converted as a linear programming model. The proposed model and the solution method are then examined with an illustrative case study to determine the number and the types of new aircraft that should be purchased at every time period. The results show that the proposed methodology is viable in providing the optimal solution.A rticle Outline1. Introduction2. Nomenclature3. Problem formulation3.1. Probabilistic dynamic programming model3.1.1. Stage, state variables and optimal decision3.1.2. Constraints3.1.3. Objective function3.1.4. The probable phenomena, s 1, … , s k3.1.5.The optimization model4. Solution method5. An illustrative case study5.1. Benchmark scenario5.1.1. Other scenarios6. Results and discussion7. Conclusions References431 Aerodynamic seal for reduction of noise generated on aircraft control surfacesSealing Technology , Volume 2009, Issue 2, February2009, Page 13Close preview | Related articles | Related reference w ork articles Abstract | Figures/TablesNo abstract is available for this article. Purchase $ 31.50432 Partial substitution of hydrogen for conventional fuel in an aircraft by utilizing unused cargo compartmentspace Original Research ArticleInternational Journal of Hydrogen Energy , Volume 35,Issue 3, February 2010, Pages 1463-1473Enis T. Turgut, Marc A. Rosen Close preview | Related articles | Related reference w ork articles Abstract | Figures/Tables | ReferencesAbstractOptions are being actively sought in aviation to switch from petroleum-based fuels to alternative fuels, of which hydrogen is a promising candidate, despite challenges associated with its production and storage. The possibility is Purchase $ 31.50demonstrated in this study of using hydrogen in place of some mission fuel without making substantial aircraft modifications and while utilizing only available unused baggage space in the lower-deck cargo compartments of aircraft. The environmental impact reduction and weight increase are obtained accounting for a broad range of factors including aircraft model, seat capacity, passenger and baggage load factors, annual landing and take off cycles, container type, and costs of metal hydride and gaseous hydrogen storage units of various sizes. It is found that, while there may be a cost increase, CO2 emissions are substantially reduced, by 25,000–570,000 tonnes annually in several cases and by up to 1.1 million tonnes annually for the 10 types of aircraft considered. It is also determined that with present technology, despite the low density of hydrogen, the weight of storage systems constitutes more of a challenge than their volume in aviation. Large-body aircraft are found to have more difficulties than the narrow-body aircraft regarding storage system weight. For the most frequently used narrow- and large-body aircraft considered, the number of the available containers within the required limits of weight and volume respectively are found to be 3 and 4 for the B 737-800 aircraft and 2 and 10 for the A 340-300 aircraft. Overall, the combined usage of hydrogen and kerosene investigated here may be feasible in the future, but is a challenging option with present technology and aircraft due to various factors.A rticle OutlineNomenclature1. Introduction2. Approach3. Hydrogen storage4. Lower-deck hydrogen storage5. Results and discussion 5.1. Case study 15.2. Case study 26. ConclusionsAcknowledgementsAppendixReferencesVitae433Noise radiation of aircraft panels subjected toboundary layer pressure fluctuationsOriginal ResearchArticleJournal of Sound and Vibration, Volume 314, Issues 3-5,22 July 2008, Pages 693-711Bilong LiuClose preview | Related articles | Related reference w ork articlesAbstract | Figures/Tables | ReferencesAbstractIn this paper, a method which predicts the sound radiation of aircraft panelssubjected to turbulent boundary layer excitation is described. The method isthe extension of an earlier deterministic approach, where the modal expansionand modal receptance methods were used to predict random noisetransmission through curved aircraft panels with stringer and ring frameattachments. Here, with implementation of the Corcos and Efimtsov models tocharacterize the dynamic surface pressure cross-spectra, closed-formsolutions for the panel displacements, radiation and transmission pressuresare derived. Numerical examples are presented to illustrate the effects of thestringers, ring frames, hydrodynamic coincidence, curvature, in-plane tension,structural dissipation and composite material on the structural and acousticresponse of the panel.Purchase$ 31.50A rticle Outline1. Introduction2. TBL-induced noise for a rectangular panel with stringer attachments2.1. Governing equations and velocity response2.2. Slightly curved panels with stringer attachments2.3. The radiated power spectrum3. Numerical study3.1. Solution for a typical aircraft panel3.2. Effects of the ring frame attachments3.3. Effects of the stringer attachments3.4. Damping influence3.5. Curvature and in-plane tension influence3.6. Metallic panel vs composite panel4.Concluding remarksAcknowledgementsAppendix A. The modal excitation termsReferences434 A computational study of Benders decomposition for the integrated aircraft routing and crew schedulingproblem Original Research Article Computers & Operations Research , Volume 32, Issue 6, June 2005, Pages 1451-1476Anne Mercier, Jean-François Cordeau, François Soumis Close preview | Related articles | Related reference w ork articles Abstract | Figures/Tables | ReferencesAbstractThe integrated aircraft routing and crew scheduling problem consists indetermining a minimum-cost set of aircraft routes and crew pairings such that Purchase$ 31.50each flight leg is covered by one aircraft and one crew, and side constraints are satisfied. While some side constraints involve only crews or aircraft, linking constraints impose minimum connection times for crews that depend on aircraft connections. We propose an enhanced model incorporating robustness to handle these linking constraints and compare two Benders decomposition methods—one with the aircraft routing problem as the master problem and one with the crew pairing problem. We also study the impact of generating Pareto-optimal cuts on the speed of convergence of these methods. Computational experiments performed on test instances provided by two major airlines show that the proposed approach yields high-quality solutions in reasonable computing times.A rticle Outline0. Introduction1. Mathematical formulation1.1. Aircraft routing networks1.2. Crew pairing networks1.3. Model2. Solution methodology2.1. Benders decomposition using the crew pairing problem as a master problem2.1.1. Three-phase algorithm2.2. Identifying strong cuts2.3. Step-by-step description of the improved algorithm3. Computational experiments3.1. Description of data sets3.2. Summary of computational experiments3.3. Comparisons with the extended crew pairing approach of Cohn andBarnhart [2]4. Conclusion AcknowledgementsReferences435Sample preparation and characterization of artificiallyaged aircraft coatings for microstructuralanalysis Original Research ArticleMaterials Characterization, Volume 55, Issue 3,September 2005, Pages 179-189Lisa M. Farrier, Stephen L. SzarugaClose preview | Related articles | Related reference work articlesAbstract | Figures/Tables | ReferencesAbstractThe paper describes a method for metallographic preparation of artificiallyaged aircraft coatings. In order to better understand the in-serviceperformance and identify degradation mechanisms of an aircraft coating,complete characterization of the microstructure is essential. This paperdiscusses metallographic sample preparation and subsequent microscopytechniques (light optical microscopy, scanning electron microscopy) forcharacterization of a standard polyurethane aircraft coating system. Thepreparation method has proven to produce good, consistent results on a widerange of laboratory-produced simulated environmental exposures.The military specification coating system under study (MIL-PRF-85285C andMIL-PRF-23377G) degrades severely after accelerated weathering. Typicaldegradation includes deterioration of the polyurethane-based resin system inthe topcoat and is observed as a visible change in the color. Increasedporosity and some physical deterioration were also observed. In addition,some inorganic pigments in the primer appear to migrate into the topcoatPurchase$ 31.50during simulated exposure.A rticle Outline1. Introduction2.Experiment procedure2.1. Coating application2.2. Proposed metallographic method2.3. Method justification3. Results and discussion3.1. Microstructural characterization3.2. Free film transmissivity4. SummaryAcknowledgementsReferences436 Intelligent adaptive nonlinear flight control for a high performance aircraft with neural networksOriginal Research ArticleISA Transactions , Volume 45, Issue 2, April 2006, Pages 225-247Aydogan Savran, Ramazan Tasaltin, Yasar Becerikli Close preview | Related articles | Related reference w ork articles Abstract | ReferencesAbstractThis paper describes the development of a neural network (NN) based adaptive flight control system for a high performance aircraft. The main contribution of this work is that the proposed control system is able to compensate the system uncertainties, adapt to the changes in flightconditions, and accommodate the system failures. The underlying study can be considered in two phases. The objective of the first phase is to model the Purchase$ 31.50dynamic behavior of a nonlinear F-16 model using NNs. Therefore aNN-based adaptive identification model is developed for three angular rates of the aircraft. An on-line training procedure is developed to adapt the changes in the system dynamics and improve the identification accuracy. In thisprocedure, a first-in first-out stack is used to store a certain history of theinput-output data. The training is performed over the whole data in the stack atevery stage. To speed up the convergence rate and enhance the accuracy forachieving the on-line learning, the Levenberg-Marquardt optimization methodwith a trust region approach is adapted to train the NNs. The objective of thesecond phase is to develop intelligent flight controllers. A NN-based adaptivePID control scheme that is composed of an em ulator NN, an estimator NN,and a discrete time PID controller is developed. The emulator NN is used tocalculate the system Jacobian required to train the estimator NN. Theestimator NN, which is trained on-line by propagating the output error throughthe emulator, is used to adjust the PID gains. The NN-based adaptive PIDcontrol system is applied to control three angular rates of the nonlinear F-16model. The body-axis pitch, roll, and yaw rates are fed back via the PIDcontrollers to the elevator, aileron, and rudder actuators, respectively. Theresulting control system has learning, adaptation, and fault-tolerant abilities. Itavoids the storage and interpolation requirements for the too many controllerparameters of a typical flight control system. Performance of the controlsystem is successfully tested by performing several six-degrees-of-freedomnonlinear simulations.437An integrated aircraft routing, crew scheduling andflight retiming model Original Research ArticleComputers & Operations Research, Volume 34, Issue 8,August 2007, Pages 2251-2265Anne Mercier, François SoumisClose preview | Related articles | Related reference w ork articlesAbstract | Figures/Tables | ReferencesPurchase$ 31.50AbstractIn the integrated aircraft routing, crew scheduling and flight retiming problem, a minimum-cost set of aircraft routes and crew pairings must be constructed while choosing a departure time for each flight leg within a given time window. Linking constraints ensure that the same schedule is chosen for both the aircraft routes and the crew pairings, and impose minimum connection times for crews that depend on aircraft connections and departure times. We propose a compact formulation of the problem and a Benders decomposition method with a dynamic constraint generation procedure to solve it. Computational experiments performed on test instances provided by two major airlines show that allowing some flexibility on the departure times within an integrated model yields significant cost savings while ensuring the feasibility of the resulting aircraft routes and crew pairings.A rticle Outline0. Introduction1. Mathematical formulation1.1. Model1.2. A simpler formulation2. Solution methodology2.1. Benders decomposition2.1.1. Three-phase algorithm2.1.2. Dynamic constraint generation3. Computational experiments3.1. Description of data sets3.2. Summary of computational experiments4. ConclusionAcknowledgements References438 Low-crested coastal defence structures as artificialhabitats for marine life: Using ecological criteria indesign Original Research ArticleCoastal Engineering, Volume 52, Issues 10-11,November2005, Pages 1053-1071P.S. Moschella, M. Abbiati, P. Åberg, L. Airoldi, J.M.Anderson, F. Bacchiocchi, F. Bulleri, G.E. Dinesen, M.Frost, E. Gacia, L. Granhag, P.R. Jonsson, M.P. Satta, A.Sundelöf, R.C. Thompson, S.J. HawkinsClose preview | Related articles | Related reference w ork articlesAbstract | Figures/Tables | ReferencesAbstractCoastal defence structures to protect sedimentary coastlines from erosion andflooding are increasingly common throughout Europe. They will become more widespread over the next 10–30 years in response to rising and stormier seasand accelerating economic development of the coastal zone. Building coastaldefences results in the loss and fragmentation of sedimentary habitats andtheir replacement by artificial rocky habitats that become colonised by algaeand marine animals. The engineering design and construction of thesestructures have received considerable attention. However, the ecologicalconsequences of coastal defences have been less extensively investigated. Furthermore, due to their rapid proliferation, there is a growing need tounderstand the role of these man-made habitats in the coastal ecosystems inorder to implement impact minimisation and/or mitigation measures.As part of the DELOS project, targeted studies were carried out throughoutEurope to assess the ecological similarity of low-crested coastal defencestructures (LCS) to natural rocky shores and to investigate the influence ofLCS design features on the colonising marine epibiota. LCSs can bePurchase$ 35.95considered as a relatively poor surrogate of natural rocky shores. Epibiotic communities were qualitatively similar to those on natural rocky shores as both habitats are regulated by the same physical and biological factors. However, there were quantitative differences in the diversity and abundance of epibiota on artificial structures. Typically, epibiotic assemblages were less diverse than rocky shore communities. Also, LCSs offered less structurally complex habitats for colonisation and in some locations experienced higher disturbance than natural shores. We propose several criteria that can be integrated into the design and construction of LCSs to minimise ecological impacts and allow targeted management of diversity and natural living resources.A rticle Outline1. Introduction2. Material and methods2.1. Study sites and general methodological approach2.2. Comparison of diversity between low crested structures and natural rocky shores2.3. Effects of LCS design features on epibiotic assemblages2.3.1. Orientation2.3.2. Location on the shore2.3.3. Habitat complexity2.3.4. Disturbance: structural stability and scouring of LCS3. Results3.1. Epibiota of LCSs versus rocky shore communities3.2. Effects of LCS design features on epibiota3.2.1. Orientation3.2.2. Location on the shore (tidal height)3.2.3. Habitat complexity3.2.4. Disturbance: structural stability and scouring4. Discussion 4.1. Can LCSs be viewed as artificial rocky shores?4.2. Influence of engineering design on epibiota: what can and cannot be modified5. ConclusionsAcknowledgementsReferences439 Carbon fiber reinforced plastics in aircraft constructionOriginal Research ArticleMaterials Science and Engineering: A , Volume 412, Issues1-2, 5 December 2005, Pages 171-176 C. Soutis Show preview | Related articles | Related reference w ork articles Purchase $ 41.95440Vibration-based skin damage statistical detection and restoration assessment in a stiffened aircraftpanel Original Research Article Mechanical Systems and Signal Processing , Volume 22,Issue 2, February 2008, Pages 315-337D.D. Rizos, S.D. Fassois, Z.P. Marioli-Riga, A.N. Karanika Show preview | Related articles | Related reference w ork articlesPurchase$ 35.95 441Estimation of directivity and sound power levels emitted by aircrafts during taxiing, for outdoor noiseprediction purpose Original Research Article Applied Acoustics , Volume 68, Issue 10, October 2007, Pages 1263-1279C. Asensio, I. Pavón, M. Ruiz, R. Pagan, M. Recuero Show preview | Related articles | Related reference w ork articles Purchase$ 41.95 442Strategic planning in fractional aircraft ownership programs Original Research ArticleEuropean Journal of Operational Research , Volume 189,Issue 2, 1 September 2008, Pages 526-539Yufeng Yao, Özlem Ergun, Ellis Johnson, William Schultz, J.M. Singleton Show preview | Related articles | Related reference w ork articlesPurchase $ 31.50。

统计学硕士研究生培养方案（2013级研究生开始使用）一、专业学科、学制、学习方式一级学科名称：统计学（代码： 0270 ）二级学科名称：金融统计与风险管理（代码： 027001 ）二级学科名称：经济统计（代码： 027002 ）二级学科名称：应用统计（代码： 027003 ）二级学科名称：管理统计与决策（代码： 027004 ）二级学科名称：数量金融与保险精算（代码： 027005 ）学制：三年学习方式：全日制二、本学科情况介绍1、本学科建设时间较长，师资力量雄厚，科研实力强。

我校统计学学科作为应用数学的一个分支，建立于1958年，1994年经广州市人民政府批准成立了广州市系统工程研究所，以社会、经济、科教、环境等领域中的复杂大系统为研究对象，开展一系列统计分析研究工作。

1997年开始招收硕士生，2009年开始招收博士生。

2011年我校统计学被批准为一级学科博士和硕士授予权。

本学科现有教授11人，副教授8人，其中博士18人，博士生导师7人。

先后主持国家自然科学基金、国家社科基金、国家统计局以及国家软科学基金等30余项，省部级项目30项，合计获得科研经费1000多万元，获省部级以上科研奖励3项。

出版学术专著9部，教材12部。

在《Biometrika》《Statistica Sinica》、《中国科学》、《统计研究》、《金融研究》等重要刊物上发表了一系列重要论文。

2、主要研究方向稳定，特色鲜明，学科带头人影响大。

学科带头人长期从事数理统计、经济统计理论与方法、管理统计与决策分析的研究，取得了一批重要成果，在国内外有重要影响。

如，本学科在时间序列分析领域已成为国内主要研究中心之一，在国际上具有重大影响的第八届泛华统计国际学术会议就是2010年在我校召开的。

3、学术交流频繁，学术地位高。

五年来，本学科组织了国际国内学术会议10次，与美国、加拿大、英国、德国、香港等海外10多所高校与研究所开展了频繁的国际交流与合作，先后有20多人次参加国际会议并做大会报告。

通类名词力学mechanics牛顿力学Newtonian mechanics经典力学classical mechanics静力学statics运动学kinematics动力学dynamics动理学kinetics宏观力学macroscopic mechanics,macromechanics细观力学mesomechanics微观力学microscopic mechanics,micromechanics一般力学general mechanics固体力学solid mechanics流体力学fluid mechanics理论力学theoretical mechanics应用力学applied mechanics工程力学engineering mechanics实验力学experimental mechanics计算力学computational mechanics理性力学rational mechanics物理力学physical mechanics 地球动力学geodynamics力force作用点point of action作用线line of action力系system of forces力系的简化reduction of force system 等效力系equivalent force system刚体rigid body力的可传性transmissibility of force 平行四边形定则parallelogram rule 力三角形force triangle力多边形force polygon零力系null-force system平衡equilibrium力的平衡equilibrium of forces平衡条件equilibrium condition平衡位置equilibrium position平衡态equilibrium state分析力学analytical mechanics拉格朗日乘子Lagrange multiplier拉格朗日[量] Lagrangian拉格朗日括号Lagrange bracket循环坐标cyclic coordinate循环积分cyclic integral哈密顿[量] Hamiltonian哈密顿函数Hamiltonian function正则方程canonical equation正则摄动canonical perturbation正则变换canonical transformation正则变量canonical variable哈密顿原理Hamilton principle作用量积分action integral哈密顿--雅可比方程Hamilton-Jacobi equation作用--角度变量action-angle variables阿佩尔方程Appell equation劳斯方程Routh equation拉格朗日函数Lagrangian function诺特定理Noether theorem泊松括号poisson bracket边界积分法boundary integral method并矢dyad运动稳定性stability of motion轨道稳定性orbital stability李雅普诺夫函数Lyapunov function 渐近稳定性asymptotic stability结构稳定性structural stability久期不稳定性secular instability弗洛凯定理Floquet theorem倾覆力矩capsizing moment自由振动free vibration固有振动natural vibration暂态transient state环境振动ambient vibration反共振anti-resonance衰减attenuation库仑阻尼Coulomb damping同相分量in-phase component非同相分量out-of -phase component超调量overshoot参量[激励]振动parametric vibration模糊振动fuzzy vibration临界转速critical speed of rotation阻尼器damper半峰宽度half-peak width集总参量系统lumped parameter system 相平面法phase plane method相轨迹phase trajectory等倾线法isocline method跳跃现象jump phenomenon负阻尼negative damping达芬方程Duffing equation希尔方程Hill equationKBM方法KBM method, Krylov-Bogoliu-bov-Mitropol'skii method马蒂厄方程Mathieu equation平均法averaging method组合音调combination tone解谐detuning耗散函数dissipative function硬激励hard excitation硬弹簧hard spring, hardening spring谐波平衡法harmonic balance method久期项secular term自激振动self-excited vibration分界线separatrix亚谐波subharmonic软弹簧soft spring ,softening spring软激励soft excitation 邓克利公式Dunkerley formula瑞利定理Rayleigh theorem分布参量系统distributed parameter system 优势频率dominant frequency模态分析modal analysis固有模态natural mode of vibration同步synchronization超谐波ultraharmonic范德波尔方程van der pol equation频谱frequency spectrum基频fundamental frequencyWKB方法WKB method, Wentzel-Kramers-Brillouin method缓冲器buffer风激振动aeolian vibration嗡鸣buzz倒谱cepstrum颤动chatter蛇行hunting阻抗匹配impedance matching机械导纳mechanical admittance机械效率mechanical efficiency机械阻抗mechanical impedance随机振动stochastic vibration, random vibration隔振vibration isolation减振vibration reduction应力过冲stress overshoot喘振surge摆振shimmy起伏运动phugoid motion起伏振荡phugoid oscillation驰振galloping陀螺动力学gyrodynamics陀螺摆gyropendulum陀螺平台gyroplatform陀螺力矩gyroscoopic torque陀螺稳定器gyrostabilizer陀螺体gyrostat惯性导航inertial guidance姿态角attitude angle方位角azimuthal angle舒勒周期Schuler period机器人动力学robot dynamics多体系统multibody system 多刚体系统multi-rigid-body system机动性maneuverability凯恩方法Kane method转子[系统]动力学rotor dynamics转子[一支承一基础]系统rotor-support-foundation system静平衡static balancing动平衡dynamic balancing静不平衡static unbalance动不平衡dynamic unbalance现场平衡field balancing不平衡unbalance不平衡量unbalance互耦力cross force挠性转子flexible rotor分频进动fractional frequency precession半频进动half frequency precession油膜振荡oil whip转子临界转速rotor critical speed自动定心self-alignment亚临界转速subcritical speed涡动whirl固体力学弹性力学elasticity弹性理论theory of elasticity均匀应力状态homogeneous state of stress 应力不变量stress invariant应变不变量strain invariant应变椭球strain ellipsoid均匀应变状态homogeneous state of strain应变协调方程equation of strain compatibility拉梅常量Lame constants各向同性弹性isotropic elasticity旋转圆盘rotating circular disk楔wedge开尔文问题Kelvin problem布西内斯克问题Boussinesq problem艾里应力函数Airy stress function克罗索夫--穆斯赫利什维利法Kolosoff-Muskhelishvili method基尔霍夫假设Kirchhoff hypothesis板Plate矩形板Rectangular plate圆板Circular plate环板Annular plate 波纹板Corrugated plate加劲板Stiffened plate,reinforced Plate中厚板Plate of moderate thickness弯[曲]应力函数Stress function of bending 壳Shell扁壳Shallow shell旋转壳Revolutionary shell球壳Spherical shell[圆]柱壳Cylindrical shell锥壳Conical shell环壳Toroidal shell封闭壳Closed shell波纹壳Corrugated shell扭[转]应力函数Stress function of torsion 翘曲函数Warping function半逆解法semi-inverse method瑞利--里茨法Rayleigh-Ritz method松弛法Relaxation method莱维法Levy method松弛Relaxation量纲分析Dimensional analysis自相似[性] self-similarity影响面Influence surface接触应力Contact stress赫兹理论Hertz theory协调接触Conforming contact滑动接触Sliding contact滚动接触Rolling contact压入Indentation各向异性弹性Anisotropic elasticity颗粒材料Granular material散体力学Mechanics of granular media 热弹性Thermoelasticity超弹性Hyperelasticity粘弹性Viscoelasticity对应原理Correspondence principle褶皱Wrinkle塑性全量理论Total theory of plasticity 滑动Sliding微滑Microslip粗糙度Roughness非线性弹性Nonlinear elasticity大挠度Large deflection突弹跳变snap-through 有限变形Finite deformation格林应变Green strain阿尔曼西应变Almansi strain弹性动力学Dynamic elasticity运动方程Equation of motion准静态的Quasi-static气动弹性Aeroelasticity水弹性Hydroelasticity颤振Flutter弹性波Elastic wave简单波Simple wave柱面波Cylindrical wave水平剪切波Horizontal shear wave 竖直剪切波Vertical shear wave体波body wave无旋波Irrotational wave畸变波Distortion wave膨胀波Dilatation wave瑞利波Rayleigh wave等容波Equivoluminal wave勒夫波Love wave界面波Interfacial wave边缘效应edge effect塑性力学Plasticity可成形性Formability金属成形Metal forming耐撞性Crashworthiness结构抗撞毁性Structural crashworthiness拉拔Drawing破坏机构Collapse mechanism回弹Springback挤压Extrusion冲压Stamping穿透Perforation层裂Spalling塑性理论Theory of plasticity安定[性]理论Shake-down theory运动安定定理kinematic shake-down theorem静力安定定理Static shake-down theorem率相关理论rate dependent theorem载荷因子load factor加载准则Loading criterion加载函数Loading function 加载面Loading surface塑性加载Plastic loading塑性加载波Plastic loading wave 简单加载Simple loading比例加载Proportional loading 卸载Unloading卸载波Unloading wave冲击载荷Impulsive load阶跃载荷step load脉冲载荷pulse load极限载荷limit load中性变载nentral loading拉抻失稳instability in tension 加速度波acceleration wave本构方程constitutive equation 完全解complete solution名义应力nominal stress过应力over-stress真应力true stress等效应力equivalent stress流动应力flow stress应力间断stress discontinuity应力空间stress space主应力空间principal stress space静水应力状态hydrostatic state of stress对数应变logarithmic strain工程应变engineering strain等效应变equivalent strain应变局部化strain localization应变率strain rate应变率敏感性strain rate sensitivity应变空间strain space有限应变finite strain塑性应变增量plastic strain increment累积塑性应变accumulated plastic strain永久变形permanent deformation内变量internal variable应变软化strain-softening理想刚塑性材料rigid-perfectly plastic Material刚塑性材料rigid-plastic material理想塑性材料perfectl plastic material材料稳定性stability of material应变偏张量deviatoric tensor of strain应力偏张量deviatori tensor of stress 应变球张量spherical tensor of strain应力球张量spherical tensor of stress路径相关性path-dependency线性强化linear strain-hardening应变强化strain-hardening随动强化kinematic hardening各向同性强化isotropic hardening强化模量strain-hardening modulus幂强化power hardening塑性极限弯矩plastic limit bending Moment 塑性极限扭矩plastic limit torque弹塑性弯曲elastic-plastic bending弹塑性交界面elastic-plastic interface弹塑性扭转elastic-plastic torsion粘塑性Viscoplasticity非弹性Inelasticity理想弹塑性材料elastic-perfectly plastic Material极限分析limit analysis极限设计limit design极限面limit surface上限定理upper bound theorem上屈服点upper yield point下限定理lower bound theorem下屈服点lower yield point界限定理bound theorem初始屈服面initial yield surface后继屈服面subsequent yield surface屈服面[的]外凸性convexity of yield surface 截面形状因子shape factor of cross-section 沙堆比拟sand heap analogy屈服Yield屈服条件yield condition屈服准则yield criterion屈服函数yield function屈服面yield surface塑性势plastic potential能量吸收装置energy absorbing device能量耗散率energy absorbing device塑性动力学dynamic plasticity塑性动力屈曲dynamic plastic buckling塑性动力响应dynamic plastic response塑性波plastic wave运动容许场kinematically admissible Field 静力容许场statically admissible Field流动法则flow rule速度间断velocity discontinuity滑移线slip-lines滑移线场slip-lines field移行塑性铰travelling plastic hinge塑性增量理论incremental theory of Plasticity米泽斯屈服准则Mises yield criterion普朗特--罗伊斯关系prandtl- Reuss relation特雷斯卡屈服准则Tresca yield criterion洛德应力参数Lode stress parameter莱维--米泽斯关系Levy-Mises relation亨基应力方程Hencky stress equation赫艾--韦斯特加德应力空间Haigh-Westergaard stress space洛德应变参数Lode strain parameter德鲁克公设Drucker postulate盖林格速度方程Geiringer velocity Equation 结构力学structural mechanics结构分析structural analysis结构动力学structural dynamics拱Arch三铰拱three-hinged arch抛物线拱parabolic arch圆拱circular arch穹顶Dome空间结构space structure空间桁架space truss雪载[荷] snow load风载[荷] wind load土压力earth pressure地震载荷earthquake loading弹簧支座spring support支座位移support displacement支座沉降support settlement超静定次数degree of indeterminacy 机动分析kinematic analysis结点法method of joints截面法method of sections结点力joint forces共轭位移conjugate displacement影响线influence line三弯矩方程three-moment equation 单位虚力unit virtual force 刚度系数stiffness coefficient柔度系数flexibility coefficient力矩分配moment distribution力矩分配法moment distribution method力矩再分配moment redistribution分配系数distribution factor矩阵位移法matri displacement method单元刚度矩阵element stiffness matrix单元应变矩阵element strain matrix总体坐标global coordinates贝蒂定理Betti theorem高斯--若尔当消去法Gauss-Jordan elimination Method屈曲模态buckling mode复合材料力学mechanics of composites复合材料composite material纤维复合材料fibrous composite单向复合材料unidirectional composite泡沫复合材料foamed composite颗粒复合材料particulate composite层板Laminate夹层板sandwich panel正交层板cross-ply laminate斜交层板angle-ply laminate层片Ply多胞固体cellular solid膨胀Expansion压实Debulk劣化Degradation脱层Delamination脱粘Debond纤维应力fiber stress层应力ply stress层应变ply strain层间应力interlaminar stress比强度specific strength强度折减系数strength reduction factor 强度应力比strength -stress ratio横向剪切模量transverse shear modulus 横观各向同性transverse isotropy正交各向异Orthotropy剪滞分析shear lag analysis短纤维chopped fiber长纤维continuous fiber纤维方向fiber direction 纤维断裂fiber break纤维拔脱fiber pull-out纤维增强fiber reinforcement致密化Densification最小重量设计optimum weight design网格分析法netting analysis混合律rule of mixture失效准则failure criterion蔡--吴失效准则Tsai-W u failure criterion达格代尔模型Dugdale model断裂力学fracture mechanics概率断裂力学probabilistic fracture Mechanics格里菲思理论Griffith theory线弹性断裂力学linear elastic fracture mechanics, LEFM弹塑性断裂力学elastic-plastic fracture mecha-nics, EPFM断裂Fracture脆性断裂brittle fracture解理断裂cleavage fracture蠕变断裂creep fracture延性断裂ductile fracture晶间断裂inter-granular fracture准解理断裂quasi-cleavage fracture穿晶断裂trans-granular fracture裂纹Crack裂缝Flaw缺陷Defect割缝Slit微裂纹Microcrack折裂Kink椭圆裂纹elliptical crack深埋裂纹embedded crack[钱]币状裂纹penny-shape crack预制裂纹Precrack短裂纹short crack表面裂纹surface crack裂纹钝化crack blunting裂纹分叉crack branching裂纹闭合crack closure裂纹前缘crack front裂纹嘴crack mouth裂纹张开角crack opening angle,COA裂纹张开位移crack opening displacement, COD 裂纹阻力crack resistance裂纹面crack surface裂纹尖端crack tip裂尖张角crack tip opening angle, CTOA裂尖张开位移crack tip opening displacement, CTOD裂尖奇异场crack tip singularity Field裂纹扩展速率crack growth rate稳定裂纹扩展stable crack growth定常裂纹扩展steady crack growth亚临界裂纹扩展subcritical crack growth裂纹[扩展]减速crack retardation止裂crack arrest止裂韧度arrest toughness断裂类型fracture mode滑开型sliding mode张开型opening mode撕开型tearing mode复合型mixed mode撕裂Tearing撕裂模量tearing modulus断裂准则fracture criterionJ积分J-integralJ阻力曲线J-resistance curve断裂韧度fracture toughness应力强度因子stress intensity factorHRR场Hutchinson-Rice-Rosengren Field守恒积分conservation integral有效应力张量effective stress tensor应变能密度strain energy density能量释放率energy release rate内聚区cohesive zone塑性区plastic zone张拉区stretched zone热影响区heat affected zone, HAZ延脆转变温度brittle-ductile transition temperature剪切带shear band剪切唇shear lip无损检测non-destructive inspection双边缺口试件double edge notched specimen, DEN specimen单边缺口试件single edge notched specimen, SEN specimen三点弯曲试件three point bending specimen, TPB specimen 中心裂纹拉伸试件center cracked tension specimen, CCT specimen中心裂纹板试件center cracked panel specimen, CCP specimen紧凑拉伸试件compact tension specimen, CT specimen大范围屈服large scale yielding小范围攻屈服small scale yielding韦布尔分布Weibull distribution帕里斯公式paris formula空穴化Cavitation应力腐蚀stress corrosion概率风险判定probabilistic risk assessment, PRA损伤力学damage mechanics损伤Damage连续介质损伤力学continuum damage mechanics细观损伤力学microscopic damage mechanics累积损伤accumulated damage脆性损伤brittle damage延性损伤ductile damage宏观损伤macroscopic damage细观损伤microscopic damage微观损伤microscopic damage损伤准则damage criterion损伤演化方程damage evolution equation 损伤软化damage softening损伤强化damage strengthening损伤张量damage tensor损伤阈值damage threshold损伤变量damage variable损伤矢量damage vector损伤区damage zone疲劳Fatigue低周疲劳low cycle fatigue应力疲劳stress fatigue随机疲劳random fatigue蠕变疲劳creep fatigue腐蚀疲劳corrosion fatigue疲劳损伤fatigue damage疲劳失效fatigue failure疲劳断裂fatigue fracture疲劳裂纹fatigue crack疲劳寿命fatigue life疲劳破坏fatigue rupture 疲劳强度fatigue strength疲劳辉纹fatigue striations疲劳阈值fatigue threshold交变载荷alternating load交变应力alternating stress应力幅值stress amplitude应变疲劳strain fatigue应力循环stress cycle应力比stress ratio安全寿命safe life过载效应overloading effect循环硬化cyclic hardening循环软化cyclic softening环境效应environmental effect裂纹片crack gage裂纹扩展crack growth, crack Propagation 裂纹萌生crack initiation循环比cycle ratio实验应力分析experimental stress Analysis 工作[应变]片active[strain] gage基底材料backing material应力计stress gage零[点]飘移zero shift, zero drift应变测量strain measurement应变计strain gage应变指示器strain indicator应变花strain rosette应变灵敏度strain sensitivity机械式应变仪mechanical strain gage 直角应变花rectangular rosette引伸仪Extensometer应变遥测telemetering of strain横向灵敏系数transverse gage factor 横向灵敏度transverse sensitivity焊接式应变计weldable strain gage 平衡电桥balanced bridge粘贴式应变计bonded strain gage粘贴箔式应变计bonded foiled gage 粘贴丝式应变计bonded wire gage 桥路平衡bridge balancing电容应变计capacitance strain gage 补偿片compensation technique补偿技术compensation technique基准电桥reference bridge 电阻应变计resistance strain gage温度自补偿应变计self-temperature compensating gage半导体应变计semiconductor strain Gage集流器slip ring应变放大镜strain amplifier疲劳寿命计fatigue life gage电感应变计inductance [strain] gage光[测]力学Photomechanics光弹性Photoelasticity光塑性Photoplasticity杨氏条纹Young fringe双折射效应birefrigent effect等位移线contour of equal Displacement暗条纹dark fringe条纹倍增fringe multiplication干涉条纹interference fringe等差线Isochromatic等倾线Isoclinic等和线isopachic应力光学定律stress- optic law主应力迹线Isostatic亮条纹light fringe光程差optical path difference热光弹性photo-thermo -elasticity光弹性贴片法photoelastic coating Method光弹性夹片法photoelastic sandwich Method动态光弹性dynamic photo-elasticity空间滤波spatial filtering空间频率spatial frequency起偏镜Polarizer反射式光弹性仪reflection polariscope残余双折射效应residual birefringent Effect 应变条纹值strain fringe value应变光学灵敏度strain-optic sensitivity应力冻结效应stress freezing effect应力条纹值stress fringe value应力光图stress-optic pattern暂时双折射效应temporary birefringent Effect脉冲全息法pulsed holography透射式光弹性仪transmission polariscope实时全息干涉法real-time holographic interferometry网格法grid method 全息光弹性法holo-photoelasticity全息图Hologram全息照相Holograph全息干涉法holographic interferometry全息云纹法holographic moire technique全息术Holography全场分析法whole-field analysis散斑干涉法speckle interferometry散斑Speckle错位散斑干涉法speckle-shearing interferometry, shearography散斑图Specklegram白光散斑法white-light speckle method云纹干涉法moire interferometry[叠栅]云纹moire fringe[叠栅]云纹法moire method云纹图moire pattern离面云纹法off-plane moire method参考栅reference grating试件栅specimen grating分析栅analyzer grating面内云纹法in-plane moire method脆性涂层法brittle-coating method条带法strip coating method坐标变换transformation of Coordinates计算结构力学computational structural mechanics加权残量法weighted residual method有限差分法finite difference method有限[单]元法finite element method配点法point collocation里茨法Ritz method广义变分原理generalized variational Principle最小二乘法least square method胡[海昌]一鹫津原理Hu-Washizu principle赫林格-赖斯纳原理Hellinger-Reissner Principle修正变分原理modified variational Principle约束变分原理constrained variational Principle混合法mixed method杂交法hybrid method边界解法boundary solution method有限条法finite strip method半解析法semi-analytical method协调元conforming element 非协调元non-conforming element混合元mixed element杂交元hybrid element边界元boundary element强迫边界条件forced boundary condition 自然边界条件natural boundary condition 离散化Discretization离散系统discrete system连续问题continuous problem广义位移generalized displacement广义载荷generalized load广义应变generalized strain广义应力generalized stress界面变量interface variable节点node, nodal point[单]元Element角节点corner node边节点mid-side node内节点internal node无节点变量nodeless variable杆元bar element桁架杆元truss element梁元beam element二维元two-dimensional element一维元one-dimensional element三维元three-dimensional element轴对称元axisymmetric element板元plate element壳元shell element厚板元thick plate element三角形元triangular element四边形元quadrilateral element四面体元tetrahedral element曲线元curved element二次元quadratic element线性元linear element三次元cubic element四次元quartic element等参[数]元isoparametric element超参数元super-parametric element亚参数元sub-parametric element节点数可变元variable-number-node element拉格朗日元Lagrange element 拉格朗日族Lagrange family巧凑边点元serendipity element巧凑边点族serendipity family无限元infinite element单元分析element analysis单元特性element characteristics刚度矩阵stiffness matrix几何矩阵geometric matrix等效节点力equivalent nodal force节点位移nodal displacement节点载荷nodal load位移矢量displacement vector载荷矢量load vector质量矩阵mass matrix集总质量矩阵lumped mass matrix相容质量矩阵consistent mass matrix阻尼矩阵damping matrix瑞利阻尼Rayleigh damping刚度矩阵的组集assembly of stiffness Matrices载荷矢量的组集consistent mass matrix质量矩阵的组集assembly of mass matrices 单元的组集assembly of elements局部坐标系local coordinate system局部坐标local coordinate面积坐标area coordinates体积坐标volume coordinates曲线坐标curvilinear coordinates静凝聚static condensation合同变换contragradient transformation 形状函数shape function试探函数trial function检验函数test function权函数weight function样条函数spline function代用函数substitute function降阶积分reduced integration零能模式zero-energy modeP收敛p-convergenceH收敛h-convergence掺混插值blended interpolation等参数映射isoparametric mapping双线性插值bilinear interpolation小块检验patch test非协调模式incompatible mode 节点号node number单元号element number带宽band width带状矩阵banded matrix变带状矩阵profile matrix带宽最小化minimization of band width波前法frontal method子空间迭代法subspace iteration method 行列式搜索法determinant search method 逐步法step-by-step method纽马克法Newmark威尔逊法Wilson拟牛顿法quasi-Newton method牛顿-拉弗森法Newton-Raphson method 增量法incremental method初应变initial strain初应力initial stress切线刚度矩阵tangent stiffness matrix割线刚度矩阵secant stiffness matrix模态叠加法mode superposition method 平衡迭代equilibrium iteration子结构Substructure子结构法substructure technique超单元super-element网格生成mesh generation结构分析程序structural analysis program前处理pre-processing后处理post-processing网格细化mesh refinement应力光顺stress smoothing组合结构composite structure流体力学流体动力学fluid dynamics连续介质力学mechanics of continuous media介质medium流体质点fluid particle无粘性流体nonviscous fluid, inviscid fluid连续介质假设continuous medium hypothesis流体运动学fluid kinematics水静力学hydrostatics液体静力学hydrostatics支配方程governing equation伯努利方程Bernoulli equation 伯努利定理Bernonlli theorem毕奥-萨伐尔定律Biot-Savart law欧拉方程Euler equation亥姆霍兹定理Helmholtz theorem开尔文定理Kelvin theorem涡片vortex sheet库塔-茹可夫斯基条件Kutta-Zhoukowski condition布拉休斯解Blasius solution达朗贝尔佯廖d'Alembert paradox雷诺数Reynolds number施特鲁哈尔数Strouhal number随体导数material derivative不可压缩流体incompressible fluid质量守恒conservation of mass动量守恒conservation of momentum能量守恒conservation of energy动量方程momentum equation能量方程energy equation控制体积control volume液体静压hydrostatic pressure涡量拟能enstrophy压差differential pressure流[动] flow流线stream line流面stream surface流管stream tube迹线path, path line流场flow field流态flow regime流动参量flow parameter流量flow rate, flow discharge 涡旋vortex涡量vorticity涡丝vortex filament涡线vortex line涡面vortex surface涡层vortex layer涡环vortex ring涡对vortex pair涡管vortex tube涡街vortex street卡门涡街Karman vortex street 马蹄涡horseshoe vortex 对流涡胞convective cell卷筒涡胞roll cell涡eddy涡粘性eddy viscosity环流circulation环量circulation速度环量velocity circulation偶极子doublet, dipole驻点stagnation point总压[力] total pressure总压头total head静压头static head总焓total enthalpy能量输运energy transport速度剖面velocity profile库埃特流Couette flow单相流single phase flow单组份流single-component flow 均匀流uniform flow非均匀流nonuniform flow二维流two-dimensional flow三维流three-dimensional flow准定常流quasi-steady flow非定常流unsteady flow, non-steady flow 暂态流transient flow周期流periodic flow振荡流oscillatory flow分层流stratified flow无旋流irrotational flow有旋流rotational flow轴对称流axisymmetric flow不可压缩性incompressibility不可压缩流[动] incompressible flow浮体floating body定倾中心metacenter阻力drag, resistance减阻drag reduction表面力surface force表面张力surface tension毛细[管]作用capillarity来流incoming flow自由流free stream自由流线free stream line外流external flow 进口entrance, inlet出口exit, outlet扰动disturbance, perturbation分布distribution传播propagation色散dispersion弥散dispersion附加质量added mass ,associated mass 收缩contraction镜象法image method无量纲参数dimensionless parameter 几何相似geometric similarity运动相似kinematic similarity动力相似[性] dynamic similarity平面流plane flow势potential势流potential flow速度势velocity potential复势complex potential复速度complex velocity流函数stream function源source汇sink速度[水]头velocity head拐角流corner flow空泡流cavity flow超空泡supercavity超空泡流supercavity flow空气动力学aerodynamics低速空气动力学low-speed aerodynamics 高速空气动力学high-speed aerodynamics 气动热力学aerothermodynamics亚声速流[动] subsonic flow跨声速流[动] transonic flow超声速流[动] supersonic flow锥形流conical flow楔流wedge flow叶栅流cascade flow非平衡流[动] non-equilibrium flow细长体slender body细长度slenderness钝头体bluff body钝体blunt body翼型airfoil 翼弦chord薄翼理论thin-airfoil theory构型configuration后缘trailing edge迎角angle of attack失速stall脱体激波detached shock wave波阻wave drag诱导阻力induced drag诱导速度induced velocity临界雷诺数critical Reynolds number 前缘涡leading edge vortex附着涡bound vortex约束涡confined vortex气动中心aerodynamic center气动力aerodynamic force气动噪声aerodynamic noise气动加热aerodynamic heating离解dissociation地面效应ground effect气体动力学gas dynamics稀疏波rarefaction wave热状态方程thermal equation of state喷管Nozzle普朗特-迈耶流Prandtl-Meyer flow瑞利流Rayleigh flow可压缩流[动] compressible flow可压缩流体compressible fluid绝热流adiabatic flow非绝热流diabatic flow未扰动流undisturbed flow等熵流isentropic flow匀熵流homoentropic flow兰金-于戈尼奥条件Rankine-Hugoniot condition状态方程equation of state量热状态方程caloric equation of state完全气体perfect gas拉瓦尔喷管Laval nozzle马赫角Mach angle马赫锥Mach cone马赫线Mach line马赫数Mach number马赫波Mach wave当地马赫数local Mach number 冲击波shock wave激波shock wave正激波normal shock wave斜激波oblique shock wave头波bow wave附体激波attached shock wave 激波阵面shock front激波层shock layer压缩波compression wave反射reflection折射refraction散射scattering衍射diffraction绕射diffraction出口压力exit pressure超压[强] over pressure反压back pressure爆炸explosion爆轰detonation缓燃deflagration水动力学hydrodynamics液体动力学hydrodynamics泰勒不稳定性Taylor instability 盖斯特纳波Gerstner wave斯托克斯波Stokes wave瑞利数Rayleigh number自由面free surface波速wave speed, wave velocity 波高wave height波列wave train波群wave group波能wave energy表面波surface wave表面张力波capillary wave规则波regular wave不规则波irregular wave浅水波shallow water wave深水波deep water wave重力波gravity wave椭圆余弦波cnoidal wave潮波tidal wave涌波surge wave破碎波breaking wave船波ship wave 非线性波nonlinear wave孤立子soliton水动[力]噪声hydrodynamic noise水击water hammer空化cavitation空化数cavitation number空蚀cavitation damage超空化流supercavitating flow水翼hydrofoil水力学hydraulics洪水波flood wave涟漪ripple消能energy dissipation海洋水动力学marine hydrodynamics 谢齐公式Chezy formula欧拉数Euler number弗劳德数Froude number水力半径hydraulic radius水力坡度hvdraulic slope高度水头elevating head水头损失head loss水位water level水跃hydraulic jump含水层aquifer排水drainage排放量discharge壅水曲线back water curve压[强水]头pressure head过水断面flow cross-section明槽流open channel flow孔流orifice flow无压流free surface flow有压流pressure flow缓流subcritical flow急流supercritical flow渐变流gradually varied flow急变流rapidly varied flow临界流critical flow异重流density current, gravity flow 堰流weir flow掺气流aerated flow含沙流sediment-laden stream降水曲线dropdown curve沉积物sediment, deposit 沉[降堆]积sedimentation, deposition沉降速度settling velocity流动稳定性flow stability不稳定性instability奥尔-索末菲方程Orr-Sommerfeld equation 涡量方程vorticity equation泊肃叶流Poiseuille flow奥辛流Oseen flow剪切流shear flow粘性流[动] viscous flow层流laminar flow分离流separated flow二次流secondary flow近场流near field flow远场流far field flow滞止流stagnation flow尾流wake [flow]回流back flow反流reverse flow射流jet自由射流free jet管流pipe flow, tube flow内流internal flow拟序结构coherent structure 猝发过程bursting process表观粘度apparent viscosity 运动粘性kinematic viscosity 动力粘性dynamic viscosity泊poise厘泊centipoise厘沱centistoke剪切层shear layer次层sublayer流动分离flow separation层流分离laminar separation 湍流分离turbulent separation 分离点separation point附着点attachment point再附reattachment再层流化relaminarization起动涡starting vortex驻涡standing vortex涡旋破碎vortex breakdown 涡旋脱落vortex shedding 压[力]降pressure drop压差阻力pressure drag压力能pressure energy型阻profile drag滑移速度slip velocity无滑移条件non-slip condition壁剪应力skin friction, frictional drag壁剪切速度friction velocity磨擦损失friction loss磨擦因子friction factor耗散dissipation滞后lag相似性解similar solution局域相似local similarity气体润滑gas lubrication液体动力润滑hydrodynamic lubrication浆体slurry泰勒数Taylor number纳维-斯托克斯方程Navier-Stokes equation 牛顿流体Newtonian fluid边界层理论boundary later theory边界层方程boundary layer equation。

数学专业词汇英汉对照汇编数学英汉词汇AAbelian group 阿贝尔群abscissa axis 横轴absolute continuity 绝对连续absolute convergence 绝对收敛absolute value 绝对值abstract algebra 抽象代数addition 加法affine 仿射Aleph-zero 阿列夫零algebra topology 代数拓扑algorithm 算法almost everywhere ⼏乎处处almost surely ⼏乎必然alternative 互斥性analogy 类似analytic expression 解析式anomalous 反常的apex 顶点approximate calculation 近似计算associative law 结合律asymmetric line 渐近线axiom of choice 选择公理axis of abscissas 横坐标轴axis of imaginary 虚轴axis of real 实轴Bbase number 底数base of logarithmic function 对数函数的底base vectors 基向量basic element 基元素bijection 双射bilinear 双线性binary ⼆元的binomial ⼆项式biunivocal ⼀对⼀的bondage 约束boundary compact space 有界紧空间boundary condition 边界条件bounded continuous function有界连续函数bounded interval 有界区间brace ⼤括号Ccalculus of proposition 命题演算canonical form 标准型cardinal number of set 集的基数Cartesian product 笛卡尔积catalog ⽬录category of a space 空间的筹数causality 因果律center of compression 压缩中⼼central limit theorem 中⼼极限定理certain event 必然事件characteristic equation 特征⽅程characteristic value 特征值chart 图check procedure 检验步骤circumscribed 外切的class field 类域closure axioms 闭包公理cluster point 聚点丛点coefficient of autocorrelation ⾃相关系数coefficient of regression 回归系数cofactor of a determinant ⾏列式的余⼦式cohomology 上同调collinear 共线column matrix 列矩阵column rank 列秩common factor 公因⼦commutative law 交换律commutative law of addition 加法交换律commutative law of multiplication乘法交换律compact convex set 紧凸集complement of a set 集的余集complement law 补余律complete matrix space 完备度量空间complete orthogonal system 完全正交系complex analysis 复分析complex conjugate 复共轭complex field 复数域compound function 复合函数concave 凹的conclusion 结论condition 条件conditional of inequality 条件不等式conditional of equivalence 等价条件conditional of integrability 可积条件conditional convergence 条件收敛confidence interval 置信区间conjugate 共轭connectivity 连通性consistency principle ⼀致原则constant factor 常数因⼦construction 作图构造continuous function 连续函数contradiction ⽭盾contrary propositions 相反命题convergence almost everywhere⼏乎处处收敛convergence in measure 依测度收敛convergence rate 收敛速度convergence region 收敛区域converse proposition 逆命题convex closure 凸包coplanar 共⾯cosine law 余弦定律countable additivity 可列可加性critical point 临界点Ddecision theory 决策论degenerate quadratic form 退化⼆次型dense 稠密derivate 导数differential 微分dimensionality 维数discriminant 判别式disjoint 不相交的distributive law 分配率divergent infinite series 发散⽆穷级数double integral ⼆重积分dual operations 对偶运算Eefficiency estimation 有效估计eigenelement 本征元素elementary event 基本事件endomorphism ⾃同态envelope 包络equivalence class 等价类equivalent relation 等价关系even number 偶数existence and uniqueness 存在且唯⼀性expansion in series 级数展开exponent 指数extreme point 极值点Ffeasible solution 可⾏解finite additivity 有限可加性fraction 分数frequency 频率fundamental assumption 基本假定fuzzy 模糊game theory 对策论general remark ⼀般说明generalized derivatives ⼴义导数geometric significance ⼏何意义global convergence 全局收敛Hharmonic analysis 调和分析harmonics 调和函数homology 同调homotopy 同伦homomorphism 同态hyperbolic plane 双曲平⾯hypothesis 假设Iideal 理想identical element 单位元identity law 同⼀律illustrate 说明阐释implicit function 隐函数in like manner 同理in the large 全局的in the small 局部的inclusion of sets 集的包含关系incompatible 互斥的不相容的independence test 独⽴性检验independent variable ⾃变量induction 归纳法归纳infinite ⽆穷⼤integral calculus 积分学integral divisor 整因⼦integrate 积分interior mapping 开映射inverse of matrix 矩阵的逆irrational root ⽆理根irreducible fraction 不可约分数irreducible polynomial 不可约多项式isolated point 孤⽴点isometric 等距的iteration method 迭代法Jjoint 连接jump discontinuity 跳跃不连续性Kkernel 核knee 拐点known quantity 已知量large by comparison 远⼤于latent vector 特征向量law of association 结合律law of causality因果律law of commutation 交换律law of contradiction ⽭盾律law of distribution 分配律law of mean 中值定理leader ⾸项limited function 有界函数linear dependence 线性相关logarithm 对数lower bound 下界Mmapping space 映射空间marginal value 临界值measure 测度metric space 度量空间monotone 单调multinomial 多项式multiplication 乘法mutual correlation 互相关mutually conjugate 相互共轭的mutually disjoint 互不相交的mutually inverse 互逆的mutually prime 互素的Nnatural logarithm ⾃然对数negate 取否定negative index 负指数negate proposition 否定命题nest of intervals 区间套neutral element 零元nonhomogeneous differential equation⾮齐次微分⽅程nonlinear boundary value problem⾮线性边值问题non-vanishing vector ⾮零向量normal space 正规空间normalized form标准型n-th power n次幂numerical analysis 数值分析Oobjective function ⽬标函数oblique line 斜线odd number 奇数odd symmetry 奇对称odevity 奇偶性one degree of freedom ⼀个⾃由度onto mapping ⾃⾝映射open covering 开覆盖opposite sign 异号optimal solution 最优解optimize 最优化order of infinity ⽆穷⼤的解ordered pair 有序偶ordinary differential equation常微分⽅程ordinary solution 通常解ordinate 纵坐标oriented circle 有向圆oriented segment 有向线段orthogonal 正交orthonormal basis 标准正交基outer measure 外测度overfield 扩张域overview 概述Ppairwise orthogonal 两两正交parabolic asymptotes 渐进抛物线parabolic curve 抛物曲线parallel 平⾏线parametric equation 参数⽅程parity 奇偶性partial sum 部分和passive 被动的path curve 轨线periodicity 周期性permutation 排列perpendicular line 垂直线piecewise 分段的plus 加point of intersection 交点population 总体的positive definite quadratic form 正定⼆次型positive number 正数potential 位势power formula 乘⽅公式prime ideal 素理想prime ring 素环primitive equation 本原⽅程primitive term 原始项principal factor method 主因⼦法principal minor 主⼦式principal of the point of accumulation聚点定理prior estimate 先验估计probability curve 概率曲线probability distribution 概率分布process of iteration 迭代法proper polynomial 特征多项式proper subset 真⼦集pure imaginary 纯虚数QQED(quod erat demonstrandum) 证毕quadrant 象限quadratic ⼆次quadratic equation with one unknown⼀元⼆次⽅程quadratic root 平⽅根qualitative analysis 定性分析quantitative analysis 定量分析quarter 四分之⼀queue discipline 排队规则quotation 引⽤引证quotient group 商群Rradial deviation 径向偏差radian 弧度radical sign 根号radius of a circle 圆的半径radius of convergence 收敛半径radius of curvature 曲率半径radix point ⼩数点random distribution 随机分布random sampling distribution 随机抽样分布randomness test 随机性检验rang of distribution 分布域rang of points 点列rank of linear mapping 线性映射的秩rank of quadratic form ⼆次型的秩rarefaction 稀疏rational proper fraction 有理真分式rationalizing denominators 有理化分母raw data 原始数据real analytic function 实解析函数real axis 实轴real variable function 实变函数reasoning by analogy 类⽐推理相似推理reciprocal 倒数reciprocal automorphism 反⾃同构reciprocal ratio 反⽐rectangular coordinates 直⾓坐标recurrence formula 递推公式recurrence relations 关系recursive function 递归函数reduction of a fraction 约分reduction to a common denominator 通分reduction to absurdity 反证法reference system 参考系reflection 反射region convergence 收敛区域regression analysis 回归分析regular function 正则函数rejection region 拒绝域relation of equivalence 等价关系relative error 相对误差relative minimum 相对极⼩值repeated integral 累积分residue class 剩余类resolution 分解reverse theorem 逆定理reversible transformation 可逆变换rigorous upper bound 严格的上界rotation axis 旋转轴roundoff error 舍⼊误差Ssample average 样本均值satisfy 满⾜scalar multiplication 数乘secondary ⼆次的辅助的次级的section 截⾯截线截点sectionally smooth 分段光滑self-conjugate subgroup 正规⼦群self-evident 显然不证⾃明semi-closure 半闭sensitivity 灵敏度separability 可分性sequence 序列series 级数series expansion 级数展开series of positive terms 正项级数shaded region 阴影区域significance level 显著性⽔平significant digits 有效数字similarity isomorphic 相似同构的simply connected region 单连通区域simulated data 模拟数据simultaneous inequalities 联⽴不等式sine curve 正弦曲线singular element 奇元素退化元素skew matrices 斜对称矩阵solid figure ⽴体形spanning set ⽣成集spherical neighborhood 球形领域stability condition 稳定性条件standard deviation 标准差stationary curve 平稳曲线statistical dependence 统计相关stochastic allocation 随机分配subadditivity 次可加性subbasis ⼦基subsequence ⼦列subtotalling 求部分和successive approximation 逐次逼近法sufficient and necessary condition 充要条件supplementary set 补集surface 曲⾯surplus variable 剩余变量symbolic function 符号函数symmetric center 对称中⼼symmetry transformation 对称变换synchronism 同步synthetic proof 综合证明Ttable of random numbers 随机数表tangent line 切线tends to infinity 趋于⽆穷term by term differentiation 逐项微分terminal check 最后校验termwise integration 逐项积分totally bounded 完全有界的transcendental equation 超越⽅程transposition 转置transverse surface 横截⾯triangle computations 三⾓形解法trisection 三等分Uultimate 最后的最终的极限unbiased estimation ⽆偏估计unconditional stability ⽆条件稳定uniform boundness ⼀致有界unilateral limits 单侧极限unique solution 唯⼀解universal proposition 全称命题unordered ⽆序的unreduced 不可约的untrivial solution ⾮零解upper integral 上积分Vvalid 有效真确valuation 赋值variance ⽅差偏差vector of unit length 单位向量velocity-time graph 速度-时间图verify 检验校验versal 通⽤的vertex 极点顶vertical 垂直的vibration 震动visual proof 图像证明直观证明volume 体积Wwave form 波形weak boundary condition 弱边界条件weighted arithmetic mean 加权算术平均whole number 整数Xx-axis x轴x-component x分量x-coordinate x坐标x-direction x⽅向Yyield estimation 合格率估计Zzero correlation 零相关zero divisor 零因⼦zone of preference for acceptance 合格域zoom up 放⼤。

标量随机延迟微分方程Euler-Maruyama方法的均方稳定性分析王琦【摘要】Abstract: Mean-square stability of Euler-Maruyama method is studied for scalar stochastic delay differential equations with multiplicative noise under the condition of analytical mean-square stability. It is proven that the numerical.solution is mean-square stable when the stepsize satisfies certain restrictions. Numerical examples verify the theoretical results.%在解析解均方稳定的条件下研究带有乘性噪声的标量随机延迟微分方程Euler-Maruyama方法的均方稳定性.证明了当步长满足一定限制时,数值解是均方稳定的.数值算例验证了理论结果的正确性.【期刊名称】《广东工业大学学报》【年(卷),期】2011(028)001【总页数】4页(P50-53)【关键词】随机延迟微分方程;Euler-Maruyama方法;均方稳定性【作者】王琦【作者单位】广东工业大学,应用数学学院,广东,广州,510006【正文语种】中文【中图分类】O241.8近年来，随机延迟微分方程理论引起了人们的普遍关注，作为一种重要的数学模型，这类方程不仅考虑了随机因素的影响，同时也反映了时滞因素的影响.所以，随机延迟微分方程能够更加真实地模拟实际问题，在金融学、计算生物学以及基因工程等研究领域均有广泛的应用.本文主要考虑下述随机延迟微分方程其中τ＞0为时间延迟，系数 a1，a2，a3，a4为实常数，W(t)是Brown运动或者说是Wiener过程，ξ(t)是初始函数.关于随机延迟微分方程的全面介绍和基本结论，Mao X R［1］已经做了详尽阐释.随机延迟微分方程本身以及解析解的最新研究成果，参见文献［2-3］.由于随机延迟微分方程的解析解很难求出或者根本求不出，所以人们对这类方程的数值解产生了浓厚的兴趣.文献［4］给出了方程解析解均方稳定的条件，在此基础上研究了Euler-Maruyama方法的均方稳定性.文献［5］证明了带跳跃的随机延迟微分方程Euler方法在比线性增长条件和全局Lipschitz条件还弱的条件下数值解是收敛于解析解的理论结果.文献［6］研究了方程(1)的半隐式Euler方法的收敛性和稳定性，给出了数值方法均方稳定和广义均方稳定的条件.文献［7］讨论了随机比例延迟微分方程解的存在唯一性以及半隐式Euler方法的收敛性.关于随机延迟微分方程数值解更多的结论参见文献［8-10］.本文拟将Euler-Maruyama方法用于求解方程(1)，在解析解均方稳定的条件下，研究数值解的均方稳定性.1 解析解的均方稳定性设(Ω，F，P)是完备的概率空间，在方程(1)中，初始函数ξ(t)是F0可测的，并且满足E|ξ|2＜∞.因为方程(1)自然满足Lipschitz条件和线性增长条件，所以方程(1)的解存在且唯一［11-12］.下面的引理1将对本文主要定理的证明起到关键作用. 引理1［6］如果方程(1)的系数 a1，a2，a3，a4 满足那么方程(1)的零解是均方稳定的，即2 数值解的均方稳定性分析Maruyama［13］是最早讨论数值求解随机常微分方程收敛性问题的学者之一.他将Euler方法用于求解Ito随机常微分方程，证明了数值方法在均方意义下的收敛阶为1/2.从此，将应用于随机微分方程的 Euler方法称为 Euler-Maruyama方法［14］.将Euler-Maruyama方法应用到方程(1)，得到差分格式其中Xn是对解析解x(tn)的数值近似，并假设它是Ftn可测的，步长 h满足τ=mh，m∈Z+，tn=nh，Wiener过程的增量ΔWn=W(tn+1)－W(tn)是一系列服从高斯分布的彼此独立的随机变量.定义1 设条件(2)成立，如果存在步长h*＞0，使得对任意的h∈(0，h*)，一种数值方法应用于方程(1)所得到的数值解Xn都满足那么称这种数值方法应用于方程(1)时是均方稳定的.下面给出本文的主要结果.定理1 假设条件(2)成立，那么应用于方程(1)的Euler-Maruyama方法是均方稳定的.证明由差分方程(3)得从而，定理得证.当a3=0时，方程(1)就变成了文献［4］所讨论的方程，所以文献［4］中的主要结果可以看作是本文定理1中a3=0时的特例.3 数值算例本节将从系数和步长两方面对数值解稳定性的影响进行试验(见图1～图4).在方程(1)中，令τ =1，ξ(t)=t+1，(t∈［－1，0］).在图1 和图2 中，试验方程(1)的系数分别选为a1=－3，a2=1.5，a3=0.1，a4= －0.5 和a1=0.2，a2=2，a3=1，a4=－1，取h=1/2048.比较两图不难看出，两组系数下的数值解的稳定性有很大的差异，事实上，在图1中，试验方程的系数满足条件(2)，而在图2中，试验方程的系数却不满足这一条件.图1 数值解的稳定性Ⅰ图2 数值解的稳定性Ⅱ另一方面，在图3和图4中，试验方程(1)的系数选为a1= －8，a2=2，a3= －1，a4= －0.5，步长分别取为h=1/64和h=1/4.由(11)式计算得h^=1/8，虽然试验方程的系数满足条件(2)，但由于有1/64∈(0，1/8)和1/4∉(0，1/8)，所以根据定理1，数值解分别为稳定和不确定的(h∈(0，h^)只是充分条件，本例图4中数值解为不稳定情形)，从而验证了理论结果与数值算例的一致性.4 结论本文利用Euler-Maruyama方法求解了一类随机延迟微分方程，研究了数值方法对解析解均方稳定性的保持性质，给出了数值解均方稳定的充分条件，推广了现有结论，今后将考虑多维问题和T-稳定性问题.图3 数值解的稳定性Ⅲ图4 数值解的稳定性Ⅳ参考文献:［1］Mao X R.Stochastic differential equations and applications［M］.Second Edition.Chichester:Harwood，2007.147-234.［2］Zhou S B，Wang Z Y，Feng D.Stochastic functional differential equations with infinite delay［J］.J Math Anal Appl，2009(357):416-426. ［3］Li C X，Sun J T，Sun R Y.Stability analysis of a class of stochastic differential delay equations with nonlinear impulsive effects［J］.Journal of the Franklin Institute，2010(347):1186-1198.［4］Cao W R，Liu M Z，Fan Z C.MS-stability of the Euler-Maruyama method for stochastic differential delay equations［J］.Appl Math Comput，2004(159):127-135.［5］Li R H，Meng H B，Dai Y H.Convergence of numerical solutions to stochastic delay differential equations with jumps［J］.Appl Math Comput，2006(172):584-602.［6］Liu M Z，Cao W R，Fan Z C.Convergence and stability of the semi-implicit Euler method for a linear stochastic differential delay equation ［J］.J Comput Appl Math，2004(170):255-268.［7］Fan Z C，Liu M Z，Cao W R.Existence and uniqueness of the solutions and convergence of semi-implicit Euler methods for stochastic pantograph equations［J］.J Math Anal Appl，2007(325):1142-1159.［8］Zhang H M，Gan S Q，Hu L.The split-step backward Euler methodfor linear stochastic delay differential equations［J］.J Comput Appl Math，2009(225):558-568.［9］谭英贤，甘四清.中立型随机比例延迟微分方程平衡半隐式Euler方法的均方收敛性［J］.数学理论与应用，2009，29(4):47-51.［10］Zhao G H，Song M H，Liu M Z.Exponential stability of Euler-Maruyama solutions for impulsive stochastic differential equations with delay［J］.Appl Math Comput，2010(215):3425-3432.［11］Mohammed S E A.Stochastic functional differential equations ［M］.Boston:Pitman Advanced Publishing Program，1984.［12］Mao X R.Exponential stability of stochastic differential equations ［M］.New York:Marcel Dekker，1994.147-290.［13］Maruyama G.Continuous Markov processes and stochastic equations［J］.Rend Circolo Math Palermo，1955(4):48-90.［14］曹婉容.随机延迟微分方程几种数值方法的收敛性和稳定性［D］.哈尔滨:哈尔滨工业大学博士学位论文，2004.。

力学词汇英汉对译通类名词力学 mechanics牛顿力学 Newtonian mechanics经典力学 classical mechanics静力学 statics运动学 kinematics动力学 dynamics动理学 kinetics宏观力学 macroscopic mechanics,macromechanics细观力学 mesomechanics微观力学 microscopic mechanics,micromechanics一般力学 general mechanics固体力学 solid mechanics流体力学 fluid mechanics理论力学 theoretical mechanics应用力学 applied mechanics工程力学 engineering mechanics实验力学 experimental mechanics计算力学 computational mechanics理性力学 rational mechanics物理力学 physical mechanics地球动力学 geodynamics力 force作用点 point of action作用线 line of action力系 system of forces力系的简化 reduction of force system等效力系 equivalent force system刚体 rigid body力的可传性 transmissibility of force平行四边形定则 parallelogram rule力三角形 force triangle力多边形 force polygon零力系 null-force system平衡 equilibrium力的平衡 equilibrium of forces平衡条件 equilibrium condition平衡位置 equilibrium position平衡态 equilibrium state分析力学 analytical mechanics拉格朗日乘子 Lagrange multiplier拉格朗日[量] Lagrangian拉格朗日括号 Lagrange bracket循环坐标 cyclic coordinate循环积分 cyclic integral哈密顿[量] Hamiltonian哈密顿函数 Hamiltonian function正则方程 canonical equation正则摄动 canonical perturbation正则变换 canonical transformation正则变量 canonical variable哈密顿原理 Hamilton principle作用量积分 action integral哈密顿--雅可比方程 Hamilton-Jacobi equation作用--角度变量 action-angle variables 阿佩尔方程 Appell equation劳斯方程 Routh equation拉格朗日函数 Lagrangian function诺特定理 Noether theorem泊松括号 poisson bracket边界积分法 boundary integral method 并矢 dyad运动稳定性 stability of motion轨道稳定性 orbital stability李雅普诺夫函数 Lyapunov function渐近稳定性 asymptotic stability结构稳定性 structural stability久期不稳定性 secular instability弗洛凯定理 Floquet theorem倾覆力矩 capsizing moment自由振动 free vibration固有振动 natural vibration暂态 transient state环境振动 ambient vibration反共振 anti-resonance衰减 attenuation库仑阻尼 Coulomb damping同相分量 in-phase component非同相分量 out-of -phase component超调量 overshoot参量[激励]振动 parametric vibration 模糊振动 fuzzy vibration临界转速 critical speed of rotation阻尼器 damper半峰宽度 half-peak width集总参量系统 lumped parameter system相平面法 phase plane method相轨迹 phase trajectory等倾线法 isocline method跳跃现象 jump phenomenon负阻尼 negative damping达芬方程 Duffing equation希尔方程 Hill equationKBM方法 KBM method, Krylov-Bogoliu-bov-Mitropol'skii method 马蒂厄方程 Mathieu equation平均法 averaging method组合音调 combination tone解谐 detuning耗散函数 dissipative function硬激励 hard excitation硬弹簧 hard spring, hardening spring谐波平衡法 harmonic balance method久期项 secular term自激振动 self-excited vibration分界线 separatrix亚谐波 subharmonic软弹簧 soft spring ,softening spring软激励 soft excitation邓克利公式 Dunkerley formula瑞利定理 Rayleigh theorem分布参量系统 distributed parameter system优势频率 dominant frequency模态分析 modal analysis固有模态 natural mode of vibration同步 synchronization超谐波 ultraharmonic范德波尔方程 van der pol equation频谱 frequency spectrum基频 fundamental frequencyWKB方法 WKB method, Wentzel-Kramers-Brillouin method 缓冲器 buffer风激振动 aeolian vibration嗡鸣 buzz倒谱 cepstrum颤动 chatter蛇行 hunting阻抗匹配 impedance matching机械导纳 mechanical admittance机械效率 mechanical efficiency机械阻抗 mechanical impedance随机振动 stochastic vibration, random vibration隔振 vibration isolation减振 vibration reduction应力过冲 stress overshoot喘振 surge摆振 shimmy起伏运动 phugoid motion起伏振荡 phugoid oscillation驰振 galloping陀螺动力学 gyrodynamics陀螺摆 gyropendulum陀螺平台 gyroplatform.陀螺力矩 gyroscoopic torque陀螺稳定器 gyrostabilizer陀螺体 gyrostat惯性导航 inertial guidance姿态角 attitude angle方位角 azimuthal angle舒勒周期 Schuler period机器人动力学 robot dynamics多体系统 multibody system多刚体系统 multi-rigid-body system机动性 maneuverability凯恩方法 Kane method转子[系统]动力学 rotor dynamics转子[一支承一基础]系统 rotor-support-foundation system 静平衡 static balancing动平衡 dynamic balancing静不平衡 static unbalance动不平衡 dynamic unbalance现场平衡 field balancing不平衡 unbalance不平衡量 unbalance互耦力 cross force挠性转子 flexible rotor分频进动 fractional frequency precession半频进动 half frequency precession油膜振荡 oil whip转子临界转速 rotor critical speed自动定心 self-alignment亚临界转速 subcritical speed涡动 whirl固体力学弹性力学 elasticity弹性理论 theory of elasticity均匀应力状态 homogeneous state of stress应力不变量 stress invariant应变不变量 strain invariant应变椭球 strain ellipsoid均匀应变状态 homogeneous state of strain应变协调方程 equation of strain compatibility拉梅常量 Lame constants各向同性弹性 isotropic elasticity旋转圆盘 rotating circular disk楔 wedge开尔文问题 Kelvin problem布西内斯克问题 Boussinesq problem艾里应力函数 Airy stress function克罗索夫--穆斯赫利什维利法 Kolosoff-Muskhelishvili method 基尔霍夫假设 Kirchhoff hypothesis板 Plate矩形板 Rectangular plate圆板 Circular plate环板 Annular plate波纹板 Corrugated plate加劲板 Stiffened plate,reinforced Plate 中厚板 Plate of moderate thickness弯[曲]应力函数 Stress function of bending 壳 Shell扁壳 Shallow shell旋转壳 Revolutionary shell球壳 Spherical shell[圆]柱壳 Cylindrical shell锥壳 Conical shell环壳 Toroidal shell封闭壳 Closed shell波纹壳 Corrugated shell扭[转]应力函数 Stress function of torsion 翘曲函数 Warping function半逆解法 semi-inverse method瑞利--里茨法 Rayleigh-Ritz method松弛法 Relaxation method莱维法 Levy method松弛 Relaxation量纲分析 Dimensional analysis自相似[性] self-similarity影响面 Influence surface接触应力 Contact stress赫兹理论 Hertz theory协调接触 Conforming contact滑动接触 Sliding contact滚动接触 Rolling contact压入 Indentation各向异性弹性 Anisotropic elasticity颗粒材料 Granular material散体力学 Mechanics of granular media热弹性 Thermoelasticity超弹性 Hyperelasticity粘弹性 Viscoelasticity对应原理 Correspondence principle褶皱 Wrinkle塑性全量理论 Total theory of plasticity 滑动 Sliding微滑 Microslip粗糙度 Roughness非线性弹性 Nonlinear elasticity大挠度 Large deflection突弹跳变 snap-through有限变形 Finite deformation格林应变 Green strain阿尔曼西应变 Almansi strain弹性动力学 Dynamic elasticity运动方程 Equation of motion准静态的 Quasi-static气动弹性 Aeroelasticity水弹性 Hydroelasticity颤振 Flutter弹性波 Elastic wave简单波 Simple wave柱面波 Cylindrical wave水平剪切波 Horizontal shear wave竖直剪切波 Vertical shear wave体波 body wave无旋波 Irrotational wave畸变波 Distortion wave膨胀波 Dilatation wave瑞利波 Rayleigh wave等容波 Equivoluminal wave勒夫波 Love wave界面波 Interfacial wave边缘效应 edge effect塑性力学 Plasticity可成形性 Formability金属成形 Metal forming耐撞性 Crashworthiness结构抗撞毁性 Structural crashworthiness 拉拔 Drawing破坏机构 Collapse mechanism回弹 Springback挤压 Extrusion冲压 Stamping穿透 Perforation层裂 Spalling塑性理论 Theory of plasticity安定[性]理论 Shake-down theory运动安定定理 kinematic shake-down theorem 静力安定定理 Static shake-down theorem率相关理论 rate dependent theorem载荷因子 load factor加载准则 Loading criterion加载函数 Loading function加载面 Loading surface塑性加载 Plastic loading塑性加载波 Plastic loading wave简单加载 Simple loading比例加载 Proportional loading卸载 Unloading卸载波 Unloading wave冲击载荷 Impulsive load阶跃载荷 step load脉冲载荷 pulse load极限载荷 limit load中性变载 nentral loading拉抻失稳 instability in tension加速度波 acceleration wave本构方程 constitutive equation完全解 complete solution名义应力 nominal stress过应力 over-stress真应力 true stress等效应力 equivalent stress流动应力 flow stress应力间断 stress discontinuity应力空间 stress space主应力空间 principal stress space静水应力状态 hydrostatic state of stress对数应变 logarithmic strain工程应变 engineering strain等效应变 equivalent strain应变局部化 strain localization应变率 strain rate应变率敏感性 strain rate sensitivity应变空间 strain space有限应变 finite strain塑性应变增量 plastic strain increment累积塑性应变 accumulated plastic strain永久变形 permanent deformation内变量 internal variable应变软化 strain-softening理想刚塑性材料 rigid-perfectly plastic Material 刚塑性材料 rigid-plastic material理想塑性材料 perfectl plastic material材料稳定性 stability of material应变偏张量 deviatoric tensor of strain应力偏张量 deviatori tensor of stress应变球张量 spherical tensor of strain应力球张量 spherical tensor of stress路径相关性 path-dependency线性强化 linear strain-hardening应变强化 strain-hardening随动强化 kinematic hardening各向同性强化 isotropic hardening强化模量 strain-hardening modulus幂强化 power hardening塑性极限弯矩 plastic limit bending Moment塑性极限扭矩 plastic limit torque弹塑性弯曲 elastic-plastic bending弹塑性交界面 elastic-plastic interface弹塑性扭转 elastic-plastic torsion粘塑性 Viscoplasticity非弹性 Inelasticity理想弹塑性材料 elastic-perfectly plastic Material 极限分析 limit analysis极限设计 limit design极限面 limit surface上限定理 upper bound theorem上屈服点 upper yield point下限定理 lower bound theorem下屈服点 lower yield point界限定理 bound theorem初始屈服面 initial yield surface后继屈服面 subsequent yield surface屈服面[的]外凸性 convexity of yield surface截面形状因子 shape factor of cross-section沙堆比拟 sand heap analogy屈服 Yield屈服条件 yield condition屈服准则 yield criterion屈服函数 yield function屈服面 yield surface塑性势 plastic potential能量吸收装置 energy absorbing device能量耗散率 energy absorbing device塑性动力学 dynamic plasticity塑性动力屈曲 dynamic plastic buckling塑性动力响应 dynamic plastic response塑性波 plastic wave运动容许场 kinematically admissible Field静力容许场 statically admissible Field流动法则 flow rule速度间断 velocity discontinuity滑移线 slip-lines滑移线场 slip-lines field移行塑性铰 travelling plastic hinge塑性增量理论 incremental theory of Plasticity米泽斯屈服准则 Mises yield criterion普朗特--罗伊斯关系 prandtl- Reuss relation特雷斯卡屈服准则 Tresca yield criterion洛德应力参数 Lode stress parameter莱维--米泽斯关系 Levy-Mises relation亨基应力方程 Hencky stress equation赫艾--韦斯特加德应力空间 Haigh-Westergaard stress space 洛德应变参数 Lode strain parameter德鲁克公设 Drucker postulate盖林格速度方程 Geiringer velocity Equation结构力学 structural mechanics结构分析 structural analysis结构动力学 structural dynamics拱 Arch三铰拱 three-hinged arch抛物线拱 parabolic arch圆拱 circular arch穹顶 Dome空间结构 space structure空间桁架 space truss雪载[荷] snow load风载[荷] wind load土压力 earth pressure地震载荷 earthquake loading弹簧支座 spring support支座位移 support displacement支座沉降 support settlement超静定次数 degree of indeterminacy机动分析 kinematic analysis结点法 method of joints截面法 method of sections结点力 joint forces共轭位移 conjugate displacement影响线 influence line三弯矩方程 three-moment equation单位虚力 unit virtual force刚度系数 stiffness coefficient柔度系数 flexibility coefficient力矩分配 moment distribution力矩分配法 moment distribution method 力矩再分配 moment redistribution分配系数 distribution factor矩阵位移法 matri displacement method单元刚度矩阵 element stiffness matrix单元应变矩阵 element strain matrix总体坐标 global coordinates贝蒂定理 Betti theorem高斯--若尔当消去法 Gauss-Jordan elimination Method 屈曲模态 buckling mode复合材料力学 mechanics of composites复合材料 composite material纤维复合材料 fibrous composite单向复合材料 unidirectional composite泡沫复合材料 foamed composite颗粒复合材料 particulate composite层板 Laminate夹层板 sandwich panel正交层板 cross-ply laminate斜交层板 angle-ply laminate层片 Ply多胞固体 cellular solid膨胀 Expansion压实 Debulk劣化 Degradation脱层 Delamination脱粘 Debond纤维应力 fiber stress层应力 ply stress层应变 ply strain层间应力 interlaminar stress比强度 specific strength强度折减系数 strength reduction factor强度应力比 strength -stress ratio横向剪切模量 transverse shear modulus横观各向同性 transverse isotropy正交各向异 Orthotropy剪滞分析 shear lag analysis短纤维 chopped fiber长纤维 continuous fiber纤维方向 fiber direction纤维断裂 fiber break纤维拔脱 fiber pull-out纤维增强 fiber reinforcement致密化 Densification最小重量设计 optimum weight design网格分析法 netting analysis混合律 rule of mixture失效准则 failure criterion蔡--吴失效准则 Tsai-W u failure criterion达格代尔模型 Dugdale model断裂力学 fracture mechanics概率断裂力学 probabilistic fracture Mechanics格里菲思理论 Griffith theory线弹性断裂力学 linear elastic fracture mechanics, LEFM 弹塑性断裂力学 elastic-plastic fracture mecha-nics, EPFM 断裂 Fracture脆性断裂 brittle fracture解理断裂 cleavage fracture蠕变断裂 creep fracture延性断裂 ductile fracture晶间断裂 inter-granular fracture准解理断裂 quasi-cleavage fracture穿晶断裂 trans-granular fracture裂纹 Crack裂缝 Flaw缺陷 Defect割缝 Slit微裂纹 Microcrack折裂 Kink椭圆裂纹 elliptical crack深埋裂纹 embedded crack[钱]币状裂纹 penny-shape crack预制裂纹 Precrack短裂纹 short crack表面裂纹 surface crack裂纹钝化 crack blunting裂纹分叉 crack branching裂纹闭合 crack closure裂纹前缘 crack front裂纹嘴 crack mouth裂纹张开角 crack opening angle,COA裂纹张开位移 crack opening displacement, COD裂纹阻力 crack resistance裂纹面 crack surface裂纹尖端 crack tip裂尖张角 crack tip opening angle, CTOA裂尖张开位移 crack tip opening displacement, CTOD裂尖奇异场 crack tip singularity Field 裂纹扩展速率 crack growth rate稳定裂纹扩展 stable crack growth定常裂纹扩展 steady crack growth亚临界裂纹扩展 subcritical crack growth 裂纹[扩展]减速 crack retardation止裂 crack arrest止裂韧度 arrest toughness断裂类型 fracture mode滑开型 sliding mode张开型 opening mode撕开型 tearing mode复合型 mixed mode撕裂 Tearing撕裂模量 tearing modulus断裂准则 fracture criterionJ积分 J-integralJ阻力曲线 J-resistance curve断裂韧度 fracture toughness应力强度因子 stress intensity factor HRR场 Hutchinson-Rice-Rosengren Field 守恒积分 conservation integral有效应力张量 effective stress tensor应变能密度 strain energy density能量释放率 energy release rate内聚区 cohesive zone塑性区 plastic zone张拉区 stretched zone热影响区 heat affected zone, HAZ延脆转变温度 brittle-ductile transition temperature剪切带 shear band剪切唇 shear lip无损检测 non-destructive inspection双边缺口试件 double edge notched specimen, DEN specimen单边缺口试件 single edge notched specimen, SEN specimen三点弯曲试件 three point bending specimen, TPB specimen中心裂纹拉伸试件 center cracked tension specimen, CCT specimen 中心裂纹板试件 center cracked panel specimen, CCP specimen紧凑拉伸试件 compact tension specimen, CT specimen大范围屈服 large scale yielding小范围攻屈服 small scale yielding韦布尔分布 Weibull distribution帕里斯公式 paris formula空穴化 Cavitation应力腐蚀 stress corrosion概率风险判定 probabilistic risk assessment, PRA损伤力学 damage mechanics损伤 Damage连续介质损伤力学 continuum damage mechanics细观损伤力学 microscopic damage mechanics累积损伤 accumulated damage脆性损伤 brittle damage延性损伤 ductile damage宏观损伤 macroscopic damage细观损伤 microscopic damage微观损伤 microscopic damage损伤准则 damage criterion损伤演化方程 damage evolution equation 损伤软化 damage softening损伤强化 damage strengthening损伤张量 damage tensor损伤阈值 damage threshold损伤变量 damage variable损伤矢量 damage vector损伤区 damage zone疲劳 Fatigue低周疲劳 low cycle fatigue应力疲劳 stress fatigue随机疲劳 random fatigue蠕变疲劳 creep fatigue腐蚀疲劳 corrosion fatigue疲劳损伤 fatigue damage疲劳失效 fatigue failure疲劳断裂 fatigue fracture疲劳裂纹 fatigue crack疲劳寿命 fatigue life疲劳破坏 fatigue rupture疲劳强度 fatigue strength疲劳辉纹 fatigue striations疲劳阈值 fatigue threshold交变载荷 alternating load交变应力 alternating stress应力幅值 stress amplitude应变疲劳 strain fatigue应力循环 stress cycle应力比 stress ratio安全寿命 safe life过载效应 overloading effect循环硬化 cyclic hardening循环软化 cyclic softening环境效应 environmental effect裂纹片 crack gage裂纹扩展 crack growth, crack Propagation 裂纹萌生 crack initiation循环比 cycle ratio实验应力分析 experimental stress Analysis 工作[应变]片 active[strain] gage基底材料 backing material应力计 stress gage零[点]飘移 zero shift, zero drift应变测量 strain measurement应变计 strain gage应变指示器 strain indicator应变花 strain rosette应变灵敏度 strain sensitivity机械式应变仪 mechanical strain gage直角应变花 rectangular rosette引伸仪 Extensometer应变遥测 telemetering of strain横向灵敏系数 transverse gage factor横向灵敏度 transverse sensitivity焊接式应变计 weldable strain gage平衡电桥 balanced bridge粘贴式应变计 bonded strain gage粘贴箔式应变计 bonded foiled gage粘贴丝式应变计 bonded wire gage桥路平衡 bridge balancing电容应变计 capacitance strain gage补偿片 compensation technique补偿技术 compensation technique基准电桥 reference bridge电阻应变计 resistance strain gage温度自补偿应变计 self-temperature compensating gage 半导体应变计 semiconductor strain Gage集流器 slip ring应变放大镜 strain amplifier疲劳寿命计 fatigue life gage电感应变计 inductance [strain] gage光[测]力学 Photomechanics光弹性 Photoelasticity光塑性 Photoplasticity杨氏条纹 Young fringe双折射效应 birefrigent effect等位移线 contour of equal Displacement暗条纹 dark fringe条纹倍增 fringe multiplication干涉条纹 interference fringe等差线 Isochromatic等倾线 Isoclinic等和线 isopachic应力光学定律 stress- optic law主应力迹线 Isostatic亮条纹 light fringe光程差 optical path difference热光弹性 photo-thermo -elasticity光弹性贴片法 photoelastic coating Method光弹性夹片法 photoelastic sandwich Method动态光弹性 dynamic photo-elasticity空间滤波 spatial filtering空间频率 spatial frequency起偏镜 Polarizer反射式光弹性仪 reflection polariscope残余双折射效应 residual birefringent Effect应变条纹值 strain fringe value应变光学灵敏度 strain-optic sensitivity应力冻结效应 stress freezing effect应力条纹值 stress fringe value应力光图 stress-optic pattern暂时双折射效应 temporary birefringent Effect脉冲全息法 pulsed holography透射式光弹性仪 transmission polariscope实时全息干涉法 real-time holographic interferometry 网格法 grid method全息光弹性法 holo-photoelasticity全息图 Hologram全息照相 Holograph全息干涉法 holographic interferometry全息云纹法 holographic moire technique全息术 Holography全场分析法 whole-field analysis散斑干涉法 speckle interferometry散斑 Speckle错位散斑干涉法 speckle-shearing interferometry, shearography 散斑图 Specklegram白光散斑法 white-light speckle method云纹干涉法 moire interferometry[叠栅]云纹 moire fringe[叠栅]云纹法 moire method云纹图 moire pattern离面云纹法 off-plane moire method参考栅 reference grating试件栅 specimen grating分析栅 analyzer grating面内云纹法 in-plane moire method脆性涂层法 brittle-coating method条带法 strip coating method坐标变换 transformation of Coordinates计算结构力学 computational structural mechanics加权残量法 weighted residual method有限差分法 finite difference method有限[单]元法 finite element method配点法 point collocation里茨法 Ritz method广义变分原理 generalized variational Principle最小二乘法 least square method胡[海昌]一鹫津原理 Hu-Washizu principle赫林格-赖斯纳原理 Hellinger-Reissner Principle 修正变分原理 modified variational Principle约束变分原理 constrained variational Principle 混合法 mixed method杂交法 hybrid method边界解法 boundary solution method有限条法 finite strip method半解析法 semi-analytical method协调元 conforming element非协调元 non-conforming element混合元 mixed element杂交元 hybrid element边界元 boundary element强迫边界条件 forced boundary condition自然边界条件 natural boundary condition离散化 Discretization离散系统 discrete system连续问题 continuous problem广义位移 generalized displacement广义载荷 generalized load广义应变 generalized strain广义应力 generalized stress界面变量 interface variable节点 node, nodal point[单]元 Element角节点 corner node边节点 mid-side node内节点 internal node无节点变量 nodeless variable杆元 bar element桁架杆元 truss element梁元 beam element二维元 two-dimensional element一维元 one-dimensional element三维元 three-dimensional element轴对称元 axisymmetric element板元 plate element壳元 shell element厚板元 thick plate element三角形元 triangular element四边形元 quadrilateral element四面体元 tetrahedral element曲线元 curved element二次元 quadratic element线性元 linear element三次元 cubic element四次元 quartic element等参[数]元 isoparametric element超参数元 super-parametric element亚参数元 sub-parametric element节点数可变元 variable-number-node element 拉格朗日元 Lagrange element拉格朗日族 Lagrange family巧凑边点元 serendipity element巧凑边点族 serendipity family无限元 infinite element单元分析 element analysis单元特性 element characteristics刚度矩阵 stiffness matrix几何矩阵 geometric matrix等效节点力 equivalent nodal force节点位移 nodal displacement节点载荷 nodal load位移矢量 displacement vector载荷矢量 load vector质量矩阵 mass matrix集总质量矩阵 lumped mass matrix相容质量矩阵 consistent mass matrix阻尼矩阵 damping matrix瑞利阻尼 Rayleigh damping刚度矩阵的组集 assembly of stiffness Matrices 载荷矢量的组集 consistent mass matrix质量矩阵的组集 assembly of mass matrices单元的组集 assembly of elements局部坐标系 local coordinate system局部坐标 local coordinate面积坐标 area coordinates体积坐标 volume coordinates曲线坐标 curvilinear coordinates静凝聚 static condensation合同变换 contragradient transformation形状函数 shape function试探函数 trial function检验函数 test function权函数 weight function样条函数 spline function代用函数 substitute function降阶积分 reduced integration零能模式 zero-energy modeP收敛 p-convergenceH收敛 h-convergence掺混插值 blended interpolation等参数映射 isoparametric mapping双线性插值 bilinear interpolation小块检验 patch test非协调模式 incompatible mode节点号 node number单元号 element number带宽 band width带状矩阵 banded matrix变带状矩阵 profile matrix带宽最小化 minimization of band width 波前法 frontal method子空间迭代法 subspace iteration method 行列式搜索法 determinant search method 逐步法 step-by-step method纽马克法 Newmark威尔逊法 Wilson拟牛顿法 quasi-Newton method牛顿-拉弗森法 Newton-Raphson method增量法 incremental method初应变 initial strain初应力 initial stress切线刚度矩阵 tangent stiffness matrix割线刚度矩阵 secant stiffness matrix模态叠加法 mode superposition method平衡迭代 equilibrium iteration子结构 Substructure子结构法 substructure technique超单元 super-element网格生成 mesh generation结构分析程序 structural analysis program前处理 pre-processing后处理 post-processing网格细化 mesh refinement应力光顺 stress smoothing组合结构 composite structure流体力学流体动力学 fluid dynamics连续介质力学 mechanics of continuous media 介质 medium流体质点 fluid particle无粘性流体 nonviscous fluid, inviscid fluid 连续介质假设 continuous medium hypothesis 流体运动学 fluid kinematics水静力学 hydrostatics液体静力学 hydrostatics支配方程 governing equation伯努利方程 Bernoulli equation伯努利定理 Bernonlli theorem毕奥-萨伐尔定律 Biot-Savart law欧拉方程 Euler equation亥姆霍兹定理 Helmholtz theorem开尔文定理 Kelvin theorem涡片 vortex sheet库塔-茹可夫斯基条件 Kutta-Zhoukowski condition 布拉休斯解 Blasius solution达朗贝尔佯廖 d'Alembert paradox雷诺数 Reynolds number施特鲁哈尔数 Strouhal number随体导数 material derivative不可压缩流体 incompressible fluid质量守恒 conservation of mass动量守恒 conservation of momentum能量守恒 conservation of energy动量方程 momentum equation能量方程 energy equation控制体积 control volume液体静压 hydrostatic pressure涡量拟能 enstrophy压差 differential pressure流[动] flow流线 stream line流面 stream surface流管 stream tube迹线 path, path line流场 flow field流态 flow regime流动参量 flow parameter流量 flow rate, flow discharge 涡旋 vortex涡量 vorticity涡丝 vortex filament涡线 vortex line涡面 vortex surface涡层 vortex layer涡环 vortex ring涡对 vortex pair涡管 vortex tube涡街 vortex street卡门涡街 Karman vortex street 马蹄涡 horseshoe vortex对流涡胞 convective cell卷筒涡胞 roll cell涡 eddy涡粘性 eddy viscosity环流 circulation环量 circulation速度环量 velocity circulation 偶极子 doublet, dipole驻点 stagnation point总压[力] total pressure总压头 total head静压头 static head总焓 total enthalpy能量输运 energy transport速度剖面 velocity profile库埃特流 Couette flow单相流 single phase flow单组份流 single-component flow均匀流 uniform. flow非均匀流 nonuniform. flow二维流 two-dimensional flow三维流 three-dimensional flow准定常流 quasi-steady flow非定常流 unsteady flow, non-steady flow 暂态流 transient flow周期流 periodic flow振荡流 oscillatory flow分层流 stratified flow无旋流 irrotational flow有旋流 rotational flow轴对称流 axisymmetric flow不可压缩性 incompressibility不可压缩流[动] incompressible flow浮体 floating body定倾中心 metacenter阻力 drag, resistance减阻 drag reduction表面力 surface force表面张力 surface tension毛细[管]作用 capillarity来流 incoming flow自由流 free stream自由流线 free stream line外流 external flow进口 entrance, inlet出口 exit, outlet扰动 disturbance, perturbation分布 distribution传播 propagation色散 dispersion弥散 dispersion附加质量 added mass ,associated mass 收缩 contraction镜象法 image method无量纲参数 dimensionless parameter 几何相似 geometric similarity运动相似 kinematic similarity动力相似[性] dynamic similarity平面流 plane flow势 potential势流 potential flow速度势 velocity potential复势 complex potential复速度 complex velocity流函数 stream function源 source汇 sink速度[水]头 velocity head拐角流 corner flow空泡流 cavity flow超空泡 supercavity超空泡流 supercavity flow空气动力学 aerodynamics低速空气动力学 low-speed aerodynamics 高速空气动力学 high-speed aerodynamics 气动热力学 aerothermodynamics亚声速流[动] subsonic flow跨声速流[动] transonic flow超声速流[动] supersonic flow锥形流 conical flow楔流 wedge flow叶栅流 cascade flow非平衡流[动] non-equilibrium flow细长体 slender body细长度 slenderness钝头体 bluff body钝体 blunt body翼型 airfoil翼弦 chord薄翼理论 thin-airfoil theory构型 configuration后缘 trailing edge迎角 angle of attack失速 stall脱体激波 detached shock wave波阻 wave drag诱导阻力 induced drag诱导速度 induced velocity临界雷诺数 critical Reynolds number前缘涡 leading edge vortex附着涡 bound vortex约束涡 confined vortex综合类：广义连续统力学 generalized continuum mechanics简单物质 simple material纯力学物质 purely mechanical material微分型物质 material of differential type积分型物质 material of integral type混合物组份 constituents of a mixture非协调理论 incompatibility theory微极理论 micropolar theory决定性原理 principle of determinism等存在原理 principle of equipresence局部作用原理 principle of objectivity客观性原理 principle of objectivity电磁连续统理论 theory of electromagnetic continuum 内时理论 endochronic theory非局部理论 nonlocal theory混合物理论 theory of mixtures里夫林-矣里克森张量 Rivlin-Ericksen tensor声张量 acoustic tensor半向同性张量 hemitropic tensor各向同性张量 isotropic tensor应变张量 strain tensor伸缩张量 stretch tensor连续旋错 continuous dislination连续位错 continuous dislocation动量矩平衡 angular momentum balance余本构关系 complementary constitutive relations共旋导数 co-rotational derivative, Jaumann derivative 非完整分量 anholonomic component爬升效应 climbing effect协调条件 compatibility condition错综度 complexity当时构形 current configuration能量平衡 energy balance变形梯度 deformation gradient有限弹性 finite elasticity熵增 entropy production标架无差异性 frame. indifference弹性势 elastic potential熵不等式 entropy inequality极分解 polar decomposition低弹性 hypoelasticity参考构形 reference configuration响应泛函 response functional动量平衡 momentum balance奇异面 singular surface贮能函数 stored-energy function内部约束 internal constraint物理分量 physical components本原元 primitive element普适变形 universal deformation速度梯度 velocity gradient测粘流动 viscometric flow当地导数 local derivative岩石力学 rock mechanics原始岩体应力 virgin rock stress构造应力 tectonic stress三轴压缩试验 three-axial compression test 三轴拉伸试验 three-axial tensile test三轴试验 triaxial test岩层静态应力 lithostatic stress吕荣 lugeon地压强 geostatic pressure水力劈裂 hydraulic fracture咬合[作用] interlocking内禀抗剪强度 intrinsic shear strength循环抗剪强度 cyclic shear strength残余抗剪强度 residual shear strength土力学 soil mechanics孔隙比 void ratio内磨擦角 angle of internal friction休止角 angle of repose孔隙率 porosity围压 ambient pressure渗透系数 coefficient of permeability [抗]剪切角 angle of shear resistance渗流力 seepage force表观粘聚力 apparent cohesion粘聚力 cohesion稠度 consistency固结 consolidation主固结 primary consolidation次固结 secondary consolidation固结仪 consolidometer浮升力 uplift扩容 dilatancy有效应力 effective stress絮凝[作用] flocculation主动土压力 active earth pressure 被动土压力 passive earth pressure 土动力学 soil dynamics应力解除 stress relief次时间效应 secondary time effect 贯入阻力 penetration resistance 沙土液化 liquefaction of sand泥流 mud flow多相流 multiphase flow马格努斯效应 Magnus effect韦伯数 Weber number环状流 annular flow泡状流 bubble flow层状流 stratified flow平衡流 equilibrium flow二组份流 two-component flow冻结流 frozen flow均质流 homogeneous flow二相流 two-phase flow。

第29卷增2岩石力学与工程学报V ol.29 Supp.2 2010年9月Chinese Journal of Rock Mechanics and Engineering Sept.，2010 一种三维均质土坡滑动面搜索的新方法邓东平，李亮，赵炼恒(中南大学土木建筑学院，湖南长沙 410075)摘要：通过提出一种新的随机方法生成2条随机曲线：一条为三维滑体滑动方向上与坡面线围成断面尺寸最大的随机母线；另一条为垂直滑动方向上与坡面线围成断面尺寸最大的随机准线。

三维滑动面是由准线在母线上移动或母线在准线上移动形成的，这时，形成的三维滑动面容易将滑体划分为n×m个铅垂条块，进而能够方便地实现三维安全系数公式中的土块微分量计算以及三维滑动面图形的生成。

在此过程中，针对随机搜索精度不高的问题，通过将随机母线和随机准线用近似母线和近似准线来代替进行改进，通过搜索由近似母线和近似准线形成的任意三维滑动面来找到最危险滑动面。

三维均质土坡算例表明：用本文方法计算的安全系数比其他方法计算的小，因而偏于安全，证明了所提方法的有效性。

滑体长度对土坡稳定性影响的研究表明：三维边坡的稳定性高于二维边坡，随着滑体长度的增加，三维稳定性趋于二维稳定性；从准线形状分析结果以及母线与二维滑动面的对比分析证明，对于无限长三维均质土坡，其滑动面简化为二维圆弧是可行性的。

关键词：边坡工程；三维均质土坡；母线；准线；滑动面；稳定性分析；安全系数中图分类号：P 642.22 文献标识码：A 文章编号：1000–6915(2010)增2–3719–09A NEW METHOD FOR SEARCHING SLIDING SURFACE OFTHREE-DIMENSIONAL HOMOGENEOUS SOIL SLOPEDENG Dongping，LI Liang，ZHAO Lianheng(College of Civil and Architectural Engineering，Central South University，Changsha，Hunan410075，China)Abstract：Two random curves are generated through a stochastic method：One is mother line，by which and slope line cross-section size on sliding direction of three-dimensional sliding body surrounded is the largest；and the other is standard line，by which and slope line cross-section size on perpendicular sliding direction surrounded is the largest. The three-dimensional sliding surface is generated through standard line moving on the mother line or mother line moving on the standard line. So sliding body is easily divided into n×m vertical stripes in the generation process of three-dimensional sliding surface，and it facilitates calculation of micro-components of stripes in formula of three-dimensional factor of safety and to generate graphic of three-dimensional sliding surface. In this process，in order to improve the accuracy of random search，the approximate mother line and approximate standard line are used instead of random mother line and random standard line respectively. Then the most dangerous sliding surface is found by searching arbitrary three-dimensional sliding surface generated by approximate mother line and approximate standard line. Examples of three-dimensional homogeneous slopes show that results computed by this method are smaller than those of previous studies，so it is safer and also proved that this method is practicable. At the same time，by studying the impact of length of sliding body on stability of slope，it shows that stability of three-dimensional slope is better than that of two-dimensional slope；with increasing收稿日期：2009–11–04；修回日期：2010–01–18基金项目：西部交通建设科技项目(2006318802111)；中南大学优秀博士学位论文扶植项目(2008yb004)；铁道部科技研究开发计划重点资助项目(2008G032–3)；湖南省交通厅科技项目(2007–29)作者简介：邓东平(1985–)，男，2007年毕业于中南大学土木工程专业，现为硕士研究生，主要从事道路与铁道工程方面的研究工作。