Ch7 Similitude and Dimensional Analysis

浅谈卢卡奇的物化理论思想
卢卡奇（Luca Pacioli），意大利数学家兼修道士，被称为现代会计学的
开创者，他最有名的著作是《比稀数学》（Summa de arithmetica），被认为是现代会计学的始祖。

他的物化理论思想关乎着理解事物的本质，他认为，物质都有可以统计的幅度，通过分析其可量化的幅度可以更深入地理解总体结构。

卢卡奇物化理论认为，物质的表面特征可以通过静态和动态分析来全面理解，
动态分析可以揭示实体间原子间运动关系，而静态分析可以深入到实体间更细微的关系，而这一理论强调以知识和技能结合来理解深层物质的本质，进而研究物体的总体结构。

卢卡奇的物化理论对于现代科学的发展有重要意义。

它强调以综合的角度理解
物质的本质，从而使科学发展走上了一个更加精确和有效的道路，而它们也是现代科学发展的重要基础。

此外，卢卡奇的物化理论在高等教育中也受到了高度重视。

此理论的发展规范
了高等教育体系的思想观念。

高校在开展教学和研究时，都会把卢卡奇的物化学理论作为其重要的参考标准，从而使学术研究有所改进。

总之，卢卡奇的物化理论提供了科学发展和高等教育体系思想观念方面的独特
贡献，深深影响着今日数学、计算机和教育研究，成为重要的参考标准和指导思想。

organic chemistry john mcmurry 7h edition解答Organic Chemistry John McMurry 7th Edition解答Organic chemistry is the branch of chemistry that deals with the structure, properties, composition, reactions, and synthesis of organic compounds. It plays a vital role in various scientific disciplines, including biology, medicine, pharmacy, and materials science. In order to understand the fundamental principles and concepts of organic chemistry, many students and professionals refer to the textbook "Organic Chemistry" by John McMurry, specifically the 7th edition. This article aims to provide a comprehensive overview of the key topics covered in this edition.1. Introduction to Organic ChemistryThe 7th edition of John McMurry's "Organic Chemistry" begins with an introduction to the basic principles and fundamental concepts of organic chemistry. It covers topics such as the structure and bonding in organic compounds, the classification of organic compounds, and the various types of chemical reactions that occur in organic compounds. This section provides a solid foundation for understanding the subsequent chapters in the book.2. Structure and Properties of Organic CompoundsOne of the main focuses of organic chemistry is the study of the structure and properties of organic compounds. McMurry's 7th edition explores the different types of organic compounds, including hydrocarbons,alcohols, aldehydes, ketones, carboxylic acids, and esters. It explains the methods for determining the structure of organic compounds, such as spectroscopy and molecular modeling. The book also discusses the physical and chemical properties of organic compounds, including their boiling points, melting points, solubility, and reactivity.3. Organic Reactions and MechanismsUnderstanding organic reactions and their mechanisms is crucial for mastering organic chemistry. The 7th edition of McMurry's book comprehensively covers various types of organic reactions, including substitution, elimination, addition, and rearrangement reactions. It explains the mechanisms behind these reactions and provides numerous examples and practice problems to enhance the reader's understanding. Additionally, the book discusses important topics such as stereochemistry, reactive intermediates, and catalysis.4. Organic SynthesisSynthesis is a key aspect of organic chemistry, as it involves the preparation of new organic compounds from simpler starting materials. McMurry's textbook devotes a significant portion of the 7th edition to the topic of organic synthesis. It covers topics such as retrosynthetic analysis, reaction planning, and the use of protective groups. The book also provides numerous examples of multi-step synthesis and highlights the importance of yield, purity, and selectivity in organic synthesis.5. Spectroscopy and SpectrometryThe 7th edition of "Organic Chemistry" also includes a comprehensive section on spectroscopy and spectrometry. It introduces various spectroscopic techniques, such as infrared spectroscopy (IR), nuclear magnetic resonance spectroscopy (NMR), mass spectrometry (MS), and ultraviolet-visible spectroscopy (UV-Vis). The book explains the principles behind these techniques and provides guidance on interpreting spectroscopic data to determine the structure of organic compounds.6. Organic Chemistry in Biology and MedicineOrganic chemistry plays a critical role in understanding the chemical processes that occur in living organisms and the development of pharmaceutical drugs. McMurry's textbook addresses the applications of organic chemistry in biology and medicine, exploring topics such as bioorganic chemistry, natural product synthesis, drug design, and drug action. It highlights the importance of organic chemistry in the fields of biochemistry, pharmacology, and medicinal chemistry.In conclusion, the 7th edition of John McMurry's "Organic Chemistry" provides a comprehensive and in-depth exploration of the fundamental principles, concepts, and applications of organic chemistry. It covers topics ranging from the structure and properties of organic compounds to organic reactions, synthesis, spectroscopy, and the role of organic chemistry in biology and medicine. This textbook serves as an invaluable resource for students, professionals, and researchers seeking to enhance their knowledge and understanding of organic chemistry.。

范希尔理论的几何思维水平研究综述范希尔理论是一种用来解释自然界各种物理现象的有效模型，它应用于几何思维水平方面有显著的价值，其研究实质上是对自然界的几何思维活动的深入解析和总结。

本文以《范希尔理论的几何思维水平研究综述》为标题，在前人的工作基础上，从范希尔理论的几何思维水平研究的三个主要领域（几何结构、空间观念和论据驱动思维）展开研究，深入剖析范希尔理论对几何思维水平研究的影响，以及具体表现出来的一些特点，并指出其存在的一些不足，从而为研究者以及未来在几何思维水平领域开展更深层次研究提供参考。

范希尔理论最初由威廉范希尔在20世纪30年代提出，被认为是20世纪最重要的物理理论之一。

范希尔理论的几何思维水平的研究着重于研究自然界中的几何结构对一些物理现象的影响，其研究可以分为三大类：几何结构，空间观念和论据驱动思维。

第一，几何结构部分主要是在运用范希尔理论来分析物理系统的空间结构，研究范希尔理论如何建立数学模型来描述物理现象，从而推导出空间结构的特征，以及其与物理现象之间的关系。

比如，研究者发现，宇宙中的物体行星等，只有在空间结构的质心关系和轨道非线性调节的情况下，才可能实现运动;而当物体在重力场中处于某些特定的空间结构位置时，力学角动量守恒定律就会得到很好的应用。

第二，空间观念方面，范希尔理论着重于提出物体在空间中的运动、运动特性以及精确测量物体的行动所需的空间参考系，而这些参考系又与物体的特性有关，它们有助于更准确地描述物体的物理运动现象，从而帮助物理学家更准确地预测物理现象。

第三，论据驱动思维。

范希尔理论中强调，在推理过程中，要有规律、全面、准确地把握物体结构，需要考虑两个因素：一是有效证据，二是准确地从证据中得出结论。

结合以上三种观点，范希尔理论可以帮助理解物体的运动规律、推理其变化的规律，从而更好地控制物体的变化状态，使物体的运动更加有序和精确。

范希尔理论对于几何思维水平的研究有所帮助，但也存在一些不足。

LEOPOLD-FRANZENS UNIVERSITYChair of Engineering Mechanicso.Univ.-Prof.Dr.-Ing.habil.G.I.Schu¨e ller,Ph.D.G.I.Schueller@uibk.ac.at Technikerstrasse13,A-6020Innsbruck,Austria,EU Tel.:+435125076841Fax.:+435125072905 mechanik@uibk.ac.at,http://mechanik.uibk.ac.atIfM-Publication2-407G.I.Schu¨e ller.Developments in stochastic structural mechanics.Archive of Applied Mechanics,published online,2006.Archive of Applied Mechanics manuscript No.(will be inserted by the editor)Developments in Stochastic Structural MechanicsG.I.Schu¨e llerInstitute of Engineering Mechanics,Leopold-Franzens University,Innsbruck,Aus-tria,EUReceived:date/Revised version:dateAbstract Uncertainties are a central element in structural analysis and design.But even today they are frequently dealt with in an intuitive or qualitative way only.However,as already suggested80years ago,these uncertainties may be quantified by statistical and stochastic procedures. In this contribution it is attempted to shed light on some of the recent advances in the now establishedfield of stochastic structural mechanics and also solicit ideas on possible future developments.1IntroductionThe realistic modeling of structures and the expected loading conditions as well as the mechanisms of their possible deterioration with time are un-doubtedly one of the major goals of structural and engineering mechanics2G.I.Schu¨e ller respectively.It has been recognized that this should also include the quan-titative consideration of the statistical uncertainties of the models and the parameters involved[56].There is also a general agreement that probabilis-tic methods should be strongly rooted in the basic theories of structural en-gineering and engineering mechanics and hence represent the natural next step in the development of thesefields.It is well known that modern methods leading to a quantification of un-certainties of stochastic systems require computational procedures.The de-velopment of these procedures goes in line with the computational methods in current traditional(deterministic)analysis for the solution of problems required by the engineering practice,where certainly computational pro-cedures dominate.Hence,their further development within computational stochastic structural analysis is a most important requirement for dissemi-nation of stochastic concepts into engineering practice.Most naturally,pro-cedures to deal with stochastic systems are computationally considerably more involved than their deterministic counterparts,because the parameter set assumes a(finite or infinite)number of values in contrast to a single point in the parameter space.Hence,in order to be competitive and tractable in practical applications,the computational efficiency of procedures utilized is a crucial issue.Its significance should not be underestimated.Improvements on efficiency can be attributed to two main factors,i.e.by improved hard-ware in terms of ever faster computers and improved software,which means to improve the efficiency of computational algorithms,which also includesDevelopments in Stochastic Structural Mechanics3 utilizing parallel processing and computer farming respectively.For a con-tinuous increase of their efficiency by software developments,computational procedure of stochastic analysis should follow a similar way as it was gone in the seventieth and eighties developing the deterministic FE approach. One important aspect in this fast development was the focus on numerical methods adjusted to the strength and weakness of numerical computational algorithms.In other words,traditional ways of structural analysis devel-oped before the computer age have been dropped,redesigned and adjusted respectively to meet the new requirements posed by the computational fa-cilities.Two main streams of computational procedures in Stochastic Structural Analysis can be observed.Thefirst of this main class is the generation of sample functions by Monte Carlo simulation(MCS).These procedures might be categorized further according to their purpose:–Realizations of prescribed statistical information:samples must be com-patible with prescribed stochastic information such as spectral density, correlation,distribution,etc.,applications are:(1)Unconditional simula-tion of stochastic processes,fields and waves.(2)Conditional simulation compatible with observations and a priori statistical information.–Assessment of the stochastic response for a mathematical model with prescribed statistics(random loading/system parameters)of the param-eters,applications are:(1)Representative sample for the estimation of the overall distribution.4G.I.Schu¨e ller Indiscriminate(blind)generation of samples.Numerical integration of SDE’s.(2)Representative sample for the reliability assessment by gen-erating adverse rare events with positive probability,i.e.by:(a)variance reduction techniques controlling the realizations of RV’s,(b)controlling the evolution in time of sampling functions.The other main class provides numerical solutions to analytical proce-dures.Grouping again according to the respective purpose the following classification can be made:Numerical solutions of Kolmogorov equations(Galerkin’s method,Finite El-ement method,Path Integral method),Moment Closure Schemes,Compu-tation of the Evolution of Moments,Maximum Entropy Procedures,Asymp-totic Stability of Diffusion Processes.In the following,some of the outlined topics will be addressed stressing new developments.These topics are described within the next six subject areas,each focusing on a different issue,i.e.representation of stochastic processes andfields,structural response,stochastic FE methods and parallel processing,structural reliability and optimization,and stochastic dynamics. In this context it should be mentioned that aside from the MIT-Conference series the USNCCM,ECCM and WCCM’s do have a larger part of sessions addressing computational stochastic issues.Developments in Stochastic Structural Mechanics5 2Representation of Stochastic ProcessesMany quantities involving randomfluctuations in time and space might be adequately described by stochastic processes,fields and waves.Typical ex-amples of engineering interest are earthquake ground motion,sea waves, wind turbulence,road roughness,imperfection of shells,fluctuating prop-erties in random media,etc.For this setup,probabilistic characteristics of the process are known from various measurements and investigations in the past.In structural engineering,the available probabilistic characteristics of random quantities affecting the loading or the mechanical system can be often not utilized directly to account for the randomness of the structural response due to its complexity.For example,in the common case of strong earthquake motion,the structural response will be in general non-linear and it might be too difficult to compute the probabilistic characteristics of the response by other means than Monte Carlo simulation.For the purpose of Monte Carlo simulation sample functions of the involved stochastic pro-cess must be generated.These sample functions should represent accurately the characteristics of the underlying stochastic process orfields and might be stationary and non-stationary,homogeneous or non-homogeneous,one-dimensional or multi-dimensional,uni-variate or multi-variate,Gaussian or non-Gaussian,depending very much on the requirements of accuracy of re-alistic representation of the physical behavior and on the available statistical data.6G.I.Schu¨e ller The main requirement on the sample function is its accurate represen-tation of the available stochastic information of the process.The associ-ated mathematical model can be selected in any convenient manner as long it reproduces the required stochastic properties.Therefore,quite different representations have been developed and might be utilized for this purpose. The most common representations are e.g.:ARMA and AR models,Filtered White Noise(SDE),Shot Noise and Filtered Poisson White Noise,Covari-ance Decomposition,Karhunen-Lo`e ve and Polynomial Chaos Expansion, Spectral Representation,Wavelets Representation.Among the various methods listed above,the spectral representation methods appear to be most widely used(see e.g.[71,86]).According to this procedure,samples with specified power spectral density information are generated.For the stationary or homogeneous case the Fast Fourier Transform(FFT)techniques is utilized for a dramatic improvements of its computational efficiency(see e.g.[104,105]).Advances in thisfield provide efficient procedures for the generation of2D and3D homogeneous Gaus-sian stochasticfields using the FFT technique(see e.g.[87]).The spectral representation method generates ergodic sample functions of which each ful-fills exactly the requirements of a target power spectrum.These procedures can be extended to the non-stationary case,to the generation of stochastic waves and to incorporate non-Gaussian stochasticfields by a memoryless nonlinear transformation together with an iterative procedure to meet the target spectral density.Developments in Stochastic Structural Mechanics7 The above spectral representation procedures for an unconditional simula-tion of stochastic processes andfields can also be extended for Conditional simulations techniques for Gaussianfields(see e.g.[43,44])employing the conditional probability density method.The aim of this procedure is the generation of Gaussian random variates U n under the condition that(n−1) realizations u i of U i,i=1,2,...,(n−1)are specified and the a priori known covariances are satisfied.An alternative procedure is based on the so called Kriging method used in geostatistical application and applied also to con-ditional simulation problems in earthquake engineering(see e.g.[98]).The Kriging method has been improved significantly(see e.g.[36])that has made this method theoretically clearer and computationally more efficient.The differences and similarities of the conditional probability density methods and(modified)Kriging methods are discussed in[37]showing the equiva-lence of both procedures if the process is Gaussian with zero mean.A quite general spectral representation utilized for Gaussian random pro-cesses andfields is the Karhunen-Lo`e ve expansion of the covariance function (see e.g.[54,33]).This representation is applicable for stationary(homoge-neous)as well as for non-stationary(inhomogeneous)stochastic processes (fields).The expansion of a stochastic process(field)u(x,θ)takes the formu(x,θ)=¯u(x)+∞i=1ξ(θ) λiφi(x)(1)where the symbolθindicates the random nature of the corresponding quan-tity and where¯u(x)denotes the mean,φi(x)are the eigenfunctions andλi the eigenvalues of the covariance function.The set{ξi(θ)}forms a set of8G.I.Schu¨e ller orthogonal(uncorrelated)zero mean random variables with unit variance.The Karhunen-Lo`e ve expansion is mean square convergent irrespective of its probabilistic nature provided it possesses afinite variance.For the im-portant special case of a Gaussian process orfield the random variables{ξi(θ)}are independent standard normal random variables.In many prac-tical applications where the random quantities vary smoothly with respectto time or space,only few terms are necessary to capture the major part of the randomfluctuation of the process.Its major advantage is the reduction from a large number of correlated random variables to few most important uncorrelated ones.Hence this representation is especially suitable for band limited colored excitation and stochastic FE representation of random me-dia where random variables are usually strongly correlated.It might also be utilized to represent the correlated stochastic response of MDOF-systems by few most important variables and hence achieving a space reduction.A generalization of the above Karhunen-Lo`e ve expansion has been proposed for application where the covariance function is not known a priori(see[16, 33,32]).The stochastic process(field)u(x,θ)takes the formu(x,θ)=a0(x)Γ0+∞i1=1a i1(x)Γ1(ξi1(θ))+∞i1=1i1i2=1a i1i2(x)Γ2(ξi1(θ),ξi2(θ))+ (2)which is denoted as the Polynomial Chaos Expansion.Introducing a one-to-one mapping to a set with ordered indices{Ψi(θ)}and truncating eqn.2Developments in Stochastic Structural Mechanics9 after the p th term,the above representations reads,u(x,θ)=pj=ou j(x)Ψj(θ)(3)where the symbolΓn(ξi1,...,ξin)denotes the Polynomial Chaos of order nin the independent standard normal random variables.These polynomialsare orthogonal so that the expectation(or inner product)<ΨiΨj>=δij beingδij the Kronecker symbol.For the special case of a Gaussian random process the above representation coincides with the Karhunen-Lo`e ve expan-sion.The Polynomial Chaos expansion is adjustable in two ways:Increasingthe number of random variables{ξi}results in a refinement of the random fluctuations,while an increase of the maximum order of the polynomialcaptures non-linear(non-Gaussian)behavior of the process.However,the relation between accuracy and numerical efforts,still remains to be shown. The spectral representation by Fourier analysis is not well suited to describe local feature in the time or space domain.This disadvantage is overcome in wavelets analysis which provides an alternative of breaking a signal down into its constituent parts.For more details on this approach,it is referred to[24,60].In some cases of applications the physics or data might be inconsistent with the Gaussian distribution.For such cases,non-Gaussian models have been developed employing various concepts to meet the desired target dis-tribution as well as the target correlation structure(spectral density).Cer-tainly the most straight forward procedures is the above mentioned memo-ryless non-linear transformation of Gaussian processes utilizing the spectralrepresentation.An alternative approach utilizes linear and non-linearfil-ters to represent normal and non-Gaussian processes andfields excited by Gaussian white noise.Linearfilters excited by polynomial forms of Poisson white noise have been developed in[59]and[34].These procedures allow the evaluation of moments of arbitrary order without having to resort to closure techniques. Non-linearfilters are utilized to generate a stationary non-Gaussian stochas-tic process in agreement with a givenfirst-order probability density function and the spectral density[48,15].In the Kontorovich-Lyandres procedure as used in[48],the drift and diffusion coefficients are selected such that the solutionfits the target probability density,and the parameters in the solu-tion form are then adjusted to approximate the target spectral density.The approach by Cai and Lin[15]simplifies this procedure by matching the spec-tral density by adjusting only the drift coefficients,which is the followed by adjusting the diffusion coefficient to approximate the distribution of the pro-cess.The latter approach is especially suitable and computationally highly efficient for a long term simulation of stationary stochastic processes since the computational expense increases only linearly with the number n of dis-crete sample points while the spectral approach has a growth rate of n ln n when applying the efficient FFT technique.For generating samples of the non-linearfilter represented by a stochastic differential equations(SDE), well developed numerical procedures are available(see e.g.[47]).3Response of Stochastic SystemsThe assessment of the stochastic response is the main theme in stochastic mechanics.Contrary to the representation of of stochastic processes and fields designed tofit available statistical data and information,the output of the mathematical model is not prescribed and needs to be determined in some stochastic sense.Hence the mathematical model can not be selected freely but is specified a priori.The model involves for stochastic systems ei-ther random system parameters or/and random loading.Please note,due to space limitations,the question of model validation cannot be treated here. For the characterization of available numerical procedures some classifi-cations with regard to the structural model,loading and the description of the stochastic response is most instrumental.Concerning the structural model,a distinction between the properties,i.e.whether it is determinis-tic or stochastic,linear or non-linear,as well as the number of degrees of freedom(DOF)involved,is essential.As a criterion for the feasibility of a particular numerical procedure,the number of DOF’s of the structural system is one of the most crucial parameters.Therefore,a distinction be-tween dynamical-system-models and general FE-discretizations is suggested where dynamical systems are associated with a low state space dimension of the structural model.FE-discretization has no essential restriction re-garding its number of DOF’s.The stochastic loading can be grouped into static and dynamic loading.Stochastic dynamic loading might be charac-terized further by its distribution and correlation and its independence ordependence on the response,resulting in categorization such as Gaussian and non-Gaussian,stationary and non-stationary,white noise or colored, additive and multiplicative(parametric)excitation properties.Apart from the mathematical model,the required terms in which the stochastic re-sponse should be evaluated play an essential role ranging from assessing thefirst two moments of the response to reliability assessments and stabil-ity analysis.The large number of possibilities for evaluating the stochas-tic response as outlined above does not allow for a discussion of the en-tire subject.Therefore only some selected advances and new directions will be addressed.As already mentioned above,one could distinguish between two main categories of computational procedures treating the response of stochastic systems.Thefirst is based on Monte Carlo simulation and the second provides numerical solutions of analytical procedures for obtaining quantitative results.Regarding the numerical solutions of analytical proce-dures,a clear distinction between dynamical-system-models and FE-models should be made.Current research efforts in stochastic dynamics focus to a large extent on dynamical-system-models while there are few new numerical approaches concerning the evaluation of the stochastic dynamic response of e.g.FE-models.Numerical solutions of the Kolmogorov equations are typical examples of belonging to dynamical-system-models where available approaches are computationally feasible only for state space dimensions one to three and in exceptional cases for dimension four.Galerkin’s,Finite El-ement(FE)and Path Integral methods respectively are generally used tosolve numerically the forward(Fokker-Planck)and backward Kolmogorov equations.For example,in[8,92]the FE approach is employed for stationary and transient solutions respectively of the mentioned forward and backward equations for second order systems.First passage probabilities have been ob-tained employing a Petrov-Galerkin FE method to solve the backward and the related Pontryagin-Vitt equations.An instructive comparison between the computational efforts using Monte Carlo simulation and the FE-method is given e.g.in an earlier IASSAR report[85].The Path Integral method follows the evolution of the(transition)prob-ability function over short time intervals,exploiting the fact that short time transition probabilities for normal white noise excitations are locally Gaus-sian distributed.All existing path integration procedures utilize certain in-terpolation schemes where the probability density function(PDF)is rep-resented by values at discrete grid points.In a wider sense,cell mapping methods(see e.g.[38,39])can be regarded as special setups of the path integral procedure.As documented in[9],cumulant neglect closure described in section7.3 has been automated putational procedures for the automated generation and solutions of the closed set of moment equations have been developed.The method can be employed for an arbitrary number of states and closed at arbitrary levels.The approach,however,is limited by available computational resources,since the computational cost grows exponentially with respect to the number of states and the selected closurelevel.The above discussed developments of numerical procedures deal with low dimensional dynamical systems which are employed for investigating strong non-linear behavior subjected to(Gaussian)white noise excitation. Although dynamical system formulations are quite general and extendible to treat non-Gaussian and colored(filtered)excitation of larger systems,the computational expense is growing exponentially rendering most numerical approaches unfeasible for larger systems.This so called”curse of dimen-sionality”is not overcome yet and it is questionable whether it ever will be, despite the fast developing computational possibilities.For this reason,the alternative approach based on Monte Carlo simu-lation(MCS)gains importance.Several aspects favor procedures based on MCS in engineering applications:(1)Considerably smaller growth rate of the computational effort with dimensionality than analytical procedures.(2) Generally applicable,well suited for parallel processing(see section5.1)and computationally straight forward.(3)Non-linear complex behavior does not complicate the basic procedure.(4)Manageable for complex systems.Contrary to numerical solutions of analytical procedures,the employed structural model and the type of stochastic loading does for MCS not play a deceive role.For this reason,MCS procedures might be structured ac-cording to their purpose i.e.where sample functions are generated either for the estimation of the overall distribution or for generating rare adverse events for an efficient reliability assessment.In the former case,the prob-ability space is covered uniformly by an indiscriminate(blind)generationof sample functions representing the random quantities.Basically,at set of random variables will be generated by a pseudo random number generator followed by a deterministic structural analysis.Based on generated random numbers realizations of random processes,fields and waves addressed in section2,are constructed and utilized without any further modification in the following structural analysis.The situation may not be considered to be straight forward,however,in case of a discriminate MCS for the reliability estimation of structures,where rare events contributing considerably to the failure probability should be gener-ated.Since the effectiveness of direct indiscriminate MCS is not satisfactory for producing a statistically relevant number of low probability realizations in the failure domain,the generation of samples is restricted or guided in some way.The most important class are the variance reduction techniques which operate on the probability of realizations of random variables.The most widely used representative of this class in structural reliability assess-ment is Importance Sampling where a suitable sampling distribution con-trols the generation of realizations in the probability space.The challenge in Importance Sampling is the construction of a suitable sampling distribu-tion which depends in general on the specific structural system and on the failure domain(see e.g.[84]).Hence,the generation of sample functions is no longer independent from the structural system and failure criterion as for indiscriminate direct MCS.Due to these dependencies,computational procedures for an automated establishment of sampling distributions areurgently needed.Adaptive numerical strategies utilizing Importance Direc-tional sampling(e.g.[11])are steps in this direction.The effectiveness of the Importance sampling approach depends crucially on the complexity of the system response as well as an the number of random variables(see also section5.2).Static problems(linear and nonlinear)with few random vari-ables might be treated effectively by this approach.Linear systems where the randomness is represented by a large number of RVs can also be treated efficiently employingfirst order reliability methods(see e.g.[27]).This ap-proach,however,is questionable for the case of non-linear stochastic dynam-ics involving a large set of random variables,where the computational effort required for establishing a suitable sampling distribution might exceed the effort needed for indiscriminate direct MCS.Instead of controlling the realization of random variables,alternatively the evolution of the generated sampling can be controlled[68].This ap-proach is limited to stochastic processes andfields with Markovian prop-erties and utilizes an evolutionary programming technique for the genera-tion of more”important”realization in the low probability domain.This approach is especially suitable for white noise excitation and non-linear systems where Importance sampling is rather difficult to apply.Although the approach cannot deal with spectral representations of the stochastic processes,it is capable to make use of linearly and non-linearlyfiltered ex-citation.Again,this is just contrary to Importance sampling which can be applied to spectral representations but not to white noisefiltered excitation.4Stochastic Finite ElementsAs its name suggests,Stochastic Finite Elements are structural models rep-resented by Finite Elements the properties of which involve randomness.In static analysis,the stiffness matrix might be random due to unpredictable variation of some material properties,random coupling strength between structural components,uncertain boundary conditions,etc.For buckling analysis,shape imperfections of the structures have an additional impor-tant effect on the buckling load[76].Considering structural dynamics,in addition to the stiffness matrix,the damping properties and sometimes also the mass matrix might not be predictable with certainty.Discussing numerical Stochastic Finite Elements procedures,two cat-egories should be distinguished clearly.Thefirst is the representation of Stochastic Finite Elements and their global assemblage as random structural matrices.The second category addresses the evaluation of the stochastic re-sponse of the FE-model due to its randomness.Focusingfirst on the Stochastic FE representation,several representa-tions such as the midpoint method[35],the interpolation method[53],the local average method[97],as well as the Weighted-Integral-Method[94,25, 26]have been developed to describe spatial randomfluctuations within the element.As a tendency,the midpoint methods leads to an overestimation of the variance of the response,the local average method to an underestima-tion and the Weighted-Integral-Method leads to the most accurate results. Moreover,the so called mesh-size problem can be resolved utilizing thisrepresentation.After assembling all Finite Elements,the random structural stiffness matrix K,taken as representative example,assumes the form,K(α)=¯K+ni=1K Iiαi+ni=1nj=1K IIijαiαj+ (4)where¯K is the mean of the matrix,K I i and K II ij denote the determinis-ticfirst and second rate of change with respect to the zero mean random variablesαi andαj and n is the total number of random variables.For normally distributed sets of random variables{α},the correlated set can be represented advantageously by the Karhunen-Lo`e ve expansion[33]and for non-Gaussian distributed random variables by its Polynomial chaos ex-pansion[32],K(θ)=¯K+Mi=0ˆKiΨi(θ)(5)where M denotes the total number of chaos polynomials,ˆK i the associated deterministicfluctuation of the matrix andΨi(θ)a polynomial of standard normal random variablesξj(θ)whereθindicates the random nature of the associated variable.In a second step,the random response of the stochastic structural system is determined.The most widely used procedure for evaluating the stochastic response is the well established perturbation approach(see e.g.[53]).It is well adapted to the FE-formulation and capable to evaluatefirst and second moment properties of the response in an efficient manner.The approach, however,is justified only for small deviations from the center value.Since this assumption is satisfied in most practical applications,the obtainedfirst two moment properties are evaluated satisfactorily.However,the tails of the。

《实验力学》英语单词振动vibration, oscillation, 机械振动mechanical vibration,稳定性stability, 简谐振动simple harmonic vibration,振幅amplitude, 固有频率natural frequency,角频率angular frequency, 相角phase angle,相差phase difference, 周期性periodicity,阻尼damping, 响应response,传递函数transfer function, 驱动力driving force,共振resonance, 振动摸态mode of vibration,自由度degree of freedom, 模态分析modal analysis,固有振型natural mode of vibration, 频谱frequency spectrum,机械阻抗mechanical impedance, 随机振动random vibration拉伸tension, 压缩compression,弯曲bending, 剪切shear,扭转torsion, 冲击impact,疲劳fatigue, 应力stress,应变strain, 载荷load,模量modulus, 泊松比Poisson ratio,柔度compliance, 韧性toughness,延性ductility, 脆性brittleness,延伸率specific elongation, 布氏硬度Brinell hardness,维氏硬度Vickers hardness, 洛氏硬度Rockwell hardness,颈缩necking, 破裂rupture,破坏fracture, failure, 蠕变creep,应变能strain energy, 桡度deflection,扭曲twist, 屈曲buckling,位移displacement, 刚度stiffness,柔度flexibility, 圣维南原理Saint-Venant plinciple,裂纹crack, 缺陷defect,损伤damage实验应力分析experimental stress analysis, 实验力学experimental mechanics, 背景background, 量纲dimension,单位unit, 相似similitude,相似变换sililar transformation, 量纲分析dimensional analysis,相似方程homogeneous equation, 统计学statistics,电阻应变计resistence strain gage, 工作片active gage,补偿片compensating gage, 温度补偿temperature compensating,应变花strain rosette, 应变灵敏度strain sensitivity,横向效应transverse gage factor, 箔式应变计foiled gage,丝式应变计wire gage, 粘贴应变计bonded strain gage,引伸计extensometer, 桥平衡bridge balancing,集电环slip ring, 应变放大器strain amplifier,光弹性photoelasticity, 双折射效应birefrigent effect,亮条纹light fringe, 暗条纹dark fringe,干涉条纹interference fringe,偏振Polaroid,四分之一波片quarter wave plate, 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,光程差optical path difference, 光弹性贴片法photoelastic coating method, 动态光弹性dynamic photo-elasticity, 起偏镜polarizer,应变条纹值strain fringe value, 透射式光弹性仪transmission polariscope,全息照相holograph, 全息干涉法holographic interferometry,散斑speckle, 全场分析法whole-field analysis,云纹法moiré method,参考栅reference grating,试件栅specimen grating, 分析栅analyzer grating,离面云纹法off-plane moiré method, 面内云纹法in-plane moiré method。