Asymptotic Analysis of Periodic Structures
01 Jun 1979-Journal of Applied Mechanics (American Society of Mechanical Engineers)-Vol. 46, Iss: 2, pp 477-477
About: This article is published in Journal of Applied Mechanics.The article was published on 1979-06-01 and is currently open access. It has received 5038 citations till now. The article focuses on the topics: Asymptotic analysis.
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TL;DR: In this article, the authors present a methodology for optimal shape design based on homogenization, which is related to modern production techniques and consists of computing the optimal distribution in space of an anisotropic material that is constructed by introducing an infimum of periodically distributed small holes in a given homogeneous, i.i.
Abstract: Optimal shape design of structural elements based on boundary variations results in final designs that are topologically equivalent to the initial choice of design, and general, stable computational schemes for this approach often require some kind of remeshing of the finite element approximation of the analysis problem. This paper presents a methodology for optimal shape design where both these drawbacks can be avoided. The method is related to modern production techniques and consists of computing the optimal distribution in space of an anisotropic material that is constructed by introducing an infimum of periodically distributed small holes in a given homogeneous, i~otropic material, with the requirement that the resulting structure can carry the given loads as well as satisfy other design requirements. The computation of effective material properties for the anisotropic material is carried out using the method of homogenization. Computational results are presented and compared with results obtained by boundary variations.
5,858 citations
TL;DR: In this article, various ways of removing this discrete nature of the problem by the introduction of a density function that is a continuous design variable are described. But none of these methods can be used for shape optimization in a general setting.
Abstract: Shape optimization in a general setting requires the determination of the optimal spatial material distribution for given loads and boundary conditions. Every point in space is thus a material point or a void and the optimization problem is a discrete variable one. This paper describes various ways of removing this discrete nature of the problem by the introduction of a density function that is a continuous design variable. Domains of high density then define the shape of the mechanical element. For intermediate densities, material parameters given by an artificial material law can be used. Alternatively, the density can arise naturally through the introduction of periodically distributed, microscopic voids, so that effective material parameters for intermediate density values can be computed through homogenization. Several examples in two-dimensional elasticity illustrate that these methods allow a determination of the topology of a mechanical element, as required for a boundary variations shape optimization technique.
3,434 citations
06 May 2002
TL;DR: Some of the greatest scientists including Poisson, Faraday, Maxwell, Rayleigh, and Einstein have contributed to the theory of composite materials Mathematically, it is the study of partial differential equations with rapid oscillations in their coefficients Although extensively studied for more than a hundred years, an explosion of ideas in the last five decades has dramatically increased our understanding of the relationship between the properties of the constituent materials, the underlying microstructure of a composite, and the overall effective moduli which govern the macroscopic behavior as mentioned in this paper.
Abstract: Some of the greatest scientists including Poisson, Faraday, Maxwell, Rayleigh, and Einstein have contributed to the theory of composite materials Mathematically, it is the study of partial differential equations with rapid oscillations in their coefficients Although extensively studied for more than a hundred years, an explosion of ideas in the last five decades (and particularly in the last three decades) has dramatically increased our understanding of the relationship between the properties of the constituent materials, the underlying microstructure of a composite, and the overall effective (electrical, thermal, elastic) moduli which govern the macroscopic behavior This renaissance has been fueled by the technological need for improving our knowledge base of composites, by the advance of the underlying mathematical theory of homogenization, by the discovery of new variational principles, by the recognition of how important the subject is to solving structural optimization problems, and by the realization of the connection with the mathematical problem of quasiconvexification This 2002 book surveys these exciting developments at the frontier of mathematics
2,455 citations
Cites background or methods from "Asymptotic Analysis of Periodic Str..."
...Homogenization in random media is described with varying degrees of rigor in the books of Beran (1968); Bensoussan, Lions, and Papanicolaou (1978); Bakhvalov and Panasenko (1989); and Zhikov, Kozlov, and Oleinik (1994), and in the papers of Kozlov (1978), Papanicolaou and Varadhan (1982), and…...
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...The results of this multiple-scale analysis can be verified by more rigorous methods [see, for example, Bensoussan, Lions, and Papanicolaou (1978); Tartar (1978); Nguetseng (1989); Allaire (1992); Allaire and Briane (1996); and Murat and Tartar (1997)]....
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...Periodic homogenization is described in the books of Bensoussan, Lions, and Papanicolaou (1978); Sanchez-Palencia (1980); Bakhvalov and Panasenko (1989); Persson, Persson, Svanstedt, and Wyller (1993); and Zhikov, Kozlov, and Oleinik (1994)....
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TL;DR: In this article, the authors define a notion of "two-scale" convergence, which is aimed at a better description of sequences of oscillating functions, and prove that bounded sequences in $L^2 (Omega )$ are relatively compact with respect to this new type of convergence.
Abstract: Following an idea of G. Nguetseng, the author defines a notion of “two-scale” convergence, which is aimed at a better description of sequences of oscillating functions. Bounded sequences in $L^2 (\Omega )$ are proven to be relatively compact with respect to this new type of convergence. A corrector-type theorem (i.e., which permits, in some cases, replacing a sequence by its “two-scale” limit, up to a strongly convergent remainder in $L^2 (\Omega )$) is also established. These results are especially useful for the homogenization of partial differential equations with periodically oscillating coefficients. In particular, a new method for proving the convergence of homogenization processes is proposed, which is an alternative to the so-called energy method of Tartar. The power and simplicity of the two-scale convergence method is demonstrated on several examples, including the homogenization of both linear and nonlinear second-order elliptic equations.
2,279 citations
Cites background from "Asymptotic Analysis of Periodic Str..."
...This is very helpful in homogenization theory, where such asymptotic expansions are frequently used in a heuristical way (see [10], [40])....
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...…recovering previous well-known results from a new point of view, we establish a new form of the limit problem, that we call the two-scale homogenized problem, and which is simply a combination of the usual homogenized problem and the cell problem (see [10], [40] for an introduction to the topic)....
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TL;DR: This paper studies a multiscale finite element method for solving a class of elliptic problems arising from composite materials and flows in porous media, which contain many spatial scales and proposes an oversampling technique to remove the resonance effect.
1,825 citations
Cites background from "Asymptotic Analysis of Periodic Str..."
...It is proved in [8] that a is symmetric and positive de nite....
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...1) has been rigorously justi ed in [8]....
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...By the homogenization theory [8], the solution of (2....
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...However, their range of applications is usually limited by restrictive assumptions on the media, such scale separation and periodicity [8]....
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References
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01 Jan 1950
TL;DR: A First Course in Probability (8th ed.) by S. Ross is a lively text that covers the basic ideas of probability theory including those needed in statistics.
Abstract: Office hours: MWF, immediately after class or early afternoon (time TBA). We will cover the mathematical foundations of probability theory. The basic terminology and concepts of probability theory include: random experiments, sample or outcome spaces (discrete and continuous case), events and their algebra, probability measures, conditional probability A First Course in Probability (8th ed.) by S. Ross. This is a lively text that covers the basic ideas of probability theory including those needed in statistics. Theoretical concepts are introduced via interesting concrete examples. In 394 I will begin my lectures with the basics of probability theory in Chapter 2. However, your first assignment is to review Chapter 1, which treats elementary counting methods. They are used in applications in Chapter 2. I expect to cover Chapters 2-5 plus portions of 6 and 7. You are encouraged to read ahead. In lectures I will not be able to cover every topic and example in Ross, and conversely, I may cover some topics/examples in lectures that are not treated in Ross. You will be responsible for all material in my lectures, assigned reading, and homework, including supplementary handouts if any.
10,221 citations
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TL;DR: The merite as discussed by the authors is a date marque une date dans le progres des mathematiques and de la physique en levant l'ambiguite que constituait le succes des methodes de calcul symbolique aupres des physiciens and l'inacceptabilite de leurs formules au regard de la rigueur mathematiques.
Abstract: Ce traite a marque une date dans le progres des mathematiques et de la physique en levant l’ambiguite que constituait le succes des methodes de calcul symbolique aupres des physiciens et l’inacceptabilite de leurs formules au regard de la rigueur mathematiques Le merite revient a Laurent Schwartz d’avoir englobe dans une theorie qui est a la fois une synthese et une simplifications, des procedes heterogenes et souvent incorrects utilises dans des domaines tres divers
Une definition correcte et une etude systematique de ces etres nouveaux, les distributions, leur ont donne droit de cite dans l’usage courant Leur utilisation extensive dans de nombreuses branches des mathematiques pures et appliquees, de la physique et des sciences de l’ingenieur fait de ce livre un classique des mathematiques modernes
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