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Non-Homogeneous Media and Vibration Theory

TL;DR: In this article, a spectral perturbation of spectral families and applications to self-adjoint eigenvalue problems are discussed, as well as the Trotter-Kato theorem and related topics.
Abstract: Distributions and Sobolev spaces.- Operators in Banach spaces.- Examples of boundary value problems.- Semigroups and laplace transform.- Homogenization of second order equations.- Homogenization in elasticity and electromagnetism.- Fluid flow in porous media.- Vibration of mixtures of solids and fluids.- Examples of perturbations for elliptic problems.- The Trotter-Kato theorem and related topics.- Spectral perturbation. Case of isolated eigenvalues.- Perturbation of spectral families and applications to selfadjoint eigenvalue problems.- Stiff problems in constant and varialbe domains.- Averaging and two-scale methods.- Generalities and potential method.- Functional methods.- Scattering problems depending on a parameter.
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Journal ArticleDOI
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

Journal ArticleDOI
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

MonographDOI
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 from "Non-Homogeneous Media and Vibration..."

  • ...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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  • ...…which has obvious applications to the management of oil and water reservoirs and to understanding the seepage of waste fluids (Scheidegger 1974; Sanchez-Palencia 1980); geometrical questions such as the microstructures of rocks (Pittman 1984) and dense random packings of hard spheres (Cargill…...

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Journal ArticleDOI
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

Journal ArticleDOI
TL;DR: Metamaterials are typically engineered by arranging a set of small scatterers or apertures in a regular array throughout a region of space, thus obtaining some desirable bulk electromagnetic behavior as mentioned in this paper.
Abstract: Metamaterials are typically engineered by arranging a set of small scatterers or apertures in a regular array throughout a region of space, thus obtaining some desirable bulk electromagnetic behavior. The desired property is often one that is not normally found naturally (negative refractive index, near-zero index, etc.). Over the past ten years, metamaterials have moved from being simply a theoretical concept to a field with developed and marketed applications. Three-dimensional metamaterials can be extended by arranging electrically small scatterers or holes into a two-dimensional pattern at a surface or interface. This surface version of a metamaterial has been given the name metasurface (the term metafilm has also been employed for certain structures). For many applications, metasurfaces can be used in place of metamaterials. Metasurfaces have the advantage of taking up less physical space than do full three-dimensional metamaterial structures; consequently, metasurfaces offer the possibility of less-lossy structures. In this overview paper, we discuss the theoretical basis by which metasurfaces should be characterized, and discuss their various applications. We will see how metasurfaces are distinguished from conventional frequency-selective surfaces. Metasurfaces have a wide range of potential applications in electromagnetics (ranging from low microwave to optical frequencies), including: (1) controllable “smart” surfaces, (2) miniaturized cavity resonators, (3) novel wave-guiding structures, (4) angular-independent surfaces, (5) absorbers, (6) biomedical devices, (7) terahertz switches, and (8) fluid-tunable frequency-agile materials, to name only a few. In this review, we will see that the development in recent years of such materials and/or surfaces is bringing us closer to realizing the exciting speculations made over one hundred years ago by the work of Lamb, Schuster, and Pocklington, and later by Mandel'shtam and Veselago.

1,819 citations


Cites methods from "Non-Homogeneous Media and Vibration..."

  • ...While period-cell averaging for the fi elds is the correct method for defi ning effective material properties (see [61, 89-93] for this type of homogenization averaging), many researchers have in practice used an approach based upon the refl ection and transmission coeffi cients of a metamaterial sample of some defi ned thickness....

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