scispace - formally typeset

Homogenization (chemistry)

About: Homogenization (chemistry) is a(n) research topic. Over the lifetime, 12172 publication(s) have been published within this topic receiving 219984 citation(s). more


Open accessJournal ArticleDOI: 10.1016/0045-7825(88)90086-2
Martin P. Bendsøe1, Noboru Kikuchi2Institutions (2)
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. more

Topics: Topology optimization (58%), Homogenization (chemistry) (54%), Finite element method (54%) more

5,045 Citations

Open accessBook
01 May 1980-
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. more

Topics: Boundary value problem (54%), Sobolev space (53%), Asymptotic homogenization (53%) more

3,326 Citations

Journal ArticleDOI: 10.1007/BF01650949
Martin P. Bendsøe1Institutions (1)
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. more

Topics: Shape optimization (64%), Topology optimization (62%), Optimization problem (58%) more

2,899 Citations

Open accessBook
19 Aug 1994-
Abstract: 1 Homogenization of Second Order Elliptic Operators with Periodic Coefficients.- 1.1 Preliminaries.- 1.2 Setting of the Homogenization Problem.- 1.3 Problems of Justification Further Examples.- 1.4 The Method of Asymptotic Expansions.- 1.5 Explicit Formulas for the Homogenized Matrix in the Two-Dimensional Case.- 1.6 Estimates and Approximations for the Homogenized Matrix.- 1.7 The Rayleigh-Maxwell Formulas.- Comments.- 2 An Introduction to the Problems of Diffusion.- 2.1 Homogenization of Parabolic Operators.- 2.2 Homogenization and the Central Limit Theorem.- 2.3 Stabilization of Solutions of Parabolic Equations.- 2.4 Diffusion in a Solenoidal Flow.- 2.5 Diffusion in an Arbitrary Periodic Flow.- 2.6 Spectral Approach to the Asymptotic Problems of Diffusion.- 2.7 Diffusion with Absorption.- Comments.- 3 Elementary Soft and Stiff Problems.- 3.1 Homogenization of Soft Inclusions.- 3.2 Homogenization of Stiff Inclusions.- 3.3 Virtual Mass.- 3.4 The Method of Asymptotic Expansions.- 3.5 On a Dense Cubic Packing of Balls.- 3.6 The Dirichlet Problem in a Perforated Domain.- Comments.- 4 Homogenization of Maxwell Equations.- 4.1 Preliminary Results.- 4.2 A Lemma on Compensated Compactness.- 4.3 Homogenization.- 4.4 The Problem of an Artificial Dielectric.- Comments.- 5 G-Convergence of Differential Operators.- 5.1 Basic Properties of G-Convergence.- 5.2 A Sufficient Condition of G-Convergence.- 5.3 G-Convergence of Abstract Operators.- 5.4 Compactness Theorem and Its Implications.- 5.5 G-Convergence and Duality.- 5.6 Stratified Media.- 5.7 G-Convergence of Divergent Elliptic Operators of Higher Order.- Comments.- 6 Estimates for the Homogenized Matrix.- 6.1 The Hashin-Shtrikman Bounds.- 6.2 Attainability of Bounds. The Hashin Structure.- 6.3 Extremum Principles.- 6.4 The Variational Method.- 6.5 G-Limit Media Attainment of the Bounds on Stratified Composites.- 6.6 The Method of Quasi-Convexity.- 6.7 The Method of Null Lagrangians.- 6.8 The Method of Integral Representation.- Comments.- 7 Homogenization of Elliptic Operators with Random Coefficients.- 7.1 Probabilistic Description of Non-Homogeneous Media.- 7.2 Homogenization.- 7.3 Explicit Formulas in Two-Dimensional Problems.- 7.4 Homogenization of Almost-Periodic Operators.- 7.5 The General Theorem of Individual Homogenization.- Comments.- 8 Homogenization in Perforated Random Domains.- 8.1 Homogenization.- 8.2 Remarks on Positive Definiteness of the Homogenized Matrix.- 8.3 Central Limit Theorem.- 8.4 Disperse Media.- 8.5 Criterion of Pointwise Stabilization A Refinement of the Central Limit Theorem.- 8.6 Stiff Problem for a Random Spherical Structure.- 8.7 Random Spherical Structure with Small Concentration.- Comments.- 9 Homogenization and Percolation.- 9.1 Existence of the Effective Conductivity.- 9.2 Random Structure of Chess-Board Type.- 9.3 The Method of Percolation Channels.- 9.4 Conductivity Threshold for a Random Cubic Structure in ?3.- 9.5 Resistance Threshold for a Random Cubic Structure in ?3.- 9.6 Central Limit Theorem for Random Motion in an Infinite Two-Dimensional Cluster.- Comments.- 10 Some Asymptotic Problems for a Non-Divergent Parabolic Equation with Random Stationary Coefficients.- 10.1 Preliminary Remarks.- 10.2 Auxiliary Equation A*p = 0 on a Probability Space.- 10.3 Homogenization and the Central Limit Theorem.- 10.4 Criterion of Pointwise Stabilization.- Comments.- 11 Spectral Problems in Homogenization Theory.- 11.1 Spectral Properties of Abstract Operators Forming a Sequence.- 11.2 On the Spectrum of G-Convergent Operators.- 11.3 The Sturm-Liouville Problem.- 11.4 Spectral Properties of Stratified Media.- 11.5 Density of States for Random Elliptic Operators.- 11.6 Asymptotics of the Density of States.- Comments.- 12 Homogenization in Linear Elasticity.- 12.1 Some General Facts from the Theory of Elasticity.- 12.2 G-Convergence of Elasticity Tensors.- 12.3 Homogenization of Periodic and Random Tensors.- 12.4 Fourth Order Operators.- 12.5 Linear Problems of Incompressible Elasticity.- 12.6 Explicit Formulas for Two-Dimensional Incompressible Composites.- 12.7 Some Questions of Analysis on a Probability Space.- 13 Estimates for the Homogenized Elasticity Tensor.- 13.1 Basic Estimates.- 13.2 The Variational Method.- 13.3 Two-Phase Media Attainability of Bounds on Stratified Composites.- 13.4 On the Hashin Structure.- 13.5 Disperse Media with Inclusions of Small Concentration.- 13.6 Fourth Order Operators Systems of Stokes Type.- Comments.- 14 Elements of the Duality Theory.- 14.1 Convex Functions.- 14.2 Integral Functionals.- 14.3 On Two Types of Boundary Value Problems.- 14.4 Dual Boundary Value Problems.- 14.5 Extremal Relations.- 14.6 Examples of Regular Lagrangians.- Comments.- 15 Homogenization of Nonlinear Variational Problems.- 15.1 Random Lagrangians.- 15.2 Two Principal Lemmas.- 15.3 Homogenization Theorems.- 15.4 Applications to Boundary Value Problems in Perforated Domains.- 15.5 Chess Lagrangians Dychne's Formula.- Comments.- 16 Passing to the Limit in Nonlinear Variational Problems.- 16.1 Definition of ?-Convergence of Lagrangians Formulation of the Compactness Theorems.- 16.2 Convergence of Energies and Minimizers.- 16.3 Proof of the Compactness Theorems.- 16.4 Two Examples: Ulam's Problem Homogenization Problem.- 16.5 Compactness of Lagrangians in Plasticity Problems Application to Ll-Closedness.- 16.6 Remarks on Non-Convex Functionals.- Comments.- 17 Basic Properties of Abstract ?-Convergence.- 17.1 ?-Convergence of Functions on a Metric Space.- 17.2 ?-Convergence of Functions Defined in a Banach Space.- 17.3 ?-Convergence of Integral Functionals.- Comments.- 18 Limit Load.- 18.1 The Notion of Limit Load.- 18.2 Dual Definition of Limit Load.- 18.3 Equivalence Principle.- 18.4 Convergence of Limit Loads in Homogenization Problems.- 18.5 Surface Loads.- 18.6 Representation of the Functional $$\bar F$$ on BV0.- 18.7 ?-Convergence in BV0.- Comments.- Appendix A. Proof of the Nash-Aronson Estimate.- Appendix C. A Property of Bounded Lipschitz Domains.- References. more

Topics: Homogenization (chemistry) (56%), Characteristic equation (55%), Dirichlet problem (53%) more

2,546 Citations

Open accessJournal ArticleDOI: 10.1137/0523084
Grégoire Allaire1Institutions (1)
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. more

Topics: Compact convergence (65%), Convergence tests (62%), Normal convergence (60%) more

2,109 Citations

No. of papers in the topic in previous years

Top Attributes

Show by:

Topic's top 5 most impactful authors

Andrey Piatnitski

53 papers, 868 citations

P. Ponte Castañeda

49 papers, 2K citations

Grégoire Allaire

45 papers, 4.1K citations

Kenjiro Terada

32 papers, 479 citations

Marcin Kamiński

32 papers, 377 citations

Network Information
Related Topics (5)
Boundary value problem

145.3K papers, 2.7M citations

84% related
Numerical analysis

52.2K papers, 1.2M citations

83% related

53.6K papers, 1M citations

82% related
Galerkin method

21.8K papers, 494.5K citations

82% related
Finite difference

19.6K papers, 408.6K citations

82% related