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Asymptotic homogenization

About: Asymptotic homogenization is a(n) research topic. Over the lifetime, 659 publication(s) have been published within this topic receiving 22421 citation(s). more


Open accessBook
01 Jan 1978-
Abstract: This is a reprinting of a book originally published in 1978. At that time it was the first book on the subject of homogenization, which is the asymptotic analysis of partial differential equations with rapidly oscillating coefficients, and as such it sets the stage for what problems to consider and what methods to use, including probabilistic methods. At the time the book was written the use of asymptotic expansions with multiple scales was new, especially their use as a theoretical tool, combined with energy methods and the construction of test functions for analysis with weak convergence methods. Before this book, multiple scale methods were primarily used for non-linear oscillation problems in the applied mathematics community, not for analyzing spatial oscillations as in homogenization. In the current printing a number of minor corrections have been made, and the bibliography was significantly expanded to include some of the most important recent references. This book gives systematic introduction of multiple scale methods for partial differential equations, including their original use for rigorous mathematical analysis in elliptic, parabolic, and hyperbolic problems, and with the use of probabilistic methods when appropriate. The book continues to be interesting and useful to readers of different backgrounds, both from pure and applied mathematics, because of its informal style of introducing the multiple scale methodology and the detailed proofs. more

Topics: Asymptotic analysis (57%), Asymptotic homogenization (56%), Asymptotic expansion (52%) more

4,687 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

Open accessJournal ArticleDOI: 10.1016/0045-7825(90)90148-F
José M. Guedes1, Noboru Kikuchi1Institutions (1)
Abstract: This paper discusses the homogenization method to determine the effective average elastic constants of linear elasticity of general composite materials by considering their microstructure. After giving a brief theory of the homogenization method, a finite element approximation is introduced with convergence study and corresponding error estimate. Applying these, computer programs PREMAT and POSTMAT are developed for preprocessing and postprocessing of material characterization of composite materials. Using these programs, the homogenized elastic constants for macroscopic stress analysis are obtained for typical composite materials to show their capability. Finally, the adaptive finite element method is introduced to improve the accuracy of the finite element approximation. more

1,042 Citations

Journal ArticleDOI: 10.1016/S0020-7683(98)00341-2
Abstract: Although the asymptotic homogenization is known to explicitly predict the thermo-mechanical behaviors of an overall structure as well as the microstructures, the current developments in engineering fields introduce some kinds of approximation about the microstructural geometry. In order for the homogenization method for periodic media to apply for general heterogeneous ones, the problems arising from mathematical modeling are examined in the framework of representative volume element (RVE) analyses. Here, the notion of homogenization convergence allows us to eliminate the geometrical periodicity requirement when the size of RVE is sufficiently large. Then the numerical studies in this paper realize the multi-scale nature of the convergence of overall material properties as the unit cell size is increased. In addition to such dependency of the macroscopic field variables on the selected size of unit cells, the convergence nature of microscopic stress values is also studied quantitatively via the computational homogenization method. Similar discussions are made for the elastoplastic mechanical responses in both macro- and microscopic levels. In these multi-scale numerical analyses, the specific effects of the microstructural morphology are reflected by using the digital image-based (DIB) finite element (FE) modeling technique which enables the construction of accurate microstructural models. more

462 Citations

Journal ArticleDOI: 10.1016/0020-7683(94)00097-G
Abstract: This paper deals with the development of a multiple scale finite element method by combining the asymptotic homogenization theory with Voronoi cell (VCFEM) for microstructural modeling. The Voronoi cell finite element model originates from Dirichlet tessellation of a representative material element or a base cell in the microstructure. Homogenized material coefficients for a global displacement finite element model are generated by VCFEM analysis using periodic boundary conditions on the base cell. Following the macroscopic analysis, the local VCFEM analysis is implemented to depict the true evolution of microstructural stresses and strains. Various numerical examples are executed for validating the effectiveness of VCFEM macro-micro modeling of elastic materials. The effect of size, shape, orientation and distribution of heterogeneities on the local and global response are examined. more

Topics: Asymptotic homogenization (56%), Finite element method (54%), Voronoi diagram (52%) more

409 Citations

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Topic's top 5 most impactful authors

Reinaldo Rodríguez-Ramos

99 papers, 1.8K citations

Raúl Guinovart-Díaz

75 papers, 1.5K citations

Julián Bravo-Castillero

61 papers, 506 citations

Federico J. Sabina

59 papers, 1K citations

Alexander L. Kalamkarov

54 papers, 1.4K citations

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