Author
Grégoire Allaire
Other affiliations: Chicago Metropolitan Agency for Planning, Pierre-and-Marie-Curie University, French Alternative Energies and Atomic Energy Commission ...read more
Bio: Grégoire Allaire is an academic researcher from École Polytechnique. The author has contributed to research in topics: Homogenization (chemistry) & Topology optimization. The author has an hindex of 46, co-authored 204 publications receiving 12849 citations. Previous affiliations of Grégoire Allaire include Chicago Metropolitan Agency for Planning & Pierre-and-Marie-Curie University.
Papers published on a yearly basis
Papers
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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
TL;DR: A new numerical method based on a combination of the classical shape derivative and of the level-set method for front propagation, which can easily handle topology changes and is strongly dependent on the initial guess.
2,176 citations
Book•
01 Jan 2002TL;DR: In this article, a relaxed formulation for shape optimization in the context of shape optimization is presented, where the authors seek minimizers of the sum of the elastic compliance and of the weight of a solid structure under specified loading.
Abstract: In the context of shape optimization, we seek minimizers of the sum of the elastic compliance and of the weight of a solid structure under specified loading. This problem is known not to be well-posed, and a relaxed formulation is introduced. Its effect is to allow for microperforated composites as admissible designs. In a two-dimensional setting the relaxed formulation was obtained in [6] with the help of the theory of homogenization and optimal bounds for composite materials. We generalize the result to the three dimensional case. Our contribution is twofold; first, we prove a relaxation theorem, valid in any dimensions; secondly, we introduce a new numerical algorithm for computing optimal designs, complemented with a penalization technique which permits to remove composite designs in the final shape. Since it places no assumption on the number of holes cut within the domain, it can be seen as a topology optimization algorithm. Numerical results are presented for various two and three dimensional problems.
1,291 citations
TL;DR: Although this method is not specifically designed for topology optimization, it can easily handle topology changes for a very large class of objective functions and its cost is moderate since the shape is captured on a fixed Eulerian mesh.
543 citations
TL;DR: In this article, a diffuse-interface method is proposed for the simulation of interfaces between compressible fluids with general equations of state, including tabulated laws, and the interface is allowed to diffuse on a small number of computational cells.
467 citations
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01 Jan 2003
2,810 citations
Proceedings Article•
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TL;DR: In this paper, the authors describe photonic crystals as the analogy between electron waves in crystals and the light waves in artificial periodic dielectric structures, and the interest in periodic structures has been stimulated by the fast development of semiconductor technology that now allows the fabrication of artificial structures, whose period is comparable with the wavelength of light in the visible and infrared ranges.
Abstract: The term photonic crystals appears because of the analogy between electron waves in crystals and the light waves in artificial periodic dielectric structures. During the recent years the investigation of one-, two-and three-dimensional periodic structures has attracted a widespread attention of the world optics community because of great potentiality of such structures in advanced applied optical fields. The interest in periodic structures has been stimulated by the fast development of semiconductor technology that now allows the fabrication of artificial structures, whose period is comparable with the wavelength of light in the visible and infrared ranges.
2,722 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
TL;DR: A new approach to structural topology optimization that represents the structural boundary by a level set model that is embedded in a scalar function of a higher dimension that demonstrates outstanding flexibility of handling topological changes, fidelity of boundary representation and degree of automation.
2,404 citations
TL;DR: A new numerical method based on a combination of the classical shape derivative and of the level-set method for front propagation, which can easily handle topology changes and is strongly dependent on the initial guess.
2,176 citations