Shape optimization by the homogenization method

Allaire, Grégoire; Bonnetier, Éric; Francfort, Gilles A.; Jouve, François

doi:10.1007/s002110050253

Cited by 605 publications

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“…Problem (1) with design set (3) (regularized through a geometrie eonstraint) is a large seale diserete optimization problem and it has in this form only reeently been handled with sueeess eomputationally ( [6]). The most popular method for solving (1), and an approach that has been extremely sueeessful for many applieations, is to eonsider formulations in terms of eontinuous variables, with the goal of using derivative based mathematieal programming algorithms. This me ans that one ehanges the model for the material properties to a situation where the volume fraetion is allowed to take on any value between zero and one.…”

Section: Basic Problem Statementmentioning

confidence: 99%

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Topology Design of Structures, Materials and Mechanisms — Status and Perspectives

Bendsøe

2000

System Modelling and Optimization

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INTRODUCTIONThe field of variable-topology shape design in structural optimization has its origins in theoretical studies of existence of solutions in variational problems, in particular shape optimization problems, and in studies in theoretical material science of variational bounds on material properties. Progress in computational methods and the ever increasing computer power has oriented the area towards applications, with significant developments being achieved over the last decade, leading to a fairly widespread use of the methodology in industry.In this short paper we outline some of the basic ideas and methods of existing methods, but it is not our purpose to cover all work and approaches in this field. Instead we refer to existing literat ure containing rather comprehensive surveys, see e.g., [7,8,31]. Moreover, note that reference is mostly made to recent papers that include bibliographies useful for on overview of the area. Thus the presentation does not try to present a complete historie al perspective.The area of computational variable-topology shape design of continuum structures is presently dominated by methods which employ a material distribution approach for a fixed reference domain, in the spirit of the so-called 'homogenization method' for topology design ([3, 9]). That is, the geometrie representation of a structure is similar to a grey-scale rendering of an image, in discrete form corresponding to araster representation of the geometry on a fixed reference domain. The physics of the problem is also represented by boundary conditions and forcing The original version of this chapter was revised: The copyright line was incorrect. This has been corrected. The Erratum to this chapter is available at DOI: 10.1007/978-0-387-35514-6_15 2 Martin P. BendsrJe terms defined on this fixed reference domain, much in analogy to fictitious domain methods for FEM analysis.One can normally distinguish between three versions of raster based geometry models for continuum topology optimization. The basic problem is an unrestricted '0-1' integer design problem (generalized shape optimization), that is, a design specifies unambiguously whether there is solid material or void at every point in a candidate design region. Otherwise, there are no restrictions on the shape. Unfortunately, in general, this dass of problems is ill-posed in the continuum setting (cf., [11,20]). Well-posed problems can be obtained by either extending the space of admissible solutions to obtain their relaxed versions, usually by incorporating microstructure (see, e.g., [3]), or by restricting the space of admissible solutions. The latter can be accomplished by enforcing an upper bound on the perimeter of the structure (see [28], and references therein), by imposing constraints on the slopes of the parameters defining geometry (see [29], and references therein), by the introduction of a filtering function limiting the minimum scale (see [10,36] for an overview), or one can introduce a ground structure with a fixed number of design degrees of...

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Section: Basic Problem Statementmentioning

confidence: 99%

“…The continuum relaxation approach can be very involved theoretically (see for example [1,13]) and much work is still needed in this area.…”

Section: Introductionmentioning

confidence: 99%

Topology Design of Structures, Materials and Mechanisms — Status and Perspectives

Bendsøe

2000

System Modelling and Optimization

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“…The theory developed here generalizes an example given in [15], reviewed in §2, that itself stemmed from developments in the calculus of variations, the theory of topology optimization and the theory of composites: see the books [16][17][18][19]. A key component of this is the Fourier space methods developed by Tartar & Murat [7][8][9] in their theory of compensated compactness for determining the quasi-convexity of quadratic forms.…”

Section: Introductionmentioning

confidence: 98%

“…An associated problem, which we are also interested in, is to obtain sharp bounds on integrals of the form 17) in which the bar denotes complex conjugation. Observe that the operators in (1.11) and (1.17) are formal adjoints.…”

Section: Introductionmentioning

confidence: 99%

Sharp inequalities that generalize the divergence theorem: an extension of the notion of quasi-convexity

Milton

2013

Proc. R. Soc. A.

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Subject to suitable boundary conditions being imposed, sharp inequalities are obtained on integrals over a region Ω of certain special quadratic functions f (E), where E(x) derives from a potential U(x). With E = ∇U, it is known that such sharp inequalities can be obtained when f (E) is a quasi-convex function and when U satisfies affine boundary conditions (i.e. for some matrix D, U = Dx on ∂Ω). Here, we allow for other boundary conditions and for fields E that involve derivatives of a variety orders of U. We define a notion of convexity that generalizes quasi-convexity. Q * -convex quadratic functions are introduced, characterized, and an algorithm is given for generating sharply Q * -convex functions. We emphasize that this also solves the outstanding problem of finding an algorithm for generating extremal quasi-convex quadratic functions. We also treat integrals over Ω of special quadratic functions g(J), where J(x) satisfies a differential constraint involving derivatives with, possibly, a variety of orders. The results generalize an example of Kang, and the author in three spatial dimensions where J(x) is a 3 × 3 matrix-valued field satisfying ∇ · J = 0.

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“…We shall not give here a detailed presentation of the many problems and results in this very wide field, but we limit ourselves to discuss some model problems. We refer the reader interested in a deeper knowledge and analysis of this fascinating field to one of the several books on the subject ( [3], [114], [142], [146]), to the notes by L. Tartar [149], or to the recent collection of lecture notes by D. Bucur and G. Buttazzo [37].…”

Section: Introductionmentioning

confidence: 99%

Optimal Shapes and Masses, and Optimal Transportation Problems

Buttazzo

Pascale

2003

Optimal Transportation and Applications

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Summary. We present here a survey on some shape optimization problems that received particular attention in the last years. In particular, we discuss a class of problems, that we call mass optimization problems, where one wants to find the distribution of a given amount of mass which optimizes a given cost functional. The relation with mass transportation problems will be discussed, and several open problems will be presented.

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Shape optimization by the homogenization method

Cited by 605 publications

References 25 publications

Topology Design of Structures, Materials and Mechanisms — Status and Perspectives

Topology Design of Structures, Materials and Mechanisms — Status and Perspectives

Sharp inequalities that generalize the divergence theorem: an extension of the notion of quasi-convexity

Optimal Shapes and Masses, and Optimal Transportation Problems

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