Characterization of optimal shapes and masses through Monge-Kantorovich equation

Bouchitté, Guy; Buttazzo, Giuseppe

doi:10.1007/s100970000027

Cited by 120 publications

(176 citation statements)

References 19 publications

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“…The strict relation between some mass optimization problems and optimal transportation results has been recently shown by Bouchitté and Buttazzo in [23]; we shall shortly summarize the results obtained and we shall present some new challenging problems.…”

Section: Introductionmentioning

confidence: 66%

“…The link between the mass optimization problem (15) and the Monge-Kantorovich equation above has been investigated in [23]; we summarize here the result.…”

Section: Remark 34mentioning

confidence: 95%

“…where the integral is intended in the sense of convex functionals on measures, and the following result holds (see [23]). …”

Section: Remark 32mentioning

confidence: 99%

“…By denoting by Lip 1 (Ω, d Ω,Σ ) the class of functions defined in Ω that are 1-Lipschitz continuous with respect to the distance d Ω,Σ one may obtain a dual formulation of problem (35) in the form (see [23]):…”

Section: Natural Extension Of the Cost Functional Is Given Bymentioning

confidence: 99%

“…A system of partial differential equations can be associated to the MongeKantorovich problem (see for instance [23], [99], [100]). In a simplified version (i.e.…”

Section: The Pde Formulation Of the Mass Transportation Problemmentioning

confidence: 99%

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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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Section: Introductionmentioning

confidence: 66%

“…The link between the mass optimization problem (15) and the Monge-Kantorovich equation above has been investigated in [23]; we summarize here the result.…”

Section: Remark 34mentioning

confidence: 95%

“…where the integral is intended in the sense of convex functionals on measures, and the following result holds (see [23]). …”

Section: Remark 32mentioning

confidence: 99%

Section: Natural Extension Of the Cost Functional Is Given Bymentioning

confidence: 99%

“…A system of partial differential equations can be associated to the MongeKantorovich problem (see for instance [23], [99], [100]). In a simplified version (i.e.…”

Section: The Pde Formulation Of the Mass Transportation Problemmentioning

confidence: 99%

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Optimal Shapes and Masses, and Optimal Transportation Problems

Buttazzo

Pascale

2003

Optimal Transportation and Applications

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Optimal and approximate boundary controls of an elastic body with quasistatic evolution of damage

Kogut

Leugering

2014

Math Methods in App Sciences

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Communicated by M. BrokateIn this paper, we study an optimal control problem for the mixed boundary value problem for an elastic body with quasistatic evolution of an internal damage variable. We suppose that the evolution of microscopic cracks and cavities responsible for the damage is described by a nonlinear parabolic equation. A density of surface traction p acting on a part of boundary of an elastic body is taken as a boundary control. Because the initial boundary value problem of this type can exhibit the Lavrentieff phenomenon and non-uniqueness of weak solutions, we deal with the solvability of this problem in the class of weak variational solutions. Using the convergence concept in variable spaces and following the direct method in calculus of variations, we prove the existence of optimal and approximate solutions to the optimal control problem under rather general assumptions on the quasistatic evolution of damage.

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Pareto Optimal Design of Non‐Homogeneous Isotropic Material Properties for the Multiple Loading Conditions

Czarnecki

Lewiński

2017

Physica Status Solidi (b)

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The present paper concerns two methods: IMD (isotropic material design) and YMD (Young modulus material design) concerning Pareto optimal distributions of the material and its isotropic properties. The merit function is the weighted sum of compliances corresponding to n independent loading conditions. The unit cost of the design is assumed as equal to the trace of the elastic moduli tensor C. In the IMD method, the design variables are the bulk and shear moduli. In the YMD method, the Poisson ratio is assumed as given, possibly non‐uniformly distributed in the design domain, the Young modulus being the only design variable. Both the methods reduce to the auxiliary minimization problem involving statically admissible stress fields corresponding to the subsequent loading conditions. The integrand is expressed by a norm of the collection of these stresses, hence has linear growth. The numerical method proposed refers to the primal formulation: it is based on piece‐wise polynomial approximation of the set of statically admissible stresses. The exemplary Pareto optimal solutions show that admitting negative values of Poisson's ratio may contribute to a decrease of the total compliances.

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Characterization of optimal shapes and masses through Monge-Kantorovich equation

Abstract: International audienceno abstrac

Cited by 120 publications

References 19 publications

Optimal Shapes and Masses, and Optimal Transportation Problems

Optimal Shapes and Masses, and Optimal Transportation Problems

Optimal and approximate boundary controls of an elastic body with quasistatic evolution of damage

Pareto Optimal Design of Non‐Homogeneous Isotropic Material Properties for the Multiple Loading Conditions

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