Introduction to shape optimization

Sokołowski, Jan; Zolésio, Jean-Paul

doi:10.1007/978-3-642-58106-9_1

Cited by 362 publications

(427 citation statements)

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“…It stays close to the classical optimization problems and gives sufficient optimality conditions of first order, while in [7,8,19] only second order optimality conditions are stated which require more regularity of the data and regularity of the perturbation of domains. We provide a dynamic programming approach to structural optimization problems.…”

Section: Introductionmentioning

confidence: 89%

“…Following Zolésio [19], for a given Φ(v), we introduce the field (again v is a parameter, not a variable)…”

Section: Reduction Of Shape Optimization Problem To a Classical Contrmentioning

confidence: 99%

“…In combination with a fast wavelet Galerkin method to solve the boundary integral equations, some very efficient first and second order algorithms for shape problems in two and three space dimensions were developed. In the monograph [19], the material derivative method is employed to calculate both the sensitivity of solutions and the derivatives of domain depending functionals with respect to variations of the boundary of the domain occupied by the body. To formulate the necessary optimality condition for shape and topology optimization, the shape and topological derivatives are employed.…”

Section: Introductionmentioning

confidence: 99%

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Dynamic programming approach to structural optimization problem - numerical algorithm

Fulmański¹,

Nowakowski²,

Pustelnik³

2014

OpMath

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Abstract. In this paper a new shape optimization algorithm is presented. As a model application we consider state problems related to fluid mechanics, namely the Navier-Stokes equations for viscous incompressible fluids. The general approach to the problem is described. Next, transformations to classical optimal control problems are presented. Then, the dynamic programming approach is used and sufficient conditions for the shape optimization problem are given. A new numerical method to find the approximate value function is developed.

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Section: Introductionmentioning

confidence: 89%

“…Following Zolésio [19], for a given Φ(v), we introduce the field (again v is a parameter, not a variable)…”

Section: Reduction Of Shape Optimization Problem To a Classical Contrmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

Dynamic programming approach to structural optimization problem - numerical algorithm

Fulmański¹,

Nowakowski²,

Pustelnik³

2014

OpMath

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show abstract

“…The variational formulation of the system (1.3-1.8) corresponds to the following system of variational inequality and equation [5]: 14) where K t is the constraint set defined by the obstacle 15) and H 1 0 (Ω t ), H 2 0 (Ω t ) are the usual Sobolev spaces defined by…”

Section: Formulation Of the Nonlinear Obstacle Plate Problem In ω Tmentioning

confidence: 99%

“…2 is majorated by a constant independent of s. In fact, from the definition of G φ2(s) (φ 1 (s)) we obtain, for each s, 15) and choosing v φ2(s) = G φ2(s) (φ 1 (s)), using the ellipticity of B φ2(s) , the continuity of b φ2(s) and the relations between the norms of the spaces [ …”

Section: Proposition 34 If Supp µ Is Admissible Then the Set K Is mentioning

confidence: 99%

Sensitivity Analysis of a Nonlinear Obstacle Plate Problem

Figueiredo

Leal

2002

ESAIM: COCV

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Abstract.We analyse the sensitivity of the solution of a nonlinear obstacle plate problem, with respect to small perturbations of the middle plane of the plate. This analysis, which generalizes the results of [9,10] for the linear case, is done by application of an abstract variational result [6], where the sensitivity of parameterized variational inequalities in Banach spaces, without uniqueness of solution, is quantified in terms of a generalized derivative, that is the proto-derivative. We prove that the hypotheses required by this abstract sensitivity result are verified for the nonlinear obstacle plate problem. Namely, the constraint set defined by the obstacle is polyhedric and the mapping involved in the definition of the plate problem, considered as a function of the middle plane of the plate, is semi-differentiable. The verification of these two conditions enable to conclude that the sensitivity is characterized by the proto-derivative of the solution mapping associated with the nonlinear obstacle plate problem, in terms of the solution of a variational inequality.Mathematics Subject Classification. 49A29, 90C31, 74B20, 74K20.

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Sensitivity of Interfacial Cracks to Non-linear Crack Front Perturbations

Kovtunenko

2002

Zeitschrift für Angewandte Mathematik und Mechanik

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Introduction to shape optimization

Cited by 362 publications

References 0 publications

Dynamic programming approach to structural optimization problem - numerical algorithm

Dynamic programming approach to structural optimization problem - numerical algorithm

Sensitivity Analysis of a Nonlinear Obstacle Plate Problem

Sensitivity of Interfacial Cracks to Non-linear Crack Front Perturbations

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