Shape Variation and Optimization

Henrot, Antoine; Pierre, Michel

doi:10.4171/178

Cited by 191 publications

(319 citation statements)

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“…smoothness of (v, ξ) → div(A(v)∇ξ) is proved as in[11, Theorem 5.3.2] and we have indeed establishedF ∈ C 1 S × H 1 0 (Ω(u)), H −1 (Ω(u)) . By the Lax-Milgram theorem, the mapping ζ → ∂ ξ F (u, χ u )ζ = −div(σ∇ζ) is bijective from H 1 0 (Ω(u)) to H −1 (Ω(u))and thus an isomorphism due to the open mapping theorem.…”

mentioning

confidence: 56%

“…Next, due to the regularity properties of ψ u provided by Theorem 1.1, we can compute the shape derivative of the Dirichlet energy J(u) with respect to u ∈ S in a classical way [11,20]. Theorem 1.2.…”

Section: Introduction and Main Resultsmentioning

confidence: 99%

“…a classical result in shape optimization states that the shape derivative of J(O) is given by [11,20]. When the shape derivative is well-defined, it provides useful information on the Dirichlet energy itself and it is the basis for deriving first-order optimality conditions.…”

Section: Introducing the Dirichlet Integralmentioning

confidence: 99%

“…When the shape derivative is well-defined, it provides useful information on the Dirichlet energy itself and it is the basis for deriving first-order optimality conditions. However, the integral on the right-hand side of the shape derivative is only meaningful provided ϕ O has sufficient regularity (typically ϕ O ∈ H 2 (O)), which, in turn, requires sufficient regularity of the source term f and the open set O, see [11, Corollary 5.3.8] for instance. Source terms with low Sobolev regularity or depending on the admissible shape O in a non-smooth way are therefore excluded.Amongst the simplest situations featuring such a dependence is the differentiability with respect to O of the Dirichlet energy…”

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Shape Derivative of the Dirichlet Energy for a Transmission Problem

Laurençot

Walker

2020

Arch Rational Mech Anal

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For a transmission problem in a truncated two-dimensional cylinder located beneath the graph of a function u, the shape derivative of the Dirichlet energy (with respect to u) is shown to be well-defined and is computed. The main difficulties in this context arise from the weak regularity of the domain and the possible non-empty intersection of the graph of u and the transmission interface. The result is applied to establish the existence of a solution to a free boundary transmission problem for an electrostatic actuator. Introducing the Dirichlet integrala classical result in shape optimization states that the shape derivative of J(O) is given by [11,20]. When the shape derivative is well-defined, it provides useful information on the Dirichlet energy itself and it is the basis for deriving first-order optimality conditions. However, the integral on the right-hand side of the shape derivative is only meaningful provided ϕ O has sufficient regularity (typically ϕ O ∈ H 2 (O)), which, in turn, requires sufficient regularity of the source term f and the open set O, see [11, Corollary 5.3.8] for instance. Source terms with low Sobolev regularity or depending on the admissible shape O in a non-smooth way are therefore excluded.Amongst the simplest situations featuring such a dependence is the differentiability with respect to O of the Dirichlet energy

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confidence: 56%

Section: Introduction and Main Resultsmentioning

confidence: 99%

Section: Introducing the Dirichlet Integralmentioning

confidence: 99%

mentioning

confidence: 99%

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Shape Derivative of the Dirichlet Energy for a Transmission Problem

Laurençot

Walker

2020

Arch Rational Mech Anal

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“…As a matter of fact, advanced mathematical programming methods are not frequently described in the literature devoted to shape optimization based on Hadamard's method (see [35,56] for an introduction). In most contributions, where, usually, only one constraint is considered, standard Penalty and Augmented Lagrangian Methods are used for the sake of implementation simplicity [9,23].…”

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confidence: 99%

Null space gradient flows for constrained optimization with applications to shape optimization

2020

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The purpose of this article is to introduce a gradient-flow algorithm for solving equality and inequality constrained optimization problems, which is particularly suited for shape optimiza- tion applications. We rely on a variant of the ODE approach proposed by Yamashita for equality constrained problems: the search direction is a combina- tion of a null space step and a range space step, aiming to decrease the value of the minimized objective function and the violation of the constraints, respectively. Our first contribution is to propose an extension of this ODE approach to optimization problems featuring both equality and inequality constraints. Here, we solve their local combinatorial character by computing the projection of the gradient of the ob- jective function onto the cone of feasible directions. This is achieved by solving a dual quadratic programming subproblem. The solution to this problem allows to identify the inequality constraints to which the optimization tra- jectory should remain tangent. Our second contribution is a formulation of our gradient flow in the context of|in nite-dimensional|Hilbert spaces, and of even more general optimization sets such as sets of shapes, as it occurs in shape optimization within the framework of Hadamard's boundary variation method.

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Continuity Equation and Characteristic Flow for Scalar Hencky Plasticity

Babadjian

Francfort

2022

Comm Pure Appl Math

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We investigate uniqueness issues for a continuity equation arising out of the simplest model for plasticity, Hencky plasticity. The associated system is of the form curl (µσ ) = 0 where µ is a nonnegative measure and σ a two-dimensional divergence free unit vector field. After establishing the Sobolev regularity of that field, we provide a precise description of all possible geometries of the characteristic flow, as well as of the associated solutions.1. Introduction 2. Notation and preliminaries 3. Hencky plasticity 4. Sobolev regularity of the stress 5. Rigidity properties of the solutions 6. Geometry of the solutions Appendix. Bibliography

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Shape Variation and Optimization

Cited by 191 publications

References 0 publications

Shape Derivative of the Dirichlet Energy for a Transmission Problem

Shape Derivative of the Dirichlet Energy for a Transmission Problem

Null space gradient flows for constrained optimization with applications to shape optimization

Continuity Equation and Characteristic Flow for Scalar Hencky Plasticity

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