On a Bernoulli problem with geometric constraints
Antoine Laurain,Yannick Privat +1 more
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In this article, a Bernoulli free boundary problem with geometrical constraints is studied, where the domain is constrained to lie in the half space determined by the hyperplane and its boundary to contain a segment of the hyper plane where non-homogeneous Dirichlet conditions are imposed.Abstract:
A Bernoulli free boundary problem with geometrical constraints is studied. The domain $\Om$ is constrained to lie in the half space determined by $x_1\geq 0$ and its boundary to contain a segment of the hyperplane $\{x_1=0\}$ where non-homogeneous Dirichlet conditions are imposed. We are then looking for the solution of a partial differential equation satisfying a Dirichlet and a Neumann boundary condition simultaneously on the free boundary. The existence and uniqueness of a solution have already been addressed and this paper is devoted first to the study of geometric and asymptotic properties of the solution and then to the numerical treatment of the problem using a shape optimization formulation. The major difficulty and originality of this paper lies in the treatment of the geometric constraints.read more
Citations
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DissertationDOI
On shape optimization with non-linear partial differential equations
TL;DR: In this article, a generalization of the celebrated Theorem of Correa-Seeger for the special class of Lagrangian functions is presented, which simplifies the derivation of necessary optimality conditions for PDE constrained optimization problems.
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Reproducing kernel Hilbert spaces and variable metric algorithms in PDE-constrained shape optimization
Martin Eigel,Kevin Sturm +1 more
TL;DR: This paper examines the usefulness of kernel reproducing Hilbert spaces for PDE-constrained shape optimization problems and shows that radial kernels provide convenient formulas for the shape gradient that can be efficiently used in numerical simulations.
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Some inverse problems around the tokamak Tore Supra
TL;DR: In this paper, two inverse problems related to the tokamak \textsl{Tore Supra} through the study of the magnetostatic equation for the poloidal flux are considered.
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A second-order shape optimization algorithm for solving the exterior Bernoulli free boundary problem using a new boundary cost functional
TL;DR: The exterior Bernoulli problem is rephrased into a shape optimization problem using a new type of objective function called the Dirichlet-data-gap cost function which measures the distance between theDirichlet data of two state functions.
Journal ArticleDOI
On a two-phase Serrin-type problem and its numerical computation
TL;DR: In this paper, an overdetermined problem of Serrin-type with respect to an operator in divergence form with piecewise constant coefficients is considered and sufficient conditions for unique solvability near radially symmetric configurations are given by means of a perturbation argument relying on shape derivatives and the implicit function theorem.
References
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Numerical Optimization
Jorge Nocedal,Stephen J. Wright +1 more
TL;DR: Numerical Optimization presents a comprehensive and up-to-date description of the most effective methods in continuous optimization, responding to the growing interest in optimization in engineering, science, and business by focusing on the methods that are best suited to practical problems.
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Elliptic Problems in Nonsmooth Domains
TL;DR: Second-order boundary value problems in polygons have been studied in this article for convex domains, where the second order boundary value problem can be solved in the Sobolev spaces of Holder functions.
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Iterative Methods for Optimization
TL;DR: Iterative Methods for Optimization does more than cover traditional gradient-based optimization: it is the first book to treat sampling methods, including the Hooke& Jeeves, implicit filtering, MDS, and Nelder& Mead schemes in a unified way.
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Introduction to shape optimization
Jan Sokołowski,Jean-Paul Zolésio +1 more
TL;DR: This book is motivated largely by a desire to solve shape optimization problems that arise in applications, particularly in structural mechanics and in the optimal control of distributed parameter systems.