On a Bernoulli problem with geometric constraints
01 Jan 2012-ESAIM: Control, Optimisation and Calculus of Variations (EDP-Sciences)-Vol. 18, Iss: 1, pp 157-180
TL;DR: In this paper, a Bernoulli free boundary problem with geometrical constraints is studied, where the domain Ω is constrained to lie in the half space determined by x 1 ≥ 0 and its boundary to contain a segment of the hyperplane {x 1 = 0 } where non-homogeneous Dirichlet conditions are imposed.
Abstract: A Bernoulli free boundary problem with geometrical constraints is studied. The domain Ω is constrained to lie in the half space determined by x1 ≥ 0 and its boundary to contain a segment of the hyperplane {x1 =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.
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10 Mar 2015
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.
Abstract: This thesis is concerned with shape optimization problems under non-linear PDE (partial differential equation) constraints. We give a brief introduction to shape optimization and recall important concepts such as shape continuity, shape derivative and the shape differentiability. In order to review existing methods for proving the shape differentiability of PDE constrained shape functions a simple semi-linear model problem is used as constraint. With this example we illustrate the conceptual limits of each method. In the main part of this thesis a new theorem on the differentiability of a minimax function is proved. This fundamental result simplifies the derivation of necessary optimality conditions for PDE constrained optimization problems. It represents a generalization of the celebrated Theorem of Correa-Seeger for the special class of Lagrangian functions and removes the saddle point assumption. Although our method can also be used to compute sensitivities in optimal control, we mainly focus on shape optimization problems. In this respect, we apply the result to four model problems: (i) a semi-linear problem, (ii) an electrical impedance tomography problem, (iii) a model for distortion compensation in elasticity, and finally (iv) a quasi-linear problem describing electro-magnetic fields. Next, we concentrate on methods to minimise shape functions. For this we recall several procedures to put a manifold structure on the space of shapes. Usually, the boundary expression of the shape derivative is used for numerical algorithms. From the numerical point of view this expression has several disadvantages, which will be explained in more detail. In contrast, the volume expression constitutes a numerically more accurate representation of the shape derivative. Additionally, this expression allows us to look at gradient algorithms from two perspectives: the Eulerian and Lagrangian points of view. In the Eulerian approach all computations are performed on the current moving domain. On the other hand the Lagrangian approach allows to perform all calculations on a fixed domain. The Lagrangian view naturally leads to a gradient flow interpretation. The gradient flow depends on the chosen metrics of the underlying function space. We show how different metrics may lead to different optimal designs and different regularity of the resulting domains. In the last part, we give numerical examples using the gradient flow interpretation of the Lagrangian approach. In order to solve the severely ill-posed electrical impedance tomography problem (ii), the discretised gradient flow will be combined with a level-set method. Finally, the problem from example (iv) is solved using B-Splines instead of levelsets.
23 citations
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.
Abstract: In this paper we investigate and compare different gradient algorithms designed for the domain expression of the shape derivative. Our main focus is to examine the usefulness of kernel reproducing ...
15 citations
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.
Abstract: We consider two inverse problems related to the tokamak \textsl{Tore Supra} through the study of the magnetostatic equation for the poloidal flux. The first one deals with the Cauchy issue of recovering in a two dimensional annular domain boundary magnetic values on the inner boundary, namely the limiter, from available overdetermined data on the outer boundary. Using tools from complex analysis and properties of genereralized Hardy spaces, we establish stability and existence properties. Secondly the inverse problem of recovering the shape of the plasma is addressed thank tools of shape optimization. Again results about existence and optimality are provided. They give rise to a fast algorithm of identification which is applied to several numerical simulations computing good results either for the classical harmonic case or for the data coming from \textsl{Tore Supra}.
15 citations
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.
Abstract: 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
$$L^2$$
-distance between the Dirichlet data of two state functions The first-order shape derivative of the cost function is explicitly determined via the chain rule approach Using the same technique, the second-order shape derivative of the cost function at the solution of the free boundary problem is also computed The gradient and Hessian informations are then used to formulate an efficient second-order gradient-based descent algorithm to numerically solve the minimization problem The feasibility of the proposed method is illustrated through various numerical examples
10 citations
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.
Abstract: We consider an overdetermined problem of Serrin-type with respect to an operator in divergence form with piecewise constant coefficients. We give sufficient condition for unique solvability near radially symmetric configurations by means of a perturbation argument relying on shape derivatives and the implicit function theorem. This problem is also treated numerically, by means of a steepest descent algorithm based on a Kohn-Vogelius functional.
7 citations
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