46 research outputs found
On a Bernoulli problem with geometric constraints
A Bernoulli free boundary problem with geometrical constraints is studied.
The domain \Om is constrained to lie in the half space determined by and its boundary to contain a segment of the hyperplane 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
Distributed shape derivative via averaged adjoint method and applications
The structure theorem of Hadamard-Zol\'esio states that the derivative of a
shape functional is a distribution on the boundary of the domain depending only
on the normal perturbations of a smooth enough boundary. Actually the domain
representation, also known as distributed shape derivative, is more general
than the boundary expression as it is well-defined for shapes having a lower
regularity. It is customary in the shape optimization literature to assume
regularity of the domains and use the boundary expression of the shape
derivative for numerical algorithms. In this paper we describe several
advantages of the distributed shape derivative in terms of generality, easiness
of computation and numerical implementation. We identify a tensor
representation of the distributed shape derivative, study its properties and
show how it allows to recover the boundary expression directly. We use a novel
Lagrangian approach, which is applicable to a large class of shape optimization
problems, to compute the distributed shape derivative. We also apply the
technique to retrieve the distributed shape derivative for electrical impedance
tomography. Finally we explain how to adapt the level set method to the
distributed shape derivative framework and present numerical results
Domain expression of the shape derivative and application to electrical impedance tomography
The well-known structure theorem of Hadamard-Zolesio states that the derivative of a shape functional is a distribution on the boundary of the domain depending only on the normal perturbations of a smooth enough boundary. However a volume representation (distributed shape derivative) is more general than the boundary form and allows to work with shapes having a lower regularity. It is customary in the shape optimization literature to assume regularity of the domains and use the boundary expression of the shape derivative for numerical algorithm. In this paper we describe the numerous advantages of the distributed shape derivative in terms of generality, easiness of computation and numerical implementation. We give several examples of numerical applications such as the inverse conductivity problem and the level set method
A bilevel shape optimization problem for the exterior Bernoulli free boundary value problem
Dieser Beitrag ist mit Zustimmung des Rechteinhabers aufgrund einer (DFG geförderten) Allianz- bzw. Nationallizenz frei zugänglich.This publication is with permission of the rights owner freely accessible due to an Alliance licence and a national licence (funded by the DFG, German Research Foundation) respectively.A bilevel shape optimization problem with the exterior Bernoulli free boundary problem as lowerlevel problem and the control of the free boundary as the upper-level problem is considered. Using the shape of the inner boundary as the control, we aim at reaching a specific shape for the free boundary. A rigorous sensitivity analysis of the bilevel shape optimization in the infinite-dimensional setting is performed. The numerical realization using two different cost functionals presented in this paper demonstrate the efficiency of the approach.FWF, SFB F32, Mathematical Optimization and Applications in Biomedical SciencesDFG, MATHEON C37, Shape/Topology optimization methods for inverse problem
Level set method with topological derivatives in shape optimization
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