171 research outputs found
Reproducing kernel Hilbert spaces and variable metric algorithms in PDE constrained shape optimisation
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 Hilbert spaces for PDE
constrained shape optimisation problems. We show that radial kernels provide
convenient formulas for the shape gradient that can be efficiently used in
numerical simulations. The shape gradients associated with radial kernels
depend on a so called smoothing parameter that allows a smoothness adjustment
of the shape during the optimisation process. Besides, this smoothing parameter
can be used to modify the movement of the shape. The theoretical findings are
verified in a number of numerical experiments
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
Shape optimisation for a class of semilinear variational inequalities with applications to damage models
The present contribution investigates shape optimisation problems for a class
of semilinear elliptic variational inequalities with Neumann boundary
conditions. Sensitivity estimates and material derivatives are firstly derived
in an abstract operator setting where the operators are defined on polyhedral
subsets of reflexive Banach spaces. The results are then refined for
variational inequalities arising from minimisation problems for certain convex
energy functionals considered over upper obstacle sets in . One
particularity is that we allow for dynamic obstacle functions which may arise
from another optimisation problems. We prove a strong convergence property for
the material derivative and establish state-shape derivatives under regularity
assumptions. Finally, as a concrete application from continuum mechanics, we
show how the dynamic obstacle case can be used to treat shape optimisation
problems for time-discretised brittle damage models for elastic solids. We
derive a necessary optimality system for optimal shapes whose state variables
approximate desired damage patterns and/or displacement fields
Optimal actuator design based on shape calculus
An approach to optimal actuator design based on shape and topology
optimisation techniques is presented. For linear diffusion equations, two
scenarios are considered. For the first one, best actuators are determined
depending on a given initial condition. In the second scenario, optimal
actuators are determined based on all initial conditions not exceeding a chosen
norm. Shape and topological sensitivities of these cost functionals are
determined. A numerical algorithm for optimal actuator design based on the
sensitivities and a level-set method is presented. Numerical results support
the proposed methodology.Comment: 41 pages, several figure
Lagrange method in shape optimization for non-linear partial differential equations: A material derivative free approach
This paper studies the relationship between the material derivative method, the shape derivative method, the min-max formulation of Correa and Seeger, and the Lagrange method introduced by Céa. A theorem is formulated which allows a rigorous proof of the shape differentiability without the usage of material derivative; the domain expression is automatically obtained and the boundary expression is easy to derive. Furthermore, the theorem is applied to a cost function which depends on a quasi-linear transmission problem. Using a Gagliardo penalization the existence of optimal shapes is established
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