Abstract
We construct a space of vector fields that are normal to differentiable curves in the plane. Its basis functions are defined via saddle point variational problems in reproducing kernel Hilbert spac...
Highlights
Shape optimization studies how to design a domain \Omega such that a shape functional J is minimized
This information needs to be taken into account to prove convergence rates of shape Newton methods [27, 31] and has a direct consequence for numerical computations: if the discretization of domain perturbations is not chosen carefully, the matrix that results by restricting the second-order shape derivative to the discrete trial space may fail to be positive definite
We use reproducing kernel Hilbert spaces (RKHSs) to define a class of basis vector fields that are weakly normal to the boundary of a domain \Omega and whose support has nonzero measure
Summary
Shape optimization studies how to design a domain \Omega such that a shape functional J is minimized. Shape optimization problems arise naturally in numerous industrial applications. Allaire, Canc\es, and Vi\e' [1] introduced domain perturbations in terms of Hamilton--Jacobi equations. These definitions lead to the same formulas for first-order shape derivatives but differ for second-order ones. The equivalence of the three approaches for the first shape derivative stems from the fact that each approach constructs domain perturbations from vector fields, and the Taylor series expansions of these constructions agree to first order. Regardless of the definition used, both first- and second-order shape derivatives have a nontrivial kernel. \ast Received by the editors May 24, 2017; accepted for publication (in revised form) September 27, 2018; published electronically January 3, 2019
Sign-in/Register to view highlights & summary Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Schedule a callDisclaimer: All third-party content on this website/platform is and will remain the property of their respective owners and is provided on "as is" basis without any warranties, express or implied. Use of third-party content does not indicate any affiliation, sponsorship with or endorsement by them. Any references to third-party content is to identify the corresponding services and shall be considered fair use under The CopyrightLaw.
