Shape optimization for non-smooth geometry in two dimensions
type: Item
instance of: scholarly article
title: Shape optimization for non-smooth geometry in two dimensions
zbMATH Open document ID: 0978.74059
DOI: 10.1016/S0045-7825(99)00141-3
author: M. Souli
author: J. P. Zolesio
author: Abdellatif Ouahsine
published in: Computer Methods in Applied Mechanics and Engineering
publication date: 2002-01-31T00:00:00Z
review text: The derivative of a functional \(J(u,\Omega)\) with respect to the domain \(\Omega\), where \(u\) is the solution of a boundary value problem in \(\Omega\), was broadly studied when \(\Omega\) is a smooth domain. Let \(\Omega\) be a non-smooth domain in \(\mathbb{R}^2\) such that its boundary \(\Gamma\) exhibits singularities at points \(a_i\), and let \(u\) be the solution of a boundary value problem in \(\Omega\). This paper shows how to differentiate such functional, and how to deduce the contributions coming from singularities. The main advantage of this expression for the functional gradient is to avoid the computation of the curvature of the boundary, arising generally for the gradient for non-singular domains. From the numerical point of view, this method allows to compute the gradient of a functional for polygonal domains in \(\mathbb{R}^2\), using a finite element approximation. Practically, one considers a continuous gradient for a smooth domain, that one will discretize, or a gradient with respect to nodes, which is a discrete gradient for a discrete domain. Then the desired gradient lies between these two gradients, being a continuous gradient for a discrete domain.
reviewed by: Ladislav Prášek
Mathematics Subject Classification ID: 74P10
Mathematics Subject Classification ID: 74S05
zbMATH DE Number: 1506859
zbMATH Keywords: shape optimization
zbMATH Keywords: polygonal domain
zbMATH Keywords: derivative of functional
zbMATH Keywords: boundary value problem
zbMATH Keywords: non-smooth domain
zbMATH Keywords: finite element approximation
zbMATH Keywords: discrete domain
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 1506859