{"title":"Differentiation with Respect to Domains of Boundary Integral Functionals Involving Support Functions","authors":"Abdesslam Boulkhemair, Abdelkrim Chakib, Azeddine Sadik","doi":"10.1007/s00245-024-10168-9","DOIUrl":null,"url":null,"abstract":"<div><p>The aim of this paper is to establish a new formula for the computation of the shape derivative of boundary integral cost functionals using Minkowski deformation of star-shaped domains by convex ones. The formula is expressed by means of the support function of the convex domain. The proof uses some geometrical tools in addition to an analysis of star-shapedness involving gauge functions. Finally, in order to illustrate this result, the formula is applied for solving an optimal shape design problem of minimizing a surface cost functional constrained to elliptic boundary value problem, using the gradient method performed by the finite element approximation.\n</p></div>","PeriodicalId":55566,"journal":{"name":"Applied Mathematics and Optimization","volume":"90 1","pages":""},"PeriodicalIF":1.6000,"publicationDate":"2024-07-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Applied Mathematics and Optimization","FirstCategoryId":"100","ListUrlMain":"https://link.springer.com/article/10.1007/s00245-024-10168-9","RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q2","JCRName":"MATHEMATICS, APPLIED","Score":null,"Total":0}
引用次数: 0
Abstract
The aim of this paper is to establish a new formula for the computation of the shape derivative of boundary integral cost functionals using Minkowski deformation of star-shaped domains by convex ones. The formula is expressed by means of the support function of the convex domain. The proof uses some geometrical tools in addition to an analysis of star-shapedness involving gauge functions. Finally, in order to illustrate this result, the formula is applied for solving an optimal shape design problem of minimizing a surface cost functional constrained to elliptic boundary value problem, using the gradient method performed by the finite element approximation.
期刊介绍：
The Applied Mathematics and Optimization Journal covers a broad range of mathematical methods in particular those that bridge with optimization and have some connection with applications. Core topics include calculus of variations, partial differential equations, stochastic control, optimization of deterministic or stochastic systems in discrete or continuous time, homogenization, control theory, mean field games, dynamic games and optimal transport. Algorithmic, data analytic, machine learning and numerical methods which support the modeling and analysis of optimization problems are encouraged. Of great interest are papers which show some novel idea in either the theory or model which include some connection with potential applications in science and engineering.
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