{"title":"拓扑优化算法的总周长泛函的一致近似","authors":"S. Amstutz, C. Dapogny, À. Ferrer","doi":"10.1051/cocv/2022005","DOIUrl":null,"url":null,"abstract":"This article revolves around the total perimeter functional, one particular version of the perimeter of a shape [[EQUATION]] contained in a fixed computational domain D\nmeasuring the total area of its boundary [[EQUATION]], as opposed to its relative perimeter, which only takes into account the regions of [[EQUATION]] strictly inside [[EQUATION]].\nWe analyze approximate versions of the total perimeter which make sense for general ``density functions'' [[EQUATION]]. Their use in the context of density-based topology optimization is particularly convenient as they do not involve the gradient of the optimized function [[EQUATION]]. Two different constructions are proposed: while the first one involves the convolution of the function [[EQUATION]] with a smooth mollifier, the second one relies on the resolution of an elliptic boundary-value problem featuring Robin boundary conditions. The ``consistency'' of these approximations is appraised from various points of view. At first, we prove the pointwise convergence of our approximate functionals, then the convergence of their derivatives, as the level of smoothing tends to [[EQUATION]], when the considered density function [[EQUATION]] is the characteristic function of a shape [[EQUATION]]. Then, we focus on the [[EQUATION]]-convergence of the second type of approximate total perimeter functional. Several numerical examples are eventually presented.","PeriodicalId":50500,"journal":{"name":"Esaim-Control Optimisation and Calculus of Variations","volume":null,"pages":null},"PeriodicalIF":1.3000,"publicationDate":"2022-01-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"3","resultStr":"{\"title\":\"A consistent approximation of the total perimeter functional for topology optimization algorithms\",\"authors\":\"S. Amstutz, C. Dapogny, À. Ferrer\",\"doi\":\"10.1051/cocv/2022005\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"This article revolves around the total perimeter functional, one particular version of the perimeter of a shape [[EQUATION]] contained in a fixed computational domain D\\nmeasuring the total area of its boundary [[EQUATION]], as opposed to its relative perimeter, which only takes into account the regions of [[EQUATION]] strictly inside [[EQUATION]].\\nWe analyze approximate versions of the total perimeter which make sense for general ``density functions'' [[EQUATION]]. Their use in the context of density-based topology optimization is particularly convenient as they do not involve the gradient of the optimized function [[EQUATION]]. Two different constructions are proposed: while the first one involves the convolution of the function [[EQUATION]] with a smooth mollifier, the second one relies on the resolution of an elliptic boundary-value problem featuring Robin boundary conditions. The ``consistency'' of these approximations is appraised from various points of view. At first, we prove the pointwise convergence of our approximate functionals, then the convergence of their derivatives, as the level of smoothing tends to [[EQUATION]], when the considered density function [[EQUATION]] is the characteristic function of a shape [[EQUATION]]. Then, we focus on the [[EQUATION]]-convergence of the second type of approximate total perimeter functional. Several numerical examples are eventually presented.\",\"PeriodicalId\":50500,\"journal\":{\"name\":\"Esaim-Control Optimisation and Calculus of Variations\",\"volume\":null,\"pages\":null},\"PeriodicalIF\":1.3000,\"publicationDate\":\"2022-01-20\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"3\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"Esaim-Control Optimisation and Calculus of Variations\",\"FirstCategoryId\":\"100\",\"ListUrlMain\":\"https://doi.org/10.1051/cocv/2022005\",\"RegionNum\":3,\"RegionCategory\":\"数学\",\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"Q4\",\"JCRName\":\"AUTOMATION & CONTROL SYSTEMS\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"Esaim-Control Optimisation and Calculus of Variations","FirstCategoryId":"100","ListUrlMain":"https://doi.org/10.1051/cocv/2022005","RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q4","JCRName":"AUTOMATION & CONTROL SYSTEMS","Score":null,"Total":0}
A consistent approximation of the total perimeter functional for topology optimization algorithms
This article revolves around the total perimeter functional, one particular version of the perimeter of a shape [[EQUATION]] contained in a fixed computational domain D
measuring the total area of its boundary [[EQUATION]], as opposed to its relative perimeter, which only takes into account the regions of [[EQUATION]] strictly inside [[EQUATION]].
We analyze approximate versions of the total perimeter which make sense for general ``density functions'' [[EQUATION]]. Their use in the context of density-based topology optimization is particularly convenient as they do not involve the gradient of the optimized function [[EQUATION]]. Two different constructions are proposed: while the first one involves the convolution of the function [[EQUATION]] with a smooth mollifier, the second one relies on the resolution of an elliptic boundary-value problem featuring Robin boundary conditions. The ``consistency'' of these approximations is appraised from various points of view. At first, we prove the pointwise convergence of our approximate functionals, then the convergence of their derivatives, as the level of smoothing tends to [[EQUATION]], when the considered density function [[EQUATION]] is the characteristic function of a shape [[EQUATION]]. Then, we focus on the [[EQUATION]]-convergence of the second type of approximate total perimeter functional. Several numerical examples are eventually presented.
期刊介绍:
ESAIM: COCV strives to publish rapidly and efficiently papers and surveys in the areas of Control, Optimisation and Calculus of Variations.
Articles may be theoretical, computational, or both, and they will cover contemporary subjects with impact in forefront technology, biosciences, materials science, computer vision, continuum physics, decision sciences and other allied disciplines.
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in control: modeling, controllability, optimal control, stabilization, control design, hybrid control, robustness analysis, numerical and computational methods for control, stochastic or deterministic, continuous or discrete control systems, finite-dimensional or infinite-dimensional control systems, geometric control, quantum control, game theory;
in optimisation: mathematical programming, large scale systems, stochastic optimisation, combinatorial optimisation, shape optimisation, convex or nonsmooth optimisation, inverse problems, interior point methods, duality methods, numerical methods, convergence and complexity, global optimisation, optimisation and dynamical systems, optimal transport, machine learning, image or signal analysis;
in calculus of variations: variational methods for differential equations and Hamiltonian systems, variational inequalities; semicontinuity and convergence, existence and regularity of minimizers and critical points of functionals, relaxation; geometric problems and the use and development of geometric measure theory tools; problems involving randomness; viscosity solutions; numerical methods; homogenization, multiscale and singular perturbation problems.