A consistent relaxation of optimal design problems for coupling shape and topological derivatives
In this article, we introduce and analyze a general procedure for approximating a ‘black and white’ shape and topology optimization problem with a density optimization problem, allowing for the presence of ‘grayscale’ regions. Our construction relies on a regularizing operator for smearing the chara...
| Autores: | , , |
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| Tipo de documento: | artigo |
| Data de publicação: | 2018 |
| País: | España |
| Recursos: | Universitat Politècnica de Catalunya (UPC) |
| Repositório: | UPCommons. Portal del coneixement obert de la UPC |
| Idioma: | inglês |
| OAI Identifier: | oai:upcommons.upc.edu:2117/118037 |
| Acesso em linha: | https://hdl.handle.net/2117/118037 https://dx.doi.org/10.1007/s00211-018-0964-4 |
| Access Level: | Acceso aberto |
| Palavra-chave: | Structural optimization -- Mathematics Multiscale modeling--Computer simulation Level set method Material interpolation Shape derivative Relaxation Optimal design topological derivative Optimització d'estructures Modelització multiescala Classificació AMS::74 Mechanics of deformable solids::74P Optimization Classificació AMS::35 Partial differential equations::35Q Equations of mathematical physics and other areas of application Classificació AMS::49 Calculus of variations and optimal control optimization::49Q Manifolds Classificació AMS::65 Numerical analysis::65N Partial differential equations, boundary value problems optimization::49M Methods of successive approximations Àrees temàtiques de la UPC::Matemàtiques i estadística::Topologia |
| Resumo: | In this article, we introduce and analyze a general procedure for approximating a ‘black and white’ shape and topology optimization problem with a density optimization problem, allowing for the presence of ‘grayscale’ regions. Our construction relies on a regularizing operator for smearing the characteristic functions involved in the exact optimization problem, and on an interpolation scheme, which endows the intermediate density regions with fictitious material properties. Under mild hypotheses on the smoothing operator and on the interpolation scheme, we prove that the features of the approximate density optimization problem (material properties, objective function, etc.) converge to their exact counterparts as the smoothing parameter vanishes. In particular, the gradient of the approximate objective functional with respect to the density function converges to either the shape or the topological derivative of the exact objective. These results shed new light on the connections between these two different notions of sensitivities for functions of the domain, and they give rise to different numerical algorithms which are illustrated by several experiments |
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