Risk averse elastic shape optimization with parametrized fine scale geometry

• 成果类型: Article
• 署名作者: Geihe, Benedict; Lenz, Martin; Rumpf, Martin; Schultz, Ruediger
• 署名单位: University of Bonn; University of Duisburg Essen
• 刊物名称: MATHEMATICAL PROGRAMMING
• ISSN/ISSBN: 0025-5610
• DOI: 10.1007/s10107-012-0531-1
• 发表日期: 2013
• 页码: 383-403
• 关键词: level-set method; topology optimization; optimal-design

摘要

Shape optimization of the fine scale geometry of elastic objects is investigated under stochastic loading. Thus, the object geometry is described via parametrized geometric details placed on a regular lattice. Here, in a two dimensional set up we focus on ellipsoidal holes as the fine scale geometric details described by the semiaxes and their orientation. Optimization of a deterministic cost functional as well as stochastic loading with risk neutral and risk averse stochastic cost functionals are discussed. Under the assumption of linear elasticity and quadratic objective functions the computational cost scales linearly in the number of basis loads spanning the possibly large set of all realizations of the stochastic loading. The resulting shape optimization algorithm consists of a finite dimensional, constraint optimization scheme where the cost functional and its gradient are evaluated applying a boundary element method on the fine scale geometry. Various numerical results show the spatial variation of the geometric domain structures and the appearance of strongly anisotropic patterns.