Multi-Scale Shape Optimization under Uncertainty

不确定性下的多尺度形状优化

• 批准号: 25167594
• 负责人: Professor Dr. Sergio Conti
• 金额: --
• 依托单位: Institut für Angewandte Mathematik (IAM)
• 依托单位国家: 德国
• 项目类别: Priority Programmes
• 财政年份: 2006
• 资助国家: 德国
• 起止时间: 2005-12-31 至 2013-12-31
• 项目状态: 已结题
• 来源: https://gepris.dfg.de/gepris/projekt/25167594?language=en
• 关键词: Multi; Scale; Shape; Optimization; under

We will deal with micro-structured materials with macroscopic boundary conditions and stochastic loading in structural mechanics. We will bring together the analytical treatment of multi-scale problems via homogenization, the reliable numerical solution of PDE problems on complex domains described via level sets, and risk neutral and risk averse two-stage stochastic programming approaches. Homogenization and shape optimization in mechanics have already been treated extensively, but usually with deterministic objective functions and without a focus on spatially varying micro-structures. Level set methods are renowned as a flexible tool in shape and topology optimization but usually without accurate, non averaged solution of the involved problems on complex domains. There is a rich theory and algorithmics in two-stage stochastic programming in finite dimensions. But up to now, to the best of our knowledge, not in a PDE optimization context. Uncertain spatial distribution, uncertain modulation of loading and cost functions involving risk of material failure are a characteristic feature in many situations. In particular, we will consider:- two shells with perforated connecting material, loaded in bending, where the locally optimal geometry will vary along the shell,- a simplified model for micro-structured materials in nature with a hard shell filled with porous material, such as bones and micro-structured plant tissue.

我们将讨论结构力学中具有宏观边界条件和随机载荷的微结构材料。我们将汇集多尺度问题的解析处理，通过水平集描述的复杂域上PDE问题的可靠数值解，以及风险中立和风险厌恶两阶段随机规划方法。力学中的均匀化和形状优化已经得到了广泛的研究，但通常具有确定性的目标函数，而没有关注空间变化的微观结构。水平集方法在形状和拓扑优化方面是一种灵活的工具，但通常没有精确的、非平均的复杂域问题的解。有限维两阶段随机规划具有丰富的理论和算法。但到目前为止，据我们所知，还没有在PDE优化上下文中。不确定的空间分布、不确定的载荷调制和涉及材料失效风险的成本函数是许多情况下的特征。特别是，我们将考虑：-两个带有穿孔连接材料的壳，弯曲加载，其中局部最优几何形状将沿壳变化；-自然界中微结构材料的简化模型，其中硬壳填充多孔材料，如骨骼和微结构植物组织。