A non-smooth phase-field approach to shape optimization with instationary fluid flow

本项目旨在介绍、分析和数值实现具有瞬时流动的形状优化问题的适定公式。这将通过利用相场ANSAZ和相关的非光滑均匀自由能密度来实现，这产生了稳健的数值近似，并且为了有效计算，允许从自适应网格到适当的正交分解的几个层次的模型简化。此外，达西型方法允许同时优化流体域的拓扑结构。我们考虑了以下研究目标，这些目标将在柏林和汉堡小组之间进行强有力的相互交织的努力：(A)调查我们的优化框架的连续性和可微性；(B)开发和分析完全集成的自适应网格加密方法，用于高效地离散基于相场的形状和拓扑优化和瞬时流体流动的拓扑优化；(C)优化问题的分析，特别是相场模型在相场参数的零极限下的后验误差估计器。(D)为基于相场的形状和拓扑优化与瞬时流体流动的有效离散化的自适应网格细化的双加权残差误差估计的推导；(E)开发、实施和分析基于相场的瞬时流体流动的形状和拓扑优化的算法解决方案框架；(F)开发、实施和分析基于相场的瞬时流体流动的形状和拓扑优化的降阶模型。(C)至(D)项中的一揽子计划将大大受益于这两个小组以前在另外两个国家方案内的合作。

This project aims at introducing, analyzing and numerically realizing well-posed formulations of shape optimization problems with instationary fluid flow. This will be achieved by utilizing a phase-field ansatz and associated non-smooth homogeneous free energy densities, which yield robust numerical approximations and, for the sake of efficient computations, admit several levels of model reduction ranging from adaptive meshing to proper orthogonal decomposition. Moreover, a Darcy-type approach allows to simultaneously optimize the topology of the fluid domain. We consider the following research objectives, which in all items will be carried in a strongly intertwined effort between the Berlin and the Hamburg group:(a) Investigation of the continuity and differentiability properties of our optimization framework;(b) Development and analysis of fully integrated adaptive numerical methods for adaptive mesh refinement for the efficient discretization of phase-field based shape and topology optimization with instationary fluid flow;(c) Analysis of the optimization problem, in particular of the a posteriori error estimator for the phase field model in the zero-limit of the phase field paramete. (d) Derivation of dual-weighted residual based error estimates for adaptive mesh refinement for the efficient discretization of phase-field based shape and topology optimization with instationary fluid flow;(e) Development, implementation and analysis of algorithmic solution frameworks for phase-field based shape and topology optimization with instationary fluid flow;(f) Development, implementation and analysis of reduced order models for phase-field based shape and topology optimization with instationary fluid flow.Here the Berlin group takes the lead in (a)-(b), and the Hamburg group in (e)-(f). The package in (c)-(d) will strongly benefit from the previous collaborations of the two groups within two other SPPs.