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

本计画旨在介绍、分析及数值实现具非定常流体流动之形状最佳化问题之适定公式。这将是通过利用相场的ananomaly和相关的非光滑均匀的自由能密度，产生强大的数值近似，为了有效的计算，承认几个层次的模型减少范围从自适应网格适当的正交分解。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，这些研究目标在所有项目中将在柏林和汉堡小组之间紧密交织的努力中进行：（a）研究我们的优化框架的连续性和可微性;（B）开发和分析用于自适应网格细化的完全集成的自适应数值方法，用于基于相场的形状和拓扑优化的有效离散化和不稳定流体流;（c）分析优化问题，特别是在相场参数的零极限中的相场模型的后验误差估计器。(d)（e）开发、实施和分析用于具有不稳定流体流的基于相场的形状和拓扑优化的算法解决框架;（f）开发、实施和分析用于基于相场的形状和拓扑优化的降阶模型，其中柏林小组在（a）-（B）中处于领先地位，和（e）-（f）中的汉堡小组。（c）-（d）项中的一揽子措施将大大受益于这两个小组以前在另外两个SPP中的合作。

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.