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固定流体流动的形状优化问题的适定公式。这将通过利用相场ansatz和相关的非光滑均匀自由能量密度来实现，这些自由能量密度产生鲁棒的数值近似，并且为了高效计算，允许从自适应网格划分到适当正交分解的几个级别的模型缩减。此外，darcy型方法允许同时优化流体域的拓扑结构。我们考虑以下研究目标，这些目标在所有项目中都将在柏林和汉堡小组之间紧密交织的努力中进行：(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数值方法，用于自适应网格细化，以有效地离散基于相场的形状和拓扑优化与固定流体流动；(c)分析优化问题，特别是相场参数为零极限时相场模型的后验误差估计。(d)推导基于双加权残差的自适应网格优化误差估计，以实现基于相场的形状和拓扑优化与固定流体流动的有效离散化；(e)开发、实施和分析具有固定流体流动的基于相场的形状和拓扑优化的算法解决框架；(f)针对固定流体流动的基于相场的形状和拓扑优化的降阶模型的开发、实施和分析。这里，柏林组在(a)-(b)中领先，汉堡组在(e)-(f)中领先。(c)-(d)中的一揽子计划将从两个小组以前在另外两个特别方案内的合作中受益匪浅。

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.