We consider the method of mappings for performing shape optimization for unsteady fluid-structure interaction (FSI) problems. In this work, we focus on the numerical implementation. We model the optimization problem such that it takes several theoretical results into account, such as regularity requirements on the transformations and a differential geometrical point of view on the manifold of shapes. Moreover, we discretize the problem such that we can compute exact discrete gradients. This allows for the use of general purpose optimization solvers. We focus on an FSI benchmark problem to validate our numerical implementation. The method is used to optimize parts of the outer boundary and the interface. The numerical simulations build on FEniCS, dolfin-adjoint and IPOPT. Moreover, as an additional theoretical result, we show that for a linear special case the adjoint attains the same structure as the forward problem but reverses the temporal flow of information.

翻译：本文研究采用映射法进行非定常流体-结构相互作用（FSI）问题的形状优化。本工作重点关注数值实现方法。我们在建模优化问题时综合考虑了多项理论成果，包括变换的正则性要求以及形状流形上的微分几何观点。此外，我们对问题进行了离散化处理，从而能够计算精确的离散梯度，这使得通用优化求解器的应用成为可能。我们以FSI基准问题为对象验证数值实现的有效性。该方法被用于优化外部边界与交界面的部分区域。数值模拟基于FEniCS、dolfin-adjoint和IPOPT平台实现。在理论层面，我们还证明了在线性特殊情形下，伴随方程保持了与正问题相同的数学结构，但实现了信息时间流向的逆转。