Multi-scale optimization problem for the flow-induced displacement of a periodic filter medium

Abstract A design optimization task in the setting of fluid–structure interaction (FSI) with a periodic filter medium is considered. On the microscale, the thin filter has a small in-plane period and thickness $$\varepsilon $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>ε</mml:mi> </mml:math> and consist of flexural yarns in contact. Its topology, as well as its linear material properties, are dependent on a discrete design variable. A desired flow-induced displacement profile of the filter in steady-state is to be obtained by optimal choice of this variable. The governing state system is a one-way coupled, homogenized and dimension reduced FSI model, attained by the scale limit $$\varepsilon \rightarrow 0$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>ε</mml:mi> <mml:mo>→</mml:mo> <mml:mn>0</mml:mn> </mml:mrow> </mml:math> . The design variable enters in the arising macroscopic model parameters, namely the filter’s homogenized stiffness tensors and its permeability tensor. The latter are attained by the solution of cell-problems on the smallest periodic unit of the filter. The existence of optimal solutions is verified by proving the continuous dependence of these macroscopic model parameters, as well as the design-to-state operator, on changes of the design. A numerical optimization example is provided.