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Abstract
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 $$\varepsilon $$ ε 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 $$\varepsilon \rightarrow 0$$ ε → 0 . 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.
Highlights
Multi-scale fluid–structure interaction (FSI) problems are frequently encountered in research fields such as biological modeling (Mikelicet al. 2007; Jäger et al 2011; Panasenko and Stavre 2006), civil engineering (Kim and Peskin 2006; Colomés et al 2023) or filtration modeling (Iliev et al 2008, 2004)
The description of the considered flexural structures is performed on a microscale, while the length-scales of the surrounding fluid are significantly larger. This description results in in-efficient mathematical models, in the sense that a sufficiently fine resolution is required to resolve the structures in numerical methods. Utilizing asymptotic approaches such as homogenization and dimension-reduction, this pitfall can be circumvented by deriving effective macroscale models, typically replacing the complex structures by easier to handle plate and shell models
The elegance of these approaches lies in the conveying of microscopic information through the scales: microscopic designs enter the macroscopic system in terms of homogenized model parameters, attained from auxiliary cell problems formulated on representative volume elements of the structure
Summary
Multi-scale fluid–structure interaction (FSI) problems are frequently encountered in research fields such as biological modeling (Mikelicet al. 2007; Jäger et al 2011; Panasenko and Stavre 2006), civil engineering (Kim and Peskin 2006; Colomés et al 2023) or filtration modeling (Iliev et al 2008, 2004). The description of the considered flexural structures (e.g., biological tissue, parachute fabric or textile-like filter media) is performed on a microscale, while the length-scales of the surrounding fluid are significantly larger This description results in in-efficient mathematical models, in the sense that a sufficiently fine resolution is required to resolve the structures in numerical methods. It is remarked that the multi-scale topology optimization is restricted to elasticity problems, as there are numerous alternative applications e.g. in optimal heat conduction and dissipation (Yan et al 2018; Das and Sutradhar 2020) or fluid permeability (Challis et al 2012) The latter reference is of high interest in the FSI setting, as generally counteracting effects of increased permeability and reduced component stiffness are observed for reduced material usage. An example problem for the optimization of a woven PET filter is discussed
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