Adjoint-aided homogenization for flows through non-periodic membranes*

Porous membranes, thin structures that permit fluid to pass through their pores, are crucial in many industrial and biological processes. Multiscale homogenization has been successfully used to predict flows through membranes, providing a macroscopic model where the membrane is described as an infinitely thin interface between different fluid regions. The model links the flow velocity and stress on the membrane through a set of coefficients (such as permeability and slip) derived from a single pore-level solution of Stokes problems. The geometry of a single pore determines these coefficients for the entire membrane in the case of a periodic microstructure. However, real membranes usually lack periodicity. In this instance, recovering the local non-periodic membrane properties requires numerous microscopic calculations, adversely impacting the homogenized model's efficiency.

In this contribution, we evaluate, via an adjoint-based methodology, the first- and second-order sensitivities of the pore-scale solution to geometry alterations. The calculated sensitivity functions allow us to predict the non-periodic microscopic membrane properties, starting from one single microscopic calculation of a base pore-scale geometry. We test this novel methodology on a membrane with a random microstructure, finding similar accuracy to previously proposed techniques at a fraction of the computational cost.