On optimal designs using topology optimization for flow through porous media applications

TL;DR

This paper advances topology optimization for flow in porous media by incorporating nonlinear effects like pressure-dependent viscosity and inertial forces, moving beyond traditional Darcy-based models to achieve more realistic device designs.

Contribution

It introduces a general strategy for topology optimization that accounts for nonlinear flow models and proposes analytical solutions for canonical problems to validate computational methods.

Findings

Nonlinear models significantly alter optimal material layouts.

Rate of mechanical dissipation is a valid objective for nonlinear flows.

Analytical solutions aid in verifying computational implementations.

Abstract

Topology optimization (TopOpt) is a mathematical-driven design procedure to realize optimal material architectures. This procedure is often used to automate the design of devices involving flow through porous media, such as micro-fluidic devices. TopOpt offers material layouts that control the flow of fluids through porous materials, providing desired functionalities. Many prior studies in this application area have used Darcy equations for primal analysis and the minimum power theorem (MPT) to drive the optimization problem. But both these choices (Darcy equations and MPT) are restrictive and not valid for general working conditions of modern devices. Being simple and linear, Darcy equations are often used to model flow of fluids through porous media. However, two inherent assumptions of the Darcy model are: the viscosity of a fluid is a constant, and inertial effects are negligible. There is irrefutable experimental evidence that viscosity of a fluid, especially organic liquids, depends on the pressure. Given the typical small pore-sizes, inertial effects are dominant in micro-fluidic devices. Next, MPT is not a general principle and is not valid for (nonlinear) models that relax the assumptions of the Darcy model. This paper aims to overcome the mentioned deficiencies by presenting a general strategy for using TopOpt. First, we will consider nonlinear models that take into account the pressure-dependent viscosity and inertial effects, and study the effect of these nonlinearities on the optimal material layouts under TopOpt. Second, we will explore the rate of mechanical dissipation, valid even for nonlinear models, as an alternative for the objective function. Third, we will present analytical solutions of optimal designs for canonical problems; these solutions not only possess research and pedagogical values but also facilitate verification of computer implementations.