  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.
