How to pose material design problems for flow through porous media applications?: Sensitivity of dissipation rate to medium's permeability holds the key

Recent studies have advocated using the total dissipation rate under topology optimization to realize material designs involving the flow of fluids through porous media. However, these studies decided how to pose the design problem, such as maximizing the total dissipation rate for some situations while minimizing for others, by solving one-dimensional problems and justifying their choices using numerical experiments. The rigor is lacking -- a bottleneck for further scientific advancements to computational material design. This paper provides the missing theoretical justification. We identify four classes of boundary value problems using the adjoint state method and analytically calculate the sensitivity of the total dissipation rate to the permeability field. For two of those classes in which the flow of fluids is pressure-driven, the sensitivity is positive -- the total dissipation rate increases if the medium's permeability increases. While for the other two classes, in which the flow is velocity-driven, the trend is the opposite. These sensitivities provide rigorous answers to the central question: how to pose a material design problem for flow through porous media applications. The impact of our work is multi-fold. First, this study further elevates the role of the dissipation rate in posing well-posed material design problems using topology optimization. Second, besides the theoretical significance, the results benefit computational scientists and practitioners to realize optimal designs. Third, given their simplicity yet far-reaching impact, both the approach and results possess immense pedagogical value.