  First-principles calculations of thermal transport in homogeneous materials
have reached remarkable predicting power. Modeling deterministically phonon
transport in nanostructures, however, poses novel challenges; notably, it
entails solving as many algebraic equations as the number of combinations of
wave vectors in the discretized Brillouen and polarizations. We show that,
within the relaxation time approximation of the Boltzmann transport equation
(BTE), this issue is resolved by interpolating the phonon distributions in the
vectorial phonon mean free paths (MFP) space. The coupling between structure
and mode-resolved heat transport is investigated in terms of angular-resolved
bulk thermal conductivity and phonon suppression function, the latter being
associated primarily to the material's geometry. Our method, termed the
anisotropic MFP-BTE (aMFP-BTE), allows for fast and accurate thermal
conductivity calculations in nanomaterials regardless of the number of phonon
branches and wave vectors. Furthermore, it naturally blends with
first-principles thermal transport calculations, therefore allowing for
multiscale, parameter-free simulations. We apply the aMFP-BTE to compute the
mode-resolved effective thermal conductivity of porous Si membranes, achieving
up to 50x speed with respect to the case with no interpolation. The proposed
approach unlocks the engineering of novel nanostructures, with applications to
thermoelectrics and heat management.
