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ABSTRACT
The phonon Boltzmann transport equation (BTE) is an important governing equation for phonon transport at the subcontinuum scale. Numerically solving the BTE can help to study the heat transfer phenomena at microscale and nanoscale, for example, in electronic devices or nanocomposites. In this work, we developed a collocation mesh-free method to solve the BTE for different heat transfer regimes. The proposed numerical scheme does not require meshing the domain or performing numerical integration, and is thus advantageous for problems with complicated geometries. A few case studies show that our method can yield comparable results with semianalytical and finite volume methods.
Nomenclature
| C | = | volumetric heat capacity (J/K m3) |
| e | = | directional energy density (J/m3 sr) |
| e0 | = | equilibrium energy density (J/m3) |
| E | = | total nodal energy density vector |
| K | = | total coefficient matrix |
| Kn | = | Knudsen number |
| l | = | phonon mean free path (m) |
| L | = | width and height of the 2D domain (m) |
| Nθ | = | number of polar angle discretization |
| Nφ | = | number of azimuthal angle discretization |
| n | = | normal vector to surface |
| R | = | radial basis function |
| p | = | polarization |
| q″ | = | energy source of phonon (W/m3) |
| r | = | position vector |
| s | = | direction vector |
| sr | = | specular direction |
| t | = | time (s) |
| T | = | temperature (K) |
| Tb | = | temperature at boundary (K) |
| Tref | = | temperature at reference (K) |
| V | = | the computational domain |
| ∂V | = | the boundary of V |
| v | = | velocity vector |
| vg | = | phonon group velocity magnitude (m/s) |
| x | = | position vector |
| τ | = | relaxation time (s) |
| θ | = | polar angle (rad) |
| φ | = | azimuthal angle (rad) |
| Ω | = | solid angle (sr) |
| ω | = | angular frequency (/s) |
Nomenclature
| C | = | volumetric heat capacity (J/K m3) |
| e | = | directional energy density (J/m3 sr) |
| e0 | = | equilibrium energy density (J/m3) |
| E | = | total nodal energy density vector |
| K | = | total coefficient matrix |
| Kn | = | Knudsen number |
| l | = | phonon mean free path (m) |
| L | = | width and height of the 2D domain (m) |
| Nθ | = | number of polar angle discretization |
| Nφ | = | number of azimuthal angle discretization |
| n | = | normal vector to surface |
| R | = | radial basis function |
| p | = | polarization |
| q″ | = | energy source of phonon (W/m3) |
| r | = | position vector |
| s | = | direction vector |
| sr | = | specular direction |
| t | = | time (s) |
| T | = | temperature (K) |
| Tb | = | temperature at boundary (K) |
| Tref | = | temperature at reference (K) |
| V | = | the computational domain |
| ∂V | = | the boundary of V |
| v | = | velocity vector |
| vg | = | phonon group velocity magnitude (m/s) |
| x | = | position vector |
| τ | = | relaxation time (s) |
| θ | = | polar angle (rad) |
| φ | = | azimuthal angle (rad) |
| Ω | = | solid angle (sr) |
| ω | = | angular frequency (/s) |
Acknowledgments
This work is supported by the National Natural Science Foundation of China (Grant Nos. 51306111 and 11402146) and Shanghai Municipal Natural Science Foundation (Grant No. 13ZR1456000). Yongxing Shen acknowledges the financial support from the Young 1000 Talent Program.