J. Murthy Purdue University

1. ME 595M: Computational Methods for Nanoscale Thermal TransportLecture 8: Introduction to the Finite Volume Method for the Gray BTE J. Murthy Purdue University ME 595M J.Murthy

2. Gray Phonon BTE • Recall gray phonon BTE: • e” is energy per unit volume per unit solid angle and depends on direction vector s. So there are as many pde’s as there are s directions. In each direction, e” varies in space and time • The e” values in different directions are related to each other because of e0 in the scattering term: • Notice that • This implies that there is no net energy source – scattering only shifts energy from one direction to another • How would you add an net energy source to the gray BTE? ME 595M J.Murthy

3. Overview of Finite Volume Method s • Divide spatial domain into control volumes of extent xy • Divide angular domain into control angles of extent  • Divide time into steps of t - but will only do steady here for simplicity • Consider gray BTE in direction s. Integrate gray BTE over control volume and corresponding control angle. Get energy conservation statement for that direction for each spatial control volume • Do the same for all directions. • Solve each direction sequentially and iteratively • Back out “temperature” from e0 upon convergence using ME 595M J.Murthy

4. Discretization z y s x x y • Divide domain into rectangular control volumes of extent x and y. Assume 2D, so that depth into page (z) is one unit. • Divide angular domain of 4 in NxN control angles per octant. Centroid of each control angle is (i , i), extents are (, ). • For each control angle i: • Important: The directions s are 3D even though we are considering 2D ME 595M J.Murthy

5. Discretization (cont’d) • Control angle extent is • In 2D, only directions in the “front” hemisphere are necessary. Thus  ranges from 0-/2 and =0-2 • Thus, increase control angle extent to: • Define for future use: ME 595M J.Murthy

6. Formula for S ME 595M J.Murthy

7. Spatial Discretization N n W E P w e y s x S e” stored at cell centroids ME 595M J.Murthy

8. Control Volume Balance s n P w e s faces f • Integrate governing equation over control volume and control angle ME 595M J.Murthy

9. Control Volume Balance (cont’d) • Now look at RHS • Collecting terms: • Control volume balance says that net rate of energy entering the CV in direction si must be balanced by net in-scattering to the direction i in the CV ME 595M J.Murthy

10. Upwinding E P W e • e” is stored at cell centroids, but we need it on the CV faces • Need to interpolate from cell centroid to face • Can use a variety of schemes to perform interpolation • Central difference scheme • Second-order accurate, but wiggles in spatial solution • Upwind difference scheme • Computationally convenient to write ME 595M J.Murthy

11. Discussion E P W e • Upwinding, as shown, is only a first-order accurate scheme • Guaranteed smooth, bounded solutions • False diffusion • In CFD, a variety of higher-order upwind-weighted schemes have been developed which typically involve other upwind points (P, W for face e) • Will go with first-order upwind scheme for now. ME 595M J.Murthy

12. Discrete Equation • Using upwinding and collecting terms, we obtain an algebraic equation: • We obtain one such equation for each grid point P for each direction i • The b term contains e0iP • Once we have boundary conditions discretized, we can solve the set ME 595M J.Murthy

13. A Closer Look Other directions appear here • Consider a direction si with sx >0, sy>0 Point p only connected to points south and west of it Influence of other directions in b term Influence of b term increases as acoustic thickness L/(vgeff )increases Diagonally dominant ME 595M J.Murthy

14. Coefficient Structure N n W E P w e s S ME 595M J.Murthy

15. Discussion • Prefer to solve iteratively and if possible, sequentially to keep memory requirements low • For upwind scheme, diagonal dominance is guaranteed, making it possible to use iterative schemes • Conservation of energy is guaranteed regardless of spatial and angular discretization • Confirm that sum of all scattering source terms at a point is zero regardless of discretization • Any linear solver can be used – will use line-by-line tri-diagonal matrix algorithm (LBL-TDMA) for now. ME 595M J.Murthy

16. Overall Solution Algorithm 1. Initialize all e”i values for all cell centroids and directions 2. Find e0P for each point P from current e” values. 3. Start with direction i=1 4. For direction i: • Find discretized equations for direction i, assuming e0 temporarily known • Solve for e”i at all grid points using LBL-TDMA • Increment I as i=i+1 5. If (i.le.4*N*N) go to 4 6. If (i>4*N*N) check for convergence. If converged, stop. Else, go to 2. ME 595M J.Murthy

17. Conclusions • In this lecture, we discretized the gray BTE. • The discretization is guaranteed to give energy conservation regardless of the fineness of the spatial or angular discretization • The discretization guarantees diagonal dominance and is hence suitable for iterative solvers such as the LBL TDMA. • The next time, we will talk briefly of boundary conditions, and start looking at a finite volume code to solve the BTE. ME 595M J.Murthy