1. ME 595M J.Murthy1 ME 595M: Computational Methods for Nanoscale Thermal Transport Lecture 9: Introduction to the Finite Volume Method for the Gray BTE J. Murthy Purdue University

2. ME 595M J.Murthy2 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 e 0 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?

3. ME 595M J.Murthy3 Overview of Finite Volume Method 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 e 0 upon convergence using s

4. ME 595M J.Murthy4 Discretization 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 x y y x z s

5. ME 595M J.Murthy5 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:

6. ME 595M J.Murthy6 Formula for S

7. ME 595M J.Murthy7 Spatial Discretization P E N W S yy xx e w n s e” stored at cell centroids

8. ME 595M J.Murthy8 Control Volume Balance Integrate governing equation over control volume and control angle: faces f n s w e P s

9. ME 595M J.Murthy9 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 s i must be balanced by net in- scattering to the direction i in the CV

10. ME 595M J.Murthy10 Upwinding 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 P E W e

11. ME 595M J.Murthy11 Discussion 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. P E W e

12. ME 595M J.Murthy12 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 e 0 iP Once we have boundary conditions discretized, we can solve the set

13. ME 595M J.Murthy13 A Closer Look Consider a direction s i with s x >0, s y >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/(v g  eff )increases  Diagonally dominant Other directions appear here

14. ME 595M J.Murthy14 Coefficient Structure P E N W S e w n s

15. ME 595M J.Murthy15 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.

16. ME 595M J.Murthy16 Overall Solution Algorithm 1. Initialize all e ” i values for all cell centroids and directions 2. Find e 0 P 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 e 0 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.

17. ME 595M J.Murthy17 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.