1. ME 595M J.Murthy1 ME 595M: Computational Methods for Nanoscale Thermal Transport Lecture 11: Homework solution Improved numerical techniques J. Murthy Purdue University

2. ME 595M J.Murthy2 Assignment Problem Solve the gray BTE using the code in the domain shown: Investigate acoustic thickesses L/(v g  eff) =0.01,0.1,1,10,100 Plot dimensionless “temperature” versus x/L on horizontal centerline Program diffuse boundary conditions instead of specular, and investigate the same range of acoustic thicknesses. Plot dimensionless “temperature” on horizontal centerline again. Submit commented copy of user subroutines (not main code) with your plots. T=310 K T=300 K Specular or diffuse

3. ME 595M J.Murthy3 Specular Boundaries

4. ME 595M J.Murthy4 Specular Boundaries (cont’d) Notice the following about the solution  For L/v g  =0.01, we get the dimensionles temperature to be approximately 0.5 throughout the domain – why?  Notice the discontinuity in t * at the boundaries – why?  For L/v g  =10.0, we get nearly a straight line profile – why?  In the ballistic limit, we would expect a heat flux of  In the thick limit, we would expect a flux of

5. ME 595M J.Murthy5 Specular Boundaries (Cont’d) L/v g  0.01 0.1 1.0 10.0 (W/m 2 ) 2.5838e102.3787e101.4047e103.2648e9 0.99010.91150.53830.1251 0.00740.06840.40370.9383

6. ME 595M J.Murthy6 Specular Boundaries Convergence behavior (energy balance to 1%) L/v g  0.010.11.0 10.0100.0 Iterations to convergence 39 52804785000+ Why do high acoustic thicknesses take longer to converge?

7. ME 595M J.Murthy7 Diffuse Boundaries 1 do i=2,l2 2 einbot=0.0 3 eintop=0.0 4 do nf=1,nfmax 5 if(sweight(nf,2).lt.0) then 6 einbot = einbot - f(i,2,nf)*sweight(nf,2) 7 else 8 eintop = eintop + f(i,m2,nf)*sweight(nf,2) 9 endif 10 end do 11 einbot = einbot/PI 12 eintop = eintop/PI 13 do nf=1,nfmax 14 if(sweight(nf,2).lt.0) then 15 f(i,m1,nf) = eintop 16 else 17 f(i,1,nf)=einbot 18 end if 19 end do 20 end do If ray incoming to boundary, sum incoming energy Find average incoming energy Set energy of all outgoing directions to average incoming value

8. ME 595M J.Murthy8 Diffuse Boundaries

9. ME 595M J.Murthy9 Diffuse Boundaries (cont’d) Notice the following about the solution  Solution is relatively insensitive to L/v g .  We get diffusion-like solutions over the entire range of acoustic thickness - why?  Specular problem is 1D but diffuse problem is 2D

10. ME 595M J.Murthy10 Diffuse Boundaries (cont’d) All acoustic thickesses take longer to converge – why? L/v g  0.010.11.0 10.0100.0 Iterations to convergence 117 1541935935000+

11. ME 595M J.Murthy11 Convergence Issues Why do high acoustic thicknesses take long to converge? Answer has to do with the sequential nature of the algorithm Recall that the dimensionless BTE has the form As acoustic thickness increases, coupling to BTE’s in other directions becomes stronger, and coupling to spatial neighbors in the same direction becomes less important. Our coefficient matrix couples spatial neighbors in the same direction well, but since e 0 is in the b term, the coupling to other directions is not good

12. ME 595M J.Murthy12 Point-Coupled Technique A cure is to solve all BTE directions at a cell simultaneously, assuming spatial neighbors to be temporarily known Sweep through the mesh doing a type of Gauss-Seidel iteration This technique is still too slow because of the slow speed at which boundary information is swept into the interior Coupling to a multigrid method substantially accelerates the solution Mathur, S.R. and Murthy, J.Y.; Coupled Ordinate Method for Multi-Grid Acceleration of Radiation Calculations; Journal of Thermophysics and Heat Transfer, Vol. 13, No. 4, 1999, pp. 467-473.

13. ME 595M J.Murthy13 Coupled Ordinate Method (COMET) Solve BTE in all directions at a point simultaneously Use point coupled solution as relaxation sweep in multigrid method Unsteady conduction in trapezoidal cavity 4x4 angular discretization per octant 650 triangular cells Time step =  /100

14. ME 595M J.Murthy14 Accuracy Issues Ray effect  Angular domain is divided into finite control angles  Influence of small features is smeared  Resolve angle better  Higher-order angular discretization ?

15. ME 595M J.Murthy15 Accuracy Issues (cont’d) “False scattering” – also known as false diffusion in the CFD literature P S W SW 100 0 P picks up an average of S and W instead of the value at SW Can be remedied by higher-order upwinding methods

16. ME 595M J.Murthy16 Accuracy Issues (cont’d) Additional accuracy issues arise when the unsteady BTE must be solved If the angular discretization is coarse, time of travel from boundary to interior may be erroneous P 1 2 3

17. ME 595M J.Murthy17 Modified FV Method Finite angular discretization => erroneous estimation of phonon travel time for coarse angular discretizations Modified FV method e ” 1 problem solved by ray tracing; e ” 2 solved by finite volume method Conventional Modified Murthy, J.Y. and Mathur, S.R.; An Improved Computational Procedure for Sub-Micron Heat Conduction; J. Heat Transfer, vol. 125, pp. 904-910, 2003.

18. ME 595M J.Murthy18 Closure We developed the gray energy form of the BTE and developed common boundary conditions for the equation We developed a finite volume method for the gray BTE We examined the properties of typical solutions with specular and diffuse boundaries We examined a variety of BTE extensions  gray., full-dispersion, full-scattering Many new areas to pursue  How to include more exact treatments of the scattering terms using interatomic potentials  How to couple to electron transport solvers to phonon solvers  How to include interfacial transport in a BTE framework