1. J. Murthy Purdue University
ME 595M: Computational Methods for Nanoscale Thermal Transport Lecture 12: Homework solution Improved numerical techniques
J. Murthy
Purdue University
ME 595M J.Murthy

2. Assignment Problem
Solve the gray BTE using the code in the domain shown:
Investigate acoustic thickesses L/(vg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
ME 595M J.Murthy

3. Specular Boundaries
ME 595M J.Murthy

4. Specular Boundaries (cont’d)
Notice the following about the solution
For L/vg=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/vg=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
ME 595M J.Murthy

5. Specular Boundaries (Cont’d)
L/vg 
0.01
0.1
1.0
10.0
(W/m2)
2.5838e10
2.3787e10
1.4047e10
3.2648e9
0.9901
0.9115
0.5383
0.1251
0.0074
0.0684
0.4037
0.9383
ME 595M J.Murthy

6. Specular Boundaries Convergence behavior (energy balance to 1%)
L/vg 
0.01
0.1
1.0
10.0
100.0
Iterations to convergence 39 52 80
478
5000+
Why do high acoustic thicknesses take longer to converge?
ME 595M J.Murthy

7. Diffuse Boundaries
do i=2,l einbot= eintop= do nf=1,nfmax if(sweight(nf,2).lt.0) then einbot = einbot - f(i,2,nf)*sweight(nf,2) else eintop = eintop + f(i,m2,nf)*sweight(nf,2) endif end do einbot = einbot/PI eintop = eintop/PI do nf=1,nfmax if(sweight(nf,2).lt.0) then f(i,m1,nf) = eintop else f(i,1,nf)=einbot end if end do end do
ME 595M J.Murthy

8. Diffuse Boundaries
ME 595M J.Murthy

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

10. Diffuse Boundaries (cont’d)
L/vg 
0.01
0.1
1.0
10.0
100.0
Iterations to convergence
117
154
193
593
5000+
All acoustic thickesses take longer to converge – why?
ME 595M J.Murthy

11. 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 e0 is in the b term, the coupling to other directions is not good
ME 595M J.Murthy

12. 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
ME 595M J.Murthy

13. 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
ME 595M J.Murthy

14. 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 ?
ME 595M J.Murthy

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

16. 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
ME 595M J.Murthy

17. 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 , 2003.
ME 595M J.Murthy

18. 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
A variety of extensions are being pursued
How to include more exact treatments of the scattering terms
How to couple to electron transport solvers to phonon solvers
How to include confined modes in BTE framework
ME 595M J.Murthy