ME 595M: Computational Methods for
Nanoscale Thermal Transport
J. Murthy
Purdue University
ME 595M J.Murthy
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• Solve the gray BTE using the code in the domain shown:
Specular or diffuse
T=310 K T=300 K
Specular or diffuse
• 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.
ME 595M J.Murthy
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ME 595M J.Murthy
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• 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
 q
 
C v g

T left

T right

4
In the thick limit, we would expect a flux of q
 
1
3 L
Cv g
2


T left

T right

ME 595M J.Murthy
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L/v g

0.01
0.1
1.0
10.0
q

(W/m 2 ) q
 q
 ballistic
2.5838e10 2.3787e10 1.4047e10 3.2648e9
0.9901
0.9115
0.5383
0.1251
q
 q
 diffuse
0.0074
0.0684
0.4037
0.9383
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• Convergence behavior (energy balance to 1%)
L/v g

Iterations to convergen ce
0.01
39
0.1
52
1.0
80
10.0
100.0
478 5000+
Why do high acoustic thicknesses take longer to converge?
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do i=2,l2 einbot=0.0
eintop=0.0
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
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ME 595M J.Murthy
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• 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
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L/v g

0.01
0.1
1.0
10.0
100.0
Iterations to convergen ce
117 154 193 593 5000+
• All acoustic thickesses take longer to converge
– why?
ME 595M J.Murthy
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• 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
 f
 t
*
* s f
 f 0*  f *
 L v
 g eff
• 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
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• 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.
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Coupled Ordinate Method (COMET)
Acoustic Thickness
2 Bands
0.0484
0.484
4.84
48.4
484.0
Sequential COMET
CPU secs Iters CPU secs Iters
98.68
98.85
5
5
97.72
6
162.25
10
529.4
33
79.52
69.62
68.54
67.95
61.22
5
5
5
5
5
•Solve BTE in all directions at a point simultaneously
•Use point coupled solution as relaxation sweep in multigrid method
10 Bands
0.0484
0.484
4.84
48.4
484.0
484.02
6
476.23
6
772.17
9
2922.16
29
19,006.7
191
424.36
6
338.19
5
338.38
5
337.03
5
354.74
5
•Unsteady conduction in trapezoidal cavity
•4x4 angular discretization per octant
20 Bands
0.0484
0.484
4.84
48.4
484.0
970.04
6
960.8
6
1609.13
10
5701.64
34
39,333.2
225
958.01
6
882.56
6
754.18
5
828.82
6
921.48
6
•650 triangular cells
•Time step = 
/100
ME 595M J.Murthy
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• Ray effect
 Angular domain is divided into finite control angles
 Influence of small features is smeared
 Resolve angle better
 Higher-order angular discretization ?
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• “False scattering” – also known as false diffusion in the CFD literature
W
100
P
P picks up an average of
S and W instead of the value at SW
SW
S
100
0
 Can be remedied by higher-order upwinding methods
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• 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
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Modified FV Method
Conventional
• Finite angular discretization => erroneous estimation of phonon travel time for coarse angular discretizations
• Modified FV method e
 e
1 e
2
1
 e
1
 v g
 t
    
1
   e
1
 v
 g eff
1
 e
2
 v g
 t
    e
0
1
 
2 e
2
 v
 g eff
Modified
• e
”
1 problem solved by ray tracing; e solved by finite volume method
”
2
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.
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• 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
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