ME 595M J.Murthy 1

ME 595M: Computational Methods for Nanoscale Thermal Transport

Lecture 12: Homework solutionImproved numerical techniques

J. Murthy

Purdue University

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Assignment Problem• Solve the gray BTE using the code in the domain shown:

• Investigate acoustic thickesses L/(vgeff) =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

Specular or diffuse

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Specular Boundaries

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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

4 g left right

Cq v T T

21

3 g left rightq Cv T TL

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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

ballistic

q

q

diffuse

q

q

q

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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?

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Diffuse Boundaries 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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Diffuse Boundaries

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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

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Diffuse Boundaries (cont’d)

• All acoustic thickesses take longer to converge – why?

L/vg 0.01 0.1 1.0 10.0 100.0

Iterations to convergence

117

154 193 593 5000+

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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

*

* 0* **

g eff

f Lf f f

t v

s

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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.

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Coupled Ordinate Method (COMET)Sequential COMETAcoustic Thickness

CPU secs Iters CPU secs Iters2 Bands0.0484 98.68 5 79.52 50.484 98.85 5 69.62 54.84 97.72 6 68.54 548.4 162.25 10 67.95 5484.0 529.4 33 61.22 5

10 Bands0.0484 484.02 6 424.36 60.484 476.23 6 338.19 54.84 772.17 9 338.38 548.4 2922.16 29 337.03 5484.0 19,006.7 191 354.74 5

20 Bands0.0484 970.04 6 958.01 60.484 960.8 6 882.56 64.84 1609.13 10 754.18 548.4 5701.64 34 828.82 6484.0 39,333.2 225 921.48 6

•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

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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 ?

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Accuracy Issues (cont’d)• “False scattering” – also known as false diffusion in the CFD

literature

P

S

W

SW

100

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

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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

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Modified FV Method

1 2

1 11

0 02 1 2 2

2

1

1

g g eff

g g eff

e e e

e ee

v t v

e e e e

v t v

s

se

• 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.

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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