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J. Murthy Purdue University

ME 595M: Computational Methods for Nanoscale Thermal Transport Lecture 12: Homework solution Improved numerical techniques. J. Murthy Purdue University. Assignment Problem. Specular or diffuse. T=310 K. T=300 K. Specular or diffuse. Solve the gray BTE using the code in the domain shown:

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J. Murthy Purdue University

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  1. ME 595M: Computational Methods for Nanoscale Thermal TransportLecture 12: Homework solutionImproved numerical techniques J. Murthy Purdue University ME 595M J.Murthy

  2. Assignment Problem Specular or diffuse T=310 K T=300 K Specular or diffuse • 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. 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) ME 595M J.Murthy

  6. Specular Boundaries • Convergence behavior (energy balance to 1%) Why do high acoustic thicknesses take longer to converge? ME 595M J.Murthy

  7. 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 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) • 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. 467-473. 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 Can be remedied by higher-order upwinding methods 100 0 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. 904-910, 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

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