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J. Murthy Purdue University

ME 595M: Computational Methods for Nanoscale Thermal Transport Lecture 11: 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 11: 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 1 do i=2,l22 einbot=0.03 eintop=0.04 do nf=1,nfmax5 if(sweight(nf,2).lt.0) then6 einbot = einbot - f(i,2,nf)*sweight(nf,2)7 else8 eintop = eintop + f(i,m2,nf)*sweight(nf,2)9 endif 10 end do11 einbot = einbot/PI12 eintop = eintop/PI13 do nf=1,nfmax14 if(sweight(nf,2).lt.0) then15 f(i,m1,nf) = eintop16 else17 f(i,1,nf)=einbot18 end if19 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 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) W P 100 SW S 100 0 • “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 Can be remedied by higher-order upwinding methods ME 595M J.Murthy

  16. Accuracy Issues (cont’d) 2 1 3 P • 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 • 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 ME 595M J.Murthy

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