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

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

Specular Boundaries ME 595M J.Murthy

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

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

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

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

Diffuse Boundaries ME 595M J.Murthy

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

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

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

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

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

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

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

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

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

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