107 90 Denis Lemonnier Fig. Directions defining the S 4 quadrature (first octant) if the emission from the wall is given, and I m (x w,t)= p w m I m (x w,t) s m n +(1 p)i ˆm (x w,t), (48) π m s.t.s m n 0, and then all others are determined by symmetry over the full 4π steradians of solid angle. 108 Solution of the Boltzmann Equation for Phonon Transport 91 Fig. Schematic representation of quadratures S 4 to S 12 in the first octant of the sphere. All directions carrying the same number are attributed the same weight To order N, thes N quadrature comprises M = N(N + 2) directions. Hence S 2 comprises 8 directions, S 4 has 24 and S 6 has 48. However, in 2D configurations, the symmetries of the problem mean that one need only use half of the directions, multiplying each weight by 2.
PHYSIQUE DES SOLIDES by Neil William Ashcroft, N.David Mermin (Contributor), Neil-W Ashcroft, Mermin Aschcroft, N. David Mermin, N-David Mermin Hardcover. Physique Des Solides Ashcroft Mermin Pdf Editor. (Bohm and Staver, 1951; Ashcroft and Mermin, 1976 and later textbooks), where m and M denote, respectively, the masses of electrons and ions in the solid, VF is the Fermi velocity and Z is the charge of the ions.
In addition to the symmetry conditions already mentioned, 19 the weights w m must satisfy M w m = m=1 4π dω =4π, (49) in order to provide a good representation of the radiative emission at all points in the medium. Note that another desirable condition, 20 viz., M µ m w m = m=1 4π µ dω =0, (50) is automatically satisfied due to the quadrature symmetries. Moreover, we have seen that the incident fluxes on the boundary walls are calculated by integrating, not over the full 4π sr, but over just half of the space. This is why we also generally require M m s.t.µ m. 110 Solution of the Boltzmann Equation for Phonon Transport 93 Fig. Control volume for integrating the transfer equation in Cartesian coordinates of directions 21 (up to restrictions on computation time) and are reputed to reduce certain inaccuracies due to the discretisation of the propagation directions. 22 A review of the various possible quadratures can be found in 21.
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Setting up Cartesian Coordinates For clarity, we restrict the discussion here to the stationary regime 23 and 2D geometries. The extension to three dimensions is immediate, at least in Cartesian geometries. In this case, the transfer equation becomes di m µ m dx + η di m m dy + κ ωi m = κ ω Iω 0. (52) Consider a rectangular region x =(x, y) 0,L 0,H covered by a uniform mesh with I J cells of size x y. For each cell, the edges are specified with the usual conventions by indicating the four cardinal points, as shown in Fig The S N quadratures are often limited to S 20 because beyond this point, the construction rules may involve negative weights w m.
22 This concerns in particular the problem known as the ray effect (see Sect. It is less important for quadratures with constant weights, but these generally require a greater number of directions to obtain an overall accuracy equivalent to results with the S N quadratures.
23 The use of the discrete ordinate method to handle non-stationary problems has recently come to the fore in connection with studies of radiative phenomena on ultrashort time scales (shorter than the time taken for light to cross the medium). Numerical diffusion and stability problems are then generally encountered in the time scheme.
For more detail, see 20, for example. 111 94 Denis Lemonnier 4.3 Integrating the RTE over a Control Volume Integrating (52) over a control volume gives where µ m A x (I m,e I m,w )+η m A y (I m,n I m,s )+κ ω VI m,p = S P V, (53) A x = y, A y = x, V = x y, S P = κ ω I 0 ω(t ). (54) When the direction cosines µ m and η m are both positive, the direction of propagation is such that, for each control volume, the specific intensities are known on the edges W and S and unknown at the center P of the cell, and on the edges E and N (see Fig. Two further relations are thus required to eliminate I m,e and I m,n and thereby calculate I m,p explicitly.
These relations are obtained by interpolation by assuming that I m,p = I m,w + a(i m,e I m,w )=I m,s + b(i m,n I m,s ), (55) whereupon I m,e = I m,p + I m,p I m,w a, I m,n = I m,p + I m,p I m,s b. (56) The coefficients a and b vary between 1/2 and 1 and may be different from one point to another. When a =1/2 =b, interpolations are globally second order in x and y. This corresponds to the so-called diamond scheme.
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For all other values of a and b, accuracy is only first order but the stability of the scheme is better. This is the case in particular when a =1=b (step scheme), which simply imposes I m,e = I m,n = I m,p. Having eliminated I m,e and I m,n via the interpolations (56), (53) reduces to an explicit relation between the unknown intensity I m,p, the known values of I m,e and I m,n,andthesourceterms P: where I m,p = λ xi m,w + λ y I m,s + VS P λ x + λ y + λ 0, (57) λ x = µ ma x a, λ y = η ma y b, λ 0 = κ ω V. (58) Once I m,p has been calculated, the values of the other unknowns I m,e and I m,n can be deduced from the interpolation formulas (56). Hence, assuming that we are working on cell (i, j), we can move to the calculation in cell (i +1,j)using: I m,w(i+1,j) = I m,e(i,j), a value just calculated or else imposed by boundary conditions if i =0.