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5.1 Single Grid and PBTG
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Thus Tf is the distance separating near field and non-near field. For non-nearfield interactions, the Green's function of free space is slowly varying on the dense grid. We can use the coarse grid to sample it. Assume the number of sampling points on the coarse grid is smaller than that on the dense grid by a factor of n1, where n1 = integer(Re( vEl)). Thus the ith point on the coarse grid corresponds to n1 points iI, i 2, . .. , in, on the dense grid. The i with no subscript, it refers to the same coarse point i. For ip , p = 1,2, ... ,n1, it refers to the n1 dense grid points associated with the coarse grid point i. The ipth (p = 1,2, ... ,nd point is the ith dense grid point wher'e i is given by i = (i - l)n1 + p. When we calculate the convolution of ai/ and surface fields of Uj on the dense grid, the following approximation can be made.
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N N/n, n, N/n, n, ""' aiJsu(xj) = ""' ""' a!!'sc u(xJc q ) ~ ""' a!!'sc ""' u(xJc q ) ~ ~ ~ ~pJq ~ ~pJ ~
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In getting (5.5.12), we need the property that the Green's function of free space is essentially constant over an interval of n1 points. Furthermore, the elements a!!'sc of p = 1,2, ... , n1 can be found by interpolating from the coarse grid to the dense grid,
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(5.5.13) where In(i p,i+T) is the interpolation operator and T is the number of points of the coarse grid we use to interpolate. Then
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What is done is that the surface fields on the dense grid are first averaged before being convolved with the free space Green's function on the coarse grid. Then we use interpolation to find N values on the dense grid. Similarly, we can obtain:
L bi/'l/J(xj) = L
j=l T=-T
In(ip, i
+ T)hi+T
N jnl nl
hi+7 =
L bc 7)) L 1/J(X)) l+
)=1 q=1
Thus, Eq. (5.5.1) can be rewritten as
In(i p , i + T)9i+7 In(i p, i + T)hi+7 = 1/Jinc(Xi) (5.5.18)
We can write Eqs. (5.5.5) and (5.5.18) as the following matrix equations.
Z A,sdg . Usdg ZA,sdg' Usdg
+ Z B,sdg '1/Jsdg = Osdg
+ ZB,sdg 1i)sdg + ZA,sCg . USCgl.mtp + ZB'SC9 . ~scgl.mtp =
(5.5.20) l , bg), afj' and bfj consist only of banded Note that the elements of ag matrices. The main CPU requirements are to calculate the values of 9i and hi with i from 1 to N/n1. For direct matrix and column vector multiplication, it takes approximately (N/n1)2 operations. Note that N/n1 is the number of unknowns on the coarse grid. The computational steps can be further reduced using the BMIA or the FMM. In Section 2, we have illustrated the PBTG-BMIA. In the following section, we will illustrate how to combine the PBTG with the multilevel steepest descent path FMM.
5.2 Computational Complexity of the Combined Algorithm of the PBTG with the MLFMM
For the multilevel FMM, multi-sized groups are formed. At the lowest level, the N elements are decomposed into L groups. Each group includes M elements where N = M L. Then each two subgroups at the level form an upper level group (large group) until the highest level. The interactions of groups at each level are calculated only for the non-near groups at this level inside the neighboring groups of an upper level. Thus, the impedance matrix can be written as the sum of the following matrices.
+,, +Z
+ +Z
5.2 Computational Complexity of PBTG with MLFMM
=(n) where the matrix Z includes only the elements that would be computed at the nth level. As illustrated in Section 3.2, the procedure of the multilevel SDFMM is composed of three steps. First, the surface fields at each element are translated to the group centers at each level. When transferring the field from the lower level group center to the upper level group center, an interpolation is required to find the values of the fields from the coarser angles to the finer angles. Second, the interactions of group centers at each level are calculated. Only those of the non-near groups inside the neighboring groups of the upper level are calculated at each level. Lastly, the receiving fields at each group center are distributed to its subgroup centers/elements. The last step is performed from the highest level to the lowest level with anterpolation. The number of computational steps for the first step is
N 1 = 2 [ QN + ~ Q 2(n-2)