(5.3.18) Let

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(01 = large constant) (5.3.19) kXmin On the other hand, the selection of sampling ~a' must be small enough to ensure enough sampling of the integrand. The worst case and the smallest ~a' corresponds to Ixi = Xmax , where Xmax is the maximum separation in x. It is required that BW

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(5.3.20) Let

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kXmax(~a')2 = Os

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(Os = small constant) Os =------r==:::;: Ixmaxl

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(5.3.21) (5.3.22)

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Thus the number of sampling points is on the order of

Q=O(B~) =0(01 V:i: min ~) ~a Os

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Note that Q can be a fixed constant if Xmax - - = constant Xmin

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(5.3.23)

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(5.3.24)

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5 FAST METHODS FOR ROUGH SURFACE SCATTERING

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xmax/Xmin

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In the multilevel grouping, the division of groups is such that the same for each level.

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3.2 Multi-Level Impedance Matrix Decomposition and Grouping

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In the fast multipole method, the impedance matrix elements are decomposed into various levels based on multilevel grouping. To illustrate, let the number of elements in the 1st level group be M and the number of groups of the 1st level be L. Then the total number of elements is N = LM (e.g., M = 20, L = 64, then N = 1280 - We use this case as an illustration in this section. Generalization can readily be made to other values of M and L). Let L = 2P+1 , where p corresponds to the level of decomposition of the impedance matrix. The first level groups have M elements each. Beyond the first level, the number of elements increase by a factor of 2 for each higher level. Levell group has M elements, level 2 group has 2M elements, level 3 group has 22 M elements, and so on. For L = 64, P = 5 level of group 1 2 3 4 5 6

of elements M 2M 4M 8M 16M 32M

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(5.3.25)

The grouping as shown in Fig. 5.3.2. Group m at level n is denoted as m n . The impedance matrix is decomposed as (p = 5 for this case)

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=(0)

=(1)U

=(2)U

=(3)U

=(4)U

=(5)U

=(l)L

=(2)L

=(3)L

=(4)L

=(5)L

(5.3.26)

where the number in the parenthesis stands for the level of the group, superscript U and L denote the upper matrix and lower matrix respectively. The upper matrix has column index larger than row index for nonzero elements. It is the reverse for the lower matrix. =(i) Let Zmini denote the interaction of elements between group mi and group ni in level i. They are all full matrices.

=(1) Zmint =(2) =(i)

dimension M x M dimension 2M x 2M

=(i)

(5.3.27) (5.3.28)

=(2)

Zmini

Note that Zmini are defined differently from Z

. For example, Z

is of di-

302 Multi-Level Impedance Matrix Decomposition and Grouping

Figure 5.3.2 Multilevel fast multipoles structureo

mension N X N while Z~:ni is of dimension 2M x 2M The various definitions of impedance matrix elements should be distinguished.

=0(1)

. . = zero matnx 0 f d'ImenSlOn M x M

(5.3.29) (5.3.30) (5.3.31)

0(2) = =O(i) =

zero matrix of dimension 2M x 2M

. . zero matnx 0 f d'ImenslOn (OM) x (2OM) 2

Suppose we use M = 20. Then examples are

(5.3.32)

(5.3.33)

Thus Zmlnl' Zm2n2' Zmgng' etc., keep track of all the individual impedance matrix elements. The Oth level impedance matrix Z

=(0)

=(1)

=(2)

=(3)

represent interaction at level 1

5 FAST METHODS FOR ROUGH SURFACE SCATTERING

between itself and its neighbors on the two sides.

=(1) =(1) =(1) =(1)

=(1)

Zl,2,

=(1)

=(1)

=(1)

=(0) Z

Z2,l,

=(1)

Z2,2,

=(1)

Z2,3,

=(1)

=(1)

o =(1) o

Z3,2,

=(1)

Z3,3,

Z3,4,

N x N

(5.3.34)

For Z ,the impedance matrix elements can be calculated accurately by using near field integrations. In the multilevel fast multipole method, the impedance matrix of each level is generated by following rules: (i) In each level, the impedance matrices only interact with itself or its nearest neighbors. (ii) Each impedance matrix element in Z can occur only once in the matrix decomposition. If a pair had interacted previously in lower level groups, it cannot interact again in the current level. That entry has to be set to zero at the current level. To generate the next level, i.e., 1st level impedance matrix Z(l), we first apply rule (i) to get the N x N matrix

=(2) =(2) =(2) =(2) =(2) =(2)

=(0)

Z1 2 12

=(2)

Z1 2 22

=(2)

=(2)

Z2 2 12

=(2)

Z2 2 22

=(2) Z 32 2 2 0(2)

Z2 2 32

=(2) Z 32 32

Z3 2 42

(5.3.35)

0(2)

However, according to rule (ii), the diagonal matrix elements should be set to zero because they have already interacted in then gives

=(2)

Z(O).

Setting them to zero

=(2)

=(2)

Z1 2 22

=(2)

=(2)

=(2)

Z2 2 12

=(2)

=(2)

=(2)

Z2 2 32

=(2)

=(2)

Z3 2 22

=(2)

=(2)

Z3 2 42

(5.3.36)

=(2)