APPENDIX D To numerically solve the Electric Field Integral Equation obtained by using the RWG basis functions, the related surface integrals can be approximated as a discrete sum evaluated on K

149

APPENDIX D

To numerically solve the Electric Field Integral Equation obtained by using the RWG basis functions, the related surface integrals can be approximated as a discrete sum evaluated on K points (usually 7 or 9) belonging to the triangle support [28].

It is convenient to express the source point with respect to a triangle area coordinate (ψ, τ, ν) which can be defined noting that the vectors ρi in Fig. D.1 divides the triangle into three subtriangles of areas A1, A2 and A3 with l1, l2 and l3 as one of their sides respectively. The area coordinate are then defined as:

23 13 12 ; ; , P P P S S S A A A A A A ψ = τ = ν = (D.1)

where AS is the triangle area while APij is the area of the generic subtriangle which can be evaluated as:

1 1 . 1 Pij i i j j x y A det x y x y     = _{ }     (D.2)

The area coordinate system univocally defined a point belonging to the triangle domain and we can note that:

(

)

21 12 3 3 1 (1, 0, 0) , . , , (0,1, 0) , (0, 0,1) , x x y y x x y y x x y y ψ τ ν ψ τ ν + + = = =   =_ = =  _{= =}  (D.3)

The Cartesian coordinates of a generic point can be easily obtained from its area coordinates as follows:

1 2 3 1 2 3 . x x x x y y y y ψ τ ν ψ τ ν = + +   = + +  (D.4)

NUMERICAL INTEGRATION

150

Figure D.1. Triangle division for the area coordinate definition.

By using the relation (D.1)-(D.4), the surface integrals employed to evaluate the vector and scalar potentials can be written as:

( )

( )

( )

( )

1 1 , K T k k k S K T k k k k S R dS A R W R RdS A R R W α α α α = = ≈ ≈

∑

∫∫

∑

∫∫

(D.5)

where AT is the area of the source triangle while R is the distance between source and observation point.

In Table D.1 and D.2 a set of sample locations inside the triangle domain and the correspondent weights are reported for the solution of integrals (D.5) via the seven point rule and nine point one respectively.

Table D. 1. Seven point integration rule.

Samples (ψ_τ i, i) ν1,ν1 ν2,ν2 ν2,ν3 ν3,ν2 ν4,ν4 ν4,ν5 ν5,ν4 Weights wi wa wb wb wb wc wc wc

(

)

(

)

1 1 3, 2,4 6 15 21, 3,5 9 2 15 21 ν = ν = _ ν = _ ,

QUADRATURE INTEGRATION RULES

151

(

9 40

)

, ,

(

155 15

)

1200

a b c

w = ∆ w =  ∆

Table D. 2. Nine point integration rule.

Samples (ψ_τ i, i) ν1,ν2 ν2,ν 1 ν1,ν4 ν4,ν1 ν2,ν3 ν3,ν2 ν3,ν3 ν2,ν4 ν4,ν2 1 1 18, 2 2 9 , 3 7 18, 4 13 18 ν = ν = ν = ν = 9 S w = ∆