Some Analytic Formulations of Weakly Singular Integrals over Polygon for IPO Applications

Jae-Won Rim and Il-Suek Koh

Department of Electronic Engineering, Inha University, Incheon, Korea

Abstract - To treat weakly singular integrals, analytic formulations have been derived for the near-field correction of the iterative physical optics applications. They are analytically derived for a flat polygon patch based on the Stokes’ theorem and numerically verified.

Index Terms — Weakly singular integrals, Iterative physical optics, Near-field correction.

1. Introduction

The iterative physical optics (IPO) method has been applied to various electromagnetics problems. To improve the accuracy of the IPO, the treatment for singular integral kernels is essential. The singular integral kernels arise from the Green’s function, which are mainly classified as 31 / R hypersingular integrals (HSIs), 21 / R strongly singular integrals (SSIs), and 1 / R weakly singular integrals (WSIs). In this work, we focus on the formulations of WSIs. One of the most widely used techniques to treat WSIs is the Duffy’s method [1]. However, it increases the number of numerical computations and is less accurate for near-singular cases.

In this work, some analytic formulations for WSIs are derived for a flat polygon patch based on the Stokes’ theorem [2] and also verified numerically.

2. Near-field Correction for IPO

The IPO iteratively updates the surface currents on an object to calculate the multiple interaction. To update the IPO currents following surface integral should be computed.

( ) ( )0 01 2 3

0 0

4 e e mS

jZ kE G J G R R J G J R dSk jZp D

é ù= - + × + ´ê ú

ë ûòò

r r r r r r r, (1)

where 0 0 0ˆ ˆ ˆ( ) ( ) ( )R u u u v v v w w w= - + - + -r

, 2 21 0( 1G k R= - -

0 30 ) /jk Rjk R e R- , 02 2 5

2 0 0( 3 3 ) /jk RG k R jk R e R-= + + , 3 (1G = + 0 3

0 ) /jk Rjk R e R- , SD is the triangular source patch. Accurately to calculate (1) for the near-singular case, the singularity subtraction method is usually used. For example, the second term in (1) can be treated as

( )2

2 40 0

2 ,5 31

2 40 0

,5 31

3 2 8

3 + 2 8

p

i

p

i

eS

N

e iSi

N

e iSi

G R R J dS

k kG RRd JR R R

k k RRd JR R R

D

D=

D=

×

é ùæ ö» - - - W ×ê úç ÷

è øë ûé ùæ ö

+ + W ×ê úç ÷è øë û

òò

å òò

å òò

r r r

r r r

r r r

(2)

where pN is the number of the subdivided polygons. The surface current can be approximated as constant ,e iJ

r on i th

polygon, and the singular terms are subtracted by the Taylor

series expansion. The first integral terms can be numerically calculated as it is regular. The second singular integral terms should be calculated by other means. Among the singular integral terms, only WSIs are considered in this work.

3. Derivation of Formulas for WSIs

Based on the Stokes’ theorem for surface integrations over a polygon patch, we can formulate following equations [2]

; u vu vS C S C

A AA du A dvv uD D

¶ ¶= - =

¶ ¶òò ò òò òÑ Ñ , (3)

where ˆ ˆ ˆu v wA A u A v A w= + +r

is a vector defined over a surface patch SD in a local coordinate ( , , )u v w , and C is the boundary of SD . The patch plane are assumed to be on uv plane and ˆ ˆw n= , i.e. 0un = , 0vn = , and 1wn = .

The subtracted WSIs in (2) are as follows

2 20 0

1 2 33 3

20 0 0

4 53

20 0

6 7

0 0 08 9

1 ( ) ( ); ; ;

( )( ) ( ); ;

( ) ( ); ;

( ) ( )( );

S S S

S S

S S

S S

u u v vI dS I dS I dSR R Ru u v v u uI dS I dS

R Rv v u uI dS I dS

R Rv v u u v vI dS I dS

R R

D D D

D D

D D

D D

- -= = =

- - -= =

- -= =

- - -= =

òò òò òò

òò òò

òò òò

òò òò

(4)

where the observation points 0 0 0( , , )u v w is closely located not inside but outside of the N -sided source polygon patch whose vertices are ( , , )i i i ip u v w with 1,2, ,i N= L . The integrals in (4) can be analytically derived based on (3) and integral identities in [3]. For instance, 1I , 2I , and 4I can be respectively calculated into a closed-form expressions as

( )

( )

1 0

2 21 1 0 1

02 2 2

0

10 2 2 2

0 0 1

10 2 2 2

0 0

1 ln ( )

1ln

1 1

tan

tan

S C

i i i i i i ii i

i i i i i i i i

i i

i i i i

i i

i i i

I dS R u u dvR

A B v A v v R AAv BA A B v A v v R A

a cwa l w w R

a bwa l w w R

D

+ + +

-

= +

-

= = - -

é ù- + + - + +-ê úê ú+ - + + - + +ê úê ú

æ öê úç ÷= -ê úç ÷+ +ê úè ø

ê úæ öê úç ÷-ê úç ÷+ +ê úè ø

ê úë û

òò òÑ

( )( )

1

20 0

2 03

0 01 3/ 22 221

( ) ( ) ln ( )

1 11

N

S C

Ni i i i i

i ii i ii

u u u uI dS R u u dvR R

A B Av Av BR RA AA

D

+=

- -é ù= = + - -ê úë ûé ùæ ö- -ê úç ÷= - - + +ê úç ÷+ ++ê úè øë û

å

òò ò

å

Ñ

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( )( )

( )

( )

0 01 3/ 22 22

2 21 1 0 1

2 20

10 2 2 2

0 0 1

10 2 2 2

0 0

1 11

1ln

1

tan

tan

i i i i ii i

i ii

i i i i i i i

i i i i i i i

i i

i i i

i i

i i i

A B Av Av BR RA AA

A B v A v v R A

A B v A v v R A

a cwa l w w R

a bwa l w w R

+

+ + +

-

+

-

é æ ö- -ê ç ÷- - + +ê ç ÷+ ++è øêê

- + + - + +ê´êê - + + - + +êê æ öê ç ÷-ê ç ÷+ +è øêê æ ö

ç ÷-ç ÷+ +è øë

ùúúúúúúúúúúúúú

ê úê ú

û

( ) ( )

( )

0 0 04 3

01 3/ 22 2

2 21 1 1 0 1

2 20

( )( ) ( )

11 1

1ln

1

S C

i i ii i

i iN

i i i i i i i i

i i i i i i i

u u v v v vI dS dvR R

A Av BR R

A A

A B v A v v R A

A B v A v v R A

D

+

= + + +

- - -= =

é ù-- - +ê ú+ +ê úê ú= ê ú- + + - + +ê ú´ê ú- + + - + +ê úë û

òò ò

å

Ñ

(5)

where

( ) ( )

( ) ( )( )( ) ( )( )( )( ) ( )( )( )( ) ( )( )

10

1

2 2 20 0 0

2 21 1

0 1 0 1

1 0 1 0

1 1 0 1 1 0

, i ii i i i i

i i

i i i

i i i i i

i i i i i i i

i i i i i i i

i i i i i i i

u uA B u u Avv v

R u u v v w

l u u v v

a u u v v v v u u

b u u u u v v v v

c u u u u v v v v

+

+

+ +

+ +

+ +

+ + + +

-= = - +

-

= - + - +

= - + -

= - - - - -

= - - + - -

= - - + - -

(6)

with the assumption that 1 mod( , ) 1i i N+ = + .

4. Numerical Verification

To calculate the surface integrals numerically, the Gauss quadrature rule is widely used [4]. However, the accuracy is deteriorated when the distance between the observation point and the source patch is very close. Thus, in this work, we keep dividing a source patch into the sub-triangle patches and apply the low-order quadrature rule to each sub-triangle patches until the numerical value converges with a given certain tolerance. A source triangle patch is located at uv plane where the vertices are defined as 1(0.7 ,0.1 ,0)p l l , 2 (0.4 ,0.3 ,0)p l l , and

3(0.1 ,0.2 ,0)p l l , respectively. And the observation point is located at 0 0(0.4 ,0.2 , )p wl l where 0w varies from 0.01l to 1l . As the number of sub-triangle patches, triN increases, the accuracy of the numerical value improves since the sampling points also increases over the triangle patch. Q.P.N is the total number of quadrature points as shown in Fig. 1. The analytic solution of 1I is in accurate agreement with the numerical solution with the Gauss quadrature of tri 1024N = and

Q.P. 7168N = even when 0w is very small. Based on the numerical solutions of tri 1024N = and Q.P. 7168N = , the relative errors e for the analytic formulas of 1I ~ 9I in (5) are calculated (see Fig. 2). The analytic formulas have low relative errors less than 410- and are accurate for any 0w values.

Fig. 1. Verification of 1I with respect to 0w by applying the Gauss quadrature rule for subdivided triangle patches.

Fig. 2. Relative error e comparison of 1I ~ 9I with respect to numerical solutions of tri 1024N = and Q.P. 7168N = .

5. Conclusion

Some analytic formulas for WSIs are derived based on Stokes’ theorem for a flat polygon patch and they are also verified numerically. It is not only exact for any near-singular case, but also require low computations.

Acknowledgement

This work was supported by ICT R&D program of MSIP/IITP. [B0717-16-0045, Cloud based SW platform development for RF design and EM analysis]

References [1] M. G. Duffy, “Quadrature over a pyramid or cube of integrands with a

singularity at a vertex,” SIAM J. Numer. Anal., vol. 19, pp. 1260–1262,1982.

[2] Tong, Mei Song, and Weng Cho Chew. "A novel approach for evaluating hypersingular and strongly singular surface integrals in electromagnetics." Antennas and Propagation, IEEE Transactions on 58.11 (2010): 3593-3601.

[3] H. B. Dwight, "Tables of Integrals and Other Mathematical Data", 4th ed. New York: Macmillan, 1961.

[4] D. Dunavant, “High degree efficient symmetrical gaussian quadrature rules for the triangle”, Internat. J. Numer: Methods Engrg, vol. 21, June 1985.

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