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Fourier expansion to elastic vector wave functions and applications of wave bases to scattering in half-space

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Title:

Fourier expansion to elastic vector wave functions and applications of wave bases to scattering in half-space

Authors:

Po-Jen Shih

Assistant Professor

Department of Civil and Environmental Engineering, National University of Kaohsiung, Kaohsiung, Taiwan 811, R.O.C.

Tsung-Jen Teng Research Fellow

National Center for Research on Earthquake Engineering, Taipei, Taiwan 106, R.O.C.

Chau-Shioung Yeh Professor

Department of Civil Engineering and Institute of Applied Mechanics, National Taiwan University, Taipei, Taiwan 106, R.O.C.

Corresponding Author:

Po-Jen Shih

Tel: +886-7-5916592 Fax: +886-7-5919376 E-mail: pjshih@nuk.edu.tw

Address: No. 700, Kaohsiung University Rd., Kaohsiung, Taiwan 811, R.O.C.

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ABSTRACT

This paper proposes a complete basis set for analyzing elastic wave scattering in half-space. The half-space is an isotropic, linear, and homogeneous medium except for a finite inhomogeneity. The wave bases are obtained by combining buried source functions and their reflected counter-waves generated from the infinite-plane boundary. The source functions are the vector wave functions of infinite-space. Based on the source functions expressed in the Fourier expansion form, the reflected counter-waves are easily obtained by solving the infinite-plane boundary conditions.

Few representations adopt Wely’s integration, but the Fourier expansion is developed from it and applied to decouple the angular-differential terms of the vector wave functions. In addition to the scattering of the finite inhomogeneity, the transition matrix method is extended to express the surface boundary conditions. For the numerical application in this paper, the P- and the SV- waves are assumed as the incoming fields. As an example, this paper computes stress concentrations around a cavity. The steepest-descent path method yielding the optimum integral paths is used to ensure the numerical convergence of the wave bases in the Fourier expansion. The resultant patterns from these approaches are compared with those obtained from numerical simulations.

Key words:

Fourier expression, Elastic half-space, Scattering

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I. INTRODUCTION

This paper proposes a new representation that is favorable for describing the characteristics of the half-space wave scattering in a systematic way using wave basis functions although scattering problems in half-space have already been studied. There are two (2) famous integral expansion representations including the angular spectrum representation [1] and the generalized Sommerfeld integral representation [2], dealing with the scattering geometric models similar to the one proposed in this paper. All of the previous methods are based on the complete basis sets. However, the angular spectrum representation based on spherical coordinate system is inconvenient for solving planar boundary value problems. The generalized Sommerfeld integral representation presented in cylindrical coordinate system is used to solve plane- parallel problems; but it involves a highly oscillatory term, causing slow value convergence. Although these mentioned methods have been intensively addressed, the discussion on the stress concentrations around the scatterer which is a significant issue in the engineering aspect is still very few. The main objective of our work is to provide a new choice of the representation method and to demonstrate its application in solving stress concentration problems.

Firstly, we will introduce the backgrounds of these two (2) famous integral expansions. Some authors used the angular spectrum representation to analyze the electromagnetic and elastic scattering by buried inhomogeneities [3-8]. The angular spectrum representation was first introduced by Whittaker [1] for expanding scalar wave functions, and then it was expanded by Weyl to the form with a polar angle

integral along a contour path (

C

 

). For more details about the derivation of the

dilatational and the transverse waves [9,10]. The other famous representation is the

generalized Sommerfeld integral representation in the Fourier-Bessel integral form (

(4)

0

[ Bessel ]

) [2]. This representation studied elastic and electromagnetic scattering [11-13]. However, the generalized Sommerfeld integrals have a highly oscillatory term and a slowly decaying kernel so that convergence is slow in the near-source fields. For more details, see Ben-Menahem and Singh [14].

Secondly, we introduce our proposed Fourier expansion. The vector wave integral representation was an extension of the Fourier expansion (also called the

generalized Weyl integration in the form

 

  ) first given by Weyl [15]. Very few studies adopted Weyl’s form to analyze scattering because of the numerical convergence of the two-sided infinite intervals. This paper improved the steepest- decent integral path [16] to increase the numerical accuracy [17] and to overcome the convergence of the two-sided infinite intervals. For the infinite-plane boundary of the half-space, the Fourier expansion was applied to transfer the current vector wave functions to scalar-decoupled and Cartesian coordinated form. Therefore the transformed functions were easy to solve under the infinite-plane boundary conditions. Our work lets the generalized Weyl integration have a new application to the vector wave functions.

The significant feature of this study is that the proposed integral expansion is

easy to fit the plane boundary conditions of the vector wave functions when compared

with the current representations. The Fourier expansion developed and used in this

paper was started by deriving the transformation of the vector wave functions from

the generalized Sommerfeld integral representation [13]. The Fourier expansion was

previously transformed from the angular spectrum representation but was not shown

in this paper because of the long derivation. The Fourier expansion was expanded to

transfer the spherical vector wave functions [18] to the scalar-decoupled form in

(5)

which the angular-differential forms were decoupled and expanded in the Cartesian coordinate system. Each Cartesian component of the scalar-decoupled form resulted in a combination by the scalar surface harmonics with different orders. Furthermore, every order of the vector wave functions was regarded as a source in order to obtain its reflected counterpart by applying the infinite-plane boundary conditions. A wave pair combining the source and its reflected counterpart satisfied the infinite-plane boundary conditions. Moreover, the wave pairs were proposed as a basis set for solving scattering generated by a finite inhomogeneity buried in half-space. This paper did not consider multi-layer models. However, specialized layers having plane interfaces can be solved by the vector wave functions written in the Fourier expansion form with appropriate modifications. Further references can be found in Kennett (R/T method [19-21]). Our proposed integral expansion also fits the context of the transition matrix method, the well-known method in solving scattering problems. The transition matrix method, regarded as the null-field integral formulation, was first developed by Waterman [22,23] to treat acoustics wave problems. Currently, the null- field method is applied widely for obtaining biharmonics and inhomogeneity [24-26].

The surface integral around the inhomogeneity yielded the transition matrix equations, and the formal inversion of this matrix in a power series was interpreted as the multiple scattering contributions from the interface and the inhomogeneity. The similar studies in modeling elastic wave propagation of canyon and surface inhomogeneity can refer Liao’s studies [27-29].

For numerically modeling the elastic wave propagation in the ground, this paper

assumed that the inhomogeneity was a cavity. Note that any inhomogeneity could be

considered when modifying the transition matrices. The P- and the SV- plane waves

were considered incidences with various incident angles. Furthermore, the closed

forms of the incident coefficients were obtained. For correctness of the transition

(6)

matrix method, the results obtained from the presented transition matrix method were compared with the numerical simulation using the least-squares method. Moreover, for the effects of the interaction between the inhomogeneities, the stress patterns of the infinite-space given by Pao [30] were compared with the stress patterns of half- space. The stress patterns between two- and three-dimensional problems were also compared with Shyu’s results [31].

II. VECTOR WAVE BASIS FUNCTIONS

A. FOURIER EXPANSION OF THE VECTOR WAVE FUNCTIONS In an isotropic, linear, and homogeneous medium, the equations of motion with the displacement vector

u

are

(         )

u

2u

u

,

where  is the mass density and  and  are the Lamé constants. The relations between stress

τ

and displacement

u

are

( ) .

 

      

τ I u u u

The traction

t

at a surface with a unit normal vector

n

are expressed as ( )   (    )      ( ).

t u n u n u u

In the following equations, the time varying factor e

i t

is omitted. The displacement

u

satisfying Equation (Eq.) can be decomposed into three (3) vector wave functions

L

,

M

, and

N

[19]. It is emphasized that the sets with L, M, and N are the complete basis sets.

L

represents the dilatational wave propagating with the speed

( 2 ) /

c

p

     , and the wave number k

p

  / c

p

.

M

and

N

denote the transverse waves propagating with the speed c

s

   / , and the wave number

s

/

s

k   c . The three (3) vector wave functions are

(7)

(1)

1

mn mn mn

,

k

p

  

u L

(2)mn mn

(

mn

R

R

) ,

  

u M e

(3)

1

( )

mn mn mn R

s

k R

  

u N e

where 

mn

and 

mn

are the scalar-spherical dilatational and transverse wave functions. To simplify the notation in the derivation, the potentials are defined by

(2)

(

*

) ( , )

mn

h

n

k R Y

mn

 

  in which 

mn

 

mn

is used when k

*

k

p

(dilatational

wave) and   

mn

is used when k

*

k

s

(transverse waves). The symbol  mn is an abbreviation for three (3) integers:  ranges from 1 (even) to 2 (odd); m 0, 1, ,n ; and n  0, 1, ,   . h

n(2)

( ) x are the spherical Hankel functions of the second kind. The

spherical surface harmonics Y

mn

( , )   are defined by Y

1mn

( , )    P

nm

(cos ) cos  m

(when   1 ) and Y

2mn

( , )    P

nm

(cos )sin  m  (when   2 ) in which P

nm

(cos )  are the associated Legendre polynomials. From Chang and Mei’s expansions [32], the Bessel-Legendre can be expanded by

 2 * 1 * *

* * *

0

( )

m

(cos ) [

n m r m

( sgn( ) ) ( )

z

] ,

n n

k

n m r r

h k R P i P i z J k r e dk

k k

 

 

  

2

cos

0

cos ( ) cos

sin 2 sin

m r

m r ik r

m J k r i e m d

m m

 

  

 

    

   

    

where sgn( ) 1 z  is used for z  0 , and sgn( ) z   1 is used for z  0 ; 

*

k

r2

k

*2

; and k

r

k

x2

k

y2

. Furthermore, the transformations x r  cos  , y r  sin  ,

x r

cos

kk  , k

y

k

r

sin  , k r

r

cos       k x k y

x

y

, and k dk d

r r

  dk dk

x y

are

(8)

used to transform the integrand of Eq. . Another methodology for solving Bessel- Fourier transformations can be referenced from the 2-D Fourier-transform expansion [33]. Moreover, cos m  and sin m  terms are replaced by Chebyshev polynomials,

1m

( / )

x r

T k k and T

2m

( / ) k k

x r

. Combining Eqs. and obtains the scalar surface harmonics written in the integration of plane waves forms with wave number integrals,

*

*

1 *

* *

[ ( sgn( )

*

) (

2 2

)]

2

x y

x y

n m x z ik x ik y

mn n m x y

x y

z ik x ik y

mn x y

i k

P i z T e dk dk

k k k k

e dk dk

 

 

 

 

 

  

 

 

 

where

1 *

2 2

* *

( sgn( )

*

) ( / )

2

n m

mn n m x x y

i P i z T k k k

k k

 

     ;  

mn

  

mn

is used for a

dilatational wave, and  

mn

  

mn

is used for a transverse wave. Substituting Eq.

into Eq. obtains the dilatational vector wave functions,

| |

| |

( )

,

( )

,

( )

x y

x y

z ik x ik y

mn mn x y

T z ik x ik y

mn x mn y mn z x y

e dk dk

L L L e dk dk

 

 

 

 

L L

  

and the corresponding components of the amplitudes (details in Appendix A),

( ) ( 1)( 1) ( 1)( 1)

( 1)( 1) ( 1)( 1)

1 [ ( 1)

2(2 1)

( 2) ( )( 1) ] ,

mn x m n m n

m n m n

L n m

n

n m n m n m

 

 

     

      

  

 

( ) ( 1)( 1) ( 1)( 1)

( 1)( 1) ( 1)( 1)

sgn( 3/ 2)

[ ( 1)

2(2 1)

( 2) ( )( 1) ],

mn y m n m n

m n m n

L n m

n

n m n m n m

  

 

     

      

  

 

( ) ( 1) ( 1)

1 {( 1) ( ) }

(2 1)

mn z m n m n

L n m n m

n

     

   

in which    3  . Note that the three (3) components were decoupled from the

(9)

differential form in Eq. (4). Equations (11), (12), and (13) have the combination form

of the scalar surface harmonics  

mn

with upward and downward orders, m  1 , m , 1

m  , n  1 , n , and n  1 . Substituting Eq. into Eq. obtains the transverse vector wave function of

Mmn

type,

| |

| |

( )

,

( )

,

( )

.

x y

x y

z ik x ik y

mn mn x y

T z ik x ik y

mn x mn y mn z x y

e dk dk

M M M e dk dk

 

 

 

 

M M

  

The corresponding components of the transverse vector wave function are

( ) ( 1) ( 1)

sgn( 3/ 2)

[( )( 1) ] ,

mn x

2

m n m n

M

   n m n m     

  

( ) ( 1) ( 1)

1 [( )( 1) ] ,

mn y

2

m n m n

M

  n m n m     

  

( )

sgn(3 / 2 ) .

mn z mn

M

    

Here M

mn x( )

, M

mn y( )

, and M

mn z( )

are dependent relations that satisfy the conditions

mn

0

 

M

 (  sgn( ) z M

mn z( )

  ik M

x

mn x( )

ik M

y

mn y( )

). The expansion for the vector wave function

Nmn

is obtained by substituting Eq. to Eq. ,

| |

| |

( )

,

( )

,

( )

x y

x y

z ik x ik y

mn mn x y

T z ik x ik y

mn x mn y mn z x y

e dk dk

N N N e dk dk

 

 

 

 

N N

  

where the corresponding Cartesian components are obtained,

 

 

( ) 1 1 1 1

1 1 1 1

1 [ ( 1) ( 1)

2(2 1)

( 2) ( 1)( )( 1) ] ,

mn x m n m n

m n m n

N n n n n m

n

n m n n m n m

 

 

     

       

  

 

 

 

( ) 1 1 1 1

1 1 1 1

sgn(3/ 2 )

[ ( 1) ( 1)

2(2 1)

( 2) ( 1)( )( 1) ],

mn y m n m n

m n m n

N n n n n m

n

n m n n m n m

  

 

      

       

  

 

(10)

 

   

( )

1 [ 1

1

1

1

].

(2 1)

mn z m n m n

N n n m n n m

n   

   

   

Furthermore,  

Nmn

 0 , i.e.   sgn( ) z N

mn z( )

  ik N

x

mn x( )

ik N

y

mn y( )

. So far, the

components of

Mmn

and

Nmn

are decoupled from the differential forms written in Eqs. (5) and (6). They are written as simple combinations of scalar surface harmonics

mn

 

with upward and downward orders.

The tractions of the vector wave functions written as the Fourier expansions can be obtained by applying Eqs. through to Eq. :

2 2

(1)

(2 )

| |

( ) 2

[ sgn( ) ]

z ik x ik yx y

,

p s

mn

mn x x y y z mn x y

s p s s

k k

ik n ik n z n e dk dk

k k k k

 

 

       

t u n L

(2) | |

( ) 2

[ sgn( ) ]

z ik x ik yx y

,

mn

mn x x y y z mn x y

s s

ik n ik n z n e dk dk

k k

 

       

t u n N M

(3) | |

( ) 2

[ sgn( ) ]

z ik x ik yx y

mn

mn x x y y z mn x y

s s

ik n ik n z n e dk dk

k k

 

       

t u n M N

where nn

x

, n

y

, n

z

,

T

is the unit normal at an arbitrary surface.

B. RECURSIVE RELATION FOR THE REMOVABLE SINGULARITIES IN FOURIER EXPANSION

On the complex plane, it can be found that the argument of the Chebyshev

polynomials contains singular poles when the term ( k

x2

k

y2

) of T

m

( / k

x

k

x2

k

2y

) in Eq. is zero. However, the singularity can be removed by applying the following recursive relations.

The Legendre polynomials P

nm

(cos )  are given by a product form, (cos )

( )

(cos ) sin

m m m

n n

P   P  

(11)

where P

n( )m

( ) xd P dx

m n

/

m

. The equation

( ) ( )1 ( )2

2 1 1

m

cos

m m

n n n

n n m

P P P

n m  

n m  

 

  is

then used to obtain the recursive relation

( ) ( )

(cos )

,

(cos )

m m

n n m m m

P   A

P

where the coefficients

, 1, 2,

2 1 1

n m m

cos

n m m n m m

n m n

A A A

n mn m

   

  

 

  ,…, A

0,m

 1 ,

1,m

0

A

 . Moreover,

( ) 1

(2 1)!

(cos )

2 ( 1)!

m

m m

P m

 

m

 are obtained by applying the

differential to 1 3 (2 1) ( 1)

2

( ) [ ]

! 2(2 1)

m m

m

m m m

P x x x

m m

  

  

   with relative to x. It

obtains T

m

(cos ) 2 cos    T

(m1)

(cos )   T

(m2)

(cos )  and shows the results

*

* 2 *

( 1) ( 2)

* * * *

11 10 21 20

( , ) sin [ (cos )]

2sin cos ( , ) sin ( , ) ,

sin cos , 1, sin sin , 0.

m

m m

m m

T T

T T

T T T T

   

      

   

 

   

Combining the results in Eqs. , , and with Eq. obtains the expression

1

* *

* * 1 ,

(2 1)!

( , )

2 2 ( 1)!

n

mn m n m m m x

i m

A T k k

k m

 

  

 

where the coefficients are expressed as

*

, * 1, 2,

0, 1,

2 1 sgn( ) 1

( )[ ] ( ) ,

1, 0 ,

n m m n m m n m m

m m

n i z m n

A A A

n m k n m

A A

   

   

 

 

 

2 2

* * * * 2 * *

( 1) ( 2)

* *

* * * *

11 * 10 21 * 20

( , ) 2 ( , ) ( ) ( , ), ,

, 1, , 0 .

x y

x

m x m x m x

x y

k k

T k k k T k k T k k

k k

k k

T T T T

k k

  

   

(12)

III. SERIES EXPANSION AND TRANSITION MATRIX METHOD FOR SOLVING SCATTERING IN HALF-SPACE

A. A SET OF VECTOR WAVE BASIS FUNCTIONS SATISFYING THE INFINITE-PLANE BOUNDARY CONDITONS

The studying model and the geometric coordinate system are shown in Fig. 1 in which the plane surface is located at z   h . In this study,

( )mn

u

are regarded as the buried emission sources located at the origin. The displacements

rsmn( )

u

are the reflected waves generated from the infinite-plane interface by the emission sources

( ) mn

u

. Furthermore,

rsmn( )

u

can be written

( ) ( ) ( ) ( ) ( )

( ) ( ) ( )

( )

rs

mn mn mn x x mn y y mn z z

  

  

 

 

u e e e

where all of the potentials

( )mn

,

( )mn x( )

,

( )mn y( )

, and

( )mn z( )

satisfy the Helmholtz equations and the condition   ( 

( )mn x( )ex

 

( )mn y( )ey

 

( )mn z( )ez

) 0  . Furthermore, the potentials are expressed in the Fourier expansion form,

( )

( ) ( )

( )

( ) ( )

( , , ) ( , , )

,

.

x y

x y

z h ik x ik y

mn mn x y

z h ik x ik y

mn x y z mn x y z x y

e dk dk

e dk dk

 

 

 

 

 

 

 

 

Because the tractions of the emission sources

t u

(

( )mn

) are expressed in Fourier

expansion [see Eqs. to ], the reflected counterparts

t u

(

rsmn( )

) are determined by applying the infinite-plane boundary conditions. In this study, the traction-free

conditions,

t u

(

( )mn

) 

t u

(

rsmn( )

)  0, 0, 0

T

, are used to solve the reflected counterparts.

Details are given as follows:

(i)

Lmn

type of emission source (   1) ,

(13)

(1)mn

[ ( , ) 8 F k k

x y

k

r2

] / ( , ) ( F k k

x y mn

/ k e

p

)

h

,

         

(1) 2 2

( )

4 (2 ) / ( , ) ( / )

h

,

mn x

i k

y

k

r

k

s

F k k

x y mn

k e

p

       

(1) 2 2

( )

4 (2 ) / ( , ) ( / )

h

,

mn y

i k

x

k

r

k

s

F k k

x y mn

k e

p

        

(1)

( )

0;

mn z

  

(ii)

Mmn

type of emission source (   2) ,

(2) 2 2

4 (2 ) ( , )

( ) h

,

mn

k

r

k

s

F k k

x y

M

mn z

e

        

(2)

( )

{ ( , )

( )

8

( )

} [ ( , )]

h

,

mn x

F k k N

x y mn x

i k k

s y

M

mn z

k F k k

s x y

e

    

  

(2)

( )

{ ( , )

( )

8

( )

} [ ( , )]

h

,

mn y

F k k N

x y mn y

i k k

s x

M

mn z

k F k k

s x y

e

    

   

(2)

( ) ( ) h

;

mn z

N

mn z

k

s

e

   

(iii)

Nmn

type of emission source (   3) ,

(3) 2 2

4 (2 ) ( , )

( ) h

,

mn

k

r

k

s

F k k

x y

N

mn z

e

       

(3)

( )

{ ( , )

( )

8

( )

} [ ( , )]

h

,

mn x

F k k M

x y mn x

i k k

s y

N

mn z

k F k k

s x y

e

    



(3)

( )

{ ( , )

( )

8

( )

} [ ( , )]

h

,

mn y

F k k M

x y mn y

i k k

s x

N

mn z

k F k k

s x y

e

    



(3)

( ) ( )

h mn z

M

mn z

k

s

e

   

where the denominator is obtained by F k k ( , ) (2

x y

k

r2

k

s2 2

)4 k

r2

 . Because the potentials were written in the Fourier expansion form, the displacements also are

( 2 )

(1) (1) (1) (1)

( )

,

( )

,

( ) T z h ik x ik yx y

,

rs rs rs rs

mn

u

mn x

u

mn y

u

mn z

e

dk dk

x y

 

  

u

  

( 2 )

(2,3) (2,3) (2,3) (2,3)

( )

,

( )

,

( ) T z h ik x ik yx y

,

rs rs rs rs

mn

u

mn x

u

mn y

u

mn z

e

dk dk

x y

 

  

u

  

and their associated tractions are

( 2 )

(1) (1) (1) (1)

( ) ( ) ( )

(

rs

) (

rs

), (

rs

), (

rs

)

T z h ik x ik yx y

,

mn

t

x mn

t

y mn

t

z mn

e

dk dk

x y

 

  

t u uuu

(14)

( 2 )

(2,3) (2,3) (2,3) (2,3)

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

(

rs

) (

rs

) , (

rs

), (

rs

)

T z h ik x ik yx y

mn

t

x mn x

t

y mn y

t

z mn z

e

dk dk

x y

 

  

t u u

u

u

in which t

( )x

, t

( )y

, and t

( )z

are the operators of Cartesian components and the

amplitudes are shown in Appendix B. Each order of the wave pairs (

u( )mn

+

ursmn( )

)

satisfies the boundary conditions of the infinite-plane. Note that

u( )mn

are obtained

from the complete basis sets (L, M, N), and

ursmn( )

are derived from

u( )mn

by applying

the boundary conditions. So that series of the wave pairs (

u( )mn

+

ursmn( )

) which are used to represent the scattering expansion of the elastic half-space are also complete basis set.

B. SERIES EXPANSION FOR SCATTERING IN ELASTIC HALF-SPACE This paper used the series expansion method to expand the scattering in half-space in which an inhomogeneity is buried and located at the geometric center as shown in Fig. 1. Three (3) basic functions, scattering function, the incident function, and the refractive function, were used to expand the half-space field in this paper. The scattering function are series-expanded by the wave pairs with undetermined

coefficients

c   ( )m n

,

3 ( ) ( ) ( )

1

( )

s rs rs

m n m n m n

m n

        

   

     

u u u c u u

where

   

m n

is an abbreviation for summations

2

0 0 1

n

n m

   . Accordingly, the

incident functions including the emission source functions and the reflected functions

generated by the infinite-plane surface were expanded by series of regular functions,

數據

Figure 3 shows the inner steepest-descent paths on the complex   - plane with various angles  
Figure 4 shows the steepest-descent paths of the outer integrals on the complex   - -plane
TABLE  Table I. Coefficients of  a (2)  mn /  mn  m n a  (2) mn /  mn   1 m  0 n  0 0   2 0m 0  n 1 m 1( n  1)1nm1(n 1)  m 1( n  1)1m1 n2m0(n1)m0(n 1)  m 2(  n  1)   1 n ( m 0(  n  1)  m 0(  n  1) ) 2n m 0(
Table II. Coefficients of  a (3)  mn /  mn  m n a  (3) mn /  mn   1 0m n  0 01n n 1n1mn1n 01   1 n ( n 01 ) 1n n 0  n  n 2  n   1 n ( n 0  n ) 1m n m n (  m  1) n n m n (  m  1) n  n (  m  1) n   2 m  0 n  0 0
+7

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