FINITE VOLUME METHOD FOR THE CALCULATION OF ELECTROSTATIC FIELD OF THE 3-D SPHERE WITH A CUBIC CAVITY

(1)

FINITE VOLUME METHOD FOR CALCULATION OF

ELEC-TROSTATIC FIELD IN THREE-DIMENSIONAL SPHERE WITH

CU-BIC CAVITY

Patsiuk V.

Institute of Power Engineering of the Academy of Sciences of Moldova

Abstract. The paper presents the numerical approach based on finite volume method, which is used to calculate the electric field in three-dimensional environment. For this reason for formulated Dirichlet problem it is proposed the construction of computational grid based on space partition, which is known as Delaunay triangulation with the use of Voronoi cells. The numerical algorithm for calculating the potential, electric field strength and capacitance in the 3-d sphere with cubic cavity is proposed. Numerical results for potential distribution, electric field strength and capacitance in three-dimensional sphere with cubic cavity are represented. Keywords: Electric field, potential, capacitance, numerical method.

METODA VOLUMELOR FINITE PENTRU CALCULAREA CÂMPULUI ELECTROSTATIC PENTRU SFERĂ CU GOLUL CUBIC

Pa_țiuc V.

Institutul Нe EnergetiМă al AŞM

Rezumat. În luМrКrО sО ОбКmТnОКгă utТlТгărОК mОtoНОТ ЯolumОlor ПТnТtО pОntru К МКlМulК Мсmpul ОlОМtrТМ ьn

mediul tridimensional. S-a formulat problema DТrТМСlОt Мu МonstruТrОК rОţОlОТ НО НТЯТгКrО К spКţТuluТ, НОnumТtă triangulКrОК DОlКunКв şТ utТlТгКrОК ьn sМСОmК НО МКlМul numОrТМ К МОlulОТ VoronoТ. S-a propus algoritmul de calcul numeric al potОnţТКluluТ, ТntОnsТtăţТТ МсmpuluТ ОlОМtrТМ şТ МКpКМТtăţТТ sПОrОТ Мu МКЯТtКţТО НО Пormă МuЛТМă.

Sunt prОгОntКtО rОгultКtОlО МКlМulОlor rОpКrtТţТОТ potОnţТКluluТ, ТntОnsТtăţТТ МсmpuluТ ОlОМtrТМ şТ МКpКМТtăţТТ sПОrОТ Мu МКЯТtКtО НО Пormă МuЛТМă.

Cuvinte-cheie: Cсmp ОlОМtrТМ, potОnţТКl, ТntОnsТtКtОК МсmpuluТ ОlОМtrТМ, МКpКМТtКtОК, mОtoНă numОrТМă.

Ч Ч Ч

Ч

. . Ин и Эне ге икиА

.

,

. ,

.

,

.

: Э , , , ,

-.

1.

-- .

-. Э

,

, ,

Д1-3].

,

-,

Д4-6]

(2)

,

-.

,

.

--

. ,

,

-. ,

-, .

, ,

.

-, ,

-.

,

[7, 8].

2.

) , , (x y z u

,

-) , (x y

a

 - .

) , , (x y z

u 

) , , ( ) grad (

div _a u  x y z , (1)

) , , (x y z

 – . 

-, (1) div(_agradu)0.

  

  2 : 1 2, 11 0. 1

-) , , (x y z u

) , , ( )

, , (

1

z y x z

y x

u _  , (2a)

2

0

2

  



n u

, (2b)

(3)

E u u

E grad , D aE

 



 .

-: [u]0 [(D,n)]0,

, n – .

3.

   

 ,

( ).

. . 1

. 1/8 .

. ,

. Э , ,

, .

. 1. , 4501

955 .

h

T , h –

. *

h

T ,

(4)

. . 2

74. , 2, 73, 75, 80, 176, 182,

188 332, ,

. ,

,

-.

0

P , *

0

P

K . *

0

P

K Qi, ,

, P0.

. 2. 74,

-(1), (2)

-) , , (x y z

uh , 

KTh. Th uh(x,y,z)

(5)

3. K P₁P₂P₃P₄

4 3 2

1PPP

P

K ( . 3) T_h

) , , (x y z

P – .

-4 , 1 ), , ,

(x y z i

N_i , :

1 0:

      k i k i P

N_i _k , 0 , 1 ) ( . -: ) 4 ( ) 3 ( ) 2 ( ) 1 ( ) , ,

( i i i i

i x y z w x w y w z w

N     . (3)

) 1 (

i

w , wi(2),

) 3 (

i

w (4)

i

w w_i,i1,4.

-i

w 4 Awi  fi,i1,4.

-A P_i P_i(x_i,y_i,z_i),i1,4

             1 1 1 1 4 4 4 3 3 3 2 2 2 1 1 1 z y x z y x z y x z y x

A





       k i k i f f f f f f k i i i i i i , 0 , 1 , , ,

, (2) (3) (4) ( )

) 1 ( _. , ( ) ) , , (x y z i

 , i = 1, 2, …, n, n+1, …, n1, n

2 , n1 – .

) , , (x y z i

 - , . .

, Pi

(6)



   1 1 ) , , ( ) , , ( n i i i

h x y z u x y z

u . (4)

, ui (4)

Pi(xi,yi,zi): uh(xi,yi,zi)ui.

, (1), (2)

-, . (4)

(1)

) , , (x y z k  n k dV dV

u_h _k _k

agrad ) , 1,

(

div   



 



 

; (5)

k n i i ki k n i i i

a u   dV   u 









   ~ ) grad ( div 1 1 1 1

; (6)

dV dV

dV _ki _a _i _k _a _i _k

k

k           





   grad grad ) grad ( div ; ~ .

1 , (6)

k n

i

i kiu  



1

, k n

n n i i ki k

k , 1,

~ 1 1       



 

. (7)

, ,

-(5) k(x,y,z)

) ( )

( ₂

0

2  L 

W , .

) , , (x y z k

 *

h

T : _k(x,y,z)

Pk

. (5) _k(x,y,z)

n k dV dV

u_h _k _k

agrad ) , 1,

(

div   



 



 

(8)

, _k(x,y,z) *

k P K



   * * ) grad ( div k P k P K K h

a u dV dV, (9)

*

k

P

K – Pk.

-h

(7)

-. Oxyz ) , , ( ) grad (

div _a u  x y z

-*

k

P K .

, (9)



   * * ) , , ( ) grad ( div k P k P K K

a u dV x y z dV . (10)

(10)    

_



 * * grad ) grad ( div k P k P K a K

a u dV udS



       * * ) , grad ( k P k P K a K a dS n u dS n

u , (11)

* k P K  – * k P

K ; n –

-*

k

P K

 , u/n – u .

-(10)



      * * ) , , ( k P k P K K

a dS x y z dV

n u

. (12)

, (1), (2)

-(12)

-. -, -. -, - (7). * 0 P

K ( . . 2) P0 Pi,i0,8

; S_i,i1,8 – , P₀P_i ;

8 , 1 ,i

M_i – P₀P_i Si.

-*

0

P

K

 (12) :



 



_ _            8 1 ₀ 0 8 1 ) ( ) ( ) ( * 0 i i i i i a i S a K a S P P P u P u M dS n u dS n u m P , i P

P₀ – P₀P_i .

(8)

V0 – *

0

P

K . (12)

0 0 8 1 ₀ 0 ) ( ) ( ) ( )

( S P V

P P P u P u M i i i i i

a 

 



 . 0 P 0 0 8 1 0

0u(P ) u(P) (P )V

i

i 

 







; (13)



         8 1 0 0 ; 8 , 1 , ) ( i i i i i a i i P P S M . (13),

(2 ).

. ,

-. (13)

( 9 25),

-. -. -, -. 4.                           y ux u u E E E y x

grad . (14)

v(x,y)

dl n E y x v l



 ( , ) ) ,

( , (15)

n – l.

u v

-- : u/xv/y; u/yv/x[10]. ,

, v -u



__ _ _ __ _ _  ( , ) ) , ( ) , ( ) ,

( 0 0 0 0

) , ( y x y x y x y x dy x u dx y u dy y v dx x v y x

(9)

0 0, y

x – R2,

.

-, .

,

.

3

R 

 u_h(x,y,z)

u E

E E

E ( _x, _y, _z)grad .

.

: P

-P

  

2 . 2 .

) , , (x y z u_h

2

) , ( ), ,

(    

h

u , vh(,),

-) , ( 

h

u :



graduh(,),gradvh(,)



0.

(16) ,

-u v - .



O v, grad₂v



v/,v/



-

   



 / , /

grad2u u u , .

u v:



grad₂ ,grad₂



0

            

 v u v u

u

v . (17)

u , (17)

-v. 5.

C

2

1 u

u Q C



 , (18)

)

(u₁ u₂ – . Q ,

V,

S V E



   

 

S S

S d u S

d E

Q   grad 



  _ 

 

S S

dS n u dS

n

u )

(grad  . (19)

S – , ; n –

;  – .

6.

(10)

, ,

.

,

.

, Д11] ( . 4).

X

MКtlКЛ- delaunayn(X).

. ,

3. (13),

-.

:

8-10.

. CSIR

(11)

. 4.

. ,

4, P xtg y0

. ₂ P . 5 6.

v (17)

-. , ,

v 0 1.

-( . 5К).

-u , v

, v

.

1, 2 3. , u v

-. : u(,) ABC

c b a

v(,)   .

                                     3 2 1 3 3 2 2 1 1 1 1 1 u u u C B A ,                                      3 2 1 3 3 2 2 1 1 1 1 1 v v v c b a

. (20)

(17) 0                 bB aA u v u v (21) 2 2 1 1

3 v v

v   , (22)

1 2 1 2 2 1 3 2 2 3

1 , 1

) ( ) ( ) ( ) (                     B A B A

. (23)

(22) 0₁ 1.

,



3,3



, . .

(12)

. 5. v

, (22) v

,

. ( . 5Л) ,

(22), v.

( . 5c, d, e).

-.

(13)

. 6. ( )

( ) x= 0 x–y = 0

(18), (19)

. 20

10 C = 2,20 .

20 ,

:

s s

r R

r R C

 

 4 ,

Rs – , rs – ,  –

-.

,

-, ,

20 10 1,113

20 14,14 2,686

20 17,32 7,192

1.

-) , , (x y z

u

-.

2. .

(14)

3.

-,

- .

-.

4.

-, .

[1] Patsiuk V.I., Rimsky V.K. Volnovye iavlenia v neodnorodnyh sredah. .1. Teoria rasprostra-nenia uprughih i neuprughih voln. – Kishinev: Izdatelisko-poligraficheskii tsentr MoldGU, 2005. – 254 s. (in Russian)

[2] Patsiuk V.I., Rimsky V.K. Volnovye iavlenia v neodnorodnyh sredah. .2. Nestatsionarnoe deformirovanie mnogosveaznyh tel. – Kishinev: Izdatelisko-poligraficheskii tsentr MoldGU, 2005. – 239 s. (in Russian)

[3] Samarskii . ., Mihailov .P. Matematicheskoe modelirovanie: idei, metody, primery. – .: Nauka, 1997. – 320 s. (in Russian)

[4] Tihonov .N., Samarskii . . Uravnenia matematicheskoi fiziki – .: Nauka, 1977. – 736 s.

(in Russian)

[5] Samarskii . ., Nikolaev .S. Metody reshenia setochnyh uravnenii. – : Nauka, 1978. – 592s. (in Russian)

[6] Samarskii . ., Popov Iu.P. Raznostnye shemy gazovoi dinamiki. – .: Nauka, 1975. – 352 s.

(in Russian)

[7] Li R., Chen Z. and Wu W. Generalized difference methods for differential equations. Numeri-cal analysis of finite volume methods. New York-Basel: Marcel Dekker, Inc., 2000. – 459 p.

[8] Rimsky V. ., Berzan V.P., Patsiuk V.I. i dr. Volnovye iavlenia v neodnorodnyh strukturah. .5. Teoria I metody rascheta electricheskih tsepei, electromagnitnyh polei i zaschitnyh obolo-chek AES. – Kishinev: Tipografia ANM, 2008. – 664 s. (in Russian)

[9] Patsiuk V. Mathematical methods for electrical circuits and fields calculation. Chişinău,

Centrul Editorial-Poligrafic al USM, 2009.442 p.

[10] Demirchean .S., Neiman L.R., Korovkin N.V., Chechurin V.L. Teoreticheskie osnovy electrotehniki. 3: Uchebnik dlea vuzov. 4- izdanie. S.-Pb: Piter, 2004, 384 s. (in Russian)

[11] Pashkiv S.V. Frontalinye algoritmy postroenia treugolinoi setki.

http://ps300.narod.ru/fr3d/mesh.htm (in Russian)

[12] Balandin .Iu., Shurina N.P. Metody reshenia SLAU bolishoi razmernosti. – Novosi-birsk: Izd. NGTU. – 70 s. (in Russian)

.

. .- . ,

, Э

. : - ,

-, , ,

-. 140 , 9 ,