BILAYER DIFFERENCE SCHEME OF A NUMERICAL SOLUTION TO TWO-DIMENSIONAL DYNAMIC PROBLEMS OF ELASTICITY

(1)

539.3 ȼ

.B.

ɇɟɦɱɢɧɨɜ

ɎȽȻɈɍ

ȼɉɈ

«

ɆȽɋɍ

»

ȾȼɍɏɋɅɈɃɇȺə

ɊȺɁɇɈɋɌɇȺə

ɋɏȿɆȺ

ɑɂɋɅȿɇɇɈȽɈ

Ɋȿɒȿɇɂə

ɉɅɈɋɄɂɏ

ȾɂɇȺɆɂɑȿɋɄɂɏ

ɁȺȾȺɑ

ɌȿɈɊɂɂ

ɍɉɊɍȽɈɋɌɂ

Ɋɚɡɪɚɛɨɬɚɧɱɢɫɥɟɧɧɵɣɦɟɬɨɞɪɟɲɟɧɢɹɩɥɨɫɤɢɯɞɢɧɚɦɢɱɟɫɤɢɯɡɚɞɚɱɬɟɨɪɢɢɭɩɪɭɝɨɫɬɢ,

ɢɫɩɨɥɶɡɭɸɳɢɣɤɨɧɟɱɧɵɟɷɥɟɦɟɧɬɵɞɥɹɚɩɩɪɨɤɫɢɦɚɰɢɢɪɚɫɱɟɬɧɵɯɨɛɥɚɫɬɟɣɫɥɨɠɧɨɣɮɨɪɦɵ,

ɜɵɱɢɫɥɹɸɳɢɣɫɤɨɪɨɫɬɢɢɧɚɩɪɹɠɟɧɢɹɜɫɪɟɞɟɢɫɜɵɫɨɤɨɣɬɨɱɧɨɫɬɶɸɜɵɩɨɥɧɹɸɳɢɣɡɚɞɚɧ

-ɧɵɟɝɪɚɧɢɱɧɵɟɭɫɥɨɜɢɹ.

Ʉɥɸɱɟɜɵɟɫɥɨɜɚ: ɪɚɡɧɨɫɬɧɚɹɩɪɨɢɡɜɨɞɧɚɹ, ɞɜɭɯɫɥɨɣɧɚɹ ɪɚɡɧɨɫɬɧɚɹɫɯɟɦɚ, ɞɢɮɪɚɤ

-ɰɢɹɩɪɨɞɨɥɶɧɨɣɜɨɥɧɵ, ɨɤɪɭɠɧɨɫɬɶ, ɤɨɧɬɭɪɧɵɟɧɚɩɪɹɠɟɧɢɹ, ɱɢɫɥɟɧɧɨɟɦɨɞɟɥɢɪɨɜɚɧɢɟ, ɦɟ

-ɬɨɞɤɨɧɟɱɧɵɯɷɥɟɦɟɧɬɨɜ.

,

-,

y

r

x

p

t

u











; (1.1)

y

q

x

r

t

v











; (1.2)

α

p

u

v

t

x

y

∂

₌

∂

₊

∂

;

(1.3)

q

u

v

t

x

y

∂

_{= α}

∂

₊

∂

; (1.4)

r

u

v

t

y

x





∂

= β

_

+

_

∂

_

∂

_

, (1.5)

v

u

,

—

x

,

y

;

p

,

q

,

r

—

,

xx yy xy

  

₁

 

2

_;

p

c



t

—

,

;

p

c

L



—

;

_L

—

;

     

1 2 ,



c c

_s _p



2

,

c

_p

,

c

_s

—

-.

:

0 0

;

t

p

r

u

x

y









(2.1)

0 0

;

t

r

q

v

x

y









(2.2)

0 0

;

t

u

v

p

x

y





 



(2.3)

0 0

;

t

u

v

q

x

y



 



(2)

0 0

,

t

u

v

r

y

x







 

_



_







(2.5)

,

  







2 ~

2

0

u

t

u

t

u











.

(3)

,

-x

:

t

u

_t







~

. (4)

.

(2)

,

-,

,

u

:

u

t

u

t

u



2

2 ~

0

_



_





_

_



_{. (5)}

(2.1)

N

_i

,

.

0 0

∫

∂

_∂

+

∫

∂

_∂

=

⋅

∫

⋅

s i s i s i

t

N

ds

y

r

ds

N

x

p

ds

N

u

(6)

:

( ) ( )

0 0

₍

0 0

₎

0 0

∫

−

=













∂

+

∂

=

∂

+

∂

+

⋅

L i s i i s i s i s i t

dl

N

dx

r

dy

p

ds

y

N

r

x

N

p

ds

y

N

r

ds

x

N

p

ds

N

u

.

(7)

:

,

0 0 j j

_N

u



v

0

=

v

0j

Nj

,

p

0



p

0j

N

_j

,

q

0



q

0j

N

_j

,

r

0



r

0j

N

_j

.

(8)

:

,

j j t t

u

N

u



v

_t



v

_tj

N

_j

,

p

_t



p

_tj

N

_j

,

q

_t



q

_tj

N

_j

,

r

_t



r

_tj

N

_j

.

(9)

u

₍

)

0

0, 5

.

t

j j

j j j

j

p

r

u

dt u

u

dt

x

y

N

N u

dt

p

r

x

y







 

   

_



_



















_



_







(10)

(

0 0

)

0, 5

.

j i j

i j t j

s s

j j j j

i i s s j j j i i j s s j j i i s G

N

N N

ds u

N ds p

x

N

dt

ds u

dt

ds v

x

y

N

N ds r

dt

ds v

y

x

N

dt

ds u

p dy

r dx N dl

y







∂



⋅

+









∂









∂



∂





∂



+

_

_

+

α

_

_

+

∂









∂



∂





∂



+

_

_

+

β

_

_

+

∂









∂



∂



+

β





=

−

(3)

-.

(11)

-.

-,

(

).

:

∫

=

∂

_∂

∂

_∂

∂

=

∂

=

s j i ij s j i ij s j i ij

ds

x

N

x

N

C

ds

N

y

N

B

ds

N

x

N

A

,

(12)

,

∫

=

∂

_∂

∂

_∂

∂

=

∂

=

s j i ij j i ij s j i ij

ds

y

N

y

N

F

ds

x

N

y

N

E

ds

y

N

x

N

D

∫

=

s j i

ij

N

ds

M

.

:

(

)

(

)

(

0 0

)

0, 5

,

j j j j j

ij ij ij ij ij ij ij

i G

M u

A p

B r

dt

C

F u

D

E

v

p dy r dx N dl





+

_

+β

+ α

+ β

_

=

_∫

−

(13.1)

(

)

(

)

(

0 0

)

0, 5

,

j j j j j

i G

M v

A r

B q

dt

C

F v

D

E

u

r dy q dx N dl





+

_

β

+

+ + β

+ α

_

=

_∫

−

(13.2)

(

)

(

0 0

)

0, 5

,

j j j j j j

i G

M p

A u

B v

dt C p

D

E

r

F q

u dy

v dx N dl





+

+ α

+

_

+

+ α

_

=

_∫

− α

(13.3)

(

)

(

0 0

)

0, 5

,

j j j j j j

i G

M q

A u

B v

dt

C p

D

E

r

F q

u dy v dx N dl





+ α

+

_

α

+ α

+

_

=

_∫

α

−

(13.4)

M r

A v

B u

dt

E p

(

C

E

)

r

D q

(

0 0

)

0,5

.

j j j j j j

i G

v dy u dx N dl





+β

+ β

+

_

β

+ β

+β

+ β

_

=

= β

_∫

−

(13.5)

(13)

,

4-

:

,

4

ij ij i

S

M

  

m

S

—

;



ij

—

;

i

—

.







ij j ij j

0, 5

ij ij j



ij ij



j



.

Fu

 

A p



B r



dt

_



C



F u

 

D

 

E

v



_

(14)

(15)

(

0 0

)

_;

i

i t i

G

m u

=

Fu

+

∫

p dy r dx N

−

(15.1)

(

0 0

)

;

i

i t i

G

m v

=

Fu

+

∫

r dy q dx N

−

(4)

(

0 0

)

;

i

i t i

G

m p

=

Fu

+

∫

u dy

− α

v dx N

(15.3)

(

0 0

)

;

i

i t i

G

m q

=

Fq

+

∫

α

u dy

−

v dx N

(15.4)

(

0 0

)

.

i

i t i

G

m r

=

Fr

+

∫

β

v dy

−β

u dx N

(15.5)

(15)

,

(15)

.

4-

(

-):

  





1

1 ,

,

4 N

  

 

 

2

  





1

1 ,

,

4 N

  

 

 

(16)

  





3

1

1 ,

,

4 N

  

 

 

₄

  

,

1 

1 

.

4 N

  

 

 

,

(17)

.

(

)

(

)

(

)

0 0

0, 5

.

i

i t i i

G G

t t i

G

m u

Fu

p dy r dx N

Fu

pdy r dx N

dt

p dy r dx N

=

+

−

=

+

−

+

−

∫

(17)

,

-.

,

x

y

n



(

):

;

dsU

 

u dx v dy



(18.1)

;

dsV



u dy



v dx

(18.2)

2 2 2

_.

ds



dx



dy

(18.3)

(18):









2 2

,

i t t t t t

t t t t

t

u dx

v dy

u dx

v dy

b

p dy r dx dx

r dy

q dx dy

u dx

v dy

b

p dxdy r dx

r dy

q dxdy

u dx

v dy bR



  

 

 



_





_







 

 



_









_



  

 



(19)



1 2



0, 25

b



L



L dt

;

L L

1

,

2

—

,

;

R

—

;



(

)

i

0, 5(

1 2

)

i

0, 5

(

1 2

)

i

L

u

Fu

pdy

rdx N

Fu

dy

p

dx

r

ℜ = −

+

_∫

−

= −

+

−

+

(20.1)

i

.

(

)

i

0, 5(

1 2

)

i

0, 5

(

1 2

)

i

;

L

v

Fv

rdy

qdx N

Fv

dy

dy r

dx

q

ℜ = −

+

_∫

−

= −

+

−

+

(20.2)









2 2 2

;

ds R



q



p dxdy



r dx



dy









2 2

,

t t t t t t

t t t t t

u dy v dx

udy

vdx b

p dy

r dx dy

r dy

q dx dx

udy

vdx b p dy

r dxdy

r dydx

q dx

udy

vdx bQ











_





_













_





_

 





2 2

₂

2

ds Q



pdx



rdxdy



qdy

—

n

.

2 2

₂

2

_;

ds S



pdy



rdxdy



qdx

(5)

2 2

₂

2

_;

ds Q



pdx



rdxdy



qdy

(20.4)

(

)

(

2 2

)

.

2

dy

dx

r

dxdy

p

q

R

ds

=

−

+

−

(20.5)

2 :

;

i t t

m U

 

U

bR

(21.1)

.

t t

i

V

bQ

m

=

ℜ

+

(21.2)

:

ds

2

S

=

pdx

2

+

2 rdxdy

+

qdy

2

,

3 :

;

i t t

m S

   

S

b V

(21.3)

;

i t t

m Q

  

Q bV

(21.4)

,

i t t

m R

   

R

bU

(21.5)

ds U

 

 

u dx

 

v dy

;

(22.1)

ds V

 



u dy



v dx

;

(22.2)

=

ℜS

ds

2 2 2

2 ;

pdx

rdxdy

qdy



 

(22.3)

=

ℜQ

ds

2

_

_pdx

2

_{ }

₂

_rdxdy

_{ }

_qdy

2

_;

(22.4)

=

ℜR

ds

2

(

ℜ

q

−

ℜ

p

)

dxdy

+

ℜ

r

(

dx

2

−

dy

2

)

,

(22.5)

Q

—

(

);

S

—

(

-).

(22) (23)

3 -,

ɱɬɨɛɵ

ɜ

ɡɧɚɦɟɧɚɬɟɥɟ

ɜɵɪɚɠɟɧɢɣ

ɧɟ

ɦɨɝɥɨ

ɜɨɡɧɢɤɧɭɬɶ

ɞɟɥɟɧɢɟ

ɧɚ

ɧɨɥɶ

:

/

1 ;

t t

i

U

R

U

R

m

b

 













(23.1)

;

t t

i

V

Q

V

Q

m

b

 







(23.2)

,

t t

i

S

Q

S

Q

m

  

 



(23.3)

i

m

;

b



0, 25



L

₁



L dt

₂



—

-,

.

(23)

-.

,

:

0;

0. Q

R



 

x

,

y

:

;

ds u

 

Udy V dx



(24.1)

;

ds v

 

Udx Vdy



(24.2)

2 2 2

2 ;

ds p



Sdx



Rdxdy



Qdy

(24.3)

2 2

₂

2

_;

ds q



Sdy



Rdxdy



Qdx

(24.4)

(

)

(

2 2

)

.

2

_r

_S

_Q

_dxdy

_R

_dx

_dy

ds

=

−

+

−

(24.5)

[1—6].

.

-min

(6)

( )

-min

h

_{ }

2

xx

p c



 , %

1. 0,098 16 –2,425 –2,97 18,4

2. 0,049 32 –2,65 –2,97 9,9

3. 0,0327 48 –2,711 –2,97 8,7

4. 0,0314 50 –2,737 –2,97 7,7

5. 0,0224 70 –2,74 –2,97 7,7

6. 0,01382 ₇₀

₍

_hr

₌

_hr

_/

₂

₎

–2,8116 –2,97 5,3

7. — –2,97 — —

 

2

xx cp

 

-min

h

.

,

.

 

2 xx

p c





h

min( .)

(

.),

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Ȼɢɛɥɢɨɝɪɚɮɢɱɟɫɤɢɣɫɩɢɫɨɤ

1. ȻɚɪɨɧɆ.Ʌ., Ɇɟɬɶɸɫ.

// . . 1961. № 3. . 31—38.

2. Ʉɥɢɮɬɨɧ Ɋ.Ⱦɠ. //

. 1968. № 1 (107). . 103—122.

3. Ɇɭɫɚɟɜȼ.Ʉ.

// . 1980. № 1. . 167.

4. Ɇɭɫɚɟɜȼ.Ʉ.

// . . «

» : . . . : , 1983. . 51.

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5. ɋɚɛɨɞɚɲɉ.Ɏ., ɑɟɪɟɞɧɢɱɟɧɤɨɊ.Ⱥ. ,

// « »

60- . . . . : , 1974. . 617—624.

6. Clifnon R.J. A difference method for plane problems in dynamic elasticity. Quart. Appl. Mfth. 1967. Vol 25. № 1, pp. 97—116.

ɉɨɫɬɭɩɢɥɚ ɜ ɪɟɞɚɤɰɢɸ ɜ ɢɸɧɟ 2012 ɝ.

: ɇɟɦɱɢɧɨɜȼɥɚɞɢɦɢɪ ȼɚɥɟɧɬɢɧɨɜɢɱ— ,

, Ɇɵɬɢɳɢɧɫɤɢɣ ɮɢɥɢɚɥ ɎȽȻɈɍ ȼɉɈ

«Ɇɨɫɤɨɜɫɤɢɣɝɨɫɭɞɚɪɫɬɜɟɧɧɵɣɫɬɪɨɢɬɟɥɶɧɵɣɭɧɢɜɟɪɫɢɬɟɬ», ., . ,

, . 50, 8(495) 583-73-81, vvnemchinov@gmail.com.

: ɇɟɦɱɢɧɨɜȼ.B.

// . 2012. № 8. . 104—111.

V.V. Nemchinov

BILAYER DIFFERENCE SCHEME OF A NUMERICAL SOLUTION TO TWO-DIMENSIONAL DYNAMIC PROBLEMS OF ELASTICITY

Numerical modeling of dynamic problems of the theory of elasticity remains a relevant task. A complex network of waves that propagate within solid bodies, including longitudinal, transverse, conical and surface Rayleigh waves, etc., prevents the separation of wave fronts for modeling pur-poses. Therefore, it is required to apply the so-called “pass-through analysis”.

The method applied to resolve dynamic problems of the two-dimensional theory of elasticity

employs _ﬁ nite elements to approximate computational domains of complex shapes, whereby the

software calculates the speed and voltage in the medium at each step. Pre-set boundary conditions are satisﬁ ed precisely.

The resulting method is classiﬁ ed as explicit bilayer difference schemes that form special

relationships at the boundary points.

The method is based on an implicit bilayer time-difference scheme based on a system of

dynamic equations of the theory of elasticity of the ﬁ rst order, which is converted into an explicit

scheme with the help of a Taylor series in time, while basic relations are resolved with the help of the Galerkin method. The author demonstrates that the speed and voltage are calculated with the

same accuracy as the one provided by the classical ﬁ nite element method, whereby determination

of stresses has to act as a numerically differentiating displacement.

The author identiﬁ es the relations needed to calculate both the internal points of the

compu-tational domain and the boundary points. The author has also analyzed the accuracy and conver-gence of the resulting method having completed a numerical simulation of the well-known problem of diffraction of a longitudinal wave speed in a circular aperture. The problem has an analytical solution.

Key words: ﬁ nite-difference derivative, two-layer difference scheme, diffraction of a

longitudi-nal wave, circle, contour stress, numerical modeling, ﬁ nite element method.

References

1. Baron M.L., Matthews. Difraktsiya volny davleniya otnositel’no tsilindricheskoy polosti v upru-goy srede [Diffraction of a Pressure Wave with Respect to a Cylindrical Cavity in an Elastic Medium]. Prikladnaya mekhanika [Applied Mechanics]. A series, no. 3, 1961, pp. 31—38.

2. Klifton R.Dzh. Raznostnyy metod v ploskikh zadachakh dinamicheskoy uprugosti [Difference Method for Plane Problems of Dynamic Elasticity]. Mekhanika [Mechanics]. 1968, no. 1 (107), pp. 103—122.

3. Musaev V.K. Primenenie metoda konechnykh elementov k resheniyu ploskoy nestatsionarnoy dinamicheskoy zadachi teorii uprugosti [Application of the Finite Element Method to Solve a Transient Dynamic Plane Elasticity Problem]. Mekhanika tverdogo tela [Mechanics of Solids]. 1980, no. 1, p. 167.

4. Musaev V.K. Vozdeystvie prodol’noy stupenchatoy volny na podkreplennoe krugloe otverstie v uprugoy srede [Impact of the Longitudinal Steo-shaped Wave on a Supported Circular Hole in an Elastic Medium]. All-Union Conference “Modern Problems of Structural Mechanics and Strength of Aircrafts.” Collected abstracts. Moscow Institute of Aviation, 1983, p. 51.

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Materials]. Collected works on “Selected Problems of Applied Mechanics” dedicated to the 60th Anniversary of Academician V.N. Chelomey. Moscow, VINITI, pp. 617—624.

6. Clifnon R.J. A Difference Method for Plane Problems in Dynamic Elasticity. Quart. Appl. Mfth. 1967, vol. 25, no. 1, pp. 97—116.

A b o u t t h e a u t h o r: Nemchinov Vladimir Valentinovich — Candidate of Technical Sciences,

Professor, Department of Applied Mechanics and Mathematics, Mytischi Branch, Moscow State

University of Civil Engineering (MGSU), 50 Olimpiyskiy prospekt, Mytischi, Moscow Region, Russian Federation; vvnemchinov@gmail.com; +7 (495) 583-73-81.