Difference schemes for the solution of a plane dynamic problem in the theory of elasticity with mixed boundary conditions

DIFFERENCE

SCHEMES

DYNAMIC

FOR

THE

SOLUTION

OF

A PLANE

IN THE THEORY OF ELASTICITY WITH MIXED BOUNDARY CONDITIONS* PROBLEM

I.G. BELUKHINA Moscow (Received

21 February

1968;

revised

11 July

1968)

schemes for the solution in a rectangule of a plane dynamic problem in the theory of elasticity, described by a set of equations in the displacements have been considered in U-61. 1. DIFFERENCE

In 11-41 the first boundary value problem was considered. In [l-21 a difference scheme with a decomposed operator was constructed for such a problem and its convergence with speed 0 (Sj was proved for ‘t/hi = const In [31 difference schemes of alternating directions were constructed, for which convergence was proved with speed 0 ( 1h 12+ t) , and also a decomposition scheme converging with speed 0 ( 1h ( 2 + 9). In [41 to solve the same problem a scheme was used which was a modification of the scheme of alternating

directions

described

in [31.

In [51 some difference schemes were mentioned without investigation for the solution in a rectangle of dynamic and static problems of the theory of elasticity with mixed boundary conditions. A dynamic problem in the theory of elasticity was considered there converge

with mixed boundary conditions

in 161. It was stated that the difference schemes if z/h, = const. with speed 0 (9 + h2)

constructed

Difference schemes for solving a plane static problem in the theory of * Zh. vychisl.

Mat. mat. Fiz.

9, 2, 362-372,

142

1969.

143

Solution of a plane dynamic problem in the theory of efasticity

elasticity for displacements and investigated in [71.

with mixed boundary conditions were constructed

In !$, 91 a general method was given for constructing stable difference schemes for solving evolutionary operational equations of the form dU

,,=-Au+l,

d2U dt2=

Au+f,

which makes it possible to construct difference schemes for concrete p~blems, if the method of app~ximating the operator A is known. In @I, in particular, it was shown which results of the general theory are applicable for constructing difference schemes approximating differential equations of the form ( * ) with boundary conditions of the first kind. It was shown there that the results of the general theory of constructing stable difference schemes can be used in the case of other boundary conditions. In the present paper the method of regularization of difference schemes of [B--91 is used to construct and investigate difference schemes ~proximating a plane denim problem in the theory of elasticity in a rectangle with mixed boundary conditions. Difference operators, constructed and investigated in 173, are used to approximate the operator A. The absolute stability of the difference scheme is proved and for the solution of the difference problem an a priori value is homogeneous in ( h / and T, is obtained, from which the convergence of the solution of the difference problem with speed 0 ( 1h 1a f T”) , a = 3/~, 2. follows, if the required solution is sufficiently smooth. It should be noted that in the investigation of the stability of the difference problem the proof of the stability with respect to the boundary conditions presents the greatest difficulty, and, even more, no success has yet been achieved in using the results of 18, 91 for this purpose. 2. Let boundary r. I’+, rj

G = G IJ r =

(0 < xa <

t,;

a =

1, 2) be a rectangle

In addition let I? = I’* lJ I?* lj F3, where the intersection

with of

can only be the corners of the region Gand some of the lri may vanish.

Displacements, stresses and the normal component of the displacements tangential component of the stresses respectively are given on lYi, l?r, r3 We look for the solution of the problem

and

144

1. G. Belukhina

~~ _

6%

drik -ek=--F+w

i

(1)

dxi

i,k=l

with initial conditions $

u(x, 0) = uCO)(x),

in the cylinder

Qr = G X [0 <

t <

(2, 0) = u(l) (x),

T]

XEC.

(2)

The boundary conditions take the

form rik

cos(v, Xi)

i, k=l (W, I& = - f,, Here u = constants,

(ui, ue) is the elastic displacement

ek =

-

(3)

f,

o
vector,

A, p > 0

are Lame

are the components of the stress tensor, v is the inward normal to the boundary are the projections on the directions of the normal and tangent to (v)v, (4, the boundary and ck is the unit vector along the xk-axis. 3. We now introduce a network which is uniform in X, and X, 0 = {(xi,, xi*) E c, xia = i&a, i, = 0, 1, . . ., N,; h, = I,/ Na, a = 1, 2) with steps h, and h,. Let 6.1= {(xi,, xiz) E G} the set of boundary nodes.

be the set of internal nodes and y = C,\

CO

We shall denote the set of corners y, which are elements of l?z\l?~ n I’1 by where yi= {xEI’i}, y2= (5E yo. Then y = YO + yi + YZ + 173, r21

\yo,

y3 =

+ E r3w

n r31

(see figure)

We also introduce a uniform network in t in the segment

[ 0 < t < T] :

Solution

~~=={tj==y’t,

145

of e pEme dynamic problem in the theory of elasticity

j=oi,

1,...,1,

z===T/J}

withstepr.

Weshalluse

the notation

We shall make use of some of the notation of 17-93: y=

(tJi,?JZ) = y(x,i),&St) = f+= Y(x,t +.c), y_Ga = (y -

f = Y(% t---z>, y,

y(-l”)) / ha,

Y1 = ($ -Y)

Before fo~ulati~

((ii+- l)hi,i&z), 392) = (i&l,(i:!fl}h,), = Y(d**$ q, Y(*la'

= fY(+3a)-

y) /h%,

yja= (y-$)/z.

/ r,

the difference problem we introduce some definitions.

The scalar product of the vector functions y and z, which are given on w, are defined by the formula

where

H = H (2)= (W,)i,is, Note that for functions y,

2

h,

tii. =

vanishing on y,

h,2

{

t

i#O,

i#N;

i =O,

N.

146

1. G. Belukhina

The scalar product of the vector functions y, z, given on y, are defined by the formula I y, z I = 2 SYZ, XE-P

I Y, z/ = I Yl, 211 f lya, 221,

(5)

where

S G S (z) =

h 17

is =

h

il = 0, N1, il = 0, N1,

(ii+hd 12,

0, Na,

-4#O,

il # N,;

iz#O,

i,#N,;

i2 = 0, N2.

The norms llYll0 =

[Y,

Yl’“,

correspond to the scalar products (4)~(5). the formula

D,, ( y) = +

i

Dh (y)

/Y, yl’“.

i1=0

(6)

We define the quantity

( N$ Ni’ h&,y:, \+ j$

a, fkl

With the help of

IlYllol =

i*=o

iI=l

;

h,h,y&)

Dh (y)

.

by

(7)

i,=t

we define the network norm W,N: Ilyll? =

ll~llo~+&(~).

(8)

In addition we introduce the norms llYlle2 =

IIY lko =

IIYillc2 + lly2llc2,

llyllc =

2 4lY (~‘)1102~ IIY lit1= 2 VEG, P&i, Ih I2 = h12 + hs2.

maxly(zI XG

I,

f (IIY (t’) II?+ IIYf (f)

(9)

lb?

(10)

4. We formulate the difference problem corresponding to the problem (l)-(3) for the case I3 = 0, l? = lYllJ I’S By analogy with [Sl we write the factorized (E + or2A + (~~r*,&Az)yrt + /iy = @,

scheme

t’ = z, . . ., T-z,

cc&iAy,,

(11)

Solution

y

of a plane dynamic problem in the theory of elasticity

(0) = u(O), Y(T) = Y(I) ---u ‘,“?$ m(l) + f Y

where A = -h

A

=

2

Ul,

(h(O) + F ItTO), 2 C-Eo’\ y1,

f=O,.

Eyl,

.

T-T,

.)

147

(12)

(13)

is the operator defined in [71,

=

A$,

AI+

A,

= i

We take the operator

A(‘)

0

;

Ab”’

i

a = I., 2.

’

A,@) in the form ‘_ ($)p $3

= (-

i,#O,

rara’

1

a$%,,, fia

_ 1 (Jf$J_ iia a “a’

Na;

i, = 0;

(14)

ia = N,,

where a?’ = a?’ = 2p + h, af’ = a(,l) = p. The function Q is defined by the formula

@=

F,

(z, t) E Q2;

F+&f,

(z, r)ESa

/

u sot

(15)

in which f = -&-(SJ”

+ .9zf(z)) for (s, t) E

SO,

f(*), fQ) are the right-hand sides of the boundary conditions on the parts of the boundary bordering on the corner and sl, sz is an element of length on the corresponding part (see [71). 5. Theorem 1 1. Let

I’EIY,,

of the problem (llM15)

and O>

(h+2p)/2(h+3~).

for ui = 0 Ily-(2+r)

Thenforthesolutiony

the a priori limit

II* < IIY(T) II, + eV liFllo,o.

I. G. Belukhina

148

holds, where

cc-a A)

YF, Yy) +

IIY (1)IL2= v4(A (Y + PI, Y + $9+ r2 fJA 02r4 %42~~,

Y?) + (~~7 yi-).

2. The solution y of problem (llM15) converges to the solution u E CW(Q,) of the problem W-(3) with speed 0 ( 1h 12+ I?), i.e.

ily--II, M > 0

eWIh12+8,

is a constant which is independent of ( hl , z.

Proof.Let Zi be the space of network vector functions which are given on z and vanish on y. It is easy to verify that the difference operators A and 1 are self-conjugate and positive definite on Gf&,while, as can be easily shown, A and A are connected by the relation

(Av, v) < 4% 4

VEi%,

9

where c=2(h+2p)/(3\.+3p). Further proof of the theorem follows directly from [8, 91. and, for definiteness,

6. Now let I’ = ri U rz,

I?1 = (3 =

(+,

~2)

: 51

0). Let a2 be the space of network vector functions which are given on z and vanish on yl. To investigate the difference scheme (llM15) we need some properties of the operators A and 8, on A% We formulate these properties in a lemma. =

Lemma1 The operators A and R are positive definite on a2 product (4) and are self-conjugate.

in the sense of the scalar

For any function v E S&z c&h,

where cl

=

p/M&p

depend on I h I, v.

VI

<

[Av,

VI

G

c2[Av7

VI,

(16)

+ A), cp = 2, MO > 0 is a constant which does not

Solution

The self-conjugacy 8~

149

of a plane dynamic problem in the theory of elasticity

and positive definiteness

of the operator -A = A on

is proved in [71; for 1 these properties are easily verified.

Relation (16)

follows from the results of [71. We now obtain some estimates, which will be useful for the future and are established by the following lemma. Lemma 2 For any vector function

v E %2

llvlli2 < M@hb) ,

(17)

ilvllo’2 <

(18)

/I v 112 <

where E >

E II V;,pr

~2ll~II~2,

Ilo2 +

M3

(El

IIv l/I27

(19)

is any number and Mi, M2, MS are constants independent of h,

0

and h,. The evaluations (18), (19) follow from similar evaluations for scalar functions which are proved in DO1 (Lemma 5) and [ill

(Lemma 1, 2). The estimate (17) is

obtained in [71. 7. We now proceed to obtain the a priori estimate for problem (llM15). We shall assume that f(l) +

f=

!

(x, t) E 8,;

Tf(2)7

22 f(I) + s f0l

(xc, tIESo.

PO)

Such a representation of f, as will be shown, reflects the structure of the error of approximation of the difference scheme. Theorem 2 Let 02

((1+e)/4)c2,

F>O.

Then to solve problem (11)~(15), (20)

with u, = 0 the a priori estimate

IIY (t) 11*2
(21)

150

I.G, Belukhina

(22)

(231

and M = M (7’) > 0

is a constant which is independent of 1h 1, z.

We shall make use of the energy identity (64) of iSI, which in our case takes the form Proof.

By definition

If f z 0 the estimate (66) of [91 holds for y;

and therefore it remains to consider the case FE

0, y(O) = 0, y(y) = 0 f i.e.

ll~(O>!l~ =2:0).

(27)

We note first of all that the estimate

[AY, Y] 3 where M!, = lq /M,.

&lly

II?,

(28)

follows from Lemma 1 and relation (17). By means of

(16) and (28) it is not more difficult to prove for CT& ( (1 + E) / 4) cz, that

where M5 = EM&/4, Me = M5 /2, M, = rr$(2~ + h). and (27) the right-hand side of (25) takes the form

On the basis of (20)

Solution

of a plane dynamic problem in the theory of elasticity

We evaluate each term separately.

151

From relations (18) and (29) we obtain

where s2’ = ~21 Me,MS = LVZ 12~2, 172> 0. The first term on the right-hand side of (30) will first be written in the form

where v = fr +

y,

and then evaluated in the same way as (31).

As a result we obtain

where

MS = M2 /4M5s3, MrO = _~I2 I4Mjeq, s3, EL> 0. The last term in (30)

is the sum of two quantities related to the corners (0, N,) and (N,, NJ. We shall consider one of these:

From (19) and (29),

r4llYfIL2< where Mil = max (i/2, MS(l/z)

r4M11

(IIY,,,t Ilo2 + II Ytll12h

).

From the last inequality it follows that

where Mii' = M,,max(l

I MT,T214M6).

1. G. Belukk~~u

152

Thus

where42 = 1/Q, Ed’= 2~&~,

E5 >

0,

Summing (241 over t’ from 7 to t and substituting tt (1--s*Z(EZ+ &6’)1~~~{~)~~*2~~6 2 ~~~~(~‘)~~.*+

(3%(351,

we obtain Gw

t’s+

Jf13

i z (IIf(z) 0’) f’mt

llo’2 +

IIf,,,?

where Q = EQ+ 2(E2 + Es’), h&s = We take sq = ‘iz*

where

P’)

llo’2 +

Mis(tQ,

IIfo (i’)

llc2) +

82, 82, &5)>

Ml0

Ilfo, (t) llo?

0.

EZ, ~3, ~4, 8~ so that 1 - ~4 = 2T (~2 $ es‘), ~3 =

(2T)-I,

and obtain from (36)

Mi3'= 4Mis, Mid = 4Mio.

Using Lemma 2.2 of [9l we obtain the estimate rlf~,,(t)/(0’~),M

I/i(l) 1!12Q hl(l[lfl/ 12+

> 0 isa constd-

From Lemma 1 of I101 we have

Thus

Combining (27) and (37) we obtain (Zl), which completely proves the theorem. 8. We will now evaluate the speed of convergence of the difference scheme (llM15). For this we put z = y - u, where u is the solution of problem

Solution

of a plane dynamic problem in the theory of elasticity

(l)-(3) and y the solution of problem (1%(15). W-(13), we obtain for z (E+u~BA+~~~~A~A~)z~~+AZ=~*, z(0) = 0,

On substituting

~‘,z,...,T-T,

z(z) = r”‘_u(Z), z = 0,

y = z + u in

r~ii\y~, 5 E-i

153

\ y1,

z E Yl,

(3) (39) (49)

where Y* = Q, -

[(E + a+A

+

is the approximation error of the difference of problem (1%(3).

a2z4RltP2) uit + Au] scheme (llM15)

(41)

for the solution II

Theorem 3 Let the solution of problem (l)-(3)

u E U4)(QT),

Then the approximation

error of Y* can be written in the form

where

lyq <

iW(h12

+ +Y,

(42)

I*0 I < M15’G2,

(43)

19(2)l <~15(lWfz% II%(l)Ilo’, II$o,i lb f

(44)

Ml5 (I h I”!+ z”).

(45)

In addition lly(‘)-u((z)II. where Ml5 > 0 Proof.


+.G*),

is a constant independent of 1h ( and r.

Let us put q~ = F -

[(E +

a+A + ONTO&&) uZt+ Au],

(46)

154

1. G. Belukhina

We define the function or on S, as follows. & = conk Then as vz, we must take

We consider the part of the boundary

vy(1j is now defined as: *4?1) = 9

(x,t)E&;

W2h -_

y

$09

(z, tIESo.

Estimates (43) and (44) follow directly from the conditions of the theorem and the representation of *PO,90). To be convinced of the truth of (44) we note that

The function vy(1j, written in the form

gel)

zz

$ -

Au -

uTt

+

F + +- f) - $ cr%“A,A, (fi -

[z’o&,

2u + ii)] -

+ W(2),

is evaluated on S,, as can easily be verified, as follows: $(I) =

The function

alhI” ++j.

(47)

~U)T is evaluated similarly on S,.

We must carry out a more detailed investigation of the error of approximation at corners. Let us consider, for instance, v(I) at the point (il z N1, ip = N2) for the first equation (see [71):

Solution

of a plane dynamic problem in the theory of elasticity

155

where

$1= $0(9),&=$0(1).

91’= wlhp++h The quantity

m-u

$(l)~ is similarly evaluated at a corner.

Estimate (45) follows from (47M49). conditions by formula (12) we have

y(l)

-

u (z)

zz

-

For the approximation of the initial

zah(O)+ F Itzo2

(y’f’ -El

(a));

=o(e), 1 (501 22 aG

atz

aTi.i

= -g

7@-

= 0 (e).

Since I~y’l’-u(7)/p

=

@[A (Y(l)-U(~)),

zw [AlA, (y’l’

- u(q),

y(l) -u

y(l)-U(Z)] + (T)] + [fy’l’ -u

(5~) (z))~+ (y(l) - u (q)rj,

estimate (46) follows from (51) because of GO). This proves the theorem. Note.

The representation

of the error of approximation of $ in the form (41’ )

in required so that we shall not need for the solution u of problem (l)-(3) a greater smoothness than is necessary for the approximation of the operator L, the boundary and initial conditions. From Theorems 2 and 3 we obtain the following.

Let the conditions of Theorem 3 be satisfied.

Then the y of problem (llM15)

156

1. G. Belukhina

converges to the solution u of problem (l)-(3) the estimate

with speed 0 ( 1h. 1G + IT*), so that

where M > 0 is a constant independent of 1h I, z respect to 1h 1 and r.

is satisfied

uniformly with

9. For a problem with mixed boundary conditions different from those just considered, but such that the corresponding static problems have a unique solution, difference schemes are similarly constructed and investigated. Here we use the operator A, constructed [71 for a corresponding problem. If the boundary conditions are such that ‘yp= 0 (e.g. if I’ = r3 I? = I’, U IY3), the convergence with speed (21), i.e.

of the difference

scheme for u >

or

( (1 -I- e) / 4) c-2

0( lhlz + I?) in the norm )I iI*, follows from the a priori estimate

lly -4.

<‘M(lh1*

+ r*),

where M > 0 is a constant independent of I 6 I, z. In conclusion I wish to thank A. A. Samarskii and ,V. B. Andreev for their interest and help. Note.

After this paper was submitted the author obtained .results enabling

h2 [ln (I/ h) 1% to be written instead of /Z’~. Transhted

by H. F. Cleaves

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2.

KONOVALOV, A. N. Difference methods for plane problems Tr. Mat. in-ta Akad. Nauk SSSR. 74, 38-54, 1966.

3.

SAMARSKII, A. A. Economic difference schemes for a hyperbolic system of equations with compound derivatives and their application to equations in the theory of elasticity. Zh. v3hisZ. Mat. mat. Fiz. 5, 1, 34-43, 1965.

4.

BATUROV, B. A. Economic plane theory of elasticity.

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schemes

in the theory of elasticity.

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in the

Solution

157

of a plane dynamic problem in the theory of elasticity

5.

DYATLOVITSKII, L. I. The solution of a plane dynamic problem in the theory of elasticity by the method of finite differences. Prikl. Mekh. 2, 10, l-9, 1966.

6.

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ANDREEV, V. B. The convergence of difference schemes with a decomposed operator, which approximate the third boundary value problem for a parabolic equation. Zh. uy’chisl. Mat. mat. Fiz. present edition 9, 2, 1969.