About correct method of analytical solution of multipoint boundary problems of structural mechanics for systems of ordinary differential equations with piecewise constant coefficients

21 A b s t r a c t

This paper is devoted to correct method of analytical solution of multipoint boundary problems of structural mechanics for V\VWHPVRIRUGLQDU\GLIIHUHQWLDOHTXDWLRQVZLWKSLHFHZLVHFRQVWDQWFRHI¿FLHQWV,WVPDMRUSHFXOLDULWLHVLQFOXGHXQLYHUVDOLW\ FRPSXWHURULHQWHGDOJRULWKPLQYROYLQJWKHRU\RIGLVWULEXWLRQVFRPSXWDWLRQDOVWDELOLW\RSWLPDOFRQGLWLRQDOLW\RIUHVXOWDQW V\VWHPVDQGSDUWLDO-RUGDQGHFRPSRVLWLRQRIPDWUL[RIFRHI¿FLHQWVHOLPLQDWLQJQHFHVVLW\RIFDOFXODWLRQRIURRWYHFWRUV Keywords: correct analytical solution, multipoint boundary problem, discrete-continual methods, structural mechanics, V\VWHPRIRUGLQDU\GLIIHUHQWLDOHTXDWLRQVSLHFHZLVHFRQVWDQWFRHI¿FLHQWV 3$9(/$$.,029 9/$',0,516,'2529 0$5,1$/02=*$/(9$ 0RVFRZ6WDWH8QLYHUVLW\RI&LYLO(QJLQHHULQJ <DURVODYVNRH6KRVVH 0RVFRZ5XVVLD 1_{HPDLOSDYHODNLPRY#JPDLOFRP} 2_{HPDLOVLGRURYYODGLPLU#JPDLOFRP} 3_{HPDLOPDULQDPR]JDOHYD#JPDLOFRP}

ABOUT CORRECT METHOD OF ANALYTICAL

SOLUTION OF MULTIPOINT BOUNDARY PROBLEMS

OF STRUCTURAL MECHANICS FOR SYSTEMS

OF ORDINARY DIFFERENTIAL EQUATIONS WITH

PIECEWISE CONSTANT COEFFICIENTS

1. Formulation of the Problem

Piecewise invariability of physical and geometrical SDUDPHWHUVLQRQHGLPHQVLRQH[LVWVLQYDULRXVSUREOHPV of analysis of structures and their mathematical models. We should mention here in particular such vital objects as beams, strip foundations, thin-walled bars, deep EHDPV SODWHV VKHOOV KLJKULVH EXLOGLQJV H[WHQVLRQDO EXLOGLQJV SLSHOLQHV UDLOV GDPV DQG RWKHUV >  @ Analytical solution is apparently preferable in all aspects for qualitative analysis of calculation data. It allows investigator to consider boundary effects when some components of solution are rapidly varying functions. Due to the abrupt decrease inside of mesh elements in PDQ\ FDVHV WKHLU UDWH RI FKDQJH FDQ¶W EH DGHTXDWHO\ considered by conventional numerical methods while analytics enables study. Another feature of the proposing method is the absence of limitations on lengths of VWUXFWXUHV +HQFH LW DSSHDUV WKDW LQ WKLV FRQWH[W VR called discrete-continual methods of structural analysis

HVSHFLDOO\ GLVFUHWHFRQWLQXDO ¿QLWH HOHPHQW PHWKRG DUH SHFXOLDUO\ UHOHYDQW >  @ *HQHUDOO\ GLVFUHWH continual formulations are contemporary mathematical models which currently becoming available for computer realization.

Discrete-continual methods are reduced at some stage to the solution of multipoint boundary problems of structural mechanics for systems of ordinary differential equations with piecewise constant FRHI¿FLHQWV &RQYHQWLRQDO IRUPXODWLRQ RI PXOWLSRLQW boundary problem of this type has the form

y A y_k f_k x x x_kb k n k b k ( )1 _{, , ), ;} 1 ( 1, 2, ..., 1 − = ∈ ₊ = − (1.1) B y x B y x g g k n k k b k k b k k k − _{− +} + _{+ =} −₊ + − ( 0 ( 0 =2, ..., 1 ) ) , ; (1.2)     _{   } k k k n b n n b g g x y B x y B1 ( 1 0) ( 0) 1

,



22 3DYHO$$NLPRY9ODGLPLU16LGRURY0DULQD/0R]JDOHYD where

y

y x

( )

[ ( )

y x

₁

y x

₂

( )

...

y x

_n

( )]

T LVWKHGHVLUDEOHYHFWRUIXQFWLRQ

x

k

n

k b k

, =1, ...,

DUHFRRUGLQDWHVRIERXQGDU\SRLQWV A_k, k n_k–1 DUHPDWULFHVRIFRQVWDQWFRHI¿FLHQWVRIRUGHUn

f

_k

f x

_k

( )

[

f

_k,1

( )

x f

_k,2

( )

x

...

f

_{k n,}

( ) ]

x

T k n_k–1

are right-side vector functions

B

B

k

n

k k k − +

=

−

,

,

2, ...,

1

and

B

₁

,

B

nk + −

are matrices of boundary conditions of order n at point

x

_kb

;

g

_k−

,

g

_k+

,

k

=

2, ...,

n

_k

−

1

and

g

g

_n

k

1

+ −

,

are right-side vectors of boundary conditions at point

x

_kb

_;

y

( )1

y

( )1

_{( )}

x

dy dx

_/

.

Solution of multipoint boundary problem of this type in structural mechanics is accentuated by numerous factors. They include boundary effects (stiff systems) and considerable number of differential equations VHYHUDOWKRXVDQGV0RUHRYHUPDWULFHVRIFRHI¿FLHQWV of a system normally have eigenvalues of opposite signs and corresponding Jordan matrices are not diagonal. Special method of solution of multipoint boundary problems for systems of ordinary differential HTXDWLRQV ZLWK SLHFHZLVH FRQVWDQW FRHI¿FLHQWV LQ structural mechanics has been developed. Not only GRHVLWRYHUFRPHDOOGLI¿FXOWLHVPHQWLRQHGDERYHEXWLWV major peculiarities also include universality, computer-oriented algorithm, computational stability, optimal conditionality of resultant systems and partial Jordan GHFRPSRVLWLRQ RI PDWUL[ RI FRHI¿FLHQWV HOLPLQDWLQJ necessity of calculation of root vectors.

-RUGDQ'HFRPSRVLWLRQRI0DWULFHVRI&RHIÀFLHQWV -RUGDQGHFRPSRVLWLRQRIPDWUL[A_k has the form

A

T J T

k

=

k k k −1 , (2.1) where

J

J

J

J

k

{

k,1

,

k,2

,

...,

k u, k

}

 T_k LV WKH PDWUL[ RI RUGHU n, which columns are HLJHQYHFWRUVDQGURRWYHFWRUVRIPDWUL[A_kJ_k is Jordan PDWUL[RIRUGHUnJ_k,p is Jordan cell corresponding to eigenvalueȜ_k,pGLPJ_k,p m_k,p.

$V ZH KDYH DOUHDG\ PHQWLRQHG DERYH VSHFL¿FLW\ of problems of structural mechanics comprises in SUHVHQFH RI PXOWLSOH HLJHQYDOXHV RI PDWUL[ A_k and

consequently in necessity of calculation of root vectors. However at the present time there are no effective numerical method of calculation of Jordan GHFRPSRVLWLRQLQWKHJHQHUDOFDVH>@0HDQZKLOHWKH number of multiple eigenvalues in the considering type of problems is normally limited. Besides these multiple eigenvalues are generally zeros. In this connection special alternative approach to solution has been developed.

3. Partial Jordan Decomposition

Partial Jordan decomposition is based on computation RIULJKWDQGOHIWHLJHQYHFWRUVRIPDWUL[A_k.

A_k = A_k,1 + A_k,2

,1 ,1 ,1 ,1

k k k k

A T J T A_k,2 = A_k – A_k,2 T_k,1 LV WKH PDWUL[ FRQWDLQLQJ ULJKW HLJHQYHFWRUV FRUUHVSRQGLQJ WR QRQ]HUR HLJHQYDOXHV RI PDWUL[ A_k

T

_k,1 LV WKH PDWUL[ FRQWDLQLQJ OHIW HLJHQYHFWRUV FRUUHVSRQGLQJ WR QRQ]HUR HLJHQYDOXHV RI PDWUL[ A_k J_k,1 LV GLDJRQDO -RUGDQ PDWUL[ FRUUHVSRQGLQJ WR QRQ]HURHLJHQYDOXHVRIPDWUL[A_kA_k,2 is the part of PDWUL[A_k corresponding to prime and multiple zero eigenvalues. It is necessary to note here that matrices T_k,1 and T_k,1 in general case are rectangular.

4. Construction of Projectors

Eigenvalues Ȝ_k,p, p 1, ..., u_k are renumbered according to the condition

∀

=

∃

=

∀

= +

O

O

k p k p k p k

p

l

m

p

l

u

, , ,

,

,

1, ...,

1

1, ...,

k k

∃

>

1









m

_{k p},  where

l

_m p u k p k k

=

=

∑

G

1 1 , ,



where

G

_i,j is Kronecker delta.

Due to distinctive procedure, we should properly modify matrices T_k,1, T_k,1 and J_k,1.

Let P_k,1 and P_k,2 be projectors to subspaces of left DQG ULJKW HLJHQYHFWRUV DQG URRW YHFWRUV RI PDWUL[ A_k corresponding to non-zero and zero eigenvalues. They may be denoted as

P

T

T T

T

P

E

P

k,1 k,1

(

k,1 k,1

)

k,

;

k, k, 1 1 2 1

=



−



=

−

_

where ELVLGHQWLW\PDWUL[

23 $%287&255(&70(7+2'2)$1$/<7,&$/62/87,212)08/7,32,17%281'$5<352%/(062)6758&785$/0(&+$1,&6

5. Construction of Fundamental Matrix-function of System of Equations

After sorting and biorthogonalization of eigenvectors DQGHLJHQYDOXHVIXQGDPHQWDOPDWUL[IXQFWLRQ

H

_k

(x

)

of system from Eq. (1.1) for arbitrary k is constructed in the special form convenient for problems of structural mechanics PD[ ,1 ,0 ,1 1 ,2 ,2 1

( )

( )

( , 0)[

],

!

k k k k k m j j k k j

x

T

x T

x

x

P

A

j

H

H

F









¦





 where i k u i k

m

m

k , PD[ ,

PD[

_dd l  , , , ,

( ,

)

sign( ) ( Re(

) ),

0

 VLJQ 



k p k p k p k p

x

x

x

x

F

O

T

O

O

O



z

­°

®

_˜

°¯



1,

0

sign( )

 



x

x

x

!

­

®

_



¯



1,

0

( )





x

x

x

T

_®

­

!



¯

 ,0 ,1 ,1 , ,

 

^  

H[S

 

 

H[S

`

k k k k k

x

diag

x

x

x

x

H

F

O

O

F

O

O



k k l l  It should be stated that the sum in the right side of Eq. FRQWDLQVIRXURUORZHUFRPSRQHQWVDQGFRUUHVSRQGV to so-called “beam” part of solution of system.

6. General Solution of the Problem

Solution of considering problem (Eq. (1.1), Eq. (TRQWKHLQWHUYDO( , 1) b b k k x x _ LVGH¿QHG by formula 1 1 ( ) ( ( ) ( )) , ( , ), b b k k k k k k b b k k k k y x x x x x C f x x x H H H         (6.1)

where

C

_k is the vector of constants of order n

_*

is FRQYROXWLRQQRWDWLRQ 1

( )

( ) ( ,

b

,

b

)

k k k

f x

{

f x

T

x x x

_  1 1 1 1, ( , ) ( , , ) 0, ( , ). b b k k b b k k _{b b} k k x x x x x x x x x

T

   ­  ° ®  °¯ 

We can rewrite Eq. (6.1) in the form

1 ( ) ( ) , ( b, b ) k k k k k k y x E x C S x x x _  1 ( ) ( b) ( b ) k k k k k E x H xx H xx _  k k k x f S ( ) H . (6.6) 6XEVWLWXWLQJ(TLQ(TDQG(TDQG taking into account that

1 ( _k 0) _k ( _k 0), 2, ..., _k y x  y _ x  k n  ( _k 0) _k( _k 0), 1, ..., _k 1 y x  y x  k n   1

(

0)

1

(

1

)

1

( 0),

 

b b k k k k k k

E

x

h

k

n

H

H





 







(6.9)

(

0)

( 0)

(

),





b b k k k k k k

E x

h

k

n

H

H



 





(6.10) 1 , 1, ..., 1 b b b k k k k h x _ x k n  (6.11)

we have the following system of linear algebraic equations for C_k  k n_k 

G C

K , (6.12) ZKHUHPDWUL[K can be divided into so-called main K0

and additional K1 _members,

0 0 0 1,1 1 1 0 0 2,1 2,2 0 0   0 0 ... 0 0 ... 0 0 0 ... 0 0 ... ... ... k ,n K K K K K K K  0 0 1 2 1 1 ... ... ... 0 0 0 ... k k k k n ,n n ,n K _{ } K _{ } ª º « » « » « » « » « » « » « » ¬ ¼  1 1 1 1,1 1 1 1 1 2,1 2,2 1 1   0 0 ... 0 0 ... 0 0 0 ... 0 0 k ,n K K K K K K K ... ... ...  1 1 1 2 1 1 0 0 0 ... k k k k n ,n n ,n ... ... ... K _{ } K _{ } ª º « » « » « » « » « » « » « » ¬ ¼ 

24 3DYHO$$NLPRY9ODGLPLU16LGRURY0DULQD/0R]JDOHYD 0 1 1

( 0)

NN N N

K



B



H

_



 0

( 0)

k,k k k

K

B



H



 0 1,1 1

( 0)

1

K

B



H



_{} 0 1,nk 1 nk nk 1

( 0)

K





B



H





 1 1 1

(

1

)

b NN N N N

K

B



H



h

  1

(

b

)

k,k k k k

K



B



H



h

_{} 1 b 1,1 1

(

1 1

)

K



B



H



h

_{} 1 1,k 1 k k 1

(

k 1

)

b n n n n

K

_

B



H

_

h

_{ } 1 2 1 [ ] k T T T T n G G G ... G _  1 2 1 [ ] k T T T T n C C C ... C _  1 1 1 1 1 1 ( 0)   k k k k b n b n n n G g g B S x B S x             1( 0) ( 0), 2, ..., 1. k k k b b k k k k k k k G g g B S x B S x k n             (6.26)

Symbol 8 imply direct product of matrices. It is necessary to note that matrices

H

_k

( 0)



and

H

_k

( 0)



are independent of x.

:H ¿QG LW YLWDO WR QRWH WKDW GLDJRQDO EORFNV RI PDWUL[K are practically singular. This fact leads to VHYHUDOSUREOHPV,WHUDWLYHPHWKRGVRIVROXWLRQFDQ¶W be applied in particular. Gaussian elimination method with pivoting is required. It is useful to specify ways of disposal of this disadvantage.

Let us transform considering system of equation as IROORZVHDFKHTXDWLRQRIV\VWHPVLQFHWKH¿UVWDQG ¿QLVKLQJQH[WWRODVWZHZLOOUHSODFHZLWKWKHVXP of this equation with the subsequent (instead of the

LQLWLDO¿UVWHTXDWLRQZHWDNHWKHVXPLQLWLDOWKH¿UVW with initial the second, instead of initial the second – the sum initial the second with initial the third and so on). Instead of initial last equation we take the sum of WKHLQLWLDOODVWZLWKLQLWLDOWKH¿UVW)LQDOO\ZHKDYH 1,1 1,2 1 1       0 0 ... 0 0 0 ... 0 0 0 0 ... k ,n K K K K K K K K K K           1,1 1 2 1 1 0 0 0 ... ... ... ... ... ... ... ... 0 0 0 ... 0 k k k k k n n ,n n ,n K _ K _{ } K _{ } ª º « » « » « » « » « » « » « » ¬   ¼_. (6.27) Thus we removed the singularity mentioned above. Acknowledgments

7KLVZRUNZDV¿QDQFLDOO\VXSSRUWHGE\WKH*UDQWRI the President of the Russian Federation for Leading 6FLHQWL¿F 6FKRROV 66 *UDQW RI Russian Academy of Architecture and Construction 6FLHQFHV  $QDO\WLFDO 'HSDUWPHQWDO 7DUJHW 3URJUDP ³'HYHORSPHQW RI 6FLHQWL¿F 3RWHQWLDO RI +LJKHU(GXFDWLRQ´3URMHFW

References

[1] =RORWRY $% $NLPRY 3$ 6LGRURY 91 Correct 'LVFUHWH&RQWLQXDO)LQLWH(OHPHQW0HWKRGIRU7KUHH Dimensional Problems of Structural Analysis. Journal of Beijing University of Civil Engineering and $UFKLWHFWXUH9RO1R-XQ

[2] =RORWRY$%$NLPRY3$6LGRURY910R]JDOHYD M.L.: 'LVFUHWHFRQWLQXDO PHWKRGV RI VWUXFWXUDO analysis0RVFRZ$UFKLWHFWXUH±6SDJHV (in Russian).

>@ =RORWRY $% $NLPRY 3$ 6HPLDQDO\WLFDO Finite (OHPHQW 0HWKRG IRU 7ZRGLPHQVLRQDO DQG 7KUHH dimensional Problems of Structural Analysis. // Proceedings of the International Symposium LSCE 2002 organized by Polish Chapter of IASS, Warsaw, 3RODQGSS

>@ Horn A.R., Johnson C.R.: Matrix Analysis. Cambridge 8QLYHUVLW\3UHVVSDJHV