Zastosowanie stochastycznej metody elementów brzegowych w modelowaniu losowych układów mechanicznych

Z E S Z Y T Y N A U K O W E P O L IT E C H N IK I ŚLĄ SK IEJ 1994

Seria: M E C H A N IK A z. 116 N r kol. 1231

T ad eu sz B U R C Z Y Ń S K I K a te d ra M echaniki T echnicznej P olitechnika Śląska

Z A S T O S O W A N IE S T O C H A S T Y C Z N E J M E T O D Y E L E M E N T Ó W B R Z E G O W Y C H W M O D E L O W A N IU L O S O W Y C H U K Ł A D Ó W

M E C H A N IC Z N Y C H

S treszczenie. W pracy przedstaw iono podstaw ow e k o ncepcje stochastycznej m eto d y elem en tó w brzegowych. N iepew ność w układach m echanicznych m o d elo ­ w an a je s t za p o m o cą pól losowych, których m om enty opisują stochastyczne w aru n k i brzegow e, losowe w łasności m ateriału i stochastyczny kształt brzegu.

A P P L IC A T IO N O F ST O C H A ST IC B O U N D A R Y E L E M E N T M E T H O D T O M O D E L L IN G O F U N C E R T A IN M E C H A N IC A L SY STEM S S um m ary. F u n d am en tal concepts o f the stochastic bou n d ary ele m e n t m ethod a re p re s e n te d . U n certain ties in m echanical systems a re m odelled by m ean s of ra n d o m fields w hose m om ents specify stochastic bou n d ary conditions, ran d o m m a te ria l p ro p e rtie s an d stochastic sh ap e o f a boundary.

A N W E N D U N G D E R S T O C H A S T IS C H E N

R A N D E L E M E N T E N M E T H O D E B E IM M O D E L L IE R E N V O N S T O C H A S T IS C H E N M E C H A N IS C H E N S Y S T E M E N

Z usam m enfassung. In d e r A rb eit w urden die G ru n d k o n zep tio n en d er stochastischen R a n d elem en ten m eth o d e dargestellt. D ie U n sich erh eit in m ech an isch en System en wird m ittels d e r stochastischen F eld er, die M o m en te von w elchen stochastische R andbedingungen, stochastische W erkstoffeigenschaften u nd stochastische R an d fo rm beschreiben, m odelliert.

1. IN T R O D U C T IO N

T h e functionality o f m any m o d ern engineering system s d ep en d s to large ex ten t on th e ir ability to p e rfo rm ad eq u ately a n a w ith a high level o f reliability u n d er n o t absolutely c o n tro llab le conditions. In resp o n se to these problem s, co m p u ter m eth o d s have b e e n d ev elo p e d to deal w ith th e statistical n a tu re o f loads, m aterial p ro p e rtie s an d th e sh ap e o f a stru ctu re. S tochastic bou n d ary elem en t m eth o d (SB E M ), as an altern ativ e num erical tech n iq u e to th e stochastic finite elem en t m eth o d [cf. e.g. G h an em an d S panos (1991), K leib er an d H ie n (1992)] belongs to th ese m ethods.M any different in te rp re ta tio n s are possible fo r term inology o f th e SBEM . This term is used h ere to re fe r to U te'boundary e le m e n t m e th o d w hich accounts fo r u ncertainties in: bou n d ary conditions o r m aterial p ro p e rtie s o f a stru ctu re, as well as th e sh ap e o f a boundary. Such un certain ties are usually d istrib u ted on a bo u n d ary ( r ) o r w ithin a dom ain (fi) o f the stru ctu re a n d should b e m o d elled as sp atial r(x ,y ) o r spatially-tem poral r(x ,t,y ) stochastic fields, w h ere x = ( x k) d e n o te s sp atial position in fi o r r , te T is tim e, y is an elem en tary event ( y e ^ ) . Such stochastic fields a re defined on th e probability space (IT, ), w here g ’is the space of elem en tary events, & is a s-algebra o f subsets o f r£ and P is a probability [Sobczyk (1984)].

A p plications o f th e S B E M a p p e a r to have b e e n initiated in th e early 1980’s. T h e earliest ap p licatio n used th e bou n d ary integral equation m eth o d to solving stochastic bou n d ary value p ro b lem s o f elastostatics [Burczynski (1981)]. N ext the SB EM was used to stochastic p o te n tia l p ro b lem s [Burczynski (1985a)], stochastic h e a t conduction p ro b lem s [D rew niak (1985)], g ro u n d w a te r flow [C heng an d L afe (1991)] an d dynam ical p ro b lem s [Burczynski (1985b), (1988a), Burczynski and Jo h n (1985a), (1985b), (1991), Spanos an d G h an em (1991)]. S B E M w as also extended to problem s w ith random m edia [Burczynski (1986a)]

an d [E tto u n ey et al. (1989), (1993), M anolis (1993)] an d to p ro b lem s w ith u n certain b o u n d aries [N akagiri el al. (1983), N akagiri and Suzuki (1989), Burczynski (1986b), (1988b), (1992)].

A concise p re s e n ta tio n o f th e SB EM to p roblem s w ith stochastic b o u n d ary conditions, stochastic m aterial p ro p ertie s, stochastic sh ap e sensitivity an d identification p ro b lem s was given in th e form o f ch a p te rs by Burczynski (1989), (1993a), (1993b), (1993c).

2. C O M P U T A T IO N A L M E T H O D O L O G Y O F SBEM

U sing S B E M to solving bo u n d ary value problem s with ran d o m bou n d ary conditions in th e form o f prescrib ed stochastic fields o f displacem ents u (x ,y )= u °(x ,y ), x e T j, and

A p p licatio n o f S tochastic B oundary E lem en t M ethod to Modelling... 59

tractions p (x ,y )= p ° , x e r2 respectively, w here r = r1U r2 and r1f l r2=0, as a result a vector stochastic b o u n d ary integral equation is obtained:

c ( x ) u ( * , y ) = f u ( x , y ) p ( y , y ) d r ( y ) - f p ( x , y ) u ( y , y ) d r ( y ) (1)

r r

w here U (x,y) an d P(x,y) are determ inistic fundam ental solutions for elastostatics.

Practically n o t stochastic solutions o f these equations are sought b u t th e ir m om ents (i.e.

expectations, covariances). In o rd e r to solve this problem it is possible to distinguish two g en eral ap p ro a c h e s, nam ely continuous an d discrete approaches.

In th e continuous a p p ro ach determ inistic boundary integral eq u atio n s for m o m en ts are fo rm u lated . B o undary integral equations for expectations are identical to those for determ inistic systems.

B oundary integral eq u atio n s for covariances differ from the traditional ones in th a t they involve double, in stead o f single, boundary integrations [cf. Burczynski (1981), (1985a), C heng an d L afe (1991), (1993)]. This ap p ro ach can be useful to p roblem s w ith the D irichlet (essen tial) o r N eum ann (n atu ral) type o f stochastic bou n d ary conditions.

In the case o f m ixed stochastic boundary conditions it is convenient to use th e discrete a p p ro a c h in w hich stochastic b oundary integral equations are discretized into a system of ran d o m algebraic equations:

[ A ] 1 X ( y ) | = [ £ ] | T ( y ) l , (2)

w here

[A] a n d [B] a re determ inistic m atrices d ep en d en t on b oundary integrals of fu n d am en tal solutions P and U,

(X (y )} is a colum n m atrix o f unknow n random nodal values o f boundary d isplacem ents an d tractions,

(Y (y )} is a colum n m atrix o f random nodal values o f boundary displacem ents and tractio p s p rescrib ed by b oundary conditions.

F o r the b o u n d ary elem en t discretization it is necessary to app ro x im ate stochastic fields o f b o u n d ary displacem ents an d tractions into random vector rep resen tatio n s (X (y )} an d (Y (y)}. Tw o m ethods o f discretization can be proposed.

The m idpoint m ethod

T h e field value o f r(x ,y ), r= u ,p , over a boundary elem en t r e is re p re se n te d by its value a t the m idpoint xe o f the elem en t r c:

r ‘ ( Y ) .= r ( x e , y ) , r ' e r ' (3 )

The boundary averaging method

T h e value fo r a b o u n d ary elem en t r e is re p re se n te d by the spatial av erag e o f th e ran d o m field ov er th e e lem en t re:

r ‘ ( y ) = — -— f r ( x , y ) d r ( x ) , (4)

|r'|;L

w h ere I rel is the a rea o f rc.

D iscretizatio n o f ran d o m fields o f displacem ents u(x,y) an d tractions p(x,y) im plies th a t th e colum n m atrices {X(y)> and {Y(y)} consist o f variables w hose covariance d ep en d s on th e b o u n d ary e lem en t m esh.

C onsequently, th e stochastic c h aracter o f th e boundary elem en t values m ay b e expected to d e p e n d significantly on the chosen m esh geom etry.

T hus, th e im p o rta n t p ro b lem in using th e discrete ap p ro ach is th e selection o f th e m esh size. N u m erical calculations show [cf. Burczynski (1993a, 1993b, 1993c)] th a t th e m id p o in t discretization m e th o d ten d s to o v er-rep resen t th e variability w ithin th e b o u n d ary elem en t w h ereas th e sp atial averaging m eth o d tends to u n d er-rep resen t th e sam e variability. T he tw o m eth o d s coincide w hen th e ran d o m field m esh is sufficiently fine in relatio n to the co rrelatio n length. T h e system o f random algebraic eq u atio n s (2) is th e b ase for evaluating o f m om ents:

- m ean value:

(m J - [ A r ' t J H m , ) , (5)

- covariance m atrix:

[Kx] = [Ar 1 [*][**][*] 7 '[Arir. (6)

T h e cross-covariance m atrices are evaluated as follows:

[*„]- [A ]'1 [*][*,] , [*„] = [*,][*]r[A]-,r. (?)

H aving o b ta in e d probabilistic characteristics o f unknow n ran d o m values o f d isplacem ents an d tractions, to g e th e r w ith the specified values o f tractions and displacem ents, the in terio r values o f displacem ents an d stresses can be calculated [cf. Burczynski (1988a), (1993a), (1993b)].

A pplication o f S tochastic B oundary E lem ent M ethod to M odelling., 61

3. A P P L IC A T IO N O F SB EM T O D Y N A M IC SY STEM S

Tw o d ifferen t b oundary elem en t ap p ro ach es to dynamic analysis can be distinguished, nam ely: th e tim e d om ain ap p ro ach and the integral transform (L aplace o r F o u rier) dom ain a p p ro a c h . B oth can be used to stochastic dynam ic analysis b u t it seem s th a t the application o f th e F o u rier integral transform dom ain a p p ro a c h offers a special convenience consisting in th e possibility o f em ploying spectral densities in description of spatially-tem poral stochastic boundary fields o f displacem ents u(x ,t,y ) an d tractions p(x,t,y). T his sp ectral a p p ro a c h was used to b oundary elem en t analysis o f stochastic vibration o f elastic an d visco-elastic systems by Burczynski (1985b, 1988a), Burczynski and Jo h n (1985a, 1985b, 1991). A n alternative tim e dom ain ap p ro ach to stochastic vibration o f elastic system s w as also p ro p o sed by Burczynski and John (1985b).

T h e stochastic b o u n d ary integral equation for dynam ic problem s in the F o u rie r tran sfo rm dom ain has th e form:

c(x) u(x,u,y) = fi/(x,y,o))p(y,o),y)dr(y)-J'p(x,y,c})u(y,a),y)dr(y) , (8)

r r

w here u (x,o>,y) a n d p (x ,u ,y ) are the F o u rier transform o f stochastic spatially-tem poral fields o f d isp lacem en ts an d tractions, respectively:

r ( x , u , y ) = J r ( x , t , y ) e x p ( - i ( D t ) d t , r = u , p (9)

and U (x ,y ,o ) an d P (x ,y ,u ) are th e F o u rier transform o f determ inistic fu n d am en tal solutions U(x,y,t) an d P(x,y,t), respectively.

D iscrete version o f eq u atio n (8) fo r o=(g>*), 1=1,2,..L, takes the form:

[ A ( o ) ] ( Y((o,y)) = [*(« ) ] 1 Y(<o,y)| , (10)

w here

[A(g>)] an d [B(c*>)] are determ inistic com plex sq u are m atrices d e p e n d e n t on boundary integrals o f F o u rier transform s o f fundam ental solutions P and U,

{X(a>,y)} is a colum n m atrix o f unknow n F ourier transform o f random nodal values o f b o u n d ary displacem ents and tractions,

{Y(<a,y)} is a colum n m atrix o f F o u rier transform o f random b oundary displacem ents a n d tractio n s prescrib ed by boundary conditions.

G e n e ra l sp ectral density o f solution o f equation (10) has the form:

[ S x ( o l t « , ) ] = [A (G)1) r1[fi(< o1) ] [ S y(o )1,G)2) ] [ B ( o )2) ] tf[A (o )2) ] - 1'/

cot = (to * ) , k= 1,2 , / = 1,2, ...L .

C ro ss-sp ectral densities a re expressed as follows:

[$„( « , , « , ) ] = [ A ( u I)]-I[ i ( « 1) ] [ S y ( o 1,u2)] , d2)

[S^CQj.a),)] = [ Sj,((o,,m2) ] [ B ( u 2) ]w [A (<o2) ]"lw , (13)

fo r to = (o )k1), k = l ,2 , 1=1,2,..L. T h e ].]H d en o tes conjugation an d tran sp o sitio n o f the m atrix [.].

4. A P P L IC A T IO N O F SB EM T O R A N D O M C O N T IN U O U S SY STE M S

S tochastic n a tu re o f the m aterial results m ainly from the nonh o m o g en eity and in d eterm in acy o f th e m edia stru ctu re. T h e com plexity and irregularity o f th e p ro p e rtie s o f re a l m edia leads to a stochastic description o f these m edia [Sobczyk (1984)].

It is difficult to o b tain fu n d am en tal solutions fo r stochastic m edia b ecau se o f the com plexity o f th eir m ath em atical form ulations. T o solve this, p ro b lem p e rtu rb a tio n tech n iq u es can be used (cf. Burczynski (1986a), E tto u n ey ct al. (1989), (1993), M anolis (1993)]. H ow ever, this a p p ro a c h requires th e assum ption th a t ran d o m fluctuations of stochastic p ro p e rtie s o f a m edium a re small.

It is possible to apply the o th e r ap p ro ach [cf. Burczynski (1993a)] w hich d oes not req u ire such assum ption. This ap p ro ach consists in using the isotropic determ inistic fu n d a m e n ta l solutions co rresponding to a referen ce elastic m odel C °, w hose p ro p ertie s m ay be found by averaging th e stochastic m edium :

C ( * ,Y) = C» + C ( x ,y ) , x e ü , <14)

w h ere C ° = E [C (x ,y )] is the m ean value o f th e elastic m oduli tensor, an d C (x ,y ) is a ran d o m field characterizing th e m edium fluctuations with m ean value E [C (x ,y )]= 0 and the co rrelatio n m o m en t K c (x1)X2)= [K kjmnprsl(x1,X2)]. In the g en eral case fo r th e th ree- dim ensional p ro b lem s C is a 6x6 m atrix and the n u m b er o f different elastic constants in C (x ,y ) is 21.

A p p licatio n o f S tochastic B oundary E lem en t M ethod to Modelling... 63

As a resu lt th e following vector integral equation is obtained:

c(x)u(x,y) = | U(x,y)p(y)dr-J’P(x,y)u(y)dr-fR(x,y)o(y,f)dQ(y) (15)

r r q

w h ere U, P a n d R a re determ inistic fundam ental solutions for th e m edium with m aterial constants C °, an d

<j(y,y) = C(y,y)e(y) , y e Q . (16)

E q u atio n (15) is sim ilar to determ inistic problem s with only th e addition o f a new stochastic d om ain term which d ep en d s on th e elastic moduli tensor C ( x , Y ) = C kj mn(x,Y) ch aracterizing th e ran d o m fluctuations o f m edium . This term can be in teg rated over the dom ain using in tern al cells.

T h e p ro b lem is solved iteratively by finding th e first solution with a(y ,Y )= 0 an d then com puting th e ir values an d resolving th e system as m any tim es as required.

5. A P P L IC A T IO N O F SB EM T O S H A P E D ESIG N SE N SIV IT Y A N A LY SIS

O n e knows th a t bo u n d aries bounding real bodies a re very com plicated as fa r as their geo m etrical sh ap e is concerned. U sually they are uneven an d the irregularities do not easily lead to a unique determ inistic description. T h erefo re boundaries o f such bodies can be defin ed stochastically. SB EM is a very useful and natural technique for m odelling such p roblem s. In o rd e r to solve these problem s it is possible to use an a p p ro a c h based on th e id ea o f stochastic shape design sensitivity analysis [ef. Burczynski (1986b, 1988b, 1992, 1993a, 1993b, 1993c)]. T h e application o f this approach to exam ine stresses, strains, displacem ents, n atu ral frequencies and in the g eneral case an arbitrary functional with resp ect to stochastic sh ap e o f the boundary is presented. F o r the simplicity o f fu rth er considerations b o u n d ary conditions have b een assum ed determ inistic.

O n e assum es th a t stochastic sh ap e o f the boundary r* may be defined by prescribing a stochastic v ecto r field 6g(x,Y)=(<5gk(x,Y)), so that:

x ‘(y) = * + 6g(x,y) , E 5 g = 0 , (17)

w here th e determ inistic variable x is related to the baseline o f the boundary r.

T he stochastic tran sfo rm atio n field g (x,a(y)) modifies the external b oundary r, w here a (Y )= ( a r(Y)), r= l,2 ,..R , is a set o f random shape p aram eters, which specifies the actual stochastic to le ra n c e ran g e o f the structure.

T h e v ariatio n o f th e tran sfo rm atio n field 6g is expressed as

6gk = v /6a , ( y ) , ( 18)

w h ere vk = 3 g k/d a r can b e considered as a velocity tran sfo rm atio n field w hich is associated w ith a sh a p e design p a ra m e te r a r(y ). O n e assum es th a t random sh ap e p a ra m e te rs can b e expressed as follows:

a ( y ) = a o + 6 a ( y ) , £ 6 a ( y ) = 0 , (19)

w h ere v ariatio n 6a (y ) rep re se n ts fluctuation o f random p a ra m e te rs an d a u = E a ( y ) is the m e a n value o f sh a p e p a ra m e te rs a (y ).

If a (y ) h as th e G aussian distribution then it is com pletely d escribed by th e m ean value aQ an d the covariance m atrix [K ]= E [6a6aT],

D u e to sm all ra n d o m variations o f the boundary, resulting stochastic fields o f stresses a (x,y), strain s e(x,y), displacem ents u(x,y) and n atu ral frequency o>(y) can b e expressed as follows:

a ( x , y ) = o 0 ( x ) + 5 a ( x, y ) , E & o ( x , y ) = 0 , x 6 & , (20)

e ( x .y ) = e0 (x) + 6e ( x ,y ) , E 5 c ( x , y ) = 0 , xeCl , (21)

u ( x , y ) = u0 ( x ) + 5 u ( x , y ) , E 5 u ( x , y ) = 0 , x e Q or x e V , (22)

u ( y ) = u 0 + 6 u ( y ) , E 6co(y) = 0 , (23)

w h ere q0 = E q (x,y), q=cr,e,u an d (j0 =E u)(y) a re identified with the m ean value o f state fields c alc u lated fo r th e untran sfo rm ed b oundary r with the determ inistic base shape p a ra m e te rs a Q= ( a or), r= l,2 ,..R .

In th e g en e ra l case it is possible to consider an arb itrary functional

J = f

Q

r(a)

w h ere T = ( a ,e ,u ) is an arb itrary function of stresses a, strains c and displacem ents u w ithin th e do m ain fi = fl(a )i an d <p(u,p) is an arbitrary function of displacem ents u and tractio n s p on the boundary r =r(a).

T h e functional J can express global m echanical characteristics (e.g. p o ten tial or co m p lem en tary energy) as well as local stresses, strains o r displacem ents [ef. Burczynski (1992)].

D ue to stochastic sh ap e variation the functional J can be expressed as follows:

,/(a# +6a(y)) = J ( a 0) + b J ( y ) , (25)

w here th e first v ariation o f th e functional 6J ( y ) can be expressed analytically utilizing the adjoint ap p ro ach .

In d ep en d en tly o f the p rim ary system, a concept o f an adjoint system with the adjoint solution ua, ea an d o a is introduced.

T he adjoint system is an elastic body with identical configuration and physical p ro p ertie s as th e p rim ary system b u t with o th e r boundary conditions and body forces. O n the bo u n d ary r th e re are prescrib ed boundary conditions in the form:

0n ( m a ^ £ ) _ 0n (26)

dp du

and w ithin th e do m ain Si initial strain e al, and stress cral fields and body forces ba are specified:

= ( a , e , u ) gml _ d'H (o,e,u) b „ = 9 T (q,e,u) (27)

d o de. d u

T he constitutive law for th e adjoint system has the form:

a° = C ( e a - e ai) - a a‘ , (28)

w here C6= c ijkle ^1, Cjjk|=A .5ij5 kl+ / i( 5 jk6 j|+ 6 jl6jk), A and ¡1 are the L am e constants.

T he first v ariation o f the functional J can be expressed as:

67(y) = fS)r (Sa(Y)) , <29>

w here

1 M y)1 = [8a1(Y),8tf:2(Y)J...,8flf(Y),-,6aji(Y)]7', (3°)

is th e m atrix o f v ariation o f random shape p aram eters, and

f S ) = (31)

A pplication o f S tochastic B oundary E lem en t M ethod to M odelling... 65

is the sensitivity m atrix w hose elem ents Sr, r= l,2 ,..R , are expressed by total m aterial derivative o f th e functional J with respect shape p aram eters, i.e. S = D J /D a r

T h e to ta l m a te ria l derivatives o f J has th e form [cf. M roz (1986), Burczynski ( 1 9 9 2 ,1993b, 1993c)]:

S r = Y - o - e “ + b-ua + (<)> +p 'ua),„ - 2(4» + p -ua)9t]nt vt d r

\/(1M

D u o r 7 T ~ ” “ *v* D a . d T ,

■/(in

D p0 o ^r n ~ P -kVk

D a . d

r,

• / I * + p - u ‘ j v ' d L ,

w h ere in teg ran d [^ + p » u al = ( ^ + p « u a) + -(0+ p « u a)"represents th e discontinuity of (0+ p « u a) along th e curve L, w hich sep a ra te s two p a rts o f th e b o u n d ary Tj an d r2, n = ( n k) is th e u n it n o rm al vector, St is th e m ean curvature o f th e boundary.

It is seen from (32) th a t sensitivities o f J d ep en d only on b o u n d ary sta te variables o f th e p rim ary system an d th e adjoint system . T his fact gives significant adv an tag es in num erical calculations by m ean s o f th e b o u n d ary elem en t m ethod.

C o n sid er now a special kind o f sh ap e tran sfo rm atio n in th e form o f tran slatio n , ro ta tio n an d scale ch an g e (expansion o r contractions). This case is especially in terestin g w hen a body co n tain s inhom ogeneity in the form o f in tern al defects such as a crack, a cavity o r an inclusion. T h e n th e d e p e n d e n c e o f the functional J on stochastic sh a p e a n d location o f such in tern al defects can be exam ined.

O n e assum es th a t th e stochastic sh ap e tran sfo rm atio n field g (x ,a(y )) m odifies o f a given initial sh a p e o f th e in tern al defect in th e form :

- stochastic tra n sla tio n (T ), by prescribing 6gk( x ,y ) =6bk(y ), k = 1,2,3 w h ere b k(y ) a re ra n d o m tran slatio n p a ram eters,

- stochastic ro ta tio n (R ), by prescribing 6gk( x ,y ) = e kp1x16o>p(y), p = l ,2 ,3 w h e re o)p(y ) a re ra n d o m ro ta tio n p a ra m e te rs, e kpl d en o tes the p erm u tatio n tensor,

- stochastic scale change (expansion o r contraction) (S) o f th e crack by prescribing

6 g k( x , y ) = x k 6 r | ( Y ) , w h ere T ] ( y ) is a ran d o m scale change p a ra m e te r.

Now th e total m aterial derivatives D J/D a r, w here a ( y ) = ( a r(Y))H(b1,b2,b3, o j , ^ , “3,11) take th e form o f th e p a th -in d e p e n d e n t integrals:

S' mJ£ r = fz[(o,e ,u,o°,e°,u°)dr' , (33)

r = 1 , 2 , 3 , 4 , 5 ,6,7; L = T ,R ,S

w h ere

Z* = ( T r* b y + o ijU‘ k + oJ«u - a e 6kj)ny , (34)

fo r tran slatio n ( k = 1,2,3),

A pplication o f S tochastic B oundary E lem en t M ethod to Modelling.., 67

7p*

kp l 0 w E ' V / ö t y + 0 < / « “ * * ! + a u u k + ° i j “ k + ° W k x i > n j , ( 3 S )

for ro ta tio n ( p = 1,2,3), and

zl = ~ a U Ui -

f 7

fo r expansion o r contraction.

In th e last case, th e form o f Zs7 is valid 7 = 0 and <p = (p(u) is a hom ogeneous function o f o rd e r a and B= 1 for 3-D and fl= 0 for 2-D. r, is an arbitrary closed surface (or contour for 2-D ) enclosing the defect. Prim ary (u^ejj.cTjj) and adjoint ( u ^ e ^ a y 8) solutions along r, can be obtain ed using boundary elem ent procedures. Enclosing the defect by a surface r*, derivatives o f J can be determ ined by calculating the respective path-ind ependent integrals along any surface r». In particular case the surface r, can be identified with the external b o u n d ary surface r.

In tern al d efects can introduce local gradients o f displacem ents o r stresses which reach g reat o r infinite values as co m p ared to respective values w ithin the stru ctu re dom ain.

C alculations o f derivatives o f J by m eans o f p a th -in d ep en d en t integrals along fixed co n to u rs fa r from such singularities (e.g. cracks) ensure good accuracy o f th ese derivatives [Burczynski an d Polch (1993)].

T h e v ariation o f n atu ral frequency can be expressed by:

6co(y) = {S“ )7-{6a(Y)) , (37)

w here e lem en ts o f sensitivity matrix {S“ > are evaluated by total m aterial derivative of n atu ral freq u en cies with resp ect to sh ap e param eters [cf.Burczynski (1986b, 1988b, 1992, 1993a, 1993b, 1993c)]:

s r = 7 7^- = f [ o ( u ) - e ( u ) - Wg p u- u] nkv'k d r , (38) u a r 2o>0 Jr

w here p is the m ass density and u(x) is th e displacem ent am plitude.

It is in terestin g to notice th a t 6u>(y ) is expressed by the boundary integral an d dep en d s on th e n a tu ra l frequency <jQ, the m ode u(x) and the stochastic fluctuation o f shape p a ra m e te rs 6a (y ). This is very im p o rtan t in num erical calculations using boundary elem ents.

T h e m e a n value o Q can b e evaluated using the dual reciprocity a p p ro a c h fo r th e free vibration pro b lem . A fte r discretization o f the boundary r by m eans o f boundary elem ents th e follow ing e q u atio n is obtained:

[if] { u 1 = <0q [Af] 1« ) , (39)

w h ere {u} is th e colum n m atrix o f nodal values o f displacem ent am p litu d e, [M] is th e m ass m atrix.

T h e covariance o f stochastic stresses, strains and displacem ents fo r th e two fixed points Xj a n d x2 is calculated from :

= £ ([9(*i>Y)-90(*i)] [9(*2*Y)-90(*2)i b1 =

£ [ & « ( * , . Y )6? ( *2>y)] = ( S ( x ,) } [ X ] {S (x 2) }r (40)

T h e v arian ce o f th e ran d o m n atu ral frequency can b e calculated using eq u atio n : Kar( tt) = £ [ q ( y ) - u0] 2 = E [ 6 c a ( Y) ] 2 = ( S “ 1 [ K ] | S “ ) r . (41)

In o rd e r to calculate the sensitivity m atrices {S} o r {S“ > o n e should d e te rm in e th e tra n sfo rm a tio n velocity field v£, which is associated with the sh ap e p a ra m e te r a r(y ). T h e selection o f sh a p e design p a ra m e te rs is th e key elem en t in th e sh ap e sensitivity analysis an d o p tim al design.

6. N U M E R IC A L E X A M P L E S

E xam ple 1

T h e chain link (Fig. la ) m ade u p o f the visco-elastic m aterial is lo ad ed by the slow -changeable spatially-tem poral stationary stochastic field o f tractions p(x ,t,y ) w hose sp ectral density is given by

S p ( x itx 2 , u ) = 2 6 (x, - x 2) a ^ a ( P 2 + a 2) / [ x ( o 2 - (f2 - a 2) 2 +4ct2 o 2]

w here o £ = 289[N 2], a = 0 .1 5 [s‘1], B = 7[s’1].

A pplication o f S tochastic B oundary E lem en t M eth o d to Modelling.., 69

Fig. 1.

N um erical results o f sp ectral densities o f vertical displacem ents o f th e p o in t F for the M axwell (M ) a n d K elvin (K ) m odels are p resen ted in Fig. lb .

E xam ple 2

T h e p ro b lem o f stochastic sh ap e sensitivity analysis fo r a vibrating steel arch (Fig.2a) is considered. T h e b o u n d ary elem en t m odel consists o f 38 linear elem ents. It is assum ed th at the b o u n d ary u n d erg o es sm all random variation m easured along the radius w ith the covariance K (x ,y )= a2exp[(-1 x r x2 I - 1 y 1-y2 I )/a], w here ct2=15»10"6 [m2].

The m ean values o f th e first, second an d third n atu ral frequencies are cjo1 = 3034[s‘1],

<i)o2= 4 7 8 8 [s'1] a n d g>o3=7743[s"1], respectively.

Fig.2b shows th e d ep e n d e n c e o f stan d ard deviations o f n atu ral frequencies as a function of th e radius o f co rrelatio n a [m].

b) Frequency Standard Deviations Versus Correlation Length

Fig. 2 .

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S treszczenie

W pracy przedstaw iono podstaw ow e koncepcje stochastycznej m etody elem entów brzegowych w rozw iązyw aniu zad ań brzegowych teorii sprężystości z losowymi w arunkam i brzegowymi. O p isan o dwa sposoby dyskretyzacji losowych pól przem ieszczeń brzegowych i sił pow ierzchniow ych prow adzące do układu losowych rów nań algebraicznych, które są podstaw ą do określania m om entów .

O m ów iono zastosow anie stochastycznej m etody elem entów brzegowych do zagadnień dynamicznych o raz do układów o losowych w łasnościach m ateriałow ych. Przedstaw iono także zastosow anie m etody do zagadnień analizy wrażliwości, gdy brzeg układu sprężystego opisany je st z losową tolerancją. P odano przykłady num eryczne ilustrujące m etodę.