Analysis of structural response to large scale nondeterministic variations

Analysis of Structural Response to Large Scale Nondeterministic Variations ~JJOEPADOVAN

and YUEHUAGUO

Depurtments o~Mechunica1 Akron, OH 44325, U.S.A.

und Polymer

Engineering,

Uniz’ersity

ofAkron,

ABSTRACT : Employiq u concutenated w-expunsion procedure, this pqer dw~lops cl.fiwmal mutri.u imwsion scheme which con br employed to solw lur,qc SC& prohubilistic .st@ess

ruriutions.

The generulity

of thc~,fi)rnlulrltiorl

r~riteriu uw ulso ur~uiluhlr to optimix~ user defincv~ uccurucy. is po.vsible. distribution

This fktions.

Dw

~wkloud

is such that a priori conwr,q:gencr and truncution rind resource rrcluirr,r?lrnts,for

to the munner of:fhr?ulrtion,

includr~s the uhilit!, To dwnonstrutc

urc presrntc~d. The.w dcul bt,ith problwls

any rungr

to hrrndlc~ urhitrurJ~

u gictw Irwl

of’probuhilistic

of

wriution

Rirmunrr

intqrrrblc

prohubilit!~

the schcmc. the results ofscwrnl

numcricul

experiments

posscwing

wide rcm,ying prohubilit~~

rwriutions.

I. Introduction

In recent years, increased attention has been given to ascertaining the static response of mechanical systems with some form of statistical characteristics. Such nondeterministic problems have been tackled by a variety of approaches, i.e. (i) The Monte Carlo scheme as per Larson (1) and Contreras (2) ; (ii) The perturbation scheme as illustrated by the work of Southwest Research reported recently by Cruse (3) and Liu et ul. (4) : (iii) Via ml hoc schemes including various formal expansions, for example, of matrix exponential functions and Sturm LouivilleeVodicka~~Tittle series as employed by Padovan et ul. (5, 6). In the case of the Monte Carlo scheme, the problem of evaluating the response by determining the requisite ensemble moments tends to be extremely cumbersome and time consuming. More importantly, it is very difficult to establish convergence either u priori or N posteriori. Such a state of affairs also applies to the perturbation approach. This is especially true when the various system probabilities range over wide intervals. In such situations. convergence can be faulty or misleading. In the context of the foregoing, this paper will develop a formal inversion scheme which through the use of concatenated re-expansions can handle any range of probabilistic variations. Due to the manner of expansion of the probabilistic stiffness matrix, indicators are developed which provide least upper bound control on such items as convergence and truncation. Because of their N priori nature, the indicators can be employed to establish optimized work load and resource requirements before commencing the calculations.

For the current purposes. the procedure will be applied to evaluate the response of nondeterministic structural and mechanical systems to static loading. Due to the generality of the scheme, it can be applied to lumped parameter, as well as large scale finite element and difrcrence simulations. This is true for any analytically or numerically Riemann integrable probability distribution function.

II. Prohlun Dejinition In the context of the introduction, assuming that a static structure possesses probabilistic propel-tics, then the stiffness formulation is cast in the form [k’(pllY(p)

= F

(1)

where [K], Y, F and p are respectively the stiffness matrix, nodal deflection vector, nodal force vector and probability vector. Given that [K@)] can be additively . decomposed, then (2)

[k’(P)1 = [&II + [K,,(P)1 where [K,,] is the deterministic portion while [k’,,(p)] consists of the nondeterministic partitions. In this paper. [K,,(p)] is assumed a linear function of p namely. [64P)l Here, [K,,,1 delines the partitioned probability variables /I, such that

(3)

= i l,,[k;,,l. i I influence

matrices

p’ = (/‘,.I’:..

for each of the individual

. ../).I)

where ( )’ denotes matrix transposition. The solution to (I) requires the development

(4)

of a statistical

Y(P) = [K(P)]

matrix inverse.

i.e. (5)

‘F.

Due to the extensive coupling within [K(p)] ‘. it is literally impossible to tract the individual /I,; in [I. J]. For the current purposes, the inverse will be established formally. Numely, limiting the range of variation of each of the individual I’,. it follows that [K,>] can be made significantly “smaller” than [K,,]. In this context. since [K(P)1 = [&I [[Ll + [k’,,l [K(P)1

’ = K/1 + [k’,,l

(6)

’ [K,Jll

‘[K/,11 ’ [k’,Il

‘*

(7)

the binomial matrix term ([I] + [K,,] ‘[K,,]) ’ can be expanded in Taylor type series so as to yield a formal invcrsc of [K(p)]. Recalling the binomial expansion theorem, (7). its formal use yields the following matrix equivalent, namely

([II + [K,,] ![K,,l)

404

’=

1 (I, 0

I)“([/;,,]

‘[K,I])“.

(8)

In terms of the additive (113+

decomposition,

(3), it follows that (8) takes the form

[&I ’[KPI) ’ = ,,$,,(~ 1Y([Gl ’ f: P,[mY. ,= 1

To determine the mean, standard derivation, covariance moments must be ascertained. These take the form

E(( y,)“‘) = where P(p) is the joint probability statistical moment and

etc., the various statistical

(10)

P(P)( Y,(P))“’ dp

s

distribution

function.

(9)

~2 defines the order of the

PE[PI..PLl iE[l,N] such that p,, and p, are the lower and upper bounds of p and N defines the number of elements in Y. Recast in full form, (IO) yields the expressions PI E(( Yi)“‘) =

.

cPI

OP,,..

../~.,)(Y,@,r

. . . . p,,))“‘dp,...d/t,.

(11)

s

Based on (9). (I I) can be directly evaluated for a wide variety of circumstances. The major difficulty in using (9) lies in its convergence requirements. Due to the inherent structure of the series, the convergence criteria can be established by determining the spectral properties of [K,]] ‘[Kp]. Specifically, the eigenvalue problem ’ [K,>]l- = Al-

[K,,] where j, is the eigenvalue to the form

and r its associated

For a convergent (3). the maximum namely, following (8)

spectral

I 3 max (2) = max

where 0 (

j symbolizes

the eigenvalues

vector. In terms of (3). (12) reduces

radius of (I 3) must be bound

o{[[K,,] ! of (

(12)

’ i,

by I .O,

w)

i and max (

) denotes the maximum

of( ). To establish the allowable range of probabilistic variations, (I 1) must be Riemann integrable, see (9). This requires that (3) be convergent. To define the upper bound on the range of convergence, employing the definition of spectral radius, we see that based on the Courant~ Fischer theorem (10, II), max (A) < max (I),) max

p (i

[K,,]

’ ,$,

[%,I)) 405

Hence, the maximum

allowable

probabilistic

m,tT~$” ’ max

excursion

(p([K,,]

takes the form

‘C[K,,,])) ’

(14)

While (14) provides an N priori upper bound requirement on the formal expansion, it also points to its weakness. i.e. a limited range of application. This range is entirely dependent on the spectral properties of the system matrices. namely [K,,] ‘[K,,]. In the next section, the range of application will be extended so as to be unlimited. This will also include the establishment ofconvcrgence and remainder formulae which enable an N priori evaluation of the scheme.

Earlier we saw that the formal use of Maclaurin series yielded a statistical inverse with a limited range of validity. To extend the interval handled, use will be made of piccewise concatenated Taylor-type expansions. As will be seen, this yields a formal invcrsc with an unlimited range of applicability. To start the dcvclopment. since the various statistical moments generally involve Ricmann integrable functions. the overall integration requirements can be performed in a piccewise concatenated format. This requires that the overall statistical interval PE [p,. pc] bc divided into a picccwise set of increments. For the current purpose. this is achieved by introducing a one parameter family of scaling factors. In this context, (10) takes the form

E((

I’,)“‘)

=

f

i-,

P(p)( Z',(P))"' dp

(15)

where here

PJ = PC -p/

(17)

such that the X, scaling factors satisfy the identity

As will be seen later, the choice of the individual U, is established via use of succcssivc spectral radius conditions. Based on the one parameter resealing, p is recast to yield the expression P(i, I) = PI. f ‘i a,P<, + [diag (p,,)][ i I 406

(19)

where

(20)

[diag (p,,)l =

and

(21)

(-‘= (;,,i? ,.... <.,I such that ci, E[O,X,]: for V(k; I)E([l.

J];

[O,Ml).

In terms of (19), dp(L 0 = [diag

(22)

(P,JIdL

hence (15) reduces to the form E((Y,)“‘)=

f

li...

,=, s 0

P(p(i,

1))( Y,(p(i, 0)“’ fi P<>,dt’, >. . ., di,.

,= I

i‘

(23)

To complete the simplification of (23), Y,(p([, I))needs to be expanded in series about succeeding values of [ and 1. This is achieved by re-expanding successive [K(p(<./))Imatrices, namely [Np(i,

(24)

O)l = [&,I + [AK;1

where

[&,I = [&I + i P,uk O[&,l ,= I

(25)

[AK;] = 2 i;,/h,[&,l ,= I

(26)

such that (27) Employing

(24), we yield that [K(P(L 011 ’ = (14 + [Ku1 ‘LAY-I) ’ [&,I

(28)

’

where

([4+[K,,l The convergence

‘LAY11 ’ = ,z$,, (- 1)“([&,I

of (29) is contingent

(29)

’ [AfW’.

on the spectral

radius

properties

of

[K,,,] ‘[AK_]. In particular. ’ [AK_]) ) < I.

max (I’ i [K,,/] Recalling nanicly

(26). it follows that (30) is ;I

result

the following

of

[~,,.I ’ i: I,p,,,[K,,,]T = I

Rearranging

eipenvaluc

problem,

(f,I-.

(31)

= m[K,,]l-.

(32)

I

(3 I), yields ,i,

Under

(30)

the Courant

max (JJ ([K,,,]

(C,P,,,-/J1(O.o,[k;,,lr

Fischer theorem. ’ [AK.]

j ) <

max

(33) yields the inequality

(<,,p,,,,

-pi

(0,

I))

p

mx !i

i

in I

[K/II ’[k;‘,l ii (33)

In this context.

it follows that inax (&/I,,,, --/J,~(0, /)) mix (i.) < I

(34)

where after rearrangement (35) with (C”/$‘~/I(O.

I)“)

=

111;1x

(;g,,,,

~/I,,(().

I))

(36)

and i ” = Since (36) provides an upper bound vergent if and only if %, <

I

(i).

(37)

on successive

c.I,,

17,;’ 1,

max

-t/40.

I)”

!

q. it follows that (2Y) is con-

. I3

2.

As can be seen from (3X). the upper bound on the range of permissible abilistic excursions varies linearly for each of the succeeding re-expansions. x, < x2 < xj < X4..

probSince

< l,. . ,

the range of convergence for any given re-expansion is seen to increase rclativc to the base level. This property helps to signifkantly reduce the number of re-expansions for any given probabilistic interval. Note, to further reduce the work load, the re-expansion process can be performed recursively. This is achieved by recognizing that [K,,,] ’ is itself the inverse of the

preceding

expansion

:

namely

[Kwl ’ - ([K,, 11+WK;l) ’

(39)

for V IC [I, ,441. To guarantee a given level of accuracy, an upper bound on truncation needs to be ascertained. For the current purposes. this can be obtained via the series remainder. In the context of (29), we yield the expression

([rl + [&,I ’[WI) where the formal remainder

’ = ,,c,, (- ~)“([&,I ‘WW’+

R!‘+’

(40)

is defined by the relation

R \‘+I - (-l)“+‘([K,,,]

‘[AK_]).L”([fj+[<])

(.‘+”

(41)

such that the matrix [c] has elements i,.$bound in the interval ]c,,] < ([K,,,] ‘[AK:]),.,. To reduce (41) to a more usable scalar form, pre and post multiply (40) via an arbitrary vector V and divide by its inner product, namely

v’([rl+[Gl

‘[WI)

v’v Based on the properties reduced to the following

‘\‘_l_v’[K,,,l~‘[AK~lV+.,, V’V

of the Rayleigh form :

v’([fl+[Kul

quotient.

\

where here R “+ ’ is the scalar version

If

with C’E [O. ~“1. From an upper bound 6 (,,“),‘--

I= 0

of the remainder,

R,%’ 1 =(-I),\“((0

]R’+‘]

see (10, 12), (42) can now be

‘[AK:11 ;v < i ~_l~,~w,,~,+R,+,

V”V

)

(42)

(43)

i.e.

\+I

(44)

point of view. we obtain

< (E,“‘([“/7,;’ -/l(O, I)“))‘~’ ‘.

Note, once the accepted error in R \‘+I is defined, i.e. E,-, then the number terms required in the expansion can be estimated. Hence. since IR”’ I] < (j’f(<+‘P\‘_P(O / 1. n

,)*f))‘b’+r < c,

(45) of

(46)

solving for N, we yield the expression

Nl, d

log (6,.) log (/?‘l(i,“/$

--I)(O, 1)“))

-I.

(47)

Equations (38) and (47) provide excellent m priori upper bound measures which guarantee convergence as well as control of the anticipated work load associated with the concatenated re-expansion process. They can be used prior to the start of calculations so as to estimate and optimize the work load and resource usage.

IV.

Benchmark

Examples

To benchmark the probabilistic scheme developed in the preceding section. the results of several examples are presented. These will illustrate the use of the rrp~io,-i convergence and remainder formulae as well as the statistical moments resulting from probabilistic variations in various components. Figure I illustrates the geometry and material properties of the truss structure used to benchmark the scheme. In the examples which follow. some of the truss elements have selective probabilities introduced in order to ascertain the resulting sensitivities. For the current purposes. a uniform probability distribution function will be employed, in particular:

for V/j E [I>,_,/)l.]. To determine structural sensitivities to variations element. (26) can be reduced to a partitioned form, namely

in a given truss

(49) where

(50)

[K,J=

FIG. 1. Gcomctry

and material

propcrtics

of nondctcrministic

benchmark

structure.

Analysis qfStructura1

Response

Based on (49), we yield that E(Y) = ;

i

/= I,,= I

(- l)“m(I)“)([&,I

’[K,IY’

(51)

such that here

(52) Note, due to its partitioned

form, (51) can be recast as

([&,,I ’[&I)” = [K,,,l ’[K,l

. . I

(53)

(K,,,,l ’[&lY ’ . .I

where [K,,,,,] ’ is a partition of [Kc,,] ’ sized and located as per [K,,] of [K,,]. To determine theconvergence and remainder requirements of (51). the maximum spectra1 radius of [Kn] ‘[K,,] must be evaluated. Once known. (38) and (47) can be used to establish the truncation limits for succeeding re-expansions. This can be achieved through the use of the power method which yields the largest eigenvalue. Note. for the current benchmark example, convergence and remainder requirements are defined by the partitioned version of [K,,] ‘[I$]. namely [K,,,,] ‘[K,,] where here, [K,,,,] ’ is a partition of [K,J ’ which is located and sized as per [I<,,] of [IV,,]. Figures 221 I illustrate the mean and standard deviations of the nodal displacements as effected by probabilistic stiffness variations in various individual system components. As can be seen, significant changes are noted in the standard deviation and mean response behavior. Here, the probability function was ranged over the interval pt./p,_ _ 200%. Note. for the given structure, the upper levels of the probability interval lead to a saturation of the sensitivity effects. This is clearly seen in Figs 2 -1 I. Such behavior is a direct result of the generic properties of the chosen structure. Namely. past a critical difference in local element stiffness, the varied components tend to act as though they were essentially rigid. In comparing the sensitivities generated by the various truss elements, Figs 2- 1I illustrate sensitivities due to changes in such individual components. As can be seen, each component generates a different range of response sensitivities. Due to the spectral radius properties of [K,,,] ‘[AK:]. the given problem required only three re-expansions to span the 200% interval in probabilistic variation. This is in contrast to the single expansion approach where for the given problem, convergence is upper bound in the range 0(26~-29%). Note the range improvement is a direct outgrowth of the increases of stiffness afforded by the re-expansions about the upper limits of each of the subsequent intermediate probability ranges. As per the Courant Fischer theorem, the maximum eigenvalues of the succeeding [K,,,] ‘[AK,] expansion decrease in the manner described via (38). To prevent the accumulation of roundoff, [K,,,] ’ needs to be expanded to

J. Pudorun urd YLW H14uGuo

3.30

2.60

1.90

”

w

1.20

0.00

0.40

0.80 PROBABILISTTC

FIG. 3. Mean nodal response

4.00

3.30

h 5 H b

2.60

k! !i fi c 1.90 2 g z ;; *.20

0.50 0

412

.oo

to probabilistic

"AI<-IATION

variation

2.00

1.60

1.20

p

in bar 5 ; ,VE[0,200’%].

Anul~~sis

of Strwturul

Rrsponw

4.00

3.30 n E E

2.60

i Li a cl 1.90 c 8 B " w 1.20

0.50 0.40

0.

0.80

PROBABILISTIC

FIG. 4. Mean nodal response

1.20

1.60

VARIATION

2 0

P

to probabilistic

variation

in bar 7 ; PE [0.200%].

to probabilistic

variation

in bar 8 ; pi [0,200%]

4.00

FIG. 5. Mean nodal response Vol 327. NO. 3. pp. 403417. Pnntcd an Great Bntoin

1990

413

J. Padovun und Yue Hua Guo

0

FIG. 6. Mean nodal response

to probabilistic

variation

in bar 9 ; pi [0,200%]

0.600 BAR

p.

5

0.640

n

8

>

z ;

0.480

El 3 a

0.320

i " 0

0.160

0.000

I

I

I

0.40

0.00

0.80

PROBABILISTIC

FIG. 7. Standard

deviation

of nodal

I

I

I

1.20 VARIATION

response to probabilistic p E [0,200%].

I

1 2.00

p

variations

Journal

414

I

1.60

in bar

of the Franklm Pergamon

5;

Institute Press plc

Analysis

of Structural

Responsc~

0.900

n

0.640

8 c ;

0.460

i n 2

0.320

! " 0

0.160

0.000

I

0

I

I

0.40

I

I

0.60 PROBABILISTIC

FIG. 8. Standard

deviation

of nodal

I

I

1.20

I

I

I

1.60

VARIATION

2.00 p

response to probabilistic p E [OT200%]

variations

in bar

6;

0.800

fi

0.640

8 H F; w

0.480

El E ::

0.320

z z " '

0.160

I

I

I

0.40

I

0.60 PROBABILISTIC

FIG. 9. Standard

deviation

Vol.327,No.3,pp.403417. 1990 Printed in Great Britam

of nodal

I

I

I

1.20 VARIATION

response to probabilistic JJE [0,200%].

I

I

1.60

2

io

p

variations

in bar

7;

415

0.800

A

0.640

5 . ;

0.480

rl g

a

0.320

; z ” 0

0.160

I

0.000

I

I

I

I

I

I

0.60

0.40

0.00

PROBAE.JzL.ISTIC FIN;.

10.

St:~ndard

deviation

of nodal ,lE

I

I

I

I

1.60

1.20 VAR

I ATION

2.00 p

response to probabilistic [0,200%].

variations

in bar

8;

0.600

0.000

I

0

I

I

0.40

1

I

0.80

I

I

I

1.20 “*\RT*TION

FIG. I I. Standard

416

deviation

of nodal response to probabilistic /)E [0.200%].

I

I

1.60

2.00 p

variations

in bar 9;

adequate truncation levels. Once the appropriate used to establish [K,,,] ’ within 0 priori accuracy

error tolerance limits.

is set, (47) can be

V. Conclusion

In preceding sections, a formal inversion procedure was developed to determine the inverse of statistical matrices arising in structural mechanics. Due to the generality of the development, this included the generation of (I priori criteria which enable the evaluation of convergence and truncation error. In this context. the scheme can be used in conjunction with either lumped parameter, finite difference or finite element simulations of static structural problems. Due to the manner of formulation. the procedure can easily be incorporated into tnost standard finite element and difference codes. This would require little rearchitecturing. Note. beyond the purely statistical applications, the overall procedure can be used to establish structural sensitivities to parametric variations evolving out of a design study. Here. the various statistical moments (standard derivation, covariante) could be employed to quantify the said variations. In such studies, it is vital that absolutely reliable results are generated. Here. the ability to determine u priori convergence and truncation limits enables the user to both anticipate work load, as well as define an acceptable accuracy level. Acknowledgement

The first author acknowledges the infuence problems.

of Chris Chamis of NASA Lewis in rekindling

his interest in probabilistic Rqfirences

(I) H. J. Larson, “Probabilistic (2) (3) (4) (5)

(6) (7) (8) (9) (10) (I 1) (12)

Models in Engineering Scicnccs”, Vols I and 2, Wiley, New York, 1979. H. Contreras, “The stochastic finite clement method”. C‘ot?lpLlr. StI.11(.t..Vol. 12, pp. 341-348. 1980. T. Crust. “Probabilistic structural analysis”, NASA Langley Computational Mechanics Workshop, Nov. 1987. W. K. Liu. A. Mani and T. Belytschko. “Finite element methods in probabilistic mechanics”. Ptdxthilistic Etzyng Mcdt., Vol. 2. No. 4, pp. 201~~213. 1987. J. Padovan, “On the solutions to multiply connected nonself-adjoint systems subject to deterministic and nondctcrministic excitations”. ht. J. Espy Sci., Vol. 14. p. 819. 1976. J. Padovan and I. Zcid, “On the transient solution of nondeterministic systems”. J. Ftwtzklitt ht.. Vol. 308, p. 497, 1979. H. Hochstadt, “Differential Equations-A Modern Approach”, Holt. Rinehart and Winston. New York. 1964. R. S. Varga, “Matrix Iterative Analysis”, Prentice Hall, Englewood Cliffs, NJ, 1962. A. Papoulis, “Probability Random Variables and Stochastic Processes, McGraw-Hill. New York. 1965. P. Lancaster, “Theory of Matrices”, Academic Press, New York, 1970. R. Courant. “Methods of Mathematical Physics”, Interscience, New York, 1962. L. Meirovitch. “Analytical Methods in Vibrations”, Macmillan, New York, 1967.