The Rayleigh quotient iteration method for computing eigenvalue bounds of structures with bounded uncertain parameters

Pergamon

Compurm % Slrucrures Vol. 55. No. 2. pp. 221-227. 1995 CopyrightB 1995 Elsevier Science Ltd Printed in Great Britain. All rights reserved 0045-7949/95 $9.50 + 0.00

:

00457949(94)00444-7

THE RAYLEIGH

QUOTIENT

COMPUTING

ITERATION

EIGENVALUE

STRUCTURES

WITH

UNCERTAIN

METHOD

BOUNDS

FOR

OF

BOUNDED

PARAMETERS?

Zhiping Qiu, Suhuan Chen and Hongbo Jia Centre

for Computational

Mechanics,

Jilin University of Technology. Republic of China

(Received

3 I December

Changchun

130022,

People’s

1993)

Abstract-When the parameters of structures are uncertain, structural natural frequencies become uncertain. In this paper, we deal with the vibration problem of structural parameters with interval uncertainty; the eigenvalue problem of the structures with interval uncertain parameters is transformed into two different eigenvalue problems to be solved. The Rayleigh quotient iteration method is applied

to the vibration problem of the structures with interval parameters; the numerical results show that the proposed method is sufficiently accurate and requires little computational effort.

INTRODUCTION

solving interval eigenvalues parameters.

In recent years, the vibration theory for structures with deterministic parameters on geometry and material has been well developed. Researchers have gradually become interested in uncertain vibration problems and monographs have been published. Thus, the dynamics of uncertain systems will be a significant area of potential future research. In a variety of design applications to calculate response quantities, such as displacements, stresses, vibration frequencies, bucking loads and mode shapes to assess the integrity of a proposed structure, those response quantities are usually thought of as the functions of design parameters. However, due to some uncertain information (for example, manufacturing errors, errors in observations) what is the predication of structural behavior for a given design caused by the uncertainties of design parameters? It is desirable or even mandatory to find stability and/or response characteristics of a system. Because the uncertainties take on various values over a range, there exist three methods to describe the uncertainty: probability (random) description, fuzzy sets description and interval (imperfection) description. The analysis of random and fuzzy eigenvalues and eigenvectors has been a subject of several studies by mathematicians and engineers and is reviewed in [l-4]. However, few attempts have been made to study vibration problems of structures with interval parameters. The efficient method to be discussed in this paper is the Rayleigh quotient iteration method for

INTERVALS

This project is supported

Science CAS 55 2-‘

Foundation

by the National

with interval

AND THE INTERVAL EXTENSION

In this section, we shall give a brief discussion on interval analysis 12, 5, 61. Assume that f(R), I(F) and I(/?” ’ “) denote the set of all closed real intervals, interval vectors and interval matrices, respectively. X’ = [x, X] is a member of I(R) and X’ can usually be written in the following form: X’ = [X‘ - AX, x’ + AX],

(1)

in which x’ and AX denote the mean (or midpoint) value of X’ and the uncertainty (or maximum error) in X’, respectively. It follows that x’ = (.? + 9/2,

AX = (_u - .u)/2.

In terms of the interval addition, into the more useable form:

where AX’ = [-AX, AX]. Similar expressions exist A ’ = [A, a] E I(R” “I), i.e. A’=

(2)

eqn (I) can be put

X’=x’+AX’,

(3)

for an interval

A‘+AA’,

matrix

(4)

where AA’= [-AA, AA], A’ and AA denote the mean matrix of A’ and the uncertain (or the maximum error) matrix of A’ respectively. It follows that A’ = (1 + A)/2

t

in structures

or

a;, = (a,, + a,,)/2

(5)

or

Aa,, = (a,, - 5,)/2;

(6)

Natural

of China.

AA 221

= (A - A)/2

Zhiping

222

A’ = [a, A] is called a symmetric interval matrix [7]. if A is symmetric for every real matrix A E A ‘. Let .f‘be a real valued function of n real variables .Y,, .x2, , x,,; by an interval extension of,/; we mean an interval valued function JF of n interval variables X{, X{, , ii’!, with the property [3, 51: F(x, ) x*,

, x,,) =f’(x,

, x:,

Qiu et al

From the set theoretic definitions of interval arithmetic it follows that any natural interval extension of a rational function in which each variable occurs only once (if at all) and to the first power only. will necessitate computing the exact range of values. providing that no division by an interval containing zero occurs.

. x,,) GENERALIZED

for all x, E X!,

i=l,2

,...)

n.

An interval valued function, F, of the interval variables Xi, ,I’{, , Xl, is inclusion monotonic if

) n,

y;cx;, i = I, 2,

(8)

implies F(Y{, YS,.

, Y,‘,)cF(X:,X;,.

(9)

, x;,,.

Real rational functions of n real variables have natural interval extensions. Given a rational expression in real variables, we can replace the real variables by corresponding interval variables and all operations by interval operations to obtain a rational interval function which is a natural extension of the real rational function. Since rational interval functions are inclusion monotonic, we have the following theorem. Theorem. If F is a rational interval interval extension off, then

{.f(x ,,...,.

Q): x,cX:,

i=1,2 EF(X{,

function

INTERVAL

EIGENVALUE

and an

It is often desirable in a variety of applications to obtain solutions to the equation Ku = EMU, in which K and M are affected by uncertainties. One therefore becomes concerned with determining the tolerance in each component i, of the solution. knowing the tolerance inherent in elements k,, and r?z,,, i,.j = 1. 2, , II. Such a problem pertains usually to dynamic model data gathered from fields or experiment observations, which certainly lack precision. Let us consider an eigenvalue problem for the structural vibration

Ku = i.Mu subject

x;,

rz)

.x;,,.

(IO)

In other words, an interval value of F contains the range of values of the corresponding real function .f: when the real arguments of ,f‘ lie in the intervals shown. The theorem provides us with a means for the finite evaluation of upper and lower bounds on the ranges of values of real rational functions over II -dimensional rectangles. For example, consider the polynomial

Ed K$ k

or k,, < k,, GE,,, i,,j = I, 2,

M
or

m,,
,n,

where K = (k,,) is the stiffness matrix, M = (m,,) is the mass matrix, u is the mode shape and i is the square of the frequency of free vibration. Both K and M are symmetric and K is positive semi-definite and M is positive definite; K = (k,,) and R= (ii,,) are the minimum and maximum allowable stiffness matrices of the system, but K is uncertain and it ranges over eqn (15); M = (E,,) and ti = (m,) are the minimum and maxir%m allowable mass matrices of the system, but A4 is uncertain and it ranges over eqn (16). With the interval matrix notation [S. 6.81, eqns (15) and ( 16) can be written as MEM’,

(17)

in which K’= [K, q is a positive semi-dcfinitc interval matrix [87 and M’= [M, /@I is a positive definite interval matrix [7]. For the sake of simplicity, by means of eqn (I 7), eqns (14))( 16) can be expressed:

(12) K’u = E.M’u.

Computing

(16)

(11)

and with A’= [0, I] and A’ = [2, 31 we get a natural interval extension off’ as the interval polynomial F( X’. A ‘) = g,

,H (15)

KEK’,

U.Y

,f’(_x,a) = __ I -_x

(14)

to

i,j = I, 2,. ,....

PROBLEM

(7)

(18)

F([2, 31, [0, I]), we obtain

F([2, 3], [(), 1]) = ]‘; ‘] ]z[‘;:] = [ _ & 01,

( I 3)

Equation (18) is called ei.!?envalueproblem L81.

a

generalized

interval

Rayleigh iteration method The basis problem is: the given central matrices K’ = (R + K)/2 and M” = (R + M)/2 of K’ and M’, and the uncertain matrices AK= (K- K)/2 and AM = (I@ - M)/2 of K’ and M’, how to find interval width and eigenvalue ET which is the smallest encloses all possible eigenvalues, i,, satisfying Ku = J.Mu, when K and M assume all possible combinations. In other words, we seek a hull, i.e.

223 M’ = M’ + AMe,,

wheree,=[-I, I], K’=(R+K)/2,AK=(R-_K)/2, M’ = (h? + M)/2, AM = (&’ -M)/2. Substituti@ eqns (26) and (27) into eqn (25) we obtain the interval evaluation of eqn (24): i: = min ,max

u’(K’ + AKe a)u

a,c~j~,,ee, uT(K’ + AKe,)u

i’=[+,Z],

LI=[&,l,],

i=l,2

,...,

n

(27)

= min , B,< R”

L,f 0

(19)

[u’K’u - (ulrAK]u], uTK’u + lul’AK/ulJ

to the set

%:” [urMcu - IuI’AMIuJ, f ={i,:AER,Ku=/IMu,

I,

u#O, M E M’},

KEK’,

(20)

where

f

u’M’u

+ lu17AMlul]

0

(W

To compute the lower and upper bounds on a particular A,, we introduce Deifs assumption [8], i.e. the signs of the components of the associated eigenvector u’ remain unchanged, when matrices K and M range over the interval K’ = [_K,q and M’ = [M. m. Denoting S’ = diag(sgn(u’,),

, sgn(u:,))

sgn(u;),

u; # 0

in which [9, IO]

(29)

u ‘Ku ,I,(( K, M)) = min, max __. @,
(23)

we have S’u’ = IuJ( > 0.

(30)

(u( = S’u into eqn (28) we have

” = ;k’?::

[uT(K’ - S’AKS’)u,

ur(K’ + S’AKS’)u]

[u’(M’ - S’AMS’)u,

u’(M’ + S’AMS’)u]

(31)

IIf 0 By interval

,I;= yii,

division,

we have

L, d 0

rnG”,” [uT(K’ - S’AKS’)u,

u’(K’ + S’AKS’)u]

By the means of the interval can be written as u TK’~ i.,‘= min, max ___ rp,,iv ,rta, u’M’u’ IIi 0

1

u’(M’

- S’AMS’)u

1 (32)

Under the constraint conditions (15) and (I 6) let us consider the Rayleigh quotient for the structural vibration [9, IO]: u ‘Ku i, = min, max ___ *,
1

u’(M’ + S’AMS’)u’

i=1,2

,.._,

n.

(24)

Consider the positive definition of the interval matrix M’= [M. &fj, the positive semi-definition of the interval m&x K’ = [K, q and the following five inequalities: u’(K’ - S’ AKS’)u u’(M’ + S’AMS’)u

evaluation,

eqn (24)

uT(K’ - S’AKS’)u u ‘(M’ + S’ AMS’)u

i = 1.2,.

, n.

(25)

uT(K’ - S’AKS’)u

< (

u ‘(M’ - S’ AMS’)u ’

Using the central notation of the interval [5, 61, the stiffness interval and the mass interval matrices can be expressed in the following form:

u ‘(K’

- S’ A KS’)u

u’(M’

+ S’AMS’)u

u

K’ = K’ + A Kc,

<

(26)

‘(K’ + S’ A KS’)u

uT(M’ + S’AMS’)u

<

<

u’(K’+S’AKS’)u u’(M‘-

S’AMS’)u

ur(K’ + S’AKS’)u

(33) (34)

zrT(M’ + S’AMS’)u u ‘( K’ + S’ AKS’)u u’(M’

-

S’AMS’)u

uT(K’ - S’AKS’)u

(35) (36)

u7(M’ - S’ AMS’)u u’(K’ + S’AKS’)u u7(M’ - S’AMS’)u

(37)

Zhiping Qiu et al

224 and based on the definition cation, we have

of the interval

multipli-

uT(K’ - S’AKS’)u ur(K’ + S’AKS’)u E.: = min, max ~,
In terms of the following K’ - SAKS’

expressions:

= K’ + SAKS’

- 2S’AK.Y’

(39)

1

(38)

If K’ = [K, m = [K’ - AK, K’ + AK] is a positive semi-definite interval matrix, M’ = [M, Ml = [M’ AM, M’ + AM] is a positive definite interval matrix, and

M’ + S’ AMS’ = M’ - S’ AMS’ - 2s’ AMS’ (40) S’ = diag(sgn(u’,),

JulrAK - Iu( = U'S' AKS'u 20

(41)

]uJr AM]uj = urS’ AMS’u > 0,

(42)

we have

taken at the corresponding the equation

iLI=[&,,z,],

UfO

uT(K’ + S’ AKS’)u (43)

< !$,i?ft~ uT(M’ - S’ AMS’)u I<+ 0

1: = @, , I,] =

L, fO

min max ur(K’ + S’ AKS’)u

and the upper bound

u7(M’ - S’ AMS’)u

,rtb,

n.

(51)

+ S’AMS’)!,

(52)

- S’ AMS’)u,

(53)

(45)

u ‘(K’ + S’ AKS’)u ur(h”

-

S’AMS’)u

2 satisfies

(K’ + S’ AKS’)U, = l,(M’

1 (44)

u r(K’ - S’ AKS’)u 4, = min, max o,<.Q,I,,Ee, ur(M’ + S’AMS’)u

O,
,....

&, satisfies

(K’ - S’AKS’)u, =?,(M’

According to the necessary and sufficient conditions of the interval equality [5, 61, we can obtain

1, = min, max

;=I,2

in which the lower bound

min, max rp,<~,~l,Eo, u’(M’ + S’ AMS’)u ’ IIf 0 utcp,

(SO)

(43) eqn (38) yields u*(K’ -SAKS’&

&CR”’

u’ of

is constant over K’ and M’, then eigenvalue E.,of K and M, K E K’ and M E M’, ranges over the interval

u’(K’-SAKS-2S’AKS’)u = min, max o,<.v c,E0, u 7(M” + S’ AMS’ - 2s’ AMS’)u

the expression

the ith eigenvector

K’u’= E.,M’u’

mEa@: u r(M’ + S’ AMS’)u

Considering

u; # 0 (49)

u T(K’ - S’ AKS’)u !$?

, sgn(u:,)

sgn(u>) ,

(46)

,r+ll The stationary condition of the Rayleigh quotient is equivalent to the algebraic eigenvalue problem [9, lo]. Thus, the eigenvalue problem corresponding to the lower bound of eqn (45) is

THE RAYLEIGHQUOTIENT ITERATION

Despite the fact that the conclusion of the above section determines exact eigenvalue bounds, one has to solve two eigenvalue problems for each i.,. We are not suggesting that users should become accustomed to writing simple programs for their problems. We recommend the Rayleigh quotient for approximating an eigenvalue and the inverse iteration method for updating an eigenvector. It runs as follows [7]: Given two real symmetric interval matrices: K’=[K,@=[K’-AK,K’+AK] and M’=[M.iii] = [MC: AM, M’ + AM], we pick a unit vector u”, thenfork=0.1,2 ,.... take S’ = diag(sgn(u;),

sgn(ui),

, sgn(u:,))

u; # 0 (54)

(K’-

S’AKS’)!,

= .,(M’

+ S’ AMS’)u,, _

(47) and we find that

where 9, is the associated eigenvector of 2,. Similarly, from eqn (46) we can also have (K’ + S’ AKS’)ii, = 2, (M’ - S’ AMS’)&, where u, is the associated eigenvector of 2,. Thus, we obtain the following conclusion.

(1) 2 = K’ + S’ AKS”, (48)

g = M’ - S” AMS”

yields

(55)

Rayleigh iteration method

225

m,fAmS

(2) for & + , , solve (A - ;;,_B)u,+ , = _Bu,

(

(56)

1k,fAk,

m,*Am,

(3) set

k,f AL;,

(57) Repeat from (1) to end. The sequence (& , iiA ) tends to an eigenpair of a=K’+S”AKS”, B=M’-S’AMS”ask-+co.In a similar way, we &n obtain the sequence (&, uL) tending to an eigenpair of A = K’ -S’ AKSh, B=M~+s~AMs~ as k+co. In the above algorithm, we take u,. as the kth eigenvector of K’ = (R + _K)/2 and M” = (R + _M)/2, and perform two Rayleigh sequences to obtain Jk and 4,: (k=1,2 ,..., n).

k,fAk,

1-1

k,fAL,

/!A?7I Ak1

Fig. 1. The frame

of a multi-storey

structure.

NUMERICAL EXAMPLE

k; = [1400., 1420.1 For the verification of the method proposed, the reader is referred to Fig. 1 which shows a frame with five degrees of freedom which we will calculate. The interval parameters are unknown, but the bounded parameters of the system are as follows. The upper and lower bounds of the spring stiffness parameters are (unit: N/m): k; = [2000., 2020.1

k: = [IOOO., 1008.]. The upper and lower bounds are (unit: kg):

rn: = [24., 26.1 rn: = [17., 19.1. Hence,

[3800, 38701

-[1800,

1850

-[1600,

we have the stiffness

-[1600,

16301

matrix:

16301 -[1400,

[3000,3050] -[1400,

14201

[2600,2630]

14201

-[1200, The mass interval

interval

18501

[3400, 34801

K’=

of the mass parameters

rn{ = [29., 31.1 rn: = [26., 28.1 rni = [26., 28.1

k; = [1300., 1350.1

k; = [1600., 1630.1

-[1800,

k; = [1200., 1210.1

1210]

-[1200,

12101

[1200, 12101

matrix: [29., 3 1.1

M’= and the stiffness mid-point mid-point matrix are

I

[26.. 23.1

[24., 26.1

matrix

and

3835 -1825 K’ =

:

[17., 19.1

[26., 28.1 the mass

-1825 3440 -1615

-1615 3025 - 1410

-1410 2615 - 1205

-1205 1205

1

1.

Zhiping

27. 25

IS.

Qiu C[ rrl

1

The maximum uncertainty of the interval stiffness matrix is (A/C,,),,, = 40.0, the minimum uncertainty of the interval stiffness matrix is (A/k,,),,,, = 5.0. and the uncertainty of the interval mass matrix is Am,, = 1.O. The eigensolutions of the system K’u’ = i.,M’u’ are listed in Table I. Now by using the Rayleigh quotient iteration method, for simplicity, we first obtain four interval

eigenvalues

of

system.

the

eigenvalues

and

and The

the associated

upper

and

the associated

0.6166% + 01 0.318098 - 01 0.63619E - 01 0.93006E - 01 O.l1568E+OO 0.12742E + 00

*:

i.2 A, i,

step I 2 I 2 1 2 I 2

T, 0.78428E + I 0.78303E + 1 0.47818E + 2 0.47820E -t 2 0.10950E + 3 0.10940E + 3 0.173993 + 3 0.17400E + 3 Table

i,

step

i,

I

;., I, *4

I I 1

of

of eirennroblem

A; 0.440786 + 02 -0.80986E - 01 -0.11150E+OO -0.93313E -01 0.44662E ~ 01 0.13076E + 00

Table 2. The upper bounds 2, i,

Based on the invariance properties of the characteristic vectors’ entries, the approximate expressions are obtained for a generalized eigenvalue problem of structures with interval parameters. There is not any interval operations in computing interval eigenvalues, therefore, the proposed method has

are listed

I. The eigensolutions

.< i:

bounds

eigenvectors

Table

CONCIAJSION

cigenvectors

lower

in Tables 2 and 3. For comparison Tables 4 and 5 show the interval eigenvalues and the associated eigenvectors of K’u = IM’u by the exact method. The results show that the Rayleigh quotient iteration method can produce a convergent sequence if used recurrently, and that the Rayleigh quotient iteration method is sufficiently accurate, as much so as the exact method.

0.14583E + 0 0.14583E + 0 -0.39473E + 0 -0.39473E + 0 -0.52561E + 0 -0.52561E + 0 -0.43769E + 0 -0.43769E + 0

‘<

A; 0.10367E + -0.10432E + -0.416128 0.10130E-01 0.64091E -0.11716E+00

of eigenvalues

0.29513E + 0.29513E+O -0.54455E + -0.64455E + -0.20365E + -0.20365E 0.278258 + 0.27825E +

3. The lower bounds

K’u’ = i., M’u’

0 0 0 0 0 0 0

02 00 01

0.165zlE + 03 -0.89004E - 01 0.55242E - 01 0.65313E - 01 -0.13025E + 00 0.88393E - 01

01

and associated 0.440UjOE + 0.44030E + -0.317lRE + -0.31715E+O 0.50101E + 0.50lOlE 0.32451E + 0.32451E +

of the eipenvalues

eigenvectors

0 0 0 0 0 0 0

and associated

0.55791 E + 0.5579lE-tO 0.21004E + 0.21004E + 0.31069E + 0.31069E + -0.65039E + -0.65039E +

0 0 0 0 0 0 0

0.62170E 0.62169E 0.63477E 0.63477E -0.57855E -0.57855E 0.45020E 0.450203

+ + + + + + + +

0 0 0 0 0 0 0 0

0.55576E + 0 0.21753E + 0 0.31766E+O -0.65lllE+O

0.62125E 0.63684E -0.58066E 0.44187E

+ + + +

0 0 0 0

eigenvectors

fi> 0.46289E 0.4064OE 0.98185E 0.15784E

+ + + +

I 2 2 3

0.15282E + 0 -0.39444E + 0 -0.51703E +O -0.44492E + 0

0.30563E -0.54306E ~ 0.20624E 0.27614E

Table 4. The exact lower bounds i., u,

0.4616& 0.32552E 0.64346E 0.92170E 0.11247E 0.12235E

Table -I.., u,

+Ol - 01 - 01 -01 + 00 +OO

0.40623E -0.79295E -0.10888E -0.61089E 0.455lOE 0.12873E

5. The exact upper

7 0.78303k + 0.31117E-01 0.62976E 0.93953E 0.11905E + 0.13266E +

01 01 01 00 00

bounds

+ + + +

0 0 0 0

0.446:lE + 0 -0.31083E + 0 0.50206E + 0 0.32649E + 0

of eigenvalues + + +

02 01 00 01 01 00

“i 0.9318OE + 02 -0.1007lE +OO -0,41179E-01 0.99695E - 01 0.64018E - 01 -O.l1545E+OO

of eigenvalues

/., 0.47820E + -0.827526 -0.11416EfOO -0.66494E 0.44033E 0.13308E +

02 01 01 01 00

and associated

and associated

2, 0.10940E + -0.10814E + -0.41900E O.lO308E + 0.63923E -O.l1903E+OO

03 00 01 00 01

eigenvectors j.4 0.15784E + 03 -0.88518E -01 0.53757E - 01 0.64123E - 01 -O.l2761E+OO 0.851 IOE - 01

eigenvectors 2, 0.17400E + -0.89522E 0.56912E 0.66373E -O.l3303E+OO 0.92080E -

03 01 01 01 01

Rayleigh

iteration

effort. An efficient Rayleigh quotient iteration method is developed for computing the lower and upper bounds of the eigenvalues of the structure with interval parameters. A simple example problem supports the accuracy of the technique. In the interval eigenvalue problem, it is impossible to transform from the generalized eigenvalue problem into the standard eigenvalue problem [5,6,8]. For the standard eigenvalue problem of the symmetric interval matrix, Deif [S] ended up with similar formulae to the conclusion in this paper. A more general criterion was used to determine the little computational

bounds

suitable

though

it was

for an interval for more

of unrestricted

width,

intricate.

REFERENCES I.

Suhuan Chen, Matrix Perturbation Dynamics.

International

Academic

Theory in Structural Publishers (1993).

method

227

2. Y. Ben-Haim and 1. Elishakoff, Conce.~ Models qf Uncerrainry in Applied Mechanics. Elsevier, New York (1990). 3. Guangyuan Wang and Jinping Qu, Fuzzy random vibration of multi-degree-freedom hysterestic systems. Earthquake Engng Struct. Dynam. 15(5), (1987). 4. Wehnu Huang, Vibration of some structures with random parameters. AlAA J. 20, 1001-1008 (1982). 5. R. Moore, Method and Applications ofInterval Anulysis. SIAM, Philadelphia (1979). 6. G. Alefeld and J. Herzberger, Introduction to Interval Computations. Academic Press. New York (1983). 7. 2. C. Shi and W. B. Gao, A necessary and sufficient condition for the positive-definiteness of interval symmetric matrices. Inf. J. Control 43, 325-323 (1986). 8. A. Deif, Advanced Matrix Theory for Scientists and Engineers. 2nd Edn. 262-281. Abacus Press (1991). 9. J. N. Franklin, Mafrix Theory, pp. 141-193. PrenticeHall, New Jersey (1968). 10. R. Bellman, Introduction to Marrix Analysis, pp. 60-64. McGraw-Hill, New York (1960).