Abstract

*** ** and

echanical Engineerin Old Dominion University, Norfolk, VA 23529-0247, US.

This paper presents a new method for calcu- lating eigenvalues and eigenvectors ( eigenpairs ) of finite element systems in solid mechanics. For con- venience, this method is called the

. The new method is based on an interesting of finite element systems

be proven by using the theory of structural variations ( Refs.1-3 ). It can give us as many eigenpairs as we need, just like the power method. However, the Z-deformation method does not have the shortcomings the power method does: the convergence rate in the power method strongly depends on the closeness of the adjacent eigenvalues and the initial guess for the displace- ments. The Z-deformation method is not an iteration method, but a procedure of successive advances.

* Supported in part by Natural Science Foundation, Grant NO. DDM-8657917.

** Research Associate, AIAA member. *** Associate Professor, AIAA member.

1993 by T.Y. Rong and G. J.W. Hou. Pub- lished by the American Institute of Aeronautics and Astro- nautics, Inc. with permission.

Although there exist a number of methods for computing eigenpairs of finite element systems, there are still some open questions to be investigated. For example, very often in engineering analysis, only the lowest few eigenpairs are required to be deter- mined and the inverse power method has been considered the best one. However, the convergence rate of this method strongly depends on the closeness of the adjacent eigenvalues and the initial guess for the displacement vector. When the adjacent eigenva- lues are very close, this method performs very poorly in both its accuracy and its efficiency, even with the shifting technique for its improvement.

The main objective of this work is to develop a new method for calculating eigenpairs of finite element systems, which can give us as many eigen- pairs as we need, just like the power method does, but without the above mentioned shortcomings. For convenience, the new method is called the ati ion

The Z-deformation method is based on an interesting and useful property of finite element systems, which is stated as onotonousn~§

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s, and will be proven in this paper by using the theory of structural variations given in Refs. 1-3. The so-called Z-defor- mation is a technical term defined in the theory of structural variations, representing a sort of general- ized deformations of a subelement, and the term subelement is a basic concept of the theory.

Suppose we have a finite element system charac- terized by its global stiffness matrix positive-definite and assembled from its elements. While in the theory of structural variations'r2, each element is decomposed into certain number of subelements, depending on the specific element model. So, the matrix in this theory is composed of a total number, say m, of subelements. Each subelement is characterized by the so-called sub- element vector, denoted by , and stiffness mo- dulus, denoted by w,, where the subscript s stands for the subelement number, s=1,2;-.,m. For in- stance, a plane beam element may have three subele- ments, defined in its local coordinates as

p q - 1 , 0, 0 7 1 , 0 , 0 I'

= [ 0, 1, L/2, 0 , -1, L/2 IT

= [ 0, 0 , -1 , o , o , 1 I'

(2. la)

(2. lb)

(2. lb)

where E is Young's module, A the cross-section area, I the moment of inertia and L the length of the element. For other element models, the of their subelements can be found or generated by following the procedure given in Ref. 2.

To deal with the analysis problems of a finite element system, the theory of structural variations

has defined several technical terms and brou light a few important properties of finite element

them are relevant to the present

tor of subelement s is

Any basic displacement vector obtained by the structural variation method given in Refs. 1 and 2, instead of direct using eqn (2.3), which is merely a symbolical definition of it.

e Z of subelement s, formed from of subelement r, is the basic displacement vector

denoted by Z,, and defined as

Z,, = r, s,r==1,2;.., m

If r=s, then 2, is called the of subelement s.

The properties which have been brought to light in the theory of structural variation^"^ and relevant to the present subject are listed below.

atrix:

is the global basic displacement matrix of all the basic displacement vectors global stiffness modulus matrix, diagonal, of all the

subelement stiffness moduli WS, s= 1,2;..,m. We are naturally wondering if the denomina- tor ( 1 +m,Z, ) contained in eqns (2.6) and (2.8)

zero, for instance, ng=0.5 and 2,=- and 2,, would blow up. Fortunately,

it may not; the values which Z, may take are limited for all W,, OSW8< 00; this is an important feature of finite element systems, being stated as the following theorem.

ent , are

valuation denotes the varied basic displacement

vector of subelement s, AW, an arbitrary increment of W, and

tions ( for static systems ):

nite element system, the principal Z- eforma~ion %,, o any subelement s is subject to

- 1 s q s o o (2.7)

Thus, the varied %-deformation due to AW,, denoted by 2,, is given by

O S 2 , S l (3.1)

(2.8) This theorem is illustrated in Fig.1, in which 2 and W represent any principal 2-deformation and the corresponding stiffness modulus, respectively.

(2.9)

- - - _ _ -

Fig. 1 Limits and monotonousness of principal 2-deformations in static systems

Before going into the eigenpair problems, we will first show an important property of finite ele- ment systems, which has not been known before and is relevant to our present subject.

Proof. By the definition of 2-deformations and the

Explicit Decomposition Theorem on the Inverse of the Global Stiffness

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= (ZJZ + c > 0

where C stands for the sum, Le.

(zJW,Nv, > 0, r f s r=l

Thus, from eqn (3.3) we have

(Z,,)~ - 2, 4- c = 0

from which we further have

z,, = 1H 1 Jr d-T-=zz

Since 2, is a positive real number due to eqn (3.3), C must be subject to

(3.6)

Thus, from eqns (3. ) and (3.6) we arrive at the conclusion (3.1). To prove the second part of the theorem, eqn (3.2), we should use eqn (2.8). By the definition of derivatives, we have

Thus, eqns (3.1) and (3.7) lead to the conclusion (3.2).

ext, we are goi to discuss the eigenpair problems.

Suppose we have a finite element system and its eigenproblem is described by

is the mass matrix ( symmetric ), nodal displacement vector and X the parameter to be determined €or eigenvalues Xi, i= 1, 2, .. ., N. For simplicity of statement, we assume mass matrix and

0 < x, c & ... < x, (4.2)

And for convenience, we call the system described by eqn (4.1) the eigensystem.

As has been mentioned in section 2 that the is composed of m subelements in a static

system, while in a eigensystem we need to treat the ) as its global stiffness matrix, which roduction of the negative stiffness

concept into a structural system. Thus, we can think that each non-zero -AMB ( the s diagonal element of

) can be considered as the contribution o ubelement to the lobal stiffhess matrix (

). This special subelement should be called the mass- su ent. Its subelement vector modulus W, are defined as follows:

3 [ -1, 1 IT (4.3)

W,= -AM,, s=m+l,m+2,...,m+N (4.4)

where the values -1 and 1 in correspond to the two degrees of freedom ( DOFs ) of M, and the ground, respectively. We can see that the distinction of a mass-subelement from a typical subelement in static systems is that a mass-subelement may have a negative stiffness modulus, while in static systems every stiffness modulus is positive ( see eqn (2.2) ).

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ensystem is composed of m itive stiffness moduli

mass-subelements with negative moduli; the global stiffness matrix ) may and may not be nonsin- gular, depending on the value of X. However, they are all legal subelements and therefore all the formu- las and theorems established in the theory of struc- tural variation^'^^ also apply to the eigensystems, except for the case when X takes some values, i.e., eigenvalues A=&, which make singular.

To find out these special values, we can take advantage of some inherent properties of finite element systems, including those already listed in the foregoing sections, and the one being proven below.

ensystem, the principal Z-defor- mation of a mass-subelement, Z,, may take any real number with W,=-AM,, - limited. However, its monotonousness still holds almost everywhere. This important feature is stated

Proof. For convenience, use

is nonsingular. And eqn (4.4) is rewritten as

w, = A*, ( s=m+ l,m+2;..,m+N )

where

Therefore,

From eqns (2.3, (2.49, (2.9) and (5.6), we have

m m+N

m+N m m+N

dzss > 0, except for A==& (z,J2~, > Q ( i= 1,2, ..., N ) dwi!

- 0 (5.2) The last equation has shown the conclusion (5.1)- .8) holds for any real W,, it also leads to and

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new one described by

( -0

where

* e -

(6.10)

(6.11)

AT= min ,

= X,.

(6.16)

where Rayleigh's Quotient Theorem has been used again for X,.

This can be simply explained as follows. Besides, it is easy to show by the same

proving procedure as has been done for the system ( onotonousness Theorem is still hold

system of (6.10). Suppose that X; and 7 are the lowest eigenpair of eqn (6.10), then

Premultiplying eqn (6.12) by

Thus, repeating the same procedure with the eigensystem of (6.10) as has been done for X, and will yield X, and 2, and so on so forth, until the last one.

(6.12)

(6.13) llust le

) has been used, Equation (6.13) -orthogonal to each

e set of all ad- the subset of

Based on the derivations given in the previ- ous sections we can summarize and list the following steps for the Z-deformation method:

other, Use the 'ssible displacement ve

whose members ar 's Quotient Theorem, Step 1, Build up the basic displacement matrix of the

given system with X=O by the structural variation A;= min ) method presented in Refs. 1 and 2.

), then due to eqn . Therefore, it may dso be expressed

Step . Chose one BOF of the system, say s, where the mass-subelement is located and serves for calcu- lating the first eigenpair. To specify this DOF, find the Dstat, produced by the mass as the static load by using eqn (2. ). And the DOF on which DSbt has its maximum component in magnitude should be taken as this particular one.

A;= min

However, due to

)

0

Thus ,

Step 3. Take xi==O, and two other arbitrary values for xi, i==2,3, but near X-0, and evaluate the corres-

eqns (2.6) and (2.8).

. Use eqn (6.3) with a selected /3 to calculate x,, moving one step towards A,.

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. Evaluate Z, at x4 by usin eqns (2.6) and. (2.8) again.

Step 6. Use xi and Zi, i=2,3,4, to obtain x5 to move one more step towards A,, and repeat steps 4 and 5 with a selected tolerance E until xk-xk.,< E and &-+ -00, giving A 1 = x k f ~ .

Step 7. If Z, is found to be a large positive number, then pull the corresponding xk back for one step and switch to eqn (6.4) to continue the advancing proce- dure towards A,.

Step 8. On obtaining A,, use eqn (6.9) to have the corresponding ei

Step 9. After obtaining the first eigenpair A, and introduce a new mass-subelement into the eigensys- tem, whose subelement vector is corresponding W is 1 ( see eqn (6.12) ) to form a new eigensystem, and repeat steps 3-8 to obtain the next eigenpair, and so on so forth until the last one if needed.

e above steps, the plane frame .4 has been analyzed by the

2-deformation method, and the inverse power method has also been used for comparison.

M = I ,

E I = 0 . 0 0 0 1 0

0 0

0

0

.i-

An eigensystem

ensystern is made of a lumped mass e m s which have identical properties

except that their I are different but quite close. This similarity in the structure produces a pair of close eigenvalues, A,=2 e 00031476142a10'12 and Az=2 ~ 0 0 ~ 2 5 8 0 9 7 5 2 ~ ~ 0 ~ 1 2 , SO, A1/A2==8.999529.

In this example the DOF for the advancing procedure was chosen to be the vertical DOF of the mass. To obtain the lowest eigenpair, A, and [ .695772201276, -.718262517421, -. 10597228 13431T, the Z- deformation method took only 16 advancing steps with p=10 and E = lo-''. On the other hand, the power method ran 28635 iteration cycles with the

=[ 1, 1, 1 ] to reach A,= ~.0003147%7.

The CPU time ratio was about 0.06 : 22.0.

The intermediate values of the first eigenso- lution produced by the computational process of the Z-deformation method are tabulated in Table 1. At step 7, the value of x, calculated from eqn (6.3) is a little bit larger than the true A,, because the corres-

e positive value ; therefore it was pulled back for one step and switch

) for the followin

For the case of repeat envalues, the new method can still ive the proper results.

This paper has revealed the following inter- properties of finite element systems:

(1) The Evaluation Theorem of Principal Z-deforma- tions for static systems, i.e. eqn (3.1). This is important for the practical application of the theory of structural variations to the static structural analy-

that the denominator in eqns (2.6) comes zero for any value of actual

stiffness moduli.

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Table 1. ~on~putations by -deformation method

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16

.000000000000E+QQ

.2500000000003+00

.500000000000E+00

.1538872163233+01

.1942466865593+01

.1994780067473+01

.2000133762963+01

.2000392976863+01

.2000310834273+01

.2000314070123+01

.2000314694713+01

.2000314754793+01

.2000314760763+01

.2000314761363+01

.2000314761423+01

.200031476142E+01

.2000314761423+01

.250000000000~+00

.250000000000E+00

.1038872163233+01

.4035947023553+00

.5231320188183-01

.5353695489933-02

.2592139004233-03 -.8214259528813-04 .3235856513913-05 .6245888255163-06 .6008271601383-07 .5967423013433-08 .5963712666373-09 .5963859315423-10 .5966059189373-11 .6083617005873-12

-.3331622025923+01 -.333180406878 -.335006508041 -.6562182386493+04 .120750179254E+05

-.2637981475413+06 -.1493810000593+07 -.1546979820733+08 -.1556510357953+09 -.1557381908033+10 -.1557347442143+11 -.1556830623393+12 -.1529755308213+13 -.1407374883553+14

~1427942071973+00 -.1903678185573+00 -.2998460000173+01 -.29986418G6193+02 -.3016884673533+03 -.6227175878443+04 .1863720031193+05

-.2758731654663+06 -.1230011853053+07 -.1397598820673+08 -.1401812375883+09 -.1401730872243+10 -.1401609251343+11 -.1401095879183+12 -.1374072245873+13 --.1254399352733+14

(2) The Monotonousness Theorem of Principal Z- Deformations for ei ensystems, Le. eqns (5.1)-(5.3). This supplies a mathematical foundation for the Z- deformation method to deal with the eigenpair analysis.

(3) The expression of eigenvectors by basic dis- placement vectors, Le. eqn (6.5). This gives us a great convenience to find the eigenvector when the corresponding eigenvalue is known.

Based on these properties, this paper has established a new numerical method, the Z-Deforma- tion method, for calculating eigenpairs. This method is a procedure of successive advances, whose perfor- mance does not depend on the closeness of the adja- cent eigenvalues. This special feature makes the proposed method superior to the commonly used power method when the adjacent eigenvalues are close.

[l] Rong, Ting-Yu (1985)) General Theorems of Topolo- gical Variations of Elastic Structures and the Method of Topological Variation, ACTA MECHANICA SOLIDA SINICA, No.1, 1985, pp 29-43, in Chinese.

[a] Rong, Ting-Yu and Lii, An-Qi (1992)) Theory and Method of Structural Variations of Finite Element @stems in Solid Mechanics, A Collection of Technical Papers of 33rd MAA SDM Con€., April 13-15,1992, Dallas, TX, US, paper No.92-2356.

[3] Rong, Ting-Yu (1992), Explicit Formulations for Sensitivity Analysis of Structural Systems, A Collection of Technical Papers of 33rd AIAA SDM Con€. , April 13-15, 1992, Dallas, TX, US, paper No. 92-2248.

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