INTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN ENGINEERING, VOL. 6, 543-552 (1973)

MASS CONDENSATION AND SIMULTANEOUS ITERATION FOR VIBRATION PROBLEMS

ALAN JE"INGS*

Queen's University, Belfast, Northern Ireland

SUMMARY A relationship is developed between mass condensation and simultaneous iteration when used to determine the natural frequencies and vibration modes of large structural systems. A significant factor in the comparison between the methods is the extra versatility and reliability of simultaneous iteration.

INTRODUCTION

The equations of motion of a structural system with n degrees of freedom may be expressed in the form

where M and K are n x n mass and stiffness matrices normally symmetric positive definite. Equation (1) defines an eigenvalue problem in which certain natural frequencies of vibration o satisfy the equation with associated column vectors x defining particular modes of vibration. Mass condensation and simultaneous iteration are both methods of obtaining numerical solutions for the lowest frequencies and vibration modes when the number of displacement variables is large. Both methods take advantage of the fact that only a partial eigensolution is needed and also of the sparse or banded nature of the mass and stiffness matrices that arise in many formulations. However, mass condensation solves a constrained system and can therefore be classed as an approximate method, whereas simultaneous iteration is an iterative method which converges to the correct solution of the equations.

~ M x = KX (1)

MASS CONDENSATION Mass condensation, otherwise known as the eigenvalue economizer method, involves defining a reduced set of variables which sufficiently accurately yield the required vibration modes of the structure.14 The structural displacements of the system are assumed to be a linear combination of the reduced set of variables. Let these variables be represented by the m x 1 column vector z and the linear relationship by

where H is an n x m matrix. The columns of H represent deflection functions which are determined semi- automatically by first allocating manually m master displacements. It is convenient, although not necessary, to specify the master displacements before the other displacements (known as slave displacements), hence XT = [xxx:] where x, and xB are of order m x 1 and (n-m) x 1 respectively. A corresponding subdivision

x=HZ (2)

in K and M gives

and (3)

* Reader, Department of Civil Engineering.

Received 7 July 1971 Revised 8 December 1971 Revised 21 November 1972

0 1973 by John Wiley & Sons, Ltd. 543

544 ALAN JENNINGS

Each displacement function is a static deformation of the structure in which one master displacement has a unit value and the other master displacements have been constrained to zero. It can be shown5s6 that these conditions are satisfied with

When expressed in the reduced variables the equations of motion become

CiPmz = fiz

where R = HTMH and it = HTKH are k x m condensed mass and stiffness matrices. The operations required to form fi and fc can be performed with M and K in band or sparse matrix store

as long as space is allowed for any build up of elements in K during elimination. In fact, the derivation of H and E can be carried out by a standard static analysis program capable of dealing with specified displace- ments and sufficient multiple loading cases.

The matrices M and it will be symmetric, full and normally positive definite. The order of m will be such that there should be no difficulty storing these in core and the subsequent eigensolution can be performed by any of several methods. An efficient procedure is to transform the variables to

f = ETz (6) where E is the lower triangular matrix obtained from the Choleski factorization of fi, so that equation (5 ) can be written in the form

The Householder transformation method can then be used to determine the eigenvalues 1/G2 of the symmetric matrix f;-l and the corresponding mode shapes for each frequency can be obtained from the eigenvectors by means of the transformations of equations (6) and (2).

SIMULTANEOUS ITERATION Simultaneous iteration is an extension of the power method in which parallel processing of vectors gives improved efficiency and reliability? several forms of which have been derived for application to symmetric eigenvalue problems.a1o In order to apply it to structural vibrationl1-l2 the basic equation (1) is rewritten in the form

where L is the lower triangular matrix arising from the Choleski factorization of K and

y = LTX (9) A partial eigensolution is required of the symmetric matrix

A = L-1 ML-T

which is carried out in the following way. A set of m orthogonal trial vectors are chosen, where m is normally greater than the number of required

vibration modes. When compounded, these can be represented by an n x m matrix U,, the orthogonality condition being

Premultiplication

up1 = I

V, = AU,

VIBRATION PROBLEMS 545

produces a set of vectors in which the components of all the dominant eigenvectors are magnified. The dominant eigenvalues are predicted from an interaction matrix of order m x m which could be

B, = q V l = UTAU, (13) This matrix is also used to realign the trial vectors which, after being re-orthogonalized, can be adopted as the trial vectors for the second round of iteration.

An important feature of the application of simultaneous iteration to large vibration problems is that the full matrix A is only used in the premultiplication of equation (12) and need not be stored explicitly. It is possible to determine V, from U, storing only the matrices M and L in half-bandwidth form, which not only saves storage space but also computipg time.

MASTER DEFORMATIONS AS TRIAL VECTORS Consider the situation in which the master deformation modes are used as a basis for choosing trial vectors. Any mass condensation solution may be specified in the simultaneous iteration co-ordinates by using the transformations of equations (2) and (9) giving

y = LTHz (14) Hence, the set of master deformation modes correspond to a set of vectors LTH in the co-ordinates y. But as these will not be an orthogonal set it is necessary to perform a further linear transformation. The m x in matrix of inner products is given by

D = HTLLTH = HTKH (15) If a Choleski triangular factorization of D is performed such that

D = GGT

an orthogonal set of trial vectors is given by

U, = LTHG-T

Using these trial vectors for simultaneous iteration the vectors after premultiplication become

V, = ALTHG-T = L-1MG-T

B, = U;V, = G-lHTMHG-T

(18)

(19)

and the interaction matrix is

Adopting the procedure of Clint and Jenningsg estimates of the dominant eigenvalues of A are given by the eigenvalues p of B,. If p is the eigenvector of B, corresponding to p

G-' HT MHG-T p = pp (20) Using equations (15) and (16), equation (20) can be transformed to

HT MHG-T p = pHT KHG-T p

Comparing this equation with equation ( 5 ) it is apparent that

p = l/G2

and G-Tp = z

Hence, the estimates of frequencies arising from the first round of simultaneous iteration will be the same as those obtained by the mass condensation method given trial vectors derived from H according to equation (17). Also, from equations (14), (17) and (23) it follows that a predicted vibration mode is equivalent in simultaneous iteration variables to

Y = UlP (24)

546 ALAN JENNINGS

In the simultaneous iteration method the set of eigenvectors p of the interaction matrix are used to revise the vectors V, by postmultiplication followed by an orthogonalizing sequence. As it can be shown that the vectors V, must have stronger components of the dominant eigenvectors than the vectors U, have, it follows that the estimates of the vibration modes by mass condensation will not be as accurate as the corresponding estimates obtained at the end of the first round of iteration by the Clint and Jennings simultaneous iteration method.

CHOLESKI TRIANGULAR FACTORIZATION Dong and co-worker@ have proposed a simultaneous iteration method which replaces the Choleski factorization by the solution of the stiffness equations. Their paper is useful because it shows that simul- taneous iteration can be viewed as an iterative extension of condensation techniques in which the modes are chosen from a set of different load distributions. However, the justification for rejecting the Choleski factorization of K was based on a false premise that Choleski factorization is inefficient, particularly for large order systems, because it results in a fully populated matrix.

In a Choleski triangular factorization some elements originally zero do become non-zero but the pattern of such fill-in is identical to the fill-in in either the lower or upper triangle of the stiffness equations during Gaussian elimination without row or column interchange. Indeed, it can be shown that the numerical operations correspond except for factors on the elements, and that the accuracy will be similar as long as square roots are evaluated to maximum accuracy, Such is the similarity that any elimination method for the solution of symmetric positive definite equations arising from the static analysis of stable structures of any configuration or size could easily be converted to a Choleski form of solution. For instance, the variable bandwidth storage scheme has been used both with a compact elimination method14 and with Choleski fact0rizati0n.l~ Of the two, the Choleski method was found to be more versatile for use with backing store and hence was the most suitable for the analysis of large structural problems.16 This variable bandwidth formulation is not only versatile in so far as it can remain efficient while accepting almost any configuration for the stiffness equations, but also can work with a very small area of core store. These features make it particularly suitable for use with simultaneous iteration methods for large structural vibration problems, because most of the core store area can be reserved for the matrices of trial vectors.

COMPARISON OF METHODS Whereas the above discussion shows that there is an equivalence between mass condensation and the first round of iteration of simultaneous iteration under certain circumstances, the nature of the two methods defines that they should be used in different ways.

Because mass condensation is not iterative it is essential that there should be sufficient master displacements and that these should be well chosen, In doing this, account should be taken of the number and accuracy of vibration frequencies required and the degree of complexity of the structure. It would, for instance, be easier to decide on suitable master displacements for a finite element representation of a single plate under bending vibration than for unsymmetrical three-dimensional configurations> Ramsden and Stoker recommend that about 70 master displacements be used when 20 vibration frequencies are required. If r is the number of required vibration frequencies this recommendation could be generalized so that mlr = 3.5. Based on this criterion, inspection of results quoted in several p a p e r ~ ~ . ~ ' ~ ~ indicates that the highest required frequency is sometimes in error by more than 2 per cent. Much emphasis has been placed in literature on the accuracies achieved, because this has given the only means of deciding how many master displacements to use and assessing how accurate the results are likely to be. It will be noted that when these accuracies are quoted against classical solutions a small part of the error will be due to the initial finite representation of the problem.

With simultaneous iteration the choice of trial vectors is open to the operator. Good trial vectors will produce a faster convergence than poor ones. The only requirement, that the vectors be mutually orthogonal, can be satisfied merely by entering the iteration sequence part way through. Whereas one way of obtainha

VIBRATION PROBLEMS 547

trial vectors is to use the master deformation modes there are other possibilities such as transferring displace- ments from sketched mode shapes. When a number of similar analyses are being carried out, for instance with varying mass or stiffness, the eigenvectors predicted for one case may be particularly good trial vectors for the next case. However, even if good trial vectors are not used, convergence is likely to be obtained with only a few extra iterations. For cases in which computing time is not critical almost any matrix which does not have linearly dependent vectors would be suitable (for instance a random matrix or a segment of the unit matrix). If built into the computer program this will relieve the operator of having to choose suitable trial vectors.

Using a method involving complete eigensolution of the interaction matrix9. lo it is possible to reduce the errors to the required amount in one iteration. However, there is one further iteration required before the tolerance test could be satisfied and in the event of the tolerance test just failing to be satisfied there would be almost three times as much computation performed as necessary before the results were confirmed as satisfactory and iteration terminated. It is more practical to use fewer trial vectors and expect the required tolerance to be reached after a few iterations. In order to achieve two figure accuracy in about six iterations, starting from arbitrary trial vectors and not using Chebyshev acceleration, it is necessary to carry extra trial vectors such that 2, i.e. w ~ + ~ / w , . = 1.5. For this particular rate-of-convergence the error at termina- tion should be no greater than the specified tolerance. The number of extra vectors to be carried to obtain this rate-of-convergence will depend on the spread of frequencies but is likely to be in the region of m/r- 1.5. It is possible to curtail iteration with the dominant vectors once they have passed the tolerance test, hence not all the iterations need to be carried out with the full m vectors. From the m/r ratios it is seen that the matrices used in simultaneous iteration will be of smaller size than the corresponding matrices of mass condensation, although the number of matrices to be stored will be larger. Hence, if the matrices are held on backing store it will be possible to fit larger problems into a given core store area when using simultaneous iteration. Also, the number of vectors to be carried is not directly related to the accuracy of the result, so if it is reduced below the optimum value the only penalty is an increase in the number of iterations. Ways of implementing simultaneous iteration are discussed in more detail by Corr and Jennings.ls It is a feature of symmetric eigenvalue problems that, if an eigenvector is known to a certain accuracy, the corresponding eigenvalue can be obtained to about twice this accuracy by means of the Rayleigh quotient. In general, the simultaneous iteration results exhibit this relationship so that a tolerance of two figures specified for the eigenvectors should result in an accuracy of three or four figures in all the required eigen- values. With this consideration in mind it seems unlikely that the data would warrant, or the analyst require, answers for eigenvectors of much more than about two figure accuracy.

Using the above ratios of m/r for both mass condensation and simultaneous iteration, together with an estimate of six iterations for simultaneous iteration, it is possible to compare the total amounts of computa- tion based on the number of multiplications. It will be assumed that the mass and stiffness matrices have a constant bandwidth of 2b - 1 . Wright and Miles1’ specify the number of multiplications required to obtain the condensed mass and stiffness matrices* as

(4mb + 3m2 + $b2) (n - m)

and the calculation of slave displacements in each of the vibration modes after having obtained the master displacements will involve a further mr(n-m) multiplications. When n is so large that any terms without n can be discounted (so that it is unnecessary to include the eigensolution of the condensed equations) then, using the ratio m/r = 3-5, the total number of multiplications by mass condensation is estimated at

14nbr + 40nr2+ +nb2

Jennings and Orr specify the number of multiplications for simultaneous iteration. With six iterations assumed and m/r = 1.5 the significant terms in the expression for the total number of multiplications are

27nbr + 34nr2 + +nb2

* The relevant formula given by Wright and Miles is 4mk(b+ I )+3k2 m+)rn(b+ 1)8 in which the parameters have to be changed according to k -+ m, b + 1 + b and m -+ n- rn to agree with the notation of the paper,

548 ALAN JENNINGS

These approximate estimates suggest that the amount of computation is of the same order of magnitude for both methods.

Similar inferences about the relative amounts of computation may also be drawn from Dong and co-worker~’~ who say that for simultaneous iteration and mass condensation ‘the computational effort may be comparable’ and from Wright and Miles17 who estimate that mass condensation will be comparable in numerical efficiency with the power method if the number of iterations using the power method are in the region of seven for each vector. It would be difficult to carry out a detailed comparison of numerical efficiencies such that the results would be reliable, because any such comparison must take into account the accuracies achieved by both methods, the skill of the users in choosing the number and type of master displacements of starting vectors (with flossible variation from one type of problem to another) and the possible efficiencies of computer implementation. Any such comparison would be subject to modification every time there was an improvement in one of the methods or the implementation of it.

EXTENSIONS OF MASS CONDENSATION Wright and Miles17 and Geradinl9 propose extensions of the mass condensation method which involve improvement in the calculated vibration modes and also the determination of bounds for the predicted frequencies. In justifying their method against the power method, Wright and Miles state that it is unlikely that an iterative process will converge in seven iterations for each of the required roots. Geradin merely states that matrix iterations are costly. It seems probable that Geradin is also referring to the power method and therefore neither of the broad statements about iterative methods in these two papers should be con- sidered as a basis for rejecting the simultaneous iteration method.

Geradin’s method is claimed to have more accurate error bounds than the Wright and Miles’ method. Geradin’s method involves a mass condensation analysis followed by an iteration of the power method on each of the vibration modes in the full co-ordinate system using the dynamical matrix K-lM, The unimproved and improved vectors are used in Rayleigh quotient predictions to determine error bounds for the determined frequencies. In view of the power method improvement of the vibration modes there is some connection between Geradin’s extension of mass condensation and the second iteration of a simultaneous iteration solution.

The determination of error bounds for mass condensation is a definite improvement of the method, the significance of which must be assessed in relation to the closeness of the error bounds. Geradin gives results for a large and complex structure and from the error bounds on the first five symmetric and the first five antisymmetric natural frequencies it can be deduced that these frequencies are determined to three or four figure accuracy for each of two examples. Whereas this result appears to be very good, it should be expected that very accurate eigenvalues would be obtained in view of the large number of master displacements adopted (80 for the symmetric analysis and 110 for the antisymmetric analysis). It would be useful to know the error bounds for some of the higher vibration frequencies, but in the absence of more information it can only be concluded that to obtain close error bounds by the extended mass condensation method it is necessary to adopt an mlr ratio much greater than 3.5 and more in the region of 16 or 22. If the chosen master displacements do not give sufficiently accurate error bounds then it is not possible to enhance the results by iteration, it is necessary to repeat all or most of the analysis using a more appropriate (and probably larger) set of master displacements.

NUMERICAL RESULTS An ICL 1907 computer was used for some numerical tests. The simultaneous iteration method adopted was the first method given by Corr and JenningslS starting with random trial vectors.

Cantilever plate A square plate fixed on one edge and free on the other edges was idealized by 16 finite elements. Each

unrestrained mode had a deflection and two rotational displacements giving a total of 60 variables. The six

VIBRATION PROBLEMS 549

different choices of 15 master displacements shown in Figure 1 were used to obtain estimates of the non- dimensional frequencies by mass condensation shown in Table I. Also shown in Table I are the results of iterating simultaneously with 10 trial vectors. After six iterations convergence to six figure accuracy was obtained for all of the first five eigenvalues, and hence these results may be considered as the accurate solution of the finite element idealization and are close to the results for a nine element idealization given by Dawe.20 An energy solution for the cantilever plate21 is also given in Table I. This may be considered as exact and hence indicates the errors caused by finite element idealization. Even starting with random trial vectors simultaneous iteration has produced, after four iterations, frequencies to a higher accuracy than mass condensation and also the finite idealization itself. On the other hand, the errors due to mass condensa- tion tend to be larger than those due to the finite element idealization.

Simultaneous iteration was used for vibration analyses involving several thousand displacement variables. Two results with about 500 variables have been included so as to give some degree of comparison with Geradin’s mass condensation results involving two analyses of 393 and 567 variables.

M @ m (0) (b) (C)

Figure 1. Finite element analysis of a cantilever plate. Different selections of 15 master displacements: lateral displacements only, 0 lateral displacement and two rotations

Table 1. Computed non-dimensional frequencies for a cantilever plate

1 2

3.4697 8.529 3-4698 8.530 3-4700 8-536 3.4696 8.526 3.4720 8.596 3.4736 8.573

~- 3 4 5

21.82 27.43 31.97 21.86 27.20 32.02 22-34 27-53 32.92 21.74 27.21 31.62 2479 29.35 40.55 23.13 28.07 37.27

Simultaneous iterations 3rd iteration 3.4694 8.5235 22.0086 27-5080 29.1967 4th iteration 3.4694 8.5204 21.5460 26.4865 30.9100 6th iteration 3-4694 8-5204 215381 269940 30.9144

3 x 3 mesh (Dawe) 3-470 8.530 21-67 2685 30.80 Energy solution (Leissa) 3.494 8.547 21.44 27.46 31.17

Multistorey building frame The plane stiff-jointed frame of Figure 2 with three displacement variables per joint has a total of 492

degrees of freedom. Simultaneous iteration with 13 vectors took 200 seconds for a complete analysis involving

550 ALAN JENNINGS

five iterations and using a core store of 25K. Table I1 shows the convergence of the predicted eigenvalues compared with the result obtained by letting simultaneous iteration run on until the thirteenth iteration.

Figure 2. Plane multistorey frame with 492 degrees of freedom

Table 11. Simultaneous iteration eigenvalue predictions for a multistorey frame with 492 degrees of freedom (eigenvalue = l / d )

Eigenvalue number 1 2 3 4 5 6 7 8

Iteration number 1 0-558 0.506 0503 0.502 0.501 0.500 0500 0.500 2 59.027 10.892 3.921 3.907 3.412 2.005 1.697 0.576 3 59.076 11.628 4.111 2.328 1.674 1.112 1.101 0.575 4 59.076 11.628 4-198 2-203 1.656 1.133 1.028 0.548 5 59.076 11.628 4.198 2.204 1.656 1.181 0,998 0.549

13 59.076 11.628 4.198 2.204 1.656 1.191 1.004 0.555

Suspension bridge A three-dimensional suspension bridge was analysed in which 12 displacement variables were used at each

hanger station as shown in Figure 3. The number of degrees of freedom totalled 486 and iteration was carried out with 17 trial vectors. Table 111 shows the predicted eigenvalues for the first 12 iterations after which the first five vectors all satisfied a 0.001 tolerance test. The total computing time was about 400 seconds.

The number of vector iterations required to achieve a given accuracy was greater for the suspension bridge problem on account of the closeness of the eigenvalues. Howeyer, the computing time per vector iteration was slightly less due to the narrow bandwidth (average b = 14). In the multistorey frame the half-bandwidth jumped from 21 for the upper storeys to 45 for the lower storeys. The accuracy of the predicted eigenvalues

Figure 3. Displacement variables for one hanger station for a three-dimensional suspension bridge analysis

VIBRATION PROBLEMS 551

Table 111. Simultaneous iteration eigenvalue predictions for suspension bridge with 486 degrees of freedom (eigenvalue = 1 / d )

Eigenvalue number 1 2 3 4 5 6 7 8

Iteration number 1 2 3 4 5 6 7 8 9

10 1 1

> 1 1

0~0020 0.5221 3.0959 4.4828 4.5755 4.5765 4.5765 4.5765 4.5765 4.5765 4.5765 4.5765

O.OoO5 0.2446 2.4640 1.8936 2.3903 2-3988 2.3989 2.3989 2-3989 2.3989 2.3989 2.3989

0*0003 0.2252 1.4345 1.1219 1 -4009 1.3957

*1*3971 1.3971 1.3971 1.3971 1.3971 1.3971

0-0002 0.1534 1 *2304 1.0477 0.8448 1,3289 1.3399 1.341 3 1.3413 1.3413 1.3413 1.3413

0~0001 0.1256 08541 1 *oO40 0.7374 0,6304 0.63 1 1 0-6333 0.6410 0.6499 0.6507 0.6507

0*0001 0.0902 01636 08445 0.4946 0.371 1 0.5181 0.5868 0.6301 0.6271 0.6264 0.6264

0~0000 0.0774 0.1528 0.5282 04853 0.3141 0.43 15 0.5822 0.5306 0.5233 0.5233 0.5233

0.oOoO 0.0750 0.1413 04785 0-3490 0.2845 0.3986 0-3382 03506 0.3507 0.3507 0.3507

after five iterations for the multistorey frame and after 12 iterations for the suspension bridge are not only to four figure accuracy but are demonstrably so by inspection of the patterns of convergence shown in Tables I1 and 111. The bounds for the first five eigenfrequencies quoted by Geradin do not give such a close assurance of their values. Taking into account the different computers used, the computing times given by Geradin for mass condensation do not seem to be any more favourable than those for simultaneous iteration. However, as the problems analysed are significantly different this should not be taken as a conclusive comparison.

CONCLUSION There is a correspondence between the mass condensation method and the first round of iteration of a simultaneous iteration method under special circumstances. However, with simultaneous iteration it is normal to use fewer trial vectors and there is no need for these to be problem orientated. The amount of numerical computation involved in using both methods appears to be of the same order of magnitude. However, the following factors would suggest that simultaneous iteration is more versatile and reliable :

1. The accuracy of the simultaneous iteration results can be controlled by means of the specified tolerance. 2. The matrices involved have smaller dimensions and do not need to be increased in size to obtain more

3. It is not necessary to rely on the skill of the operator in choosing master displacements. 4. Any estimate of the required modes of vibration may be used to improve the efficiency. Even with the enhancements of Wright and Miles and also Geradin, it would appear that mass condensation

is not as reliable a method as simultaneous iteration because it is dependent on an initial choice of master displacements and has not such satisfactory error control. .

accurate results.

ACKNOWLEDGEMENT

The author would like to thank B. R. Corr of the Civil Engineering Department, Queen’s University, Belfast, for conducting the computer investigation.

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552 ALAN JENNINGS

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