transient shell response by numerical time integration

INTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN ENGINEERING, VOL. 7, 273-286 (1973)

TRANSIENT SHELL RESPONSE BY NUMERICAL TIME INTEGRATION

RAYMOND D. KRIEG AND SAMUEL W. KEY

Sandia Laboratories, Albuquerque, New Mexico, U.S.A.

SUMMARY In using the finite element method to compute a transient response, two choices must be made. First, some form of mass matrix must be decided upon. Either the consistent mass matrix prescribed by the finite element method can be employed or some form of diagonal mass matrix may be introduced. Secondly, some particular time integration procedure must be adopted. The procedures available divide themselves into two classes: the conditionally stable explicit schemes and the unconditionally or conditionally stable implicit schemes. The choices should be guided by both economy and accuracy. Using exact discrete solutions compared to the exact solutions of the differential equations, the results of these choices are displayed. Concrete examples of well-matched methods, as well as ill-matched methods, are identified and demonstrated. In particular, the diagonal mass matrix and the explicit central difference time integration method are shown to be a good combination in terms of accuracy and economy.

INTRODUCTION In using the finite element method to compute a transient response, two choices must be made. First, some form of mass matrix must be decided upon. Either the consistent mass matrix prescribed by the finite element method can be employed or some form of diagonal mass matrix may be introduced. Secondly, some particular time integration procedure must be adopted. The procedures available divide themselves into two classes : the conditionally stable explicit schemes and the unconditionally or conditionally stable implicit schemes. The choices should be guided by both economy and accuracy. Exploring these choices for a general shell element is difficult. However, if limiting cases of the shell are considered, exact solutions to both the differential and discrete equations can be displayed. When a shell is taken as flat, isotropic and uniformly thick, the membrane and bending behaviour uncouple. The free vibration response for both the continuous and discrete forms of the membrane and bending behaviour can be found in closed form.

These solutions with various choices of mass representations, coupled with various time integration schemes, provide a wealth of information that is used to make extensive comparisons. Concrete examples of well-matched methods, as well as ill-matched methods, are identified and demonstrated. In particular, the diagonal mass matrix and the explicit central difference time integration method are shown to be a good combination in terms of accuracy and economy for the shell problem.

The literature in this area is growing and many different opinions exist as to what is a good technique. Kriegl in an early paper studied the joint effect of both space and time discretization on membrane and bending behaviour, compare Sobel and G e e n 6 Washizu2 examined discrete spatial solutions to the free vibrating string equations and to the steady heat conduction problem in a bar with surface loss. Goudreau3 and Goudreau and TayloF have extensive studies of the behaviour of space and time discretizations and should be consulted along with this work. Nickel16 reported on the frequency distortion and damping characteristics of several popular time integration schemes.

Received 18 May 1973 @ 1973 by John Wiley & Sons, Ltd.

12 273

274 RAYMOND D. KRIEG AND SAMUEL W. KEY

FINITE ELEMENT IN SPACE

The wave equation in one dimension which characterizes the degenerate membrane behaviour is given in equation (1)

Here, u is the in-plane displacement and x and t denote differentiation in space and time, respectively. The sound speed cm is given by (E/p)f, where E is Young's modulus and p is the density. The thickness of the shell cancels out. For a homogeneous problem with homogeneous boundary conditions, Hamilton's principle can be written as shown in equation (2)

cg u,, - Uft = f(x, t ) (1)

Using an element extending from x, to xi, a linear displacement assumption can be written as

(3) I u(x, 0 = u&t> (1 - r)P + u&) (1 + r))P

r) = (x - x5)/1+ (x - X,)/Z, 1 = 5 - x,

The local co-ordinate r) runs from - 1 to + 1. For convenience, the length of the element is written as 1. The resulting element stiffness and mass matrices are given in equation (4)

u, u5 ii, u5 u, ii5

The stiffness matrix k and the consistent mass matrix m, are well-known results and are easily obtained by standard methods. The diagonal mass matrix md is the result of lumping at the nodal points.

For a uniform shell with uniform meshing, the equation at any interior node j is given by equation (5 )

where the operator notation Guj means ~5-1- 2u5 + ~ 5 + 1 . The consistent mass results are obtained with r = 6 and the diagonal mass results with r = 0. The product solution in equation (6) is assumed for the homogeneous equation

Separation of variables leads to the pair of expressions in equation (7). u5 = SjT(t) (6)

T + G 2 T = 0 ) The first of equation (7) is satisfied by the discrete harmonic solution in equation (8)

S5 = ClsinN+C2cos- Ai h j N

For a free-free or fixed-fixed boundary condition, h = nr, while for the free-fixed case, h = (2n - 1) 4 2 , the same as in the continuous case. The positive integer n is less than or equal to N, the total number of elements making up the mesh. Since the mode shapes are the same in both the discrete and continuous cases, a frequency-by-frequency comparison can be made. The continuous frequency is given by o = hcm/NZ. The result is given by equation (9) and shown

TRANSIENT SHELL RESPONSE

3

275

1 .Consistent mas, r = 1/6 PDioganol moss,r=O

1

in Figure 1.

- 1 2

- -

or13

420 -

md = I 2 -

a!i 3

- 420-

1.5 I

A cubic displacement assumption can be written as

I 1 4x9 t ) = ui hlh) + Uj h,(rl) + uxi h,(rl) 3 + uxj h,(rl) 5 hl(rl) = Hr13 - 371 + 21, h,(rl) = Hr13 - 11, - 7 + 1) h,(rl) = - th3 - 37 - 21, h,(rl) = &(.I3 + r12 - 11 - 1)

The resulting element stiffness and mass matrices are given in equation (1 1)

k =

ui ua ui ux* 18 1 18 1 151 10 151 a 1 41 1 I

10 30 10 30

- - --

- - -- --

1 -- 18 1 18 151 10 10

-- --

1 1 1 41 10 30 10 30 - -- -- -

3 mc=

iii

131 35

1112 210

91 70

1312 420

- -

- --

UX< iij 1112 91 210 70 - - l3 1312

105 420

131, 131 420 35

- -

- -

I3 1112 140 210

-- --

1

iixj 1312 420

--

13 -- 140

1112 210

--

13 - 105


With this displacement assumption, two discrete equations are obtained at each mesh point. The discrete solution is again given by equations (6) and (8). The mode shapes are again the same as those in the continuous case, so that a frequency-by-frequency comparison can be made. The details are considerably more involved than above. The description of the process is contained in Goudreau and Tayloi' and Key and Beisinger? The results are contained in Figure 2.

The beam equation which characterizes the degenerate bending behaviour is given in equation (1 3)

(13) Here, w is the lateral displacement and x and t are again differentiations in space and time, respectively. The constant cb is given by (Eh2/12p)*, where h is the thickness. For a homogeneous problem with homogeneous boundary conditions, Hamilton's principle can be written as shown in equation (14)

4 w,,,, + Wtf = A x , 0

Rotatory kinetic energy is omitted. A cubic displacement assumption can be written just as in equation (10). The resulting element stiffness and mass matrices are given in equation (15).

k =

w, 131 35

1 1 1 2

210

91

-

-

76 1312 420

--

wa 1112 210 -

13 - 105

13P 420 -

13 -- 140

wj w,, 91 1 312 70 420 - --

1312 13 - -- 420 140

131 1 1 P 35 210

1112 13

210 105

- --

-- -

The stiffness matrix k and the consistent mass matrix m, are well-known results. The same diagonal mass matrix as before is employed with 01 = 17.5 and is given in equation (12). Two discrete equations are obtained at each mesh point. The discrete solution is given by equation (1 6)

w =

The mode shapes are once more the same as those in the continuous case, so that a frequency-by- frequency comparison can be made. The description of the process is contained in Goudreau and Tayloi' and Key and Beisinger.* For the relation between h and mode number for various boundary conditions, Hurty and Rubinstein: p. 203, can be consulted. The results are contained in Figure 3.

Figures 1,2 and 3 tend to favour the consistent mass representation as being the more accurate. These are the results that would be obtained in a mode shape and frequency evaluation. If only the lower modes are being sought, there is no clear-cut preference except on the basis of computational effort and then a diagonal mass is better. In practice, a transient response is obtained by numerically integrating the equations. This further distorts the spectrum and must be considered before any judgements can be made.

TRANSIENT SHELL RESPONSE 277

1 5

3 \ I 0 13

z 0 .- L

z u C 0) 3

0.5 t

I 2.Diagonol moss 01r14.6

1.5

,3 1.0

P

13 0 .-

x 0 c 0)

$ 0 5 t

0

I . Consistent mass 2 . Diagonal mass 01 =17 5

1.0 2.0 Mode number 7

Figure 3. The ratio of discrete frequency B to exact frequency w versus mode number 7 for both a consistent mass and diagonal mass discretizations of the beam equation using cubic displacement assumptions (pin-pin 7 = n/N; for other boundary condi-

tions, see Reference 9, p. 203)

TIME INTEGRATION FOR TRANSIENT RESPONSE

The numerical time integration can be carried out by many different methods. Only one- and two-step methods are examined here.

Ordinarily, the time integrators are examined for their rate of convergence near At = 0, their stability and whether they exhibit damping. This is certainly not an exhaustive list of what should be known. The entire frequency response should be known as well as the amount of damping versus frequency, items which are covered below. Other issues of importance, but not covered here, are the consequences of extraneous roots, ease of implementation, self-starting capability, estimates of error bounds and self-adjusting time step size. In particular, the emphasis here is on the time integration methods which are presently used or modest extensions of them. A method is sought which maximizes accuracy up to a given integration time and minimizes the work required. The point of this analysis is that the best time integrator for the bar, for example, will give the best answer to the partial differential equation (1). A good time integrator will introduce errors of a type which will tend to offset the errors introduced by discretization in space.

Newmark beta method

The Newmark beta method with the explicit central difference method as a special case is the most popular method in use today and is examined first. Equation ( 5 ) can be rewritten for the Newmark beta method as follows


The superscript k is used to denote the value of the function at time step (kAt) and the operator E is defined as usual where E* uk = The central difference operator in time is then defined as Z = (E- 1)(1 -El).

A product solution for the homogeneous form (ff = 0) of equation (17) is assumed and the variables separated. The spatial equation is again equation (7), where the separation variable & is given by equation (9). The temporal equation is replaced with the discrete equation as follows

ZTk+&2At2[~(1+E-1)+y(Z-E-1)+~Z]Tk = 0 (18) This equation has a damped sinusoidal solution, but it is useful to transform with a one-to-one

and onto transformation to the complex z-plane as follows

Tk = C, exp [(a + iB) kAt] = C (19)

The resulting transformed stability polynomial is then found by substituting equation (19) into equation (18) and rearranging in decreasing powers of z

~~[4+&~At'(4fl-2y)]+~&'At~(2y- l)+&'At' = 0 (20) The solution is said to be stable if it is bounded. Since a power solution was assumed in equation (19), this means that if the magnitude of (1 +z)/(l -2) is no greater than one, then its powers are no greater than one and the solution is bounded, i.e. it is stable. In terms of z, this means that it cannot have a positive real part. The quadratic has no roots in the right half plane if all the coefficients are of the same sign, i.e. the solution is stable if

I 4 + O2 At2(4p - 27) 2 0

2y-120

Equations (21) are the usual stability results as obtained by Nickell, and Goudreau and Taylor: among others. The damping parameter a and frequency B may be obtained from the roots of equation (20) where z = p + iq

(22) 4 G A ~ = arc tan Q + arc tan -

1 +P 1 -P

For small p and q (small &At) these can be approximated by

(24) aAt~2p/( l +p2+q2)

&At229 The roots of equation (20) are

0.56ht(2y - 1) fi iJ[4 + At2(4b - 2y) - 0.25&2 At2(2y - 1)2]

4 + G2 At2(4b - 2 ~ ) z = &At

Note that the damping is controlled by y since it controls the real part of z in equation (26). Note further that it decreases stability slightly as seen in equation (21), and has a negligible effect on frequency as seen in equation (25) or in equation (22).

In any case, equation (26) may be substituted into equation (22) to obtain B/& as a function of &At. A second useful relation is the number of time steps required to reduce the amplitude by


a factor of ten versus &At. This is found from equation (26) and (22) and the expression

log0.1 aAt

N = -

The frequency response for y = 0.5 and the damping for f i = 0.25 are plotted in Figure 4. The damping, N, for other values of y is approximately 0.001 - 0.5). The functions plotted in Figure 4 are useful for examination of the time integration alone, without regard for a space discretization method.

I"

0.1 0.2 0.5 1.0 2.0 5.0 ;?At

Figure 4. Frequency response for y = 0.5 and damping for /3 = 0.25 for the Newmark beta time integrator

In application, the discretizations in time and space should be considered together. For example, the frequencies predicted by the consistent mass linear displacement bar equations are too high as seen in Figure 1. This could be compensated by a Newmark beta method which predicts frequencies which are too low. With this idea in mind, it is possible to select a best value of fi. Since y affects frequencies very little, a choice of y = 0.5 will simplify the roots of equation (26) to an imaginary pair of roots given by

2 i&At = 4 4 + &2 At 2(4fi - l)]

This is substituted into equation (22), the tangent taken of both sides, the result inverted and then squared to give


The identity cot2a+ 1 = c x 2 a is used, the result inverted and equation (9) substituted where y / N = wAx/c, is used to give the result

&At sin2 (wAx/2cm) sin2- = 2 (Ax/c, At)2 + 4[/3 - r (Axlc, At)2] sin2 (wAx/2cm)

The denominator on the right-hand side of this expression can be made equal to one if

Ax 1 and j l = r a= The result is that b = w. This fortunate result of an exact frequency spectrum for all wavelengths has been noted before in Key and KrieglO for the case of a diagonal mass matrix with the central difference time integrator, and is the same as the characteristics solution. Here a more important point follows, namely, that a whole family of error introducing spatial discretizations (for all real r ) can be matched with a family of error introducing numerical time integrators (the Newmark family) in such a way that the errors exactly compensate and perfect answers result.

Two notable examples are the diagonal mass (r = 0) when used with the explicit central difference (p = 0) time integrator, and the consistent mass (r = 6) when used with a Newmark beta method with /3 = 6. These two examples both require operation at exactly the critical time step for perfect answers.?

It is interesting to pause here and examine the full non-homogeneous discretized equation (17). It can be rearranged into the following form

The first term is just that which results from integrating a diagonal mass representation with a central difference time integrator. The second term, which makes the scheme implicit, has a zero coefficient when r, /3 and c,At/Ax are chosen to give a minimum error. This entire family of matched spatial and temporal discretizations collapses into a single method. The forcing functions for various /3 remain distinct, however. The consistent mass case has a forcing function which is smoothed in time, causing a Heaviside to be spread over three time steps. This will be investigated in a future paper.

Just as the optimum matching of spatial and temporal discretization is useful, the observation of bad matching is also of value. One bad choice would be a diagonal mass with an unconditionally stable Newmark beta method, /3 = 0-25. A second bad choice would be a consistent mass (r = Q) with the explicit central difference time integrator. Another combination worth noting is the case of a consistent mass (r = 8 ) with the /3 = Newmark beta time integrator. In this last example, the choices of spatial and temporal discretizations are individually optimal choices but each raises all of the frequencies. The combination is certainly not optimal.

The results of this section have been exclusively for the bar equations with a linear displacement assumption. These results can also be applied to a wider class of problems. In particular, the discrete temporal equation (18) can be applied directly to time integration of the equation

Mij+Kq= 0 (28) where h2 in equation (18) is one of the eigenvalues of M-lK in equation (28). The matrices M and K might arise from any number of physical problems, either continuous or discrete. Figure 4 is directly applicable to equation (28) with this interpretation.

~~~

t The notation of a ‘critical’ time step in fact is a bit misleading in this case, since one usually associates the critical time step with poor results and wild oscillations.

TRANSIENT SHELL RESPONSE 28 1

General time integrator

Consider the time integration of equation (28) with an integrator of the form

M[l +cY(~-E-’)] ZUk+KAt2[1 +yl(E- 1)+y2(l -E-l)+y3(E1-E--2 11 uk = At2[1 +yl(E- 1)+y2(1 -E-1)+y3(E-1-EE-2)]fk (29)

This integrator has, as special cases, many of the integrators in use today, as given in Table I. The product solution and power time function of the form of equation (19) is again used to give the following transformed stability polynomial.

2[4 + ~ C Y + (&At)2( - 1 + 2y1- 2y2 + 2y3)] + z2[4 + (&At)2( - 1 + 4y1- 4y3)]

+~[(&At)~(1+ 2y, + 2y2 + 2y3)] + (&At)2 = 0

Table I

Method a Y1 Ya Y3

Central difference Newmark beta Fox-Goodwin Chan-Cox-Benfield Houbolt Wilson-Farhoomand (4 = 2) (4 = %) equation (29) special case

0 0 1 0 B 0 A 0 t - t 1 1 0

-4 1

& 4

b 4 i - B -- 12

4-1 )+S &(-5+3++3+2- 4 -

-- 12 10 -2

* i

For stability, no roots may lie in the right half of the complex z-plane. The Routh-Hurwitz criteria can be applied which states that the following inequalities must be satisfied for stability.

A = 4 + ~ C Y +(&At)’( - 1 + 2 ~ 1 - 2 ~ 2 + 2y3) 2 0

B = 4+(&At)2(-1+4yl-4y3)20

C = l+2y1+2y2+2y320

where (,At) is taken to be positive. These conditions do not apply if the polynomial is in fact a quadratic, i.e. a = y3 = 0. But this case is exactly that of the Newmark beta method which has been covered in the previous section.

The present integrator with a#O or y3#0 is a questionable integrator because of the presence of three roots, two complex conjugates and a third real root, compare Nickell.6 This third undesired root may be activated in many cases and affect the solution. This will be examined in future work. Along with the extra work and core memory associated with an integrator of this type are also benefits, Unconditional stability is desirable and can be achieved by requiring the above inequalities to hold for all &At > 0. This set of inequalities is

1 +2y1+2y2+2y3> 1

(1 + 2yl-k 2y2 + 2y3) (- 1 +4y1-4y3) 2 - 1 4- 271-2y2 2y3 > O ] (30) y1-3/3 2 0.25

The Houbolt, Wilson and Farhoomand’s extension of Wilson’s method all satisfy these inequalities and are unconditionally stable.

The frequency response and damping characteristics of the methods used by Houbolt, Wilson and Farhoomand, with 4 = 8, are plotted in Figure 5. Also shown in Figure 5 are the damping


characteristics of another method (a = y1 = 0.5, 'ys = -'ya = 0.1) which has a slightly better frequency response, better damping characteristics and is also unconditionally stable. It is interesting to note the similarity between the frequency response curves for the various unconditionally stable three root methods. It is also interesting to compare Figure 4 with Figure 5 and note that none of the three root methods have a frequency response which is as accurate as the unconditionally stable Newmark method which is a two root method. This observation and the uncertainty of the behaviour of the real root in the three root methods prompts a more careful look at the two root methods.

1.0

ZI 0 8 : =l 0. a h 0 7

06

FA? Figure 5. Frequency response and damping for the Houbolt, Wilson and Farhoomand time integrators and a

special case of equation (29)

Higher-order two root methods

An obvious generalization of two root methods is to define an integrator of the form MZuk + At '[&( 1 + E-l) I+ (yI+ cAt KM-l) (1 - E-l)

+ @I+ bAt KM-l >2](Kuk--fk) = 0 (31) The added feature is the use of second-order terms, (M-lK)'. Since the eigenvectors of

(M-1 K)' are identical to those of M-l K, and the eigenvalues are the square of those of M-l K by the Cayley-Hamilton theorem, the analysis of equation (31) is straightforward. Again, using a product solution and a temporal function form and transformation given by equation (19), the transformed stability polynomial is

z'[4+ (&At)'(4/3 - 2y) + (&At)44b - 2 ~ ) ] +~[(&At)'(27 - 1) + (&At)*2~] +(&At)' = 0 (32)


For stability, no roots can lie in the right half of the complex z-plane so that the following inequalities (for positive &At) ensure stability

2+ ( c ~ A t ) ~ ( 2 / l - y ) + ( .?A~)~(2b - c) 2 0

If b = c = 0, then this is the usual Newmark beta method. However, if a or b are non-zero, then the linear damping term y must be one-half or the damping will decrease at high frequencies.

Now, for an unconditionally stable response, the inequalities must hold for all &At. The inequalities for unconditional stability for y = 4 are

( 2 ~ - 1) + 246At)’ 2 0

2b-ca.0 cao I (33)

and either /l 2 0.25 or 2b - c > 0.5@ - 0.25)2. Damping and frequency response characteristics are found by ensuring that equation (33) are

satisfied, using equation (32) to find complex conjugate roots and applying equations (22) and (23). Figure 6 is a plot of the damping and frequency response characteristics for c = 0.001 and various values of /3 and b for unconditional stability. The curve (dashed) for /I = i, y = 4, c = b = 0, is included for comparison.

1.2

g 14

2

l3

9 1.0

8 0.9 t

08

3 ;!f2

ag 8 ; = 8 33 z I%

5 6

s,o

* Gat .

Figure 6. Frequency response and damping for the Ka two root method of equation (31)

The family of higher-order integration methods can be tailored to give frequencies which are high or low depending upon the parameters chosen. Unfortunately, the advantages of this two root unconditionally stable family of integrators is offset as usual by an undesirable attribute.


The second-order term will require additional storage and computational effort. The choice of a time integrator, then, is seen to depend upon the work required to integrate to a given real time for a given mesh with a given accuracy bound.

Practical integration of membrane and bending behaviour

From a practical standpoint, the choices of a mass representation and integration scheme depend upon the work required to integrate to a given real time for a given mesh versus accuracy. Secondarily, storage requirements will also come into play. The real problem is to integrate the membrane and bending behaviour of shells which is represented here by the bar and the beam equations with cubic displacement assumptions. Figures 2 and 3 show the frequency behaviour that would be obtained if the finite element equations were integrated exactly. Two integration schemes have been selected to study. The central difference scheme is the most viable explicit conditionally stable scheme, Krieg.ll The Newmark method with beta equal to a quarter has been selected as the most viable unconditionally stable implicit scheme. This method has the lowest frequency distortion of all of the popular unconditionally stable schemes.

To put the comparison on a work basis, a typical application needs to be considered. The choices made are based on current practice. A square two-dimensional mesh is assumed. The work required to advance the solution one time step using a diagonal mass matrix and the central difference method involves a matrix times a vector for each time step; only the non-zero terms are counted. For the implicit scheme, the matrix is assumed to be symmetric and banded. Gaussian elimination is assumed for the solution of the equations at each time step. The matrix is used in triangularized form so that each time step only requires the processing of a single vector. In this

1.5

1.0 3 Z3 0 .- c e

i 0.5 t L

n " 0 1.0 d

Modanumber7)

Figure 7. The ratio of computed frequency to exact frequency versus non-dimensional mode number for central difference (CD) and Newmark beta(N/?), /3 = &, time integrations combined with consistent mass (CM) and diagonal mass (DM) discretizations of the bar equation using cubic displacement assumptions (fixed-fixed, free-free r ] = n/N; fmed-

free 7 = (2n- 1)/2N)

15

3 1.0

23 e 0 .-

2;

LL t O5

0 I 1.0 2.0

Mode number7

Figure 8. The ratio of computed frequency to exact frequency versus non-dimensional mode number for central difference (CD) and Newmark beta(N/?), /? = a, time integrations combined with consistent mass (CM) and diagonal mass (DM) discretizations of the beam equation using cubic displacement assumptions (pin-pin r ] = n/N; for other boundary

conditions, see Reference 9, p. 203)


case, the implicit schemes require four times as much work per time step. Thus, they must function with time steps at least four times that of the central difference method.

Figures 7 and 8 show the frequency response that results from the four combinations. The solid curves are all equal work curves. The dashed curves, which are for the central difference, consistent mass combination, represent the best that can be done. This combination is handicapped by both a lower critical time step and a non-diagonal mass matrix which means the solution of a set of equations at each time step. For the bar, it requires eleven times the effort of the central difference, diagonal mass combination, and fifteen times the effort for the beam. This combination cannot really be considered fruitful. In the Newmark method, if the time step is lengthened to reduce the work, both curves drop even further down. Particularly distressing is the distortion near 7 = 0 where the lower modes of response exist. Because of this and the better overall frequency response, the diagonal mass paired with the central difference method must be considered better for these two spatial discretizations. The Houbolt, Wilson and Farhoomand methods will all be below the Newmark method because of their greater frequency distortion.

CONCLUSIONS

In practical calculations where finite time, money and computer resources are available, the surest and most direct means of computation are required. Neither an examination of the error in spatial discretization nor a study of time integrators alone can give the information needed for a choice. The numerical results are a product of the two and, thus, various combinations must be examined together. Here, it has been found that explicit, conditionally stable central difference time integration used with a diagonal mass matrix for the finite element bar and beam equations provides the most practical means of computing a transient response.

There are several other alternative choices which will improve the accuracy, but only at much greater expense. For example, a non-diagonal mass representation that lies somewhere between the consistent mass and lumped mass results may be used. Then, a time integrator that has the opposite distortion curve could be used. The Newmark method with beta as a variable can be tailored to have a distortion almost opposite to that of the consistent mass behaviour. Then, with four times the effort and several times the storage, virtually flat frequency curves would be obtained.

ACKNOWLEDGEMENT

This work was supported by the U.S. Atomic Energy Commission.

REFERENCES

1. R. D. Krieg, ‘Phase velocities of elastic waves in structural computer programs’, Report, SC-DR-67-816, Sandia Laboratories, Albuquerque, New Mexico (1 967).

2. K. Washizu, ‘Some remarks on basic theory for finite element method’, in Recent Advances in Matrix Methods of Structural Analysis and Design (Eds. R. H. Gallagher, Y. Yamada and J. T. Oden), Univ. of Alabama Press, Univ. of Alabama, 1971.

3. G. L. Goudreau, ‘Evaluation of discrete methods for the linear dynamic response of elastic and visco- elastic solids’, Report, 69-15, Dept. Civil Engng, Univ. of Calif., Berkeley, Calif. (1970).

4. G. L. Goudreau and R. L. Taylor, ‘Evaluation of numerical integration methods in elastodynamics’, ASCE Mtg, Cleveland, Ohio (1972).

5. R. E. Nickell, ‘On the stability of approximation operators in problems of structural dynamics’, Znt. J. Solids Struct. 7 , 301-319 (1971).

6. L. H. Sobel and T. L. Geers, ‘Numerical study of the convergence of finite difference transient response computations’, Narl Symp. Comput. Struct. Anal. Design, George Washington Univ., Washington, D.C. (1972).

7. S. W. Key and Z. E. Beisinger, ‘The transient dynamic analysis of thin shells by the finite element method’, Proc. 3rd Con$ Matrix Meth. Struct. Mech., Wright-Patterson Air Force Base, Ohio (1971).

286 RAYMOND D. KRIEG AM) SAMUEL W. KEY

8. S. W. Key, Z. E. Beisinger and D. Slade, ‘A computer program for the dynamic analysis of thin shells’,

9. W. C. Hurty and M. F. Rubinstein, Dynamics of Structures, Prentice-Hall, New Jersey, 1964. 10. S. W. Key and R. D. Krieg, ‘Comparison of finite element and finite difference methods’, ONR Symp.

11. R. D. Krieg, ‘Unconditional stability in numerical time integration methods’, J. appl. Mech., 40, Series E,

Report, SLA-73-0079, Sandia Laboratories, Albuquerque, New Mexico (1973).

Num. Comput. Meth. Struct. Mech., Urbana, Illinois (1971).

No. 2, 417-421 (1973).

transient shell response by numerical time integration

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