SCALING AND THE PRE-ASYMPTOTIC BEHAVIOR OF THE CONDITION OF

HIGH ORDER FINITE ELEMENT STIFFNESS MATRICES

hTHUR JOHNSONt

Boston University, Boston, MA 02215, U.S.A.

(Received 14 Febnrary 1977; Received for pub~icafion I5 My 1977)

Abstraet-Scaiing is a simple ~nsformation designed to reduce the condition of the global stiffness matrix K formed by the finite element method. The effectiveness of scaling depends on the problem; that is, on the order of differentiation in the nodal values, the physical dimensions of the elements, and the order of the boundary value problem. High order elements exhibit an optimistic pre-asymptotic behavior.

tiWRODWTION TERTHINBRAM

The asymptotic behavior of both the !s and L condition numbers of the global stiffness and mass finite element matrices. and other aspects of this problem have been studied by Fried in a recent seriesfLZ.41 of papers. These studies are based on two fund~ent~ bounds on the C&K) and C_,(K) condition numbers. They are

In order to analyze the behavior of the condition number of the globat stiffness matrix for the beam, its element matrices are needed. For a cubic element written down relative to the nodal values (w,, wrx, w,, ws,) they are:

1 k =s;s

i

12

6h

-12

6h

6h -12 6h

4h2 -6h 2h2

-6h 12 -6h

2h2 -6h 4hZ

C_(K) I: NT?,- m,ax H&J.. (2)

in which k, and m, arc the element stiffness and mass matrices of the eth element, Pm, the maximum number

of elements meeting at a nodal point, m,8x ( ) the maxi-

mum choice over ah finite elements, N the number of columns or rows in K, A, the lowest exact eigenvalue of the structure considered, pl the corresponding approx, imate eigenvalue computed with finite elements and r the maximum response to a unit force, torque, etc.

r 156 22h

4h2

13h

-3h*

54 -13h

13h -3h*

156 -22h

-22h 4h2

An immediate basic result derived from eqns (1) and (2) is that with mesh size h, K can be scaled such that in problems of order 2m

C,(K) = q/t-*y (3)

where 1)~ = i in the string (membrane) problems, and nr = 2 in the beam (plate) problems.

This paper examines two important aspects of the behavior of C(K) and L,(K) in the light of eqns (1) and (2). First, the ore-usy~~tofjc behavior of the condition numbers is studied and second, the effect of scaling on the condition numbers is examined. The surprising con- clusions reached are not as suspected from the theory and are of considerable practical interest.

+Graduate student, Boston University, and engineer at US Army k&k Research aad Development Command, N&i&+ MA.

Scaling the nodal rotations according to 0 = w,h removes h from the interior of lc. and m, Then A,“‘, = O(h), A:* = 0th~‘) and C,(K)=O(h-‘), which is sup- ported by direct computation, see Fig. 1. Without scaling A,5 ZS 0th’) and G(K) s O(h-‘l= o(W) in which 6 is too huge. However, it leads us to believe that, since the unscaled K has terms o(I) and O(h*) on its diagonal it will have a larger, faster growing, condition number than that of the scaled global matrix, which has only o(1) terms on its diagonal.

F&&ion (2) can be used now to show that this com- monly held assumption is incorrect. When the beam is discretized with finite elements having w and w, as nodal values, then r in eqn (2) is, successively, the response to a unit force; for w, and a unit torque for w, whichever is larger. For the beam both these responses are finite, for any h, and C..(K)SO(~-‘) without scaling, as shown in Fig. 1.

The success or failure of scaling is seen thus to be connected to r being finite or singular. A beam can resist

(4)

CAS Vol. 9 vo I-G 97

A. JOHNSON

6

OL- ._~. . _..+_..:, ._CC_

2 5 IO 15 2025

NE

Fig. I. Log,, condition number vs log,,, number of elements (NE) for a simply supported beam. Cubic elements. A, C, condition number K not scaled; B, Cz condition number K not scaled; C, C, condition number K scaled: D, C, condition

number K scaled.

a point force and also a point torque. A string can resist a point force but not a torque, while a membrane can not even carry a point force. How the above characteristics of Green’s, or the response, function are related to scaling and then to the pre-asymptotic behavior of C,(K) and C,(K) is examined next.

SCALING WITH HIGH ORDER STRING ELEMENTS

First, consider the cubic string element with nodal values u and u, and the corresponding element mat&es

36 3h -36 3h

3h 4h* -3h -h2 1 k<=& I -36 -3h 36 -3h

3h -h* -3h 4hZ

r 156 22h 54 -13h 1 h

22h 4h2 l3h -3h’

me=420 54 13h 156 -22h

L- 13h -3h* -22h 4hZ_]

(5)

Scaling by 0 = u,h removes h from the interior of K and M, and eqn (I) readily predicts that C,(K) = O(h-*). Without scaling, eqn (2) predicts that Cz(K) = O(h-‘). This cannot be disproved with eqn (2) because r cor- responding to uX is infinite; the string being unable to carry a point torque. Does scaling influence C#) in this case? In particular, is C,(K) = O(h-‘) when K is scaled but otherwise C,(K)=O(h-*), or is the theoretical pre- diction just too pessimistic? The answer is given in Fig. 2. Asymptotically no numerical difference is detected in the rate of growth of C,(K) and C,(K), scaled and not. In both cases it is nearly C*,,(K) = Che2 as predicted by

oL_--__-_?_._, ,,___ 2 5 IO 20 30

NE

Fig. 2. Log,, condition number vs log,, number of elements (NE) for a string fixed a1 both ends. Cubic elements. A, C, condition number K not scaled; B, C, condition number K not scaled; C, C, condition number K scaled; D, C, condition

number K scaled.

Fried[4]. Scaling has a beneficial effect, nevertheless, on the constant C and asymptotically

C&K) = 4Oh-* (6)

without scaling, and

C,,,(K) = fh-’ (7)

with scaling. Surprisingly, below 10 elements, which is the range of greatest practical interest, both C,(K) and C,(K), for a scaled K are near/y constant.

The inclusion of higher order derivatives should accentuate this behavior because the response to a higher order torque is of stronger singularity. To test this the element matrices in eqn (8), for the quintic string element, formed with respect to the nodal values u, u,, II, are used to assemble K. Figure 3 shows the condition of this K and the results are still more surprising. Without scaling, C,,(K) = O(h-‘). With scaling, G.,(K) is, at least for N. < 30, nearly constant.

12

II 1

I04

% 9.

5 8. c 7- .g

5 6. 5.

Q 4. k? -I 3.

1 j._.__ 2 5 IO 20 30

NE

Fig. 3. Log,,, condition number vs log,, number of elements (NE) for a string fixed at both ends, quintic elements. A, C, condition number K not scaled; B, Cz condition number K not scaled: C, C, condition number K scaled: D, C, condition

number K scaled.

Behavior of high order finite element stiffness matrices

1800 270h ISh* -1800 270h -15h’

270h 288h* 2lh’ -270h -18h’ 6h’ 1 I

kr = 126Oh lSh2 21h’ 2h4 -IS/,’ -6h3 h’

-1800 -27irk - 1511* 18M -270h I’Sh’

270h -18h’ -6h’ -270h 288h* -2lh’

-15h2 6h’ h’ lSh2 -2lh’ 2h4 J

h me=5G

- 21720 3732h 28lh* 6000 -1812h 181h2

3732h 832h* 69h’ 1812h -532h’ 52h3

281h’ 69h’ 6h’ 181h* -52h’ Sh’

6000 1812h 181h2 21720 -3732h 281h2

-1812h -532h’ -52h3 -3732h 832h2 -69h’

181h* 52h’ 5h4 281h’ -69h3 6h’ -L (8)

To more clearly bring out the relationship between scaling and r for the problems considered above, we look at the changes scaling induces in JIK-‘II, and mfx(K,y’). Figure 4 shows that the computed approxi-

mate response to a point force or torque, max(K,;‘),

remains bounded as the number of elements increases, that iZ& = O(ZVe’) and that llKK-‘llS is increasing O(Ne), as predicted in (4). Similarly, Figs. 5 and 6 show the computed approximate response when considering the string problem with cubic and quintic elements. In addi- tion, Figs. 4-6 show the corresponding responses when scaled elements are used. It is seen that scaling actually does not change lKI_ or the rate of growth of llK-‘ll., but can significantly reduce the magnitude of llK-‘lj.. and max (K,;‘) which is the approximate counterpart to f. Fig. 5. Log,, norm, specified, vs log,, number of elements (NE)

for a string fixed at both ends. Cubic elements. A, [K-‘[. not scaled:B,max(K,‘)notscaled;C.IKCscaled;D,IK&notscaled;

E, wK-‘k scaled: F, max (K;‘) scaled.

-1

2 q . i.

1 c !c is Z.7 ---A -

li.. -IF F

NE

Fig. 4. Log,, norm, specified, vs log,, number of elements WE) for a simply supported beam. Cubic elements. A. IK!.: scaled: B,

Fig. 6. Log,, norm, specified, vs log,, number of elements (NE)

[Ki.. not scaled; C. IK-‘L. not scaled: D. m?x (K,;‘) not scaled: for a string fixed at both ends, Quintic elements. A, j/K-‘& not

E. IKK-‘k scaled; F. m?x (K,;‘) scaled.

scaled: B. mjtx MT’) not scaled: C, [K-‘k scaled; D, max (Kit)

scaled: E, UK!. scaled: F, [Klp not scaled.

100 A. JOHNSON

Acknowledgements-The author gratefully wishes to acknow- 2. 1. Fried, Condition of finite element matrices generated from ledge the support and guidance of Professor Isaac Md of nonuniform meshes. AIAA I. M(2), 219-221 (1972). Boston University. Part of this research was supported by the 3. G. Strang and G. Fix, An Analysis of the Finite Elemenr O&e of Naval Research Contract No. N~i4-76~~. M&od, 209. Prentice Hall. Englewood Cliffs. NJ (1973).

4. 1. Fried, Bounds on the spectral maxjmum norms of the finite REFERENCES efement stiffness, flexibility and mass matrices. inf. .f. Solids

1. 1. Fried, Discretization and computational errors in high order SO?Kf. 9. 1013-1034 (1973). finite elements. AIAA /. 9(10), 2071-2073 (1971).