Control of Zero-Energy-Modes in 9-Node Plane Element

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INTERNATIONAL JOURNAL FOR NUM ERICAL METHODS IN ENGINEERING, VOL. 23,863-869 (1986)

CONTROL O F ZERO-ENERGY MODES IN9-NODE PLAN E ELEMENT

BENEDICT VERHEGGHE+

Laboratorium voor M odelonde rzoek, Rijksuniversiteit Gent, Gent, Belgium

GRAHAM H. POWELL'

University of Califi,rnia, Berkeley, California, U S A .

SUMMARY

For plane stress/plane strain analysis, the 9-node quadrilateral element performs better than thecorresponding 8-node element, especially for non-rectangular shapes. For improved element flexibility andlower computer cost. 2 x 2 quadrature is generally preferable to 3 x 3 quadrature. Unfortunatelythe 9-node element contains spurious zero-energy modes when under-integrated. A method is proposed torestrain these modes without significant loss of accuracy or added cost.

I N T R O D U C T I O N

It has been repo rted by several investigators th at the 8 -node 'serendipity' element is significantly

less accurate tha n the corresponding Lagrang ian element with 9 In particular, as the

corners depart from 90 degrees or the sides become curved, the performance of the 8-nodeelement declines rapidly. The 9-node element, on the other hand, is less sensitive to shape

distortion.It is well known that use of 3 x 3 Gauss quadrature for 8- and 9-node elements tends

to make the element too stiff, and that this problem can be overcome by using reduced

i n t e g r a t i ~ n . ~ . ~se of 2 x 2 quadrature is thus preferable, but has the disadvantage of

introducing three zero-energy modes into the 9-node element (see Figure 1). The first of these

mod es (the 'hourglass' mod e) exists for 2 x 2 quadrature in both the 8- and 9-node elements.It causes n o problems because it can not exist in two adjacent elements. Th e oth er two m odes,

however, can lead t o a singular struc ture stiffness matrix if they are not restrained by displacem ent

boundary conditions.

Several methods have been proposed for controlling the zero-energy modes. Among these are:selective combining 8- and 9 -node shape f~ nc t i o ns ; ~verlaying fully integrated

elements;" stiffening the 'bubble-function' mo de of the 9-nod e element;' com bining the 8- and

9-node stiffness matrices;' an d introdu cing add itional sh ape functions com bined (in effect) witha fictitious constitutive m atrix.12,13 This last app roa ch is the most general and elegant, and has

been applied to a number of different elements. In this pape r we present a me thod which is very

similar in its result to that described in References 12 and 13. The present method. however, is

'Assistant.Professor of Civil Engineering.

0029-598 1/86/080863-07$05.000 986 by John Wiley & Sons, Ltd.

Received 1 February 1985Revised 30 September I985



864 B. V E R H E G G H E A N D G. H. POWELL

2 5 1 2 5 1

9- NODE7 47 4

8- NODE 9- NODE 8- NODE 9- NODE 9 - NODE

( a 1 ( b )

Figure 1 Zero-energy modes

simpler conceptually and rather more direct computationally, and hence we believe that i t

warrants presentation as a distinct m ethod.

ZERO-ENERGY MODES

Figure 2 shows a 9-node element. If we impose nodal displacements u,= 1 at the four corner

nodes, ui - at the four mid-side nodes, u i 0 at the central node and ZI,= 0 at all nodes,

then, using the well-known shape functions, the displacements at a p oint (g,q ) become

u =: 352$ - 2 - 2

u = o

The derivatives with respect to the natural co-ordinates are thus

All fou r derivatives a re zero a t the points 5 = k /J3 and q = i. l/J3, which are precisely thequad rature points for 2 x 2 integration. It follows tha t all strains a re zero at these points, and

no strain energy is detected f or deformations in this mode. T he sam e applies for displacements

in the y-direction. Zero-energy modes are thus defined, for an element of any shape, by the

column vectors

+,,=El 1 1 1 - 1 - 1 - 1 - 1 0 ~ 0 0 0 0 0 0 0 0 0 ] ~

= f + , 0lT

Figure 2. Nine-node element



CONTROL OE ZERO-ENERGY NODES 865

and

+ , , =CO +,IT

Since the zero-energy mode shapes are thus known, it is a simple matter to restrain them byaugmenting the basic element stiffness matrix with the stiffness matrices g+,,+z, and g+,,+z,,

where g is some generalized (scalar) stiffness. These matrices are both of rank 1, having nonzeroeigenvalues (stiffnesses) associated only with the eigenvectors+ and +, Adding these matrices

to the basic element stiffness matrix thus stiffens the rigid body modes. For small values of 9 ,

the zero-energy modes are converted to low energy modes, whereas for large values they areeffectively eliminated. Unfortunately, numerical tests will quickly show that if g is small the lo w

energy modes can still persist, leading to excessively flexible results, whereas if g is large certainother modes of deformation are also restrained, leading to excessively stiff results. This secondtype of behaviour occurs because the modes defined by Cp,, and +z u are in general not orthogonalto the modes corresponding to the rigid body motions and the constant and higher order strainstates. Some adjustments to the procedure are thus essential.

The remedy is to orthogonalize + and + with respect to the rigid body and constant strainmodes of the element. Fortunately this can be done with little additional computation. Theprocedure is described in the following section.

ORTHOGONALIZATION

For an element of any shape, the rigid body and constant strain modes are defined by the columnvectors

in which 1 and O=vectors of unit and zero values, x=vector of nodal x co-ordinates andy=vector of nodal y co-ordinates. By combining vectors we get the following equivalentorthogonal base:

Remembering that the two zero-energy modes are defind by

where

Cp,=[l 1 11

- 1- 1 - 1

- 1O]',

we see that it suffices to orthogonalize vector +, with respect to vectors 1, x and y, and to usethe resulting vector, +, to calculate stiffness matrices for use in place of g+,,+T, and yCp,,+~,.

This means simply computing the 9 x 9 matrix g+,,+zn, and adding its terms at appropriatelocations in the element stiffness matrix (each term added at two locations). The orthogonaliLationis performed by the standard Gram-Schmidt procedure.

Up to this point, little has been said about the value of the scalar y, which is a generalizedstiffness for the deformation modes [4,,0] and [O+ , , ] '. This stiffness must be large enough torestrain the zero-energy modes, but not so large as to muse numerical problems in solving theequilibrium equations. We have found that the spurious modes are effectively restrained if y is



866 B. VERHEGGHE AND G . H. POWELL

of the same order of magnitude as the largest diagona1 element in the original stiffness matrix.We have thus used this largest element as the value of g (although larger values did not lead tosubstantial changes in the results).

The above procedure eliminates the two troubling zero-energy modes and ensures that theelement satisfies the patch test in all cases, even when it becomes distorted or when the ‘midside’nodes are not at the true midpoints of the sides.

It is possible to improve the element further, by orthogonalizing @z with respect to vectorscontaining x2, y2 and xy.Exact results could then be obtained for cases involving linear strainvariations, but only for elements which have straight edges, with the ‘midside’ and interior nodesat the true midpoints. We have not explored this aspect.

EXAMPLES

The three element patches shown in Figure 3 were subjected to the load conditions shown inFigure 4. Three degrees-of-freedom were restrained to prevent rigid body motions. For the firsttwo load conditions (constant strain), exact displacements and stresses were obtained at allpoints. The third load condition (constant bending) was modelled exactly only by the first mesh,the other two giving approximate results. For the fourth load condition (linear bending) all three

( b )

Figure 3. Element patches

( b ) ( C )

Figure 4. Loads on element patches

(a 1 f b )

Figure 5. Example with sen sitivity to low energy modes



CONTROL OF ZERO-ENERGY NODES 867

meshes gave only approximate results, the accuracy being much better for the first mesh. In allcases where approximate values were obtained, the stresses at the quadrature points were closeto the exact values, while at other points there were significant deviations (see Table I). These

deviations became larger for progressively higher values of the generalized stiffness 9. It shouldbe emphasized that the 9-node element with 2 x 2 quadrature and no restraining measures (i.e.g = 0) gives totally incorrect results for all of these examples.

Figure 5 shows an example which is likely to develop spurious deformation nodes and has

previously been studied by Stricklin et aE.,2 Cook and Zhao-Hua," and Bicanic andH i n t ~ n . ' ~or overall stability it is suflicient to restrain the vertical displacement at only oneof the three nodes at the built-in end, as shown. If this is done, however, the two zero-energynodes defined by t$zu and are not restrained. Fixing all three nodes at the end restrains thezero-energy modes in elements close to the end, but still leaves low energy modes in the elementsaway from the end, allowing substantial deformations of the form shown in Figure 5(b).

In Table I1 we compare the computed horizontal displacement at the application point of theload for different elements and integration schemes. In particular, we compare the results usingthe restraining method proposed by Cook and Zhao-Hua' (restraining Method 'C-Z) and themethod proposed in the present paper (Method 'V-P). It can be seen that Method 'C-Z' stillallows substantial low energy mode deformations even with three fixed nodes, whereas Method

'V -P does not. With no control over the zero energy modes, the displacements are very large.

Table I. Ratio of calculated to theoretical bending stress for plate in Figures 3 and 4 (A =quadraturepoint and B = corner point, as shown in Figure 3)

Elementmesh A B A B

Constant bending moment (Figure 4c) Linear bending moment (Figure 4d)

Figure 3(a) 1Ooo

Figure 3(b) 0.952Figure 3(c) 1.033

1~Ooo 1.157

1.115 0.9292.270 1.070

0555

0-5661.23 1

Table 11. Displacement at the load application point for structure in Figure 5.P = 480AE v = 0.2;displacement= (tabulated coefficient) x (10- 3,

Number of Integrationnodes order

Restraining Number of nodes fixed verticallymethod 3 1

3 x 32 x 23 x 3

2 x 22 x 22 x 2

16.79717028 17.05617.789 17.819

497.4c-z 96.13V-P 17.192 17.219

Table 111. Displacement of structure in Figure 5(a) for different restrained elements

Restrained elements 1 1, 2 1, 3 1,4 All

Displacement 21.061 17.264 19.070 18.686 17.219



868 B. VERHEGGHE AND G. H. POWELL

It has been noted by Milford and Schrobrichlo that it is sufficient to restrain the zero-energy

modes in only one element. This will indeed make the global stiffness matrix nonsingular. It can

still, however, allow low energy modes to develop. Table 111shows the computed displacements

for the structure of Figure 5 when zero-energy modes are restrained (using the ‘V-P’ method)in only one or two elements. Although the results do not change greatly, it is apparent that low

energy modes still exist.

( a 1 ( b ) (C

Figure 6. Meshes for cantilever analysis

Table IV. Comparison of results for cantilevers in Figu re 6. Displacements are multiples of 10-3PL3/3EIand stresses are multiples of 10-3PL2/121. Poisson’s ratio = 0-3

3 fixed nodes 1 fixed node

A A *B A A *B

Beam theory 40 209.8

8 node 3 x 38 n o d e 2 x 29 node 3 x 3

9 node 2 x 29 node 2 x 2, C-Z9 node 2 x 2, V-P

37.238.738.2

40.240239.0

236-9209.8239.4 38.6 234.5

209.82098209.8 40 3 209.8


8 node 3 x 38 node 2 x 29 node 3 x 39 node 2 x 29 node 2 x 2, C-Z9 node 2 x 2, V-P

6-4 12.314.5 13.231.7 171.2 31.9 173.743.0 264.337.9 211.939.5 2215 42.0 222.7


8 node 3 x 38 node 2 x 29 node 3 x 39 node 2 x 29 node 2 x 2, C-Z9 node 2 x 2, V-P

8.8 10.917.2 - 10.4295 154.4 29.8 158.038.2 2 10-937.8 202.336.8 195.9 38.2 199.9



C O N T R O L OF ZERO-ENERGY NODES 869

Finally we consider a cantilever with tip shear loading, using the three different meshes shown

in Figure 6. In Table IV we compare the tip-displacement(AA)at point A and the bending stress

(gB)at the quadrature point B. Again we make a distinction between configurations with 1 or

3 nodes fixed vertically at the built-in end. It is important to note that if all three nodes arefixed vertically, the boundary conditions along provide enough restraint to eliminate the

zero-energy modes. Consequently his case does not necessarily provide evidence that a restraining

method works effectively. Previous results have been reported only for this case.’,’’ Beam theory

values are given in Table IV only for the case with one node restrained vertically, since only

that case is strictly in accordance with simple beam theory (i.e. no restraint of Poisson’s effect

at the fixed end). In the case with three nodes fixed vertically, there is some stiffening due to

restraint of Poisson’s effect.As can be seen from Table IV, the differences are small, but only if

the zero-energy modes are sufficiently restrained.

CONCLUSIONS

The method presented herein eliminates spurious zero-energy modes in the 9-node Lagrangian

plane element. The resulting element models exactly the rigid body and constant strain modes

for all element shapes. The results obtained appear to be better than those reported for other

comparable elements, especially when the boundary conditions are not sufficient to restrain the

zero-energy nodes. The computational cost for restraining the spurious modes is not a major

disadvantage if the. orthogonalization is programmed efficiently.

The method has the promise of being applicable to other elements, such as 27-node

isoparametric solid element and an 18-node isoparametric element for shell analysis. This

extension is being explored.

REFERENCES

1. J. P. Hollings and E. L. Wilson, ‘3-9 node isop arame tric planar or axisymm etric finite element’, Report N o 78-3,Div. of

2. J. A. Stricklin, W. S. Ho, E. Q.Richardson and W . E. Haisler, ‘On isoparametric vs. linear strain triangu lar elements’,

3. J. Backlund, ‘On isoparametric elements’, Int. j. numer. methods eng., 12, 731-732 (1970).4. W . A. Cook , ‘The effect of geometric shape on two-dim ensional finite elements’,C A F E M 6, Proc. 6th Int. Seminar on

5. S. E. Pawsey and R. W . Clough, ‘Improved numerical integration of thick finite elements’,Int. . numer. methods eng.,3,

6. 0:C . Zienkiewicz, R. L. Taylo r and J. M .Too , ‘Reduced integration techniques in general analysis of plates a nd shells’,

7. T. J. R. Hughes, M. Cowen and M . Haro un, ‘Reduced and selective integration in the finite element analysis of plates’,

8. H. H. Dovey , ‘Extension of three-dimensional analysis to shell structure s using the finite element idealization’, Report

9. T. K. Heller, ‘Effectivequad rature rules for qua dratic solid isoparametric elements’, Int.j. numer. methods eng.,4,597-

10. R. V. Milford and W. C. Schnobrich, ‘Nonlinear behavior of reinforced concrete cooling towers’, Civil Engineering

11. R. D. Cook and F. Zhao-Hua,‘Control of spurio us modes in the nine-node quadrilateral element’,Int. . numer. methods

12. T. Belytschko, W . K. Liu and J. S.-J. Ong, ‘A consistent control of spurious singular modes in the 9-node Lagrange

13. T. B elytschko, J. S.-J.Ong, W. K. Liu and J. M. Kennedy, ‘Hourglass control in linear and nonlinear problems’,Comp.

14. N. Bicanic and E. Hinton, ‘Spurious modes in two dimens ional isoparametric elements’, Int. . numer. methods eng., 14,

Struct. Engr. and Struct. Mech., Univ. of California, Berkeley (1977).

Int. j . numer. methods. eng., 11, 1041-1043 (1977).

Computational Aspects of the FEM, Paris, 1981.

545-586 (1971).

Int. j . numer. methods eng., 3, 275-290 (1971).

Nucl. Eng. Design, 46, 203-222 (1978).

N o . 74-2 Div. of Struct. Engr. and Struct. Mech., Univ. of California, Berkeley (1974).

599 (1972).

Studies, Univ. of Illinois, Urbana, ill. (1984).

eng., 18, 1576-1580 (1982).

element for the Laplace and Mindlin plate equations’, Comp. Me th. Appl. Mech. Eng. ,44, 269-295 (1984).

Meth . Appl. Mech. Eng. , 43, 251-276 (1984).

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Control of Zero-Energy-Modes in 9-Node Plane Element