Analysis of the effectivity of the Zienkiewicz-Zhu error ...

Analysis of the effectivity of the

Zienkiewicz-Zhu error estimator

J.L. Oliver, F.J. Fuenmayor

Dpto. Ingenieria Mecdnica y de Materiales,

Universidad Politenica de Valencia, Cno. de Vera

s/n, 4 022 Valencia, Spain

INTRODUCTION

One necessary component in an automated, adaptive finite elementmodeling system is the error estimating algorithm. Although thereliability of the finite elements solution to engineering problems isinfluenced by several types of errors, the discretization error isperhaps the most important.

The discretization error represents the difference between theexact and the approximate solution obtained by finite elements (FE),without taking into account the round off errors. The exact evaluationof the discretization error is, of course, impossible without an exactsolution being available, which will not be for most of the practicalproblems. However, even when the exact solution is not available, it ispossible to construct quantitative estimates for the discretization error.

The usual form to measure the discretization error is in terms ofa norm, which can give the required information in terms of a scalarquantity. One natural, convenient, also frequently used norm is theenergy norm, which can be written in terms of the stresses as:

11 / 2

_

a(i)

where "a^ is the inaccurate stress field obtained by finite elements,and OEX is the accurate stress field.

In this paper the a-posteriori error estimator for finite elementanalysis proposed by Zienkiewicz and Zhu [1] is analyzed. Thisestimator is based on the comparison in energy norm between the

Transactions on the Built Environment vol 2, © 1993 WIT Press, www.witpress.com, ISSN 1743-3509

310 Optimization of Structural Systems

discontinuous and inaccurate stress field obtained from the finiteelement solution and a smoothed continuous stress field obtainedfrom the application of postprocessing techniques. The Zienkiewicz-Zhu error estimator is written as:

»«j' I

, 1 / 2

-a ^[0]"' (a* -aFE FE

(2)

where a* is the smoothed continuous stress field and \\~Q^\\ representsthe strain energy corresponding to the estimated error.

The advantages of this scheme are that it is possible to obtain theerror estimation for the whole domain, for element subdomains, andeven a pointwise definition for the estimation. It can be readilyimplemented in a finite element code and it is computationallyeffective.

The reliability of error estimation is measured by an effectivityindex, which is defined by the ratio of the predicted error and theactual error:

9 =

(3)

The error estimator is called asymptotically correct if 9 convergesto unity when the errors converge to zero. In practice, to the accuracyof finite element solutions would be in the range of the engineeringrequirements, it is convenient for 9 to be close to one.

The error estimation and its effectivity depends, of course, on theway to achieve the smoothed continuous stress field. In this paper, weanalyze several smoothing procedures that recover more accuratenodal values for derivatives (stresses) from the original finite elementsolution. The procedures can be classified in two groups: (1) Nodalaveraging of stresses, (2) Local and global least squares smoothing, orprojection techniques. Moreover, as the reliability of the errorestimator is obviously dependent on the accuracy of the recoveredsolutions and therefore the smoothing procedures, we will study theeffectivity index for the Zienkiewicz-Zhu error estimator as a functionof the smoothing procedure, the type of refinement (uniform oradaptive), the type of triangular element (linear or quadratic), and the


Optimization of Structural Systems 3 1 1

number of integration points used in the numerical integration of theerror estimation. Several two dimensional elastic problems areconsidered to test the effectivity index.

CLASSIFICATION

There are several techniques that we can use to achieve the smoothedcontinuous stress field. In this paper we have use the followingtechniques:

1) Nodal averaging of stresses. This is the simplest technique. Itconsists of averaging at each node, the stresses of all the elements thatshare it. The smoothed continuous stress field will be defined byinterpolation in each element with the corresponding shape functions.

2) Techniques based in a least squares method. These techniquesare also called techniques of projection, and they are those mainlystudied in this paper.

LEAST SQUARES METHODS

This technique was proposed initially by Hinton and Campbell [2].Firstly, we need to define the smoothing function, i.e., the polynomicalfunction for which coefficients are to be obtained. In our casecoefficients will be the smoothed nodal stresses, and we will use onesmoothing function for each stress component ( o%, Oy, t%y).

Following the usual discretization procedure at the finite elementmethod, each of those smoothing functions is made up piecewisecontinuous function in the whole domain by using shape functions:

(4)

where o\ is one component of the smoothed nodal stress at the node i,and N^ is the smoothing shape function at this node.

The values of the smoothed nodal stresses (o\, 0*%, .., a* J areobtained by minimizing the functional "squared estimated error in theL norm":

m

X = ) M e (S.n) dQ,<J

9 = 1 a(5)



where n is the total number of nodes, m is the total number ofelements and:

e (%,n) = aU.ri) - aa FE

(6)

Carrying out the minimization procedure, the smoothed nodal stressescan be obtained from the following integral condition:

(7)

that is the same one proposed by Zienkiewicz, Kui and Nakazawa [3].

Local SmoothingThis least square method is called local smoothing when performedseparately over each individual element.

The smoothed nodal stresses will be obtained from the integralcondition shown above when separately applied to each element:

a(8)

Working out the integral shown above [41, the nodal stresses ofthe continuous field may be easily found by solving for each elementthe small linear equations system shown bellow:

= 0i

(9)

where the [M] matrix is called stress smoothing matrix, and (F) vectorof weighted stresses at the element e.

Hinton and Campbell [2], who proposed this method,recommended its application with a set of weighting shape functionsthat were one order lower than those used for the primary variables(displacements).

For quadratic elements, the system of equations can be evaluated


Optimization of Structural Systems 313

on the parent element instead of doing it on each real element. Burnett[5] proves that it is possible to derive a transformation matrix [TR] thatrelates the stresses at the Gauss points in an element to the nodalstresses, no matter the real element used, making the calculationseasy. That is:

{a'} = [TR] (o'|e e G

(10)

where (a*) is the vector of stresses at the Gauss points.

The Jacobian of the coordinate transformation plays a importantrole when we are interested in the comparison of the nodal averagingand the least square method. We can say for the quadratic triangularelement that when jacobian is constant, i.e., when the elements arenot distorted, we obtain the same accuracy.

Finally, the element nodal stresses are averaged as usual at eachnode, to yield the nodal stresses of the continuous field.

Global smoothingWhen the least square method is performed over the entire problemdomain it is called global smoothing. The smoothed nodal stresses willbe obtained from the integral condition when it is applied to the wholedomain:

i

(in

Working out the above integral [4], the nodal stresses of thecontinuous field may be found by solving the linear equations systemshown bellow:

[M] . (a'} - (F) = 0(12)

where the [M] matrix and the IF) vector can be obtained from thefollowing equations:

™ ff(°V : (F}= ~ [N«] *[ D ] [ B ] dfl {u} '

e = 1 a 9 = 1 a(13)



Hinton and Campbell [21, who proposed this method,recommended its application with a set of weighting shape functions[NJ, that were as the same order as those used for the primaryvariables (displacements).

The most important trouble is that the solution effort required forsolving Equation (12) is similar to the effort required to solve theoriginal stiffness equations, since the size of [M] is one the order of thenumber of nodes and it has the sparsity pattern defined by theelement connectivity information.

There are several alternatives to reduce the cost of this process.One alternative is to develop a diagonal form of the stress smoothingmatrix [M], which reduces the effort to solve Equation (12) to onedivision operation for each stress component at each node. Anotheralternative is to use an iterative process.

Diagonal form of the stress smoothing matrix [Ml. With regard to thedevelopment of a diagonal form of the stress smoothing matrix itshould be noted that this matrix has the same form as the massmatrix, which is commonly diagonalized in many kinds of structuraldynamics problems. Therefore, it is possible to use the sameprocedures in this development.

Whichever the method used, the smoothed nodal stress field willbe obtained by solving the following system of linear equations for eachstress component:

(a* } = [M ]" {F}

(14)

where [MJ is the diagonal form of the smoothing matrix, also calledthe lumped smoothing matrix.

There are several lumping procedures, but we can highlight thefollowing ones:

a) the HRZ Lumping Scheme [6], that only uses the diagonal termsof the consistent mass matrix, but scaling them so that the total massof the element is preserved.

b) and the nodal integration, called by Cook [6], optimal lumping.

For the linear triangle element, both HRZ lumping and optimallumping, produce the same diagonal smoothing matrix. On the otherhand, for quadratic triangular elements, HRZ lumping and optimallumping are markedly different.



The optimal lumping or nodal integration can be used since allthe nodal weights are non-zero. The case of zero weights does arise inthe application of a standard Newton-Cotes procedure to the quadratictriangle, see Malkus and Plesha [7]. Since this is the specific elementtype that has proved effective for use in adaptive analysis using h-refinement, an alternative method to define a diagonal smoothingmatrix was developed by Shephard, Niu and Bahemann [8].

The approach consists of finding the weights for a numericalintegration scheme where, instead of equating the numerical integralto a polynomial that can be exactly solved (Newton-Cotes), an errorminimization procedure is used to obtain the weights that bestapproximate the polynomial of the desired order.

Solving the system bv an iterative process. The alternative to develop adiagonal form of the stress smoothing matrix is to solve the system byan iterative process using the consistent smoothing matrix.

Zienkiewicz et al. [9] propose the following iterative scheme toobtain each of the components of the smoothed nodal stress field:

,- [Mj"[ {F} - [Mjla'l,., ]

(15)

where [MJ is the lumped matrix [M], calculated by the HRZ lumpingscheme, and [MJ is the correction matrix described by:

[Ml = [M] - [M ]C L

(16)and the i subscript denotes the iteration step.

It has to be said that for the quadratic triangular element and theiterative scheme shown above, the authors have developed an originalprocedure consisting of using linear shape functions in the element,with satisfactory results.

ANALYSIS OF THE EFFECTIVITY

The reliability of the error estimator is obviously dependent on theaccuracy of the recovered solutions and therefore the smoothingprocedures. Therefore, in this paper we study the effectivity index ofthe error estimator as a function of the smoothing procedure, the typeof refinement (uniform or adaptive), the type of triangular element(linear or quadratic), and the number of integration points used in thenumerical integration of the error estimator. Several two dimensionalelastic problems are considered to test the effectivity. The thick



circular cylinder under internal pressure and the square plate withcentral hole, have smooth solutions. The short cantilever beam, the L-shaped domain, and the square plate with crack, have singularities.The exact solution provided for these examples is either the analyticalone or that provided by using a series of refined meshes and theirextrapolation.

The notation used both in figures and in tables, is the following:

TYPE OF TRIANGULAR ELEMENT (E)

Effectivities for the linear triangle (LI

Smoothing procedure

0 global smoothing; 8* nodal averaging of stresses;

Number of Gauss integration points (I)

3 points (G-3 and A-3 curves); 1 point (G-l and A-l curves)

Effectivities for the quadratic triangle fQ)

Smoothing procedure

9^ global smoothing; 0Q original procedure8 nodal averaging of stresses

9g method derived by Shephard, Niu and Bahemann

Number of Gauss integration points (I)

6 points (G-6, O-6, H-6 and A-6 curves)3 points (G-3, O-3, H-3 and A-3 curves)

EXAMPLES (EX)

With smooth solutions

Thick circular cylinder under internal pressure (TC);Square plate with central hole (SP);

With stress singularities

Short cantilever beam (SC)Square plate with crack (PC)

L-shaped domain (LS).

TYPE OF REFINEMENT (R)

uniform (U); adaptive (A)

Results and conclusionsThe following figures show the effectivity index versus the number ofdegrees of freedom in a logarithmic scale. From the effectivity indicesobtained from the examples analyzed, we can conclude the following.



0

Q-3A-3G-1A-1

80,220,240,180,23

160,290,310,250,3

600,70,780,560,72

960,790,850,640,79

2120,860,930,650,82

3640,870,920,660,79

8100,910,960,670,8

14260,930,970,70,79

31680,940,980,690,78

Figure 1.- TC problem, linear triangles, uniform refinement.

04CM

32 62 108 200 326 538 872 1354 2094

Figure 2.- TC problem, quadratic triangles, adaptive refinement.



S-4

0' ' ' ' ' ' ' '36 66 100 182 372 594 936 1364

Figure 3.- LS problem, quadratic triangles, adaptive refinement.

0,4-

0,2 - - G-1 -&- A-3

u

G-3A-3G-1A-3

130,470,510,40,49

350,690,750,570,71

1130,740,80,610,74

3580,810,810,650,77

11640,860,90.680,78

38760,890,930,70,77

Figure 4.- PC problem, linear triangles, adaptive refinement.



G-6G-4

O-6CM

S-6S-4

A-6A-4

0,2 •—

46 86 178 382 946 2352

Figure 5.- SP problem, quadratic triangles, adaptive refinement.

0-6CM

S-6 -e- A-6&4 -»~ A

40 62 96 154 202 294 390 624 900 1332 2930

Figure 6.- SC problem, quadratic triangles, adaptive refinement.



When we are solving a problem with a low number of degrees offreedom (dof), we obtain a bad error estimation and therefore loweffectivity indices, see Figure 1.

This is reasonable as one mesh with a high discretization errorcannot provide a good error estimation. However there are otherreasons for this behavior. On the one hand, the influence of thecontour definition must be considered. With not a lot of elements inthe mesh, and if there are curved contours, we will have a poordefinition problem, see to Figure 2. On the other hand, in the problemssolved with an adaptive refinement, we can see oscillations in theeffectivity indices for a low range of degrees of freedom. This effect,which is more important in problems with stress singularities, can bethe result of the influence of the initial mesh used, see Figure 3.

As the number of dof increases, the effectivity index getsestabilized. This fact is quite important, since it is possible to apply anempirical constant scale factor to this index, and therefore it ispossible to achieve the convergence of this error estimation to theactual error.

From the observation of the graphs representing effectivity valuesversus dof, a maintenance of the relative order between the differentcurves is observed: A-3, G-3, A-l and G-l for linear triangles; and O-6,S-6, A-6, O-4, S-4, A-4, G-6 and G-4 for quadratic triangles. SeeFigures 4 and 5.

It is also possible to observe how the original procedure developed(O-6 curve) always achieves the best effectivity indices for quadratictriangles.

The convergence of the effectivitv indexFrom the convergence of the effectivity indices and the tables made, wecan obtain the following conclusions.

As was explained before the error estimator is calledasymptotically correct if the effectivity index (9) converges to unitywhen the errors converge to zero. In practice, we have to require that 9must converge to a certain value no matter the problem considered.Nevertheless this value must be close to one.

For problems with smooth solution, when linear triangles areused, if the integration is carried out in three Gauss points, theeffectivity indices obtained are similar and close to unity for all theexamples. It is possible to conclude from the Table 1 that in practice itis not necessary to use any scaling factor. On the other hand, if theintegration is carried out in only one Gauss point, the effectivityindices obtained are similar for all the examples (mainly at the limit),



but the indices are always much lower than the unity. The similarindices make possible to obtain a scaling factor to be applied to get theconvergence to the unity. This factor is independent of the exampleconsidered, and it is different for 0^ y 8& because the second is alwaysbigger than the first.

Table 1.- Examples with smooth solution. Linear triangles. Influence ofthe number of integration points.

EXSPTC

SPTC

ELLLLLL

RAUAAUA

I333111

OG.94.94.96.71.69.68

0A.97.98.97.78.78.72

From the 9^ and^ effectivity indices and bearing in mind thatthe smoothing procedure based on the nodal averaging of stresses atnodes (9J use a smaller computational time, it seems that thisprocedure is the most appropriate one for this kind of problem, i.e., inproblems with smooth solutions when solved with linear triangles.

With regard to problems with stress singularities, if quadraticelements are used, the effectivity indices obtained are always lowerthan the unity as in the previous case, whether the integration iscarried out at six or four Gauss points. Hence, for this type of elementit is necessary to scale the effectivity indices and therefore theestimated errors. Refer to Table 2.

Table 2.- Examples with stress singularities. Quadratic triangles.Influence of the number of integration points

EX

SP

TC

E

Q

Q

Q

R

A

U

A

I

46

*46

9G

.43

.62

.48

.67

.43

.61

9A

:fl

:$f

:2t

90

.68

.79

.69

.78

.79

.95

9S

.67

.77

.69

.77

.78

.93

This behavior can be explained by the fact that the finite elementsolution has equilibrium defects (residuals) in the interior and on theportion of the boundary where tractions are prescribed, also jumps inthe tractions values at the inter-element boundaries. In quadraticelements the interior residuals are dominant, and the tractions' jumpsat interface are the most important source of the error for theZienkiewicz-Zhu error estimator.



On the other hand, the effectivity indices obtained are not similarat all for the different smoothing procedures and for the differentexamples. The smallest variations are presented for the globalsmoothing and the iterative procedure of Zienkiewicz et al. (0). This isvalid for both uniform and adaptive refinements. From the examplesanalyzed it is possible to conclude that for quadratic elements andproblems with smooth solutions, the use of global smoothing and theZienkiewicz et al. iterative procedure is the most effective way whenused with the corresponding scale factor, see Table 2. The numericalintegration at six Gauss points will allow the use of scale factors closerto the unity than if four Gauss points are used.

Table 3.- Example with stress singularities. Linear triangles. Influenceof the type of refinement (uniform, adaptive)

EX

SC

E

L

L

R

U

A

I

1

1

9G

.61

.74

.71

.94

9A

.71

.79

.76

.97

With regard to problems with singularities, if linear elements areused, the effectivity indices obtained are different depending on thetype of refinement. See Table 3 for the short cantilever beam example(SC). In order to obtain a minimum value for the scale factor, it isnecessary to carry out the numerical integration at three Gauss pointsand to use adaptive refinement.

Table 4.- Examples with and without (**) singularities. Lineartriangles. Adaptive procedures

EX

SC

PC

LS

SP**

CL**

E

L

L

L

L

L

L

R

A

A

A

A

U

A

I

%

1

1

1

1

1

6G

.71

.94

.70

.89

:ll.71.94

.69

.94

.68

.96

0A

.76

.97

.77

.93

:!i

:tf

.78

.98

.72

.97

From the comparison between the different examples analyzedwith linear triangles and subjected to adaptive refinements, we can



conclude that the different effectivity indices have similar values. Thiswill allow us to obtain the corresponding scale factors. This is truewhen the integration is carried out both at three Gauss points and at

one. See Table 4.

From the effectivity indices 8 we can conclude that thesmoothing procedure based on the nodal averaging of stresses (8J willbe the most computationally effective. It will need a scale factor nearerthe unity mainly when three Gauss points will be used to carry outthe numerical integration. This scale factor is about the same value forproblems with and without singularities.

Table 5.- Example with stress singularity. Quadratic triangles.Influence of the type of refinement (uniform, adaptive)

EX

SC

E

Q

Q

R

U

A

I

i

46

9G

.29

.39

.46

.61

9A

.40

.43

.62

.71

60

.45

.49

.68

.79

9S

.46

.50

.64

.73

With regard to problems with singularities, if quadratic elementsare used, again the effectivity indices obtained are not similar at all,depending whether the refinement is uniform or adaptive. Refer to

Table 5.

Table 6.- Examples with singularities. Quadratic triangles. Adaptiveprocedures

EX

SC2S

PCIS

LLIS

E

C

C

C

R

A

A

A

I

\

46

46

9G

:*f

•Jl

:H

9A

.62

.71

.74

.81

.66

.79

90

.68

.79

.81

.89

.72

.86

9S

:H

.82

.68

.81

From the comparison between the different examples analyzedwith quadratic triangles and subjected to adaptive refinements, we canconclude: (1) for problems with one singularity (IS), each of thedifferent effectivity indices are in a narrow interval, mainly if thenumerical integration is carry out at six Gauss points, see Table 6;and (2) for the short cantilever beam example, which have twosingularities (2S), until the different singularities have not beendetected by the process, it is not possible to obtain good effectivities.and many'dof are taken. Moreover, the effectivity indices obtained aresmaller than the corresponding examples with one singularity. It is



possible to check this in Figure 6, where the effectivity index (e) isplotted against the number of degrees of freedom, for an adaptiverefinement with quadratic triangles.

Therefore, for the examples with singularities solved withquadratic elements, and bearing in mind the effectivity indicesobtained, it is not possible to specially recommend any of the abovesmoothing procedures. Here it will be necessary to use the scale factorobtained for the most unfavorable situation, and also a safety factor.Therefore we will be overestimating the discretization error. Perhaps, itwould be more interesting to choose the smoothing procedure with thesmallest computational cost demanded.

In order to conclude it has to be noted, that the work made isbased on the analysis of the local effectivities. It remains for futuredevelopments to do an analysis similar to the one carried out in thispaper but considering the effect of local error estimation versus thesmoothing procedure.

REFERENCES

1. Zienkiewicz, O. C.; Zhu, J. Z. "A Simple Error Estimator and AdaptiveProcedure for Practical Engineering Analysis". Int J. Numer. Meth. Eng.,vol. 24, pp. 337-357, 1987.

2. Hinton, E., Campbell, J. S. "Local and Global Smoothing ofDiscontinuous Finite Element Functions using a Least SquaresMethod". Int. J. Numer. Methods Eng., vol. 8, pp. 461-480, 1974.

3. Zienkiewicz, O. C., Kui, L. X., Nakazawa, S. "Iterative Solution of MixedProblems and the Stress Recovery Procedures". Communications inApplied Numerical Methods., vol. 1, pp. 3-9, 1985.

4. Oliver, J. L. Delaunay's Triangulation and Adaptive Finite ElementMethods. Ph. D. Dissertation, Polytechnic University of Valencia, 1991.In Spanish.

5. Burnett, D. S. Finite Element Analysis. Addison-Wesley, 1987.

6. Cook, R. D., Malkus, D. S., Plesha, M. E. Concepts and Applications ofFinite Element Analysis. Third Edition. John Wiley & Sons, 1989.

7. Malkus, D. S., Plesha, M. E. "Zero and Negative Masses in FiniteElement Vibration and Transient Analysis". Comput. Meth. in Appl.Mech. andEng., vol. 59, pp. 281-306, 1986.

8. Shephard, M. S., Niu. Q., Baehmann, P. L. Some Results Using StressProjectors for Error Indication and Estimation. RPI, Report TR-89013.1989.

9. Zienkiewicz, O. C., Kui, L. X., Nakazawa. S. "Iterative Solution of MixedProblems and the Stress Recovery Procedures'. Communications inApplied Numerical Methods., vol. 1, pp. 3-9, 1985.


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