padé approximants applied to a non-linear finite element solution strategy

PADEÂ APPROXIMANTS APPLIED TO A NON-LINEARFINITE ELEMENT SOLUTION STRATEGY

HENK DE BOER� AND FRED VAN KEULEN

Laboratory for Engineering Mechanics, Delft University of Technology,P.O. Box 5033, 2600 GA Delft, The Netherlands

SUMMARY

The present work deals with an asymptotic numerical method, based on PadeÂ approximants. The expectedadvantage of this method is twofold. Firstly, it reduces the computational costs. Secondly, the auto-matization of the continuation process becomes easier, since the step-length can be determined a posteriori.So far, this method has only been applied to DKT elements. Here it is applied to other types of elements,namely truss elements and ®nite rotation non-linear shell elements. It will be shown that di�culties arisewhen this method is applied to ®nite rotation shell elements. # 1997 by John Wiley & Sons, Ltd.

Commun. Numer. Meth. Engng, 13, 593±602 (1997).

No. of Figures: 3 No. of Tables: 0 No. of References: 12

KEY WORDS: PadeÂ approximant; nonlinear solver; ®nite rotation; shell element

1. INTRODUCTION

In general, a standard Newton±Raphson algorithm is successful in determining a non-linearsolution path. However, the method has two disadvantages. Firstly, the computing time isusually large. Secondly, e�cient automatization of the continuation process is a di�cultproblem.1 For these reasons Cochelin et al.2;3 have developed an asymptotic numerical method(ANM) which applies PadeÂ approximants.4 It was shown that this method increases thecomputational e�ciency.3;5 Moreover, the approach has an attractive feature for determining thestep length a posteriori.

Before application of PadeÂ approximants, the non-linear problem is transformed into a set ofrecursive linear equations by substitution of truncated power series expansions for the unknowns.These expansions have a limited radius of convergence. For that reason, the power seriesexpansions are replaced by their PadeÂ approximants.4 In general, the radius of convergence ofthese PadeÂ approximants is much larger as compared to the corresponding power series.3

However, due to their very nature, these PadeÂ approximants may possess poles. This behaviourcan seriously destroy the robustness of the ANM. Therefore, Cochelin5 did not use PadeÂapproximants when applying the ANM to DKT elements. He mentioned the fact that the presentperturbation technique could be used in a Ritz reduction technique. This reduction technique isproposed and tested by Noor and Peters.6 Using this technique will require some additional

CCC 1069±8299/97/070602±10$17.50 Received 5 February 1996# 1997 by John Wiley & Sons, Ltd. Accepted 29 July 1996

COMMUNICATIONS IN NUMERICAL METHODS IN ENGINEERING, Vol. 13, 593±602 (1997)

� Correspondence to: H. de Boer, Laboratory for Engineering Mechanics, Delft University of Technology,P.O. Box 5033, 2600 GA Delft, The Netherlands.

computations but, according to Cochelin,5;7 the step length will always be larger as compared tothe polynomial or rational representation. We will come back to this topic later.Cochelin et al.3;5 did not apply their approach to ®nite rotation problems but to moderate

rotation problems using DKT elements. The present work deals with ®nite rotation problems, inparticular for shells. We have studied application of the ANM, which uses PadeÂ approximants, totypical loading cases involving nearly inextensional bending and ®nite rotations. In the presentwork no attention is paid to possibly better ways, e.g. the reduced basis technique,6 to extend thedomain of convergence of the ANM.

The present paper is organized as follows. A short description of the employed shell elementand general information on PadeÂ approximants is given in Section 2. Section 3 presents anoverview of the adopted asymptotic method for computing non-linear solution paths. Somenumerical results are given in Section 4. Finally, Section 5 presents some conclusions.

2. PRELIMINARIES

2.1. Adopted shell element

The adopted shell element is based on the so-called Morley triangle. Additional terms for themembrane deformations account for initially curved geometries and for the e�ects of changes ofcurvature on the membrane deformations.8;9 Finite rotations are described using a co-rotationalformulation. A consistent tangent operator was reported by Booij and Van Keulen.10;11 Theelement has 12 degrees of freedom, namely the displacements in the vertices of the triangle andthe rotations about the elements sides. The latter are attributed to the mid-side nodes.

2.2. PadeÂ approximants

Stated brie¯y, a PadeÂ approximant represents a function by the ratio of two polynomials.4 Thecoe�cients occurring in theses polynomials are determined by the coe�cients in the Taylor seriesexpansion of the function. Thus, assuming a power series expansion

f �z� � c0 � c1z � c2z2 � . . . �1�

the coe�cients ai , bi in the [L/M ] PadeÂ approximant of (1), de®ned as

a0 � a1z � a2z2 � . . . � aLz

L

b0 � b1z � b2z2 � . . . � bMzM� �L=M � �2�

are determined by the coe�cients ci in (1). Notice that in (2) there are L � 1 numeratorcoe�cients and M � 1 denominator coe�cients. There is a more or less irrelevant commonfactor between them, therefore, b0 � 1 is taken.5 Thus, there remain L � 1 independent numer-ator coe�cients and M independent denominator coe�cients, making altogether L �M � 1unknowns. This number suggests that the [L/M ] PadeÂ approximant ®ts the power series (1)through the orders 1; z; z2; . . . ; zL�M . It can be shown that this representation by rationalfractions is asymptotically equivalent to the corresponding power series.4

Once the Taylor series expansion (1) is known, the coe�cients ai and bi can be computed byequating (1) and (2). For more detailed information about PadeÂ approximants and their accuracythe reader is referred to the textbook of Baker and Graves-Morris,4 among others.

COMMUN. NUMER. METH. ENGNG, VOL. 13, 593±602 (1997) # 1997 by John Wiley & Sons, Ltd.

594 H. DE BOER AND F. VAN KEULEN

3. ASYMPTOTIC METHOD

The non-linear equilibrium path of the equations R�u; l� � 0 will be determined by meansof asymptotic expansions. Here u is a vector of nodal displacements and rotations and l is aload parameter. In this Section the governing equations are based on a ®nite element formu-lation.

The generalized strains ei can be expressed as analytic functions of the nodal displacementsuk as

ei � Bi �uk� �3�

In the case of linear elasticity the generalized stresses si are determined by

si � Dijej �4�

where the summation convention is applied to repeated indices. In a state of equilibrium it holdsthat

Bi;kDijBj � lf k �5�

Here f represents the external load vector, which is scaled by the load factor l. It is assumed thatthe external load vector is independent of the nodal displacements and rotations. Bi,k denotepartial derivatives of Bi with respect to uk . Now, consider the power series expansion of Baround u�:

Bi � B�i � B�i;kDuk �1

2B�i;klDukDul � . . . �6�

and similarly for its derivative

Bi;k � B�i;k � B�i;klDul �1

2B�i;klmDulDum � . . . �7�

Here uk � u�k � Duk, and B�i;k, B�i;kl denote partial derivatives of Bi with respect to uk , evaluated

for uk � u�k. Substitution of the expansions (6) and (7) in (5), using l � l� � Dl, yields

�B�i;kDijB�j;l � B�i;kls

�i �Dul � B�i;klDijB

�j;m �

1

2B�i;kDijB

�j;lm

� �DulDum

� 1

2B�i;klDijB

�j;mn

� �DulDumDun � Dlf k

�8�

Here use has been made of the fact that (5) holds for u�k. Notice that only terms containing up tosecond order derivatives of the operator B are taken into account. For many problems (8)describes exactly the state of equilibrium. This holds for problems where third and higher orderderivatives of the operator B are zero. However, for the adopted shell elements equation (8) is, ingeneral, an approximation of the equations of equilibrium (5). We will come back to thiscomplication later.

# 1997 by John Wiley & Sons, Ltd. COMMUN. NUMER. METH. ENGNG, VOL. 13, 593±602 (1997)

FINITE ELEMENT SOLUTION STRATEGY 595

At this stage it is possible to substitute power series expansions for the unknowns

Du � aDu�1� � a2Du�2� � . . . �9�Dl � aDl�1� � a2Dl�2� � . . . �10�

The following compact notations are introduced:

K �kl � B�i;kDijB�j;l � B�i;kls

�i

F 1klm �

1

2B�i;kDijB

�j;lm � B�i;klDijB

�j;m

F 2klmn �

1

2B�i;klDijB

�j;mn

�11�

Substitution of (9) and (10) in (8), and using (11), leads to the following set of recursive linearequations:

K �klDu�1�l � Dl�1�f k

K �klDu�2�l � Dl�2�f k ÿ F 1

klm Du�1�l Du�1�m

K �klDu�p�l � Dl�p�f k ÿ F 1

klm

Xpÿ1r�1

Du�r�l Du�pÿr�m

ÿ F 2klmn

X�pÿ2�r�1

X�pÿrÿ1�s�1

Du�r�l Du�s�m Du�pÿrÿs�n p > 2

�12�

These recursive linear equations can be solved by taking Dl�1� � 1 and making use of theadditional orthogonality condition

Du�p�;Du�1� � � 0 p5 2 �13�

As discussed by Cochelin et al.,3 this orthogonality condition and the choice Dl�1� � 1 deter-mines our choice of the expansion parameter a. It means that the parameter a is the projection ofthe solution Du on the tangent solution Du�1�.

After solving (12) the solution is known in the form of power series expansions. In addition,PadeÂ approximants are used to extend the domain of validity. Therefore, (9) is orthogonalized,using a classical Schmidt orthogonalization procedure.3 First an orthogonal basis Du0 �k� is builtup from the vectors Du�p�, such that

Du�p� �Xpk�1

akpDu

0 �k� akp �hDu�p�;Du0 �k�ihDu0 �k�;Du0 �k�i for 1 < k < p �14�

and app � 1, a1p � 0. This ®nally results in

Du � aDu�1� �Xnj�2

a jgj �a�Du 0 � j � where gj �a� �Xpi� j

a ji a

iÿ j �15�



Here n � round� p=2�. Now the scalar series gj (a) can be replaced by their PadeÂ approximants.The radius of convergence of this representation of u is in general signi®cantly larger as comparedto the corresponding power series representation.

Cochelin et al.3 have derived governing equations that are quite similar to those mentionedabove. However, they made use of a mixed unknown, i.e. a vector containing both the unknownnodal displacements and rotations and the unknown stresses. The advantage thereof is that thecubic terms in (8) and (12) reduce to quadratic terms. Notice that, when using a mixed unknown,all inner products mentioned above are formulated with the mixed unknown vectors. At thisstage this method is not implemented here. However, from the viewpoint of e�ciency, it shouldbe implemented.

Finally, the load parameter Dl must be evaluated. Four alternative ways to do this areinvestigated. Firstly, it can be obtained straightforwardly from its power series (10). Secondly, thescalar series (10) can be replaced by its PadeÂ approximant. Thirdly, it is possible to substitute thepredicted displacements in the governing equation (8), and project it on the solution Du�1� inorder to evaluate the load parameter. This yields

Dl � hK�Du;Du�1�i � hF �Du�;Du�1�i

hf;Du�1�i �16�

where K� is the tangent operator and the operator F takes into account the non-linear part of theequations of equilibrium (see (11)). Hence, again terms containing third and higher orderderivatives of the operator B are neglected. Finally, updated equations of equilibrium can becomputed by using the predicted de¯ections. The load parameter is then obtained by projectingDu�1� on these updated equations. Hence,

l � hBt;us;Du

�1�ihf;Du�1�i �17�

Here B,u denote partial derivatives with respect to u. The last method becomes more relevantif the governing equations (8) are no longer a good approximation. It was not used by Cochelinet al.3

So far, the above asymptotic method has only been applied to DKT elements. As mentionedbefore, this results in governing equations of equilibrium that are cubic functions of the de¯ec-tions. However, it generally appears that for ®nite rotation shell elements the governing equationsare no longer cubic, since such elements have no quadratic relations between the ®nite elementdeformations and nodal displacements and rotations, but cubic or higher. This means thatevaluation of third and higher order derivatives of the ®nite element deformations with respectto nodal displacements and rotations is required. However, evaluation of the second orderderivatives was possible for the adopted element, but was already rather troublesome.10;11

Furthermore, but without taking these higher order derivatives into account, it is possible (byusing the mixed unknown framework) to obtain governing equations that have a quadratic non-linearity. As mentioned by Cochelin et al.3 and Cochelin5, a cubic non-linearity is much morecomputationally expensive. For these reasons we have neglected the contribution of the third andhigher order derivatives.



4. RESULTS

In this Section some numerical results are shown: ®rstly, a simple two-bar frame (Figure 1);secondly, a simple plate problem (Figure 2); and ®nally, a problem dealing with ®niterotations and nearly inextensional bending (Figure 3). In all Figures the curves are numbered asfollows:

. Curve 1: The `exact' solution obtained by using an arc-length method with many smallincrements.

. Curve 2: The de¯ections and the load parameter are obtained by their power seriesexpansions (9) and (10), respectively. These expansions are truncated at order 18.

. Curve 3: The de¯ections and the load parameter are both obtained by their correspondingPadeÂ approximants. Therefore, the scalar series

gj �a� �Xpi� j

a ji a

iÿ j

are replaced by their �n ÿ j=n ÿ 1� PadeÂ approximants. Remember, that p � 18 andn � round � p=2�.

. Curve 4: The de¯ections are obtained in the same way as for curve 3. However, the loadparameter is obtained by projecting the equations of equilibrium (8) on the linear solution.

. Curve 5: Again the representation of the de¯ections remains unchanged, but the loadparameter is obtained by projecting the updated equations of equilibrium on the linearsolution.

Additional tests, dealing with several loading cases for shells, show the same characteristicfeatures as the problems discussed here. This additional set of test cases is described in DeBoer.12

4.1. Two-bar frame

The geometry of the ®rst test case is de®ned in Figure 1. Two non-linear truss elements are used,with cross-sectional areas of 1 mm2. Young's modulus and Poisson's ratio are 2� 105N=mm2

and 0.30, respectively. The frame is loaded by a vertical force F of 100 N. This load is scaled bythe parameter l.

As can be seen from Figure 1, PadeÂ approximants signi®cantly improve the domain of validityof the corresponding approximated solution. In this case it seems that the best results areobtained by using PadeÂ approximants for both the displacements and the load parameter(curve 3). The problem is that at the point of divergence large displacements occur, but due to themechanism, only a small percentage result in strains and stresses. If there occurs a relatively smallerror in the predicted displacements the corresponding stresses may be relatively large. Sincecurves 4 and 5, which compute a load parameter depending on the predicted displacement vector,take this into account it seems as if they give inferior results. In fact they are a sensitive indicatorfor the accuracy of the displacement vector.

Figure 1 also shows another important aspect of curves 4 and 5. The divergence behaviour ofthese curves is much more dramatic as compared to the behaviour of curve 3, which uses a loadparameter independent of the predicted displacements.



It can be concluded that for this problem the asymptotic method combined with PadeÂapproximants works very well. That is not surprising, since for these truss elements the equationsof equilibrium (8) are exact. Notice that hence the load parameters computed by the third andfourth method are equal.

4.2. Square plate

This problem deals with the bending of a square plate. Consider a square plate of thicknessh � 1 mm, loaded by a single vertical force at its centre, the four edges being clamped. The lengthof the plate is chosen to be 100 mm and the force 100 N. This load is scaled by the loadparameter l. The material constants are identical to the foregoing case.

The load-displacement diagram, presented in Figure 2, again shows a remarkable improve-ment by using PadeÂ approximants. As shown, curves 4 and 5 seem identical. That is what isexpected, since the equations of equilibrium governing this problem are nearly exactly describedby equation (8). In this particular case, it holds that the terms neglected in (8) are indeednegligible. Furthermore it is seen that curves 4 and 5 best ®t the exact solution and theirdivergence behaviour is not dramatic as compared to curve 3. Notice that these results are quitesimilar to those reported by Cochelin et al.3

4.3. Racketball

This case deals with a spherical shell loaded by a single force at its centre (see Figure 3). ThisFigure also presents the results. The radius of the sphere is 26.3 mm, its thickness is 4.4 mm and

Figure 1. Load-displacement diagram of a two bar problem: improvement of the asymptotic solution by usingPadeÂ approximants



Figure 2. Load-displacement diagram of bending of a square plate: considerable improvement by usingPadeÂ approximants

Figure 3. Load-displacement diagram of a racketball loaded by a single force at its centre: load parameter l versusdisplacement w at the centre



F � 1� 103N. Young's modulus is taken as 4� 103N=mm2, while Poisson's ratio is taken as0.5. The shell elements, the numbering of the curves and the order of the expansions are the sameas in the foregoing cases.

Figure 3 shows that the application of PadeÂ approximants does not improve the validity of thepredicted solution. That is explained by the fact that here the terms neglected in (8) are notnegligible. Therefore, an incorrect power series representation of the solution path is obtained.Before this series starts to diverge, it has already deviated from the exact solution path. ApplyingPadeÂ approximants to the power series results in a solution that is asymptotically equivalent tothe polynomial representation. Hence, curve 3 will not describe the solution path correctly.Curves 4 and 5 obtain the load parameter by projecting respectively the approximated or updatedequations of equilibrium on the linear solution. Both apply the same approximated incorrectdisplacement vector. Projection of the approximated equations of equilibrium will lead to anincorrect load parameter. However, projection of the updated equations of equilibrium will yielda better load parameter. Although the displacement vector is not correct, the latter projectionsmethod attempts to ®nd a correct load parameter corresponding to this displacement vector. Ofcourse, that will not be possible if the error in the predicted displacement vector is too large. Thisis shown clearly in Figure 3. Thus, use of the fourth method (projection of the updated equationsof equilibrium) to compute the load parameter yields a little larger domain of validity of thepredicted solution.

5. DISCUSSION AND CONCLUSIONS

The adopted asymptotic method3 is applied to a ®nite rotation non-linear shell element.8;9 This isin contrast to Cochelin et al.,3 who used DKT elements. Furthermore, an alternative method tocompute the load parameter is introduced, which should be more accurate than the methodspresented by Cochelin et al.3 Also the way the orthogonalizations are carried out is slightlydi�erent. These di�erences lead to essential di�erent conclusions.

When using DKT elements, the truncated power series expansions yield governing equations,which are exact around the expansion point. Undoubtedly, that does in general not hold for theadopted ®nite rotation shell elements, as applied to the above problems. This is due to the factthat, for this element, in contrast to the DKT element, the ®nite element deformations are ingeneral not quadratic functions of the nodal displacements and rotations. Notice that, for reasonsof computational e�orts and storage capacity, these higher order derivatives are not taken intoaccount.

Moreover, it is expected that the basis determined by the recursive linear equation (12) isinsu�cient for the ®nite rotation problem considered here. Therefore, application of the reducedbasis technique, as presented by Noor and Peters,6 might not give more satisfying results, in thesense that the radius of convergence will be much larger. The authors think that the advantage ofthe reduced basis technique will be merely in the area of robustness, because the problem of PadeÂapproximants becoming singular will be eliminated.

Hence, it is concluded that application of the present asymptotic method to ®nite rotationshell problems does not yield the advantages as reported by Cochelin et al.3 for moderaterotations using the DKT elements. It is interesting to use the presented method as an improvedpredictor in an incremental-iterative algorithm rather than in the way introduced by Cochelinet al.3 that avoids the correction step. This correction step can be carried out with a standard



Newton±Raphson procedure.12 In that way, a very practical continuation algorithm can bedeveloped, since the step length is determined a posteriori. However, the authors do not thinkthat in the case of ®nite rotation problems the method will be much more e�cient as compared toa standard Newton±Raphson algorithm. This is caused by the rather limited step-length incomparison with the required additional calculations.

ACKNOWLEDGEMENT

The authors appreciate the stimulating remarks made by Dr. E. Riks.

REFERENCES

1. M. A. Cris®eld, Non-linear Finite Element Analysis of Solids and Structures, Vol. 1: Essentials, Wiley.1991.

2. L. Azrar, B. Cochelin, N. Damil and M. Potier-Ferry, Àn asymptotic-numerical method to computethe postbuckling behaviour of elastic plates and shells', Int. j. numer. methods eng., 36, 1251±1277(1993).

3. B. Cochelin, N. Damil and M. Potier-Ferry, Àsymptotic-numerical methods and PadeÂ approximantsfor non-linear elastic structures', Int. j. numer. methods eng., 37, 1187±1213 (1994).

4. G. A. Baker and P. Graves-Morris, PadeÂ Approximants, Part I: Basic Theory; Encyclopedia ofMathematical and its Applications, Vol. 13, Addison-Wesley, Reading, 1981.

5. B. Cochelin, À path-following technique via an asymptotic-numerical method', Comput. Struct., 54,1181±1192 (1994).

6. A. K. Noor and J. M. Peters, `Reduced basis technique for non-linear analysis of structures', AIAA J.,18, 455±462 (1980).

7. B. Cochelin, `MeÂ thodes Asymptotiques-NumeÂ riques pour le calcul non-lineÂ aire geÂ omeÂ trique desstructures eÂ lastiques', Thesis, UniversiteÂ de Metz, 1994.

8. F. van Keulen, À geometrically nonlinear curved shell element with constant stress resultants', Comput.methods. appl. mech. eng., 106, 315±352 (1993).

9. F. van Keulen, A. Bout and L. J. Ernst, `Nonlinear thin shell analysis using a curved triangular element',Comput. methods appl. mech. eng., 103, 315±343 (1993).

10. J. Booij and F. van Keulen, `Consistent formulation of a triangular ®nite rotation shell element,Proceedings of the First International DIANA Conference on Computational Mechanics', in DIANAComputational Mechanics '94, G. M. A. Kusters, M. A. N. Hendriks (Eds.), Kluwer, 1994, pp. 235±244.

11. F. van Keulen and J. Booij, `Re®ned consistent formulation of a curved triangular ®nite rotation shellelement', Int. j. numer. methods eng., to be published.

12. H. de Boer, PadeÂ Approximants Applied to a Nonlinear Finite Element Solution Strategy, TU Delftreport LTM-1083, Internal Report, Delft, 1995.



padé approximants applied to a non-linear finite element solution strategy

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