Birth FEM Zienkiewicz

INTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN ENGINEERINGInt. J. Numer. Meth. Engng 2004; 60:310 (DOI: 10.1002/nme.951)

The birth of the nite element method andof computational mechanics

O. C. Zienkiewicz

Professor Emeritus of the Department of Civil Engineering, University of Wales Swansea, Swansea, U.K.and UNESCO Professor of Computational Mechanics, Technical University of Catalunia, Barcelona, Spain

SUMMARY

This brief paper attempts to indicate the motivation which led to the development of the nite elementmethod by engineers and shows how later this became integrated with various current mathematicalprocedures. In the opinion of the writer, the broad denition of nite elements today includes allthe known procedures of approximation for solving partial differential equations and allows the usersto include a variety of methods which are mathematically acceptable. Copyright 2004 John Wiley& Sons, Ltd.

KEY WORDS: nite element method; computational mechanics

1. INTRODUCTION

This paper was written as an introductory lecture to the Fifth World Congress on ComputationalMechanics held in Vienna in July 2002. As this was the rst conference of the 21st century,it seemed appropriate to outline the history and motivation for the development of the niteelement method which forms today the basis of computational mechanics, and to discuss howit reached the present very sophisticated form.

Part of the following story has already been presented in some detail by myself in a paperpublished in 1995 [1]. The development of computational mechanics clearly owes much to thepresence of the electronic computer which came on the scene only in the middle of the lastcentury. However, the words computer and computations are much older. In the very rstpaper on nite differences in the 20th century, Richardson, in 1910, uses the word computersto describe his assistants who were boys from the local high-school, employed to do thenumerical calculations at each iteration.

It is interesting to note that Richardson paid a price of N/18 pence per co-ordinate pointcalculation in which N was the number of digits used and, as his note says, he did not payif the computers committed errors.

Correspondence to: O. C. Zienkiewicz, Department of Civil Engineering, University of Wales Swansea, Swansea,U.K.

Received 10 May 2003Copyright 2004 John Wiley & Sons, Ltd. Accepted 1 June 2003

4 O. C. ZIENKIEWICZ

Though this characteristic differs very much from the computers we use today, it is interestingto observe that the procedures used differed in only a small manner from the calculations weperform today. For this reason, in this present paper I want to concern myself with thebackground of methodologies used in computation.

It is clear to me that the nite element method has two distinct lines of ancestry. The rstof these is the one originated for the solution of engineering systems assembled from simplecomponents. Typical here were steel structures being built at the turn of the century from simplestructural elements. Similar problems arose in the design of electrical and hydraulic networks.The second line of ancestry originated in more mathematical reasoning in which the well-deneddifferential equations, which generally were established in the 19th century, were subject to anumerical solution. At rare intervals researchers used both approaches simultaneously but moreusually the developments used were completely separate. I shall now describe both lines ofancestry separately.

2. THE DISCRETE OR LUMPED ENGINEERING SYSTEMS

In the eld of solid mechanics or engineering structures, the behaviour of simple componentssuch as rods, beams or struts, can be described very simply from rst principles. The estab-lishment of relationships between the forces acting on such components and the displacementsimposed at the end of each component is almost trivial. However, a large number of suchcomponents can be put together to build a bridge, or another similar complicated structure.Thus procedures for dealing with such systems had to be developed quite early on. The mostobvious treatment, indeed the one which is today adopted in the solution of such complexstructures, is the one known as the stiffness method in which the displacements given to theends (nodes) of an element are related to the forces acting at these ends. The linear relationshipbetween such forces and displacements is known as the stiffness. By assuming the displace-ments at the junction points as being the unknowns, the total system of displacements is madecontinuous. To preserve equilibrium the sum of all the forces exerted at the nodes has to bezero. This provides the second set of conditions from which displacements can be evaluated.Analysis based on such displacements was rst suggested by Louis Navier in 1826 [2]. It isfully described in the well-known book by Alfred Clebsch in 1862 [3] which gives details ofwhat we would describe today as the direct stiffness method. Saint-Venant includes the subjectin his lectures to the cole de Pnts et Chausses in 1838 [4]. It seems, therefore, extraordinaryto me that the Direct Stiffness Method should have needed to be revived and described so manytimes since. But still Southwell in 1936 [5] and Turner, as late as 1962 [6], referred to theStiffness Method and again formulated it in detail.

Matrices, of course, could well be used for the assembly of a large number of linearcomponents. The use of such matrices appears on the scene at a later date, viz. Fraser andDuncan, 1928 [7], Duncan and Collar, 1934 [8].

Notable in the context of matrices is the work of Argyris in 1955 [9] in which he producesa rather complex form of the stiffness method, now no longer used. This, however, served toillustrate the possibility of using similar methodology in the so-called force approach, which Ishall not discuss here.

In the context of aircraft structures, shear panels are used for the rst time and the approx-imation is not quite as simple as with bars and beams. Nevertheless this work does not yetapproach the study of continua.

Copyright 2004 John Wiley & Sons, Ltd. Int. J. Numer. Meth. Engng 2004; 60:310

THE BIRTH OF THE FINITE ELEMENT METHOD 5

The standard engineering system, identical to the one described for structures, of course,applies in other areas. For instance, the assembly of electrical components to form a circuitcan be done in precisely the same manner as the assembly of bars to form a structure. Thework of Kirchhoff shows such an approach as early as 1847 [10].

Similarly, the assembly of pipe networks, such as are used for distributing water in townsand cities, follows identical procedures. Here, generally the stiffnesses are non-linear and thesolution becomes more complex.

Before the advent of electronic computers, only a relatively small number of unknowns couldbe dealt with in the assembled equations representing the stiffness of a complex structure. But itis of interest to note that Southwell, who developed his relaxation methods as early as 1935 [5],showed that the solution of such problems could be approached iteratively. A direct solutionprocess in which elimination is used, of course, is not practical by hand calculation as humanresources would fail rapidly. However, iterative processes, in particular those of the relaxationmethods developed by Southwell [1113], allowed very large problems to be solved. I think itis worth noting that Lazarides, as early as 1952 [14], succeeded in solving a structure involvingseveral hundred unknowns. The structure was the Dome of Discovery in the Festival of Britainheld at that time. Hardy Cross [15], working in the U.S.A. produced work parallel to thatof Southwell at a somewhat earlier date. He did not explicitly form the equation used in therelaxation process, and thus his method lacks the Southwell generality. However, his momentdistribution method is known to every engineer who has studied structural frame design fromthe early 1930s onwards until it was made unnecessary by the use of the modern computer.

Now everyone knows that todays electronic computer with its enormous capacity can dealwith any discrete system efciently, irrespective of the number of unknowns. However, thequestion is what is the relation of large discrete systems to the problem of analyzing continuawhich we are addressing in computational mechanics?

The rst ideas on this that come to mind are those in which the network of discrete com-ponents, which are placed in some regular pattern, can model the behaviour of the continuumproblem which we intend to solve. Such a physical approximation to the continuum has beenused by many engineers to model, for instance, the behaviour of plates and shells. They do thisby replacing the continuum by an equivalent gridwork of beams, or by representing a simpletwo-dimensional or three-dimensional continuum by an appropriate assemblage of bars. We seesuch modelling used in the work of Hrenikoff [16] followed by that of McHenry [17]. Both ofthem proved the validity of their approximations by comparing them with the appropriate nitedifference approximations to the continuum equations, though Hrenikoff uses simple beams asthe components while McHenry uses pin-jointed bars.

However, such approximations or idealizations are obvious to those who realize that theidea of a continuum is articial and that at some stage of sub-division the continuum becomessimply an assembly of particles (or atoms) behaving in a perfectly discrete way. Perhaps sucha sub-division is a natural way of modelling the continuum, even though it does not reachatomic sizes. Certainly attempts have been made, with some considerable success, to use suchmodels to directly represent various materials. However, it is possible to approach the subjectmore rationally. A giant step forward was taken by Turner et al. [18] at the Boeing AircraftCompany. In the early 1950s they produced a physical model in which the continuum, usuallyan elastic one, was subdivided into elements of various shapes and sizes. However, this workwas not published until 1956. This paper was the rst to use the nite element approximation.The physical approach used in that paper had to wait for a few years until the subject matured


6 O. C. ZIENKIEWICZ

and it was only in 1960 that Clough [19] succeeded in showing that a mathematical approachwhich minimized potential energy for an assumed potential pattern did indeed produce someof the approximations he had earlier derived physically. Therefore a perfectly logical approachto the subject could now be made.

Before proceeding further, it is of interest to examine in detail the second approach tosolving continuum problems. This is based on a direct mathematical approximation to solvingthe differential equations which govern the physical problem.

3. THE MATHEMATICAL APPROXIMATIONS

One of the most obvious methods for approximating partial differential equations of continuais to cast these equations in simple difference form and then proceed to carry out an alge-braic solution. This is the so-called nite difference method. Such early attempts at numericalcalculations were developed at the start of the last century. The most serious of these wasdone by Richardson, published in 1910. He succeeded in solving the problem of plain stressapproximation for a masonry dam (Assuan). He solved the rather complicated nite differencebi-harmonic equation by simple iteration and succeeded in dealing with some 250 unknowns.

It was a few years later that Southwell realized that the problem of such nite differenceapproximations gave equations very similar to those obtained for discrete engineering systems.For instance, it was well known that the approximation to the Poisson or Laplace equations,which represent a stressed membrane, could be achieved by a network of strings which werepreviously tensioned and from which the structural solution could be obtained. This analogybetween a membrane and a set of strings allowed Southwell to transfer all his experience ofsolving the structural problems iteratively, by the relaxation method, to deal with nite differenceapproximations. His work in the 1940s, which culminated in two books [12, 13] showed forthe rst time that the problem of computational analysis was well advanced in solving practicalproblems. My own work of solving a dam problem, similar to that of Richardsons, allowedsome 900 equations to be solved rapidly in place of Richardsons 250.

However, there were many alternative methods of dealing with the approximation to con-tinuum PDEs. These became available in the rst half of the 20th century though none couldproduce, at that time, the complex and realistic solutions obtainable by Southwells use of nitedifferences.

The earliest of these alternative methods was used by Ritz [20] who employed trial functionswith unknown parameters to represent the unknown function. He then minimized the poten-tial energy of the system to determine the unknown parameters. The solution thus obtainedgave the best approximation for the given trial function set. In 1915 Galerkin showed thatthe same answers could be obtained by using a weighted residual method in which the sametrial functions were employed for both approximation and weighting. However, the weightedresidual process was much more powerful as other trial functions and weighting functions couldbe used. The weighted residual process developed rapidly. Such procedures as collocation (inwhich the weighting function was simply that of a Dirac delta) and sub-domain collocation (inwhich the weighting was unity over a sub-domain and zero elsewhere) became available. Asummary of such weighting procedures was presented clearly by Crandall in 1955 [21]. Todaymany possibilities are being explored using different trial functions and weighting functions.The boundary solution methods, for instance, use an approximation in which the trial functions



automatically satisfy the differential equation in the problem domain and thus the only approx-imation to be done is on the boundaries. Such a procedure was given as early as 1926 byTrefftz [22]. This led to many different types of boundary solutions.

In most of the early applications using weighted residual processes, the trial functions weredened over the whole domain by a single set of expressions which generally had to satisfythe forced boundary conditions. Much later it was realized that fully continuous expressionsfor the trial functions were unnecessary. This allowed the trial functions to be dened in apiecewise manner over sub-domains (or elements). Courant [23, 24] was the rst to realizethis. As early as 1923 he indicated that it would be possible to sub-divide a problem intotriangles and by imposing simple continuity of the function itself at interfaces, a minimizationof the total potential could be achieved. In 1943 he implemented this idea and published therst paper in which the prototype triangular element was used for solving the torsion problem(Poisson equation). Courant could indeed be credited with the invention of the nite elementand, by many mathematicians, he is. However, he did not realize that the process tted sowell into the discrete engineering approach previously discussed. Also, the computer was notavailable in his time to implement the process in a practical way.

4. THE SYNTHESIS

Cloughs classical paper, published in 1960 [19], showed for the rst time a full mathematicalreason for the success of the sub-divisions of the domain into elements. He showed thatby minimizing the total potential energy, the approximate solution would converge to theexact mathematical solution as the size of the elements decreased. At this stage both theintuition, which stemmed from the discrete engineering approach, and the purely mathematicalreasoning coincided and both approaches were united. Thus the Finite Element Method becamea legitimate and extremely useful tool for solving many engineering and physical problems.A few early pioneers now entered the eld. They included Robert Melosh [25], RichardGallagher [26], Veubeke [27] and myself [2831]. The work done on plates by Melosh and onthree-dimensional analysis by Gallagher were seminal contributions in the USA. A very earlypaper dealing with the use of rectangular elements in elasticity was published by Szmelterin 1959 [32]. Veubeke, in 1975, introduced for the rst time the concept of mixed elements.My work in the early 1960s started with an attempt to obtain solutions for complex shellshapes. It soon became apparent that a best approximation could be achieved by the use ofweighted residual processes which in structural mechanics problems were equivalent to the useof virtual work. This provided a more general approach than that which used potential energyminimization. My rst book in 1967 [31] expanded these ideas and became the basis of latervolumes in which the method was further developed. The rst and second editions of this bookwere alone in the eld until 1971 and I hope claried and explained the nite element methodto engineers and mathematicians alike.

From the simple linear elements formulated in the early days, there evolved more complex,higher-order elements. It became clear to some of us that the complex functions could, onoccasion, use a single element to solve a large problem. In the original weighted residualprocedures, only a single domain was used. It soon became apparent that all numerical meth-ods, even including the original simple nite difference methods, could always be describedas particular cases in which various trial and weighting functions were used. Therefore, all


8 O. C. ZIENKIEWICZ

numerical methods could be considered as particular cases of the weighted residual method orof the nite element method.

These generalizations inevitably led to many new developments. New elds or applicationswere conceived, and new methods of approximation and of interpolation were introduced.Two in which I was personally involved and which are of special importance, are: (1) thedevelopment of error estimation [33], and (2) the successful treatment of problems in whichnon-self-adjoint equations arise, e.g. uid mechanics.

To obtain error estimates, a particularly simple method was introduced by Zienkiewicz andZhu [34, 35]. In uid mechanics the introduction of upwinding, which was already in use innite differences, led to the development of more efcient algorithms capable of dealing witha wide range of problems. In particular, the CBS algorithm developed by Zienkiewicz andCodina [36] has been widely applicable.

Undoubtedly these areas will be addressed and many new procedures will be investigated.However, the question now arises of how long we are still justied in carrying on the researchaimed at increasing computational efciency, in view of the computer power available todaywhich is capable of solving inexpensively almost all problems, regardless of their computationalefciency. In Section 5 I will try to address this question.

5. THE COMPUTER AND THE FUTURE

Much has changed since the early 1960s when the computer entered engineering ofces anduniversities as a tool for calculation. At that time, a nite element solution involving two orthree thousand variables was considered to be large. (One of the greater problems of thatdecade was a three-dimensional nite element solution of a nuclear pressure vessel in whichsome 18 000 variables were present.)

When the rst three-dimensional solutions were attempted in Swansea, it was apparentthat in using simple linear elements, we would need several thousand variables to achieve areasonable solution for an arch dam. The largest computer then present in England was theAtlas which could crunch that many numbers. However, this machine unfortunately introduceda computational mistake randomly every 20 min or so of operation. In our case, the estimatedtime for solving a realistic problem was measured in hours and it became almost impossibleto achieve any meaningful solutions. It was at this stage that isoparametric elements wereintroduced by our group. These elements allowed a higher order of approximation to be usedand thus the number of variables was greatly reduced. Over the following decades the progressin the development, and therefore the power of available computers, was enormous. By themid-1980s it became possible to solve uid mechanics problems involving several millionvariables. This was, at last, the kind of capacity necessary to deal with the ow problems inhigh-speed aircraft using the simplest linear elements. Today, using even larger numbers ofvariables, problems can be solved and I understand that one using a billion plus variables hasbeen achieved. This possibility will be discussed further by Professor Oden in his lecture atthis Congress, in which he will refer to teraop computing. However, there exist dangersas well as opportunities in the use of this power. Referring to the dangers rst, it is alreadyevident that many users of nite element computational mechanics attempt to use inferiorcomputational methods which, although convergent, can yield acceptable results if, and onlyif, an excessive number of variables is introduced. It follows that today less emphasis is being



placed on research leading to more economical and efcient methods of computation. Indeedmany agencies which sponsor research are quite happy to accept proposals using outdatedmethodologies providing an impressive amount of computation is involved. Could this lead tothe death-knell in expansion of research in the eld of computational mechanics in searchingfor new and better methods? I hope not, but the danger exists. Therefore we should striveto show that more rened calculation is generally preferable to the use of inefcient methodswhich rely on the fact that the computer usage is cheap.

On the other hand, the possibilities offered by the speed and capabilities of the present-daycomputer open new doors. For instance, by a return to discrete models we can directly representthe behaviour of new materials. Furthermore, such new areas of exploration which are alreadyattempted today, such as the development of genetic algorithms and neural network concepts,will allow the engineer to compute complex phenomena without the need for the establishmentof detailed material models.

Similarly, in turbulence, the power of the computer is beginning already to be used for directmodelling. Thus it may well be that the present quest for improved approximation methodsmay, in future, be superseded by research in as yet unimagined elds.

REFERENCES

1. Zienkiewicz OC. Origins, milestones and directions of the nite element method. Archives of ComputationalMethods in Engineering 1995; 2(1):148.

2. Navier L. Resum des lecons sur lapplication de la Mcanique. Ecole de Ponts et Chausses, Paris, 1826.3. Clebsch A. Theorie der Elasticitt fester Krper, Berlin, 1862.4. Saint-Venant Barr de. Course lectures Ecole Polytechnique, Paris, 1838.5. Southwell RV. Stress calculation in frameworks by the method of systematic relaxation of constraints.

Proceedings of the Royal Society 1935; 15:3695.6. Turner MJ, Martin HC, Weikel RC. Further development and applications of the stiffness method. AGARD

Structures and Materials Panel, Paris, France in AGARDograph 72, 1962.7. Fraser RA, Duncan WJ, Collar AR. Elementary Matrices and some Applications to Dynamics and Differential

Equations. Cambridge University Press: Cambridge, 1938.8. Duncan WJ, Collar AR. Matrices applied to the motions of damped systems. Philosophical Magazine Series

7, vol. 19, 1935; 197.9. Argyris JH. Energy theorems and structural analysis. Aircraft Engineering 1955 (reprinted: Butterworths:

London, 1960).10. Kirchhoff GR. Annalen der Physic und Chemie 1847; 72.11. Southwell RV. Relaxation Methods in Engineering Science. Oxford University Press: Oxford, 1940.12. Southwell RV. Relaxation Methods in Theoretical Physics, vol. I. Clarendon Press: Oxford, 1946.13. Southwell RV. Relaxation Methods in Theoretical Physics, vol. II. Clarendon Press: Oxford, 1956.14. Lazarides TO. The Structural Analysis of the Dome of Discovery. Crosby, Lockwood and Sons Ltd.: London,

1952.15. Cross H. Analysis of continuous frames by distributing end moments. Transactions of the American Society

of Civil Engineers 1932; 96:110.16. Hrenikoff A. Solution of problems in elasticity by the framework method. Journal of Applied Mechanics

1941; A 8:169175.17. McHenry D. A lattice analogy for the solution of plane stress problems. Journal of the Institution of Civil

Engineers 1943; 21:5982.18. Turner MJ, Clough RW, Martin HC, Topp LJ. Stiffness and deection analysis of complex structures. Journal

of Aerosol Science 1956; 23:805823.19. Clough RW. The nite element in plan stress analysis. In: Proceedings of the 2nd ASCE Conference on

Electronic Computation, Pittsburgh, PA, 1960.20. Ritz W. Uber eine neue methode zur Losung gewissen variationsprobleme der mathematischen physik.

Journal fur die Reine und Angewandte Mathematik 1909; 135:161.


10 O. C. ZIENKIEWICZ

21. Crandall SH. Engineering Analysis. McGraw-Hill: New York, 1955.22. Trefftz E. Ein Gegenstuck zum Ritzschen Verfahren. Proceedings of the 2nd International Congress on

Applied Mechanics, Zrich, 1926.23. Courant R. On a convergence principle in calculus of variation (German). Kn Gesellschaft der Wissenschaften

zu Gtingen Wachrichten, Berlin, 1923.24. Courant R. Variational methods for the solution of problems of equilibrium and vibration. Bulletin of the

American Mathematical Society 1943; 49:123.25. Melosh RJ. Structural analysis of solids. Proceedings of the American Society of Civil Engineers 1963;

ST4:205223.26. Gallagher RH, Padlog J, Bulaard PP. Stress analysis of complex shapes. ARS Journal 1962; 700707.27. Veubeke B, Fraeij de. Displacement and equilibrium models in nite element method. In Stress Analysis,

Zienkiewicz OC, Holister GS (eds), Chapter 9. Wiley: New York, 1965; 145197.28. Zienkiewicz OC, Cheung YK. The nite element method for analysis of elastic isotropic and orthotropic

slabs. Proceedings of the Institution of Civil Engineers 1964; 28:471488.29. Zienkiewicz OC. Finite element procedures in the solution of plate and shell problems. In Stress Analysis,

Zienkiewicz OC, Holister GS (eds), Chapter 8. Wiley: New York, 1965; 120144.30. Zienkiewicz OC, Holister GS (eds). Stress Analysis. Wiley: Chicago, 1965.31. Zienkiewicz OC, Cheung YK. The Finite Element Method in Structural Mechanics. McGraw-Hill: New York,

1967; 272.32. Szmelter J. The energy method of networks of arbitrary shape in problems of the theory of elasticity. In

Proceedings of IUTAM 1958 Symposium on Non-Homogeneity in Elasticity and Plasticity, Olszak W (ed.).Pergamon Press: Oxford, 1959.

33. Zienkiewicz OC, Zhu JZ. Error estimation and adaptive renement for plate bending problems. InternationalJournal for Numerical Methods in Engineering 1989; 28:28392853.

34. Zienkiewicz OC, Zhu JZ. Superconvergent patch recovery and a-posteriori error estimation in the niteelement method, part I: a general superconvergent recovery technique. International Journal for NumericalMethods in Engineering 1992b; 33:13311364.

35. Zienkiewicz OC, Zhu JZ. Superconvergent patch recovery and a-posteriori error estimation in the niteelement method, part II: the ZienkiewiczZhu, a-posteriori error estimator. International Journal for NumericalMethods in Engineering 1992c; 33:13651381.

36. Zienkiewicz OC, Codina R. A general algorithm for compressible and incompressible ow, part I: the splitcharacteristic based scheme. International Journal for Numerical Methods in Fluids 1995; 20:869885.

37. Zienkiewicz OC. The stress distribution in gravity dams. Proceedings of the Institution of Civil Engineers1947; 27:244247.

38. Zienkiewicz OC, Taylor RL. The Finite Element Method (5th edn). Butterworth-Heinemann: Oxford, 2000.


Documents

Birth FEM Zienkiewicz