Origins, milestones and directions of the finite element...

Vol. 2, 1, 1–48 (1995) Archives of ComputationalMethods in Engineering

State of the art reviews

Origins, Milestones and Directions of the FiniteElement Method – A Personal View†

O.C. Zienkiewicz∗

Institute for Numerical Methods in Engineering. University College of Swansea, Wales (UK)

Summary

The article traces the important steps of the development of the finite element method from its origins inaircraft structural engineering to the present day, where it provides the essential tool for solution of a greatvariety of problems in engineering and physics. The emphasis and the choice of the “landmarks” stresses theaspects which are general and essentially of mathematical nature applicable to a wide range of situations.For this reason no mention is made of perhaps equally important developments to new application fields suchas metal forming, electromagnetics, geomechanics etc.

1. INTRODUCTION

It is now over thirty years since I became involved in “The Finite Element Method” whichduring most of that period dominated my research activity. The invitation to write thisarticle provides me with a most welcome opportunity to record the story of its origins andof its subsequent development, highlighting the important milestones and directions. Clearlythe latter parts are very much a personal view and hence selective. Apologies are given inadvance for omission of those who perhaps may view other steps as more important.

Since my early introduction to the possibilities offered by Numerical Approximationby Sir Richard Southwell viz his Relaxation Methods (1940,46,56) and Allen (1955) myobjective has been always that of providing solutions for otherwise intractible problems ofinterest to applied Science and Engineering. This objective indeed was shared by otherswith similar background and led to the development of the Finite Element Method in thelate fifties and sixties.

This method was only made possible by the advent of the electronic, digital, computerwhich at the time was making its entry into the field of large arithmetic processing. Indeedthe rapid widespread recognition of the methodology of Finite Elements is clearly linkedwith the development of the computer. This led to a rapid development of the methodwhich today, through various commercial and research codes, provides the key for rationaldesign of structures, study of aeronautical fluid dynamics and of electromagnetic devicesneeded by physics.

It is therefore not surprising that much of the development and direction of the Finite Ele-ment Method was provided by applied scientists (engineers) seeking to solve real problems.Though recognising the roots and the mathematical basis of the procedures, such workfrequently omitted the very rigorous proofs of the quality satisfying the pure mathematicians.It was thus of much value to the field that in the seventies more formal, mathematical,approaches were introduced, generally confirming the validity of the previous reasoning butadding a deeper understanding. However, in what follows I shall try to present a view ofthe discovery process and of the motivation which led to our present knowledge.

† This article is being published at the same time in Volume IV of the Handbook of Numerical Analysis, ed.J.L. Lions and P. Ciarlet (Elsevier Scierce).

∗ The author also holds the Unesco Chair of Numerical Methods in Engineering at the Universitat Politecnicade Catalunya, Barcelona, Spain where part of this paper was written.

c©1995 by CIMNE, Barcelona (Spain). ISSN: 1134–3060 Received: 15 November 1994

2 O.C. Zienkiewicz

2. THE ORIGINS

The search for the exact origins of the finite element method would be as fruitless as thesearch for the inventor of the wheel! The roots of the method are deeply embedded bothin the mathematics of continua and in engineering, where assemblies of discrete ‘elements’were, and continue to be, the only practicable way of dealing with complex design systems.

Indeed in the earliest days of science the ‘atomistic’ or ‘discrete’ concepts introducedby Aristotles have governed the thinking and the mathematical concept of a continuumpermitting infinite subdivision was introduced only as a convenient fiction, as late as theend of the sixteenth century by Newton and Leibnitz.

While the concept of a continuum is a useful one and has led to the fuller comprehensionof modern mechanics, the engineer and physicist have learned to understand that it must notbe pushed beyond certain limits. Thus the infinitesimal “dx” is limited in size by the problemat hand, involving for instance atomic dimensions in a study of crystals or dimensions ofseveral metres in a study of rock mass behaviour. The solution of practical problems canthus frequently be approached keeping in mind the “duality” of description in much thesame manner as prevails in physics in the description of light phenomena (the quantum andthe wave nature).

The choice of the solution procedure is, for this reason, frequently governed by mereconvenience. Examples of complex discrete structures being modelled as continua (using aprocess of homogenization) or of continua modelled by a physical discretisation process are,and were for at least a century, a common device. There is here an obvious necessity. Thefinal solution of the problem on the scale considered must be (nearly) identical whicheverapproach is used! Here indeed lies the first important intersection of the engineer’s andmathematicians approaches (though the two are by no means very different species).

As an example we can consider the modelling of an elastic continuum as a uniformstructural bar assembly by Hrenikoff (1941) and McHenry (1943) who in the early fiftieswere able to show the equivalence between this and a continuum approach to plane elasticityindicating convergence of the discretization process (or simply that as the size of discretebars decreased the solution of the continuum problem was approached).

Unfortunately the Hrenikoff–McHenry ‘discretization’ procedure was only available forrectangular mesh assemblies. To overcome this difficulty Turner together with Cloughin the aircraft industry showed that a more direct discretization process was possible byapproximating the behaviour of a continuum using “elements” of arbitrary triangular orrectangular forms. In these, simple strain states were assumed, and using such elementssolution of full scale continuum problems could be again achieved in a ‘convergent’ manner.

Although their work was presented in January 1954 at a meeting of the Institution ofAeronautical Sciences in New York, it was not published until 1956 (Turner et al. 1956).That paper to many is the start of the engineering finite element method although that namewas only first used in 1960 by Clough (1960). The rapid development of the methodologyfrom those early days is of course linked with the meteoric rise of the computer powerpermitting realistic calculations on an unprecedented scale. In the Appendix a verbatimextract form a later paper by Clough (1979) is given where he describes this exciting phaseof the developments.

Much contribution to this early engineering work was doubtless made in the early sixtiesby a systematic organisation of the computations using a matrix methodology coupled withenergy methods in structural analysis by Falkenheimer (1951), Langefors (1952) and Argyris(1955).

Indeed the last author shows in the context of aircraft skin models that a triangular panelbehaviour may well be approximated by using energy minimisation procedures – which wereshown later to provide a firm basis for finite element formulation.

Origins, Milestones and Directions of the Finite Element Method 3

Priorities are of course, as shown above, difficult to establish due to much independentwork containing the germs of the basic ideas. Indeed some of these appeared in China quiteearly, (Feng Kang 1965) and were totally unknown in the west.

3. THE “VARIATIONAL” APPROACHES VIA EXTREMUM PRINCI-PLES

Although the earliest finite element forms were derived by a direct consideration of inter–element “forces” equivalent to the internal stresses in the continuum and the “displacements”of connecting nodes, it soon became evident that a more general approach could be obtainedby seeking the minimum of the total potential energy within the constraint of an assumeddisplacement field. One of the first to adopt this approach was Szmelter (1958) but othersindependently arrived at the same conclusions.

Thus if the strain field in an elastic continuum were defined by a suitable operator Sacting on the displacements u as

εεεεεεεεεεεεεε = Su (3.1)

with the corresponding stresses given as

σσσσσσσσσσσσσσ = Dεεεεεεεεεεεεεε (3.2)

where D was a matrix of elastic constants, then the finite element solution sought could beobtained by the minimisation of the potential energy defined as

Π =12

∫Ω

εεεεεεεεεεεεεεTDεεεεεεεεεεεεεεdΩ−∫

Γt

tTudΓ−∫

Ω

bTudΩ (3.3)

in which the displacement field is approximated as

uh = Nu (3.4)

In above u are ‘nodal’ values of u (or other parameters) satisfying prescribed displacementson the boundary Γu, t are the prescribed tractions on the boundary Γt and b are the bodyforces. The functions N are given in terms of the coordinates and are variously known as‘shape’ or ‘basis’ functions.

Clearly such a minimisation will lead to the final, ‘discrete’, algebraic equations of theform

Ku = f (3.5)

whereK =

∑Ke Ke =

∫Ωe

(SN)T D (SN) dΩ

f =∑fe fe =

∫Ωe

NTbdΩ+∫

Γet

NT tdΓ(3.6)

and the simple, additive, rule of structural assembly common to standard engineeringproblems is preserved.

With Ωe (and Γe) corresponding to “elements” into which the whole continuum problemis divided, i.e.

ΩUΩe ΓeUΓet (3.7)

4 O.C. Zienkiewicz

the Eqs. (3.6) provide a convenient means of generating “element” stiffness coefficients andforces providing the approximation shape functions of Eq. (3.4) are defined on a local basis.

Clearly such a derivation of the finite element procedure showed it to be but a particularcase of the approaches introduced much earlier by Rayleigh (1870) and Ritz (1909) whichwere well known and used in engineering circles. Indeed the main difference appeears inthe computational advantage of using a local definition of the shape functions N yieldinga banded structure of the assembled stiffness matrix K of equation (3.5) and in preservingthe local assembly structure of matrix equations.

Further, the definition of the process implies immediately certain conditions which aresufficient (but not always necessary) for the convergence of the approximation. These givenfirst in the early 60’s are still valid today and are:

(i) that the displacement u is so defined by the shape functions (Eq. (3.4)) that nodiscontinuities (leading to infinite strains) develop

(ii) that, if polynomial shape functions are used, the terms leading to constant strainvalues in an element can be given any arbitrary values.

The full details of such finite element requirements were presented, together with manycorrolaries at a meeting held in Swansea in January 1964 and published formally as textin 1965 (Zienkiewicz and Holister 1965) where many seminal papers appeared, e.g. Clough(1965), Fraeijs de Veubeke (1965), Zienkiewicz (1965), Massonnet (1965). Of particularimportance is the contribution of de Veubeke who was the first to realise that other varia-tional, extremum, principles can be used in addition to the principle of minimum potentialenergy to derive approximation to problems of structural, elastic continua. Here, Fraeijsde Veubeke introduces for the first time the concept of ‘equilibrium’ finite elements bas-ing these on the maximisation of complementary potential energy. Such new elements arecapable of ensuring equilibrium of stresses and thus provide an upper bound on the strainenergy of the approximate solution while the potential energy form provides the appropriateminimum of this. Although the direct formulation of such ‘equilibrating’ elements presentsmany difficulties of stability (which were overcome much later by the introduction of thestress function approximation by Fraeijs de Veubeke and Zienkiewicz 1967) the provision ofenergy bounds was important in bracketing the error of the aproximation.

Even more important is the contribution that Fraeijs de Veubeke introduces in the so–called mixed formulations based on the Reissener variational principle, Reissner (1950),(later referred to as the Hellinger–Reissner principle by Washizu 1975). This work led tomany later publications by others and introduced the so called principle of limitation whichis of fundamental importance in judging the possible performance of the mixed methods†.This principle originally limited to elastic problems can be paraphrased as (viz Zienkiewiczand Taylor 1989) the observation that

“if mixed and irreductible approximation to the same problem can result in the sameapproximation they will do so – and both will yield idential results”.Thus for instance as stated by de Veubeke “it is useless to look for a better solution (of a

displacement form) by injecting additional degrees of freedom for the stresses...”. This fact,not realised widely, led to many false computational claims for mixed approaches.

The fact that finite element forms can be derived for any variational principle and notonly those referring to solid mechanics led to the extension of the finite element methodbeyond the domain of structural mechanics. The first applications of this extension have

† The subdivision of finite element approximations into mixed and irreducible follows the nomenclatureof Zienkiewicz and Taylor (1989). The name is associated with the differential equations from whichthe approximation starts. In the irreducible form the dependent functions are reduced to the essentialminimum (if necessary using penalty functions). The remainder is of course mixed.


been made in the context of the variational principle corresponding to the quasi–harmonicPoissons equations by Zienkiewicz and Cheung (1965a) and Zienkiewicz et al. (1966,1967)allowing problems governed by such equations in both solid, fluid mechanics, and electro–magnetics to be solved.

Indeed at that time the earlier suggestions of Courant for similar solution of the Laplaceequations were discovered. It is interesting to note that the classic paper of Courant (1943)is indeed not the first one of his contributions in which the use of triangular finite elements issuggested. These are mentioned by him as a possibility as early as 1923 (though he appearsat that time not to anticipate the importance of the discovery!).

4. SOME EARLY APPLICATIONS OF THE FINITE ELEMENTMETHOD AND ALTERNATIVES

At this stage it is perhaps of interest to show some very early examples of finite elementapplication which illustrate the versatility of the process. Here I have chosen a CivilEngineering problem – that of a dam solved circa 1963 and I believe the first real applicationof the method to design. The problem is taken from my first text on the Finite ElementMethod in 1967 in which I tried to assemble the state of the knowledge available at the time(Zienkiewicz and Cheung 1967).

Incidentally the various editions of this book (Zienkiewicz 1971,77 and Zienkiewicz andTaylor 1989,91) record the growth of the subject in which today hundreds of texts exist.However till 1971 the above texts were the sole source – fortunately for my ego!

In Figure 1 the dam, later constructed at Clywedog, Wales, is illustrated together withthe subdivision into linear triangular elements. The mesh subdivision shows how easily theFinite Element Method allows a graded mesh to be achieved solving simultaneously theproblem of foundation and the dam with largely different finite element sizes.

Figure 2 shows the stress distribution achieved and the natural treatment of non–orthogonal boundaries.

Up to the time of this example the only alternative to the solution of this realistic andimportant problem was by the use of finite differences. In 1910 L.F. Richardson presentedthe first analysis of the Aswan Dam using a mesh with some 250 nodes and solving theequations by a laborious Gauss–Seidel process. Figure 3 shows this mesh composed ofrectangles and in Figure 4 a more elaborate mesh with some 900 nodes is presented fromZienkiewicz (1947) for a somewhat similar structure. Here solution was laboriously obtainedusing the Southwell relaxation procedure with some results shown in Figure 5.

Such finite difference procedures illustrate the difficulty of grading the mesh size withrectangular meshes – and the difficulties of dealing with arbitrary boundaries.

These examples show indirectly why the finite element procedures had to wait for thedigital computer!! In the original Richardson solution and in the later relaxation solutionthe tedious calculation of residuals could only be accomplished by the device of memorisingsimple patterns of the finite difference “stencil” and using this for the residual distribution.

On the other hand, the arbitrary (though banded) structure of the finite element equa-tions preserves no simple pattern and needs to use the facilities of modern electronic com-puters to “memorise” the matrices and solve them. Little wonder that the formulationinherent in the early work of Courant was not considered by him as a practical possibility.

Another example from this era illustrated in Figure 6 is one relating to fluid mechan-ics and shows a complex problem of fluid flow in a hypothetical, anisotropic, foundation(Zienkiewicz et al. 1967). Such complex problems presented serious difficulties to the thenconventional finite difference processes – especially if abrupt material changes occured.

Clearly by the mid–sixties the start of the finite element method was firmly established.

6 O.C. Zienkiewicz

Figure 1 Finite element analysis of the Clywedog buttress dam (1963). Plane stress andplane strain assumptions with linear elasticity. Note ease of simultaneous modellingof small and large detail


8 O.C. Zienkiewicz

Figure 3 The first ‘practical’ numerical analysis. Assuan Dam by Richardson (1910) usinga stress function formulation and finite differences with 250 nodes, (a) mesh (b)vertical stresses at base

Figure 4 A relaxation solution of finite difference equations for a gravity dam by Zienkiewicz(1947) using circa 900 nodes


Figure 5 Distribution of vertical (a) and shear (b) stresses in the problem of Figure 4

Figure 6 Finite elements used for solution of seepage flow in a highly inhomogenous andanisotropic foundation, Zienkiewicz et al. (1967)

10 O.C. Zienkiewicz

5. VIRTUAL WORK AND WEIGHTED RESIDUAL APPROACHES.GENERALISED FEM – AND OTHER APPROXIMATIONS

An alternative derivation of the finite element approximation to that of using potentialenergy minimisation was developed in the early sixties (Zienkiewicz 1965, Zienkiewicz andCheung 1964). This was based on the use of the well known virtual work principle. Thisprinciple simply states that if an equilibrating system of stresses σσσσσσσσσσσσσσ, boundary traction t andbody forces b is subject to virtual displacement systemsW, then the internal and externalwork is equal.

Thus if the strains corresponding to the virtual displacementW are given by expression(3.1), i.e.

εεεεεεεεεεεεεεW = SW (5.1)

then the virtual work equality requires that∫Ω

(SW)T σσσσσσσσσσσσσσdΩ =∫

Γt

tTWdΓ +∫

Ω

bTWdΩ (5.2)

providing the systemW satisfies

W = 0 on Γu (5.3)

Of course if the virtual displacement system is made such that

W = N (5.4)

and the stress system σσσσσσσσσσσσσσ is that due to a displacement approximation of Eq. (3.4)

uh = Nu (5.5)

then, using definitions (3.1) and (3.2) the virtual work equation (5.2) will yield again analgebraic equation set identical to Eq. (3.5), i.e.

Ku = f (5.6)

with the same definition of the stiffness matrix and “forces” as given in Eq. (3.6). Now, ofcourse, if the virtual work principle is applied to a single element, equivalent, inter–elementforces can be obtained and the physical analogy with the early approximation is availableas shown in Zienkiewicz (1967). However more important corollaries follow.

(i) Virtual displacements of a form different toN can be used if desired (though of coursethe satisfaction of the condition (5.4) is optimal from the point of view of an energyminimisation)

(ii) It is possible to use non–symmetric forms of the D matrix such as arise for instancein non–associative plasticity, viz Zienkiewicz et al. (1969), Nayak and Zienkiewicz(1972), for which the potential energy can not be defined.

However, there is more. The mathematician will recognise in Eq. (5.2) the standardbilinear form used later in their approach to be finite element theory – and written for atypical scalar problem as

a (w, u) + b(w) = 0 (5.7)

Others will note that on using integration by parts the equation (5.2) can be written as∫

Ω

WT(ST σσσσσσσσσσσσσσ + b

)dΩ −

∫Γt

WT (nσσσσσσσσσσσσσσ − t) dΓ = 0 (5.8)


with σσσσσσσσσσσσσσ =DSu.This is, of course, a weighted residual form of the equilibrium equation

STσσσσσσσσσσσσσσ + b = 0 in Ω (5.9)

together with the boundary conditions

nσσσσσσσσσσσσσσ − t = 0 on Γt (5.10)

if equation (3.5) is used for approximating the unknown u.The name of ‘weighted residual’ appears to be introduced into literature of approximate

numerical solutions by Crandall (1956) as a general procedure recognising that Galerkin(1915) was the first to use it formally with the special case of weighting of Eq. (5.4) i.e.

W = N (5.11)

However point collocation, method of moments etc. present other possibilities. Today itappears customary to refer to the case of Eq. (5.11) as the Galerkin–Bubnov method whileall other possibilities are simply lumped as Petrov–Galerkin methods for which

W = N (5.12)

The original references to this classification are not clear – but ‘what’s in a name’?! Theimportant matter is that by the late 60’s the recognition of the relation between weightedresidual forms and finite element method was well established opening the doors to thesolution of most problems cast in terms of partial (ordinary) differential governing equationand boundary conditions.

Clearly it now became possible to apply the Galerkin (Bubnow) process to non–selfadjoint equation systems such as arise in fluid mechanics, viz Oden and Samogyi (1968),Oden and Wellford (1972), Oden (1973), Zienkiewicz and Taylor (1973) and finally Taylorand Hood (1973) who were the first to solve successfully the Navier-Stokes problem.

Equally clearly it became possible to interpret most other approximation procedures suchas finite differences, finite volumes, boundary methods etc. as variants of the general processimplied in Eq. (5.8). Let us write for such a process the governing equation as

L(u) = 0 in Ω (5.13)

and the boundary conditions (not automatically satisfied by the approximation) as

B(u) = 0 on Γ (5.14)

Using the approximation

u ≈ u = Nu =∑Niui i = 1, .., u (5.15)

the discrete approximation is formed as∫

Ω

WTL(u)dΩ+∫

Γ

WTB(u)dΓ = 0 (5.16)

For any such approximations the summation rule of the finite element method obviouslyapplies though of course the banded, sparse, matrix structure will arise only if a locallybased shape functions N are used.

12 O.C. Zienkiewicz

Figure 7 The interpretation of finite difference and finite volume approximations as spe-cial cases of the generalized finite method. Note shape functions N i

j depend onsubdomain i considered

With such a general statement it is easy to see that for instance we can now interpretfinite differences as a point collocation process in which

Wi = δi Dirac (5.17)

In Fig. 7 we show for instance an approximation to a one dimensional problem governed by

d2u

dx2+ q = 0 (5.18)

Here the shape functions Ni are simple parabolas determined by ui−1, ui and uu+1, whichgive a discontinuous approximation shown, violating the previously stated continuity require-ments. However the contribution of the discontinuities in the weighted form disappears andthe reader can verify that application of the Eq. (5.16) will result simply in the well knowndifference form

u1+1 − 2ui + ui−1 + h2qi = 0 (5.19)


In a similar manner the so called finite volume approximation can be interpreted as aparticular kind of a finite element. This is in fact a subdomain collocation (to use theclassification of Crandall 1955) with

Wi = 1 (5.20)

over a ‘volume’ hi shown again in Fig. 7. Now the approximation equation is again ofsimilar form to (5.17) but with the last term modified

ui+1 − 2ui + ui−1 + h2qi = 0 (5.21)

where

qi =1h

∫ 1+ 12

i− 12

qdx (5.22)

Above approximation can of course be contrasted with that obtained using simple C0

continuous linear shape functions Hhi . Here again the form is identical to that of equation

(5.21) but with qi now defined as

qi =1hi

∫ i+1

i−1

Niqdx (5.23)

Such a standard (Galerkin) finite element form is in fact optimal, delivering exact nodalvalues in this case as shown by Tong (1974). However the finite volume process has a usefulphysical interpretation and has gained widespread use in fluid mechanics where the problemsare generally non self–adjoint!. The above interpretation puts that approximation of coursein a form suitable for finite element type of computation viz Zienkiewicz and Onate et al.(1994).

Boundary methods (Trefftz 1926) are yet another alternative approximate for which manyadvantages are sometimes claimed. These procedures choose the basis (shape) functions Nin such a manner that the differential equations are satisfied identically throughout thedomain Ω by these. Now of course, for linear problems, Eq. (5.13) is

Lu ≡ 0 (5.24)

and the approximation equations (5.16) become simply∫

Γ

WTB(u)dΓ = 0 (5.25)

thus reducing the dimensionality of the problem. The possibility of deriving banded matrixsystems is of course not available but the total number of unknown parameters can well bereduced for a given accuracy. The use of such boundary methods has been extensive – andit is interesting that their proponents today refer to them as the boundary element method,(e.g., Beer and Watson 1992).

In above I attempt to show that all well known approximation methods have very similarorigins and a basic form of the finite element procedure. Taking a chauvinistic viewpointwe could thus embrace all in the name of generalised finite element procedure though thealternative of generalised Galerkin method would be equally applicable, (Fletcher 1984).The name is not important! What is, is the recognition of the educational aspect ofsimilarity of the processes and more importantly the realisation that practical advantagemay occasionally accrue by utilising these in combined computer codes.

A fairly obvious combination is the use of boundary procedures for treatment of semiinfinite domains of linear kind together with non linear finite elements of a ‘standard’ kind

14 O.C. Zienkiewicz

in other subregions. Suggestions for such a marriage a la mode made first by the authorand his colleagues (Zienkiewicz et al. 1977) has found many applications.

While the so–called spectral methods have not been specifically examined here these obvi-ously are another general application of the basic formulation of equation (5.16). These areof course mirrored in the use of very high order polynomial based functions in the standardfinite element approaches to which we shall refer later. In this context it is important torealise that “nodal values” are not the essential requirement of any finite element approxi-mation. From the earliest days ‘nodeless’ variables have been used in standard codes withsuch variables generally confined still to single elements. A ‘landmark’ in the use of suchfinite element interpolation was the introduction of the so–called hierarchical interpolation(Zienkiewicz, Irons et al. 1970).

In the hierarchic form the approximation in each element is still local and of the form(for a scalar variable)

u =Nu =n∑

i=1

Niui (5.26)

but ui parameters are no longer nodal values of u and the usual condition that∑

Ni = 1 (5.27)

does not apply. Moreover, the expression is so constructed that the shape (basis) functionsNi are independent of the number of parameters chosen for the approximation (i.e. theexpression of equation (5.26) is in a series form).

This has the advantage that not only the matrices of lower orders do not change aselements are refined by adding higher order polynomials (p–refinement) –but also the equa-tion conditioning is vastly improved. The hierarchical form is today widely adopted– andparticularly useful in adaptive refinement, (Zienkiewicz et al. 1983, Peano 1976).

While the lowest hierarchic form still preserves element variable ‘banding’ of the matrices– an alternative in which global functions are superposed on the local ones is possible. This,first introduced by Mote (1974), allows exact solutions to be used as an addition to thebasis functions with the true finite element refinement being a perturbation in this solutionand thus requiring smaller accuracy (and cost). The solution matrix no longer banded stillremains sparse and the use of the type of hierarchical form deserves to be further explored.

6. NONCONFORMING APPROXIMATION AND THE PATCH TESTAS NECESSARY AND SUFFICIENT CONDITION OF F.E.M. CON-VERGENCE

6.1 Plate Bending – Conforming and Nonconforming Variants

We have already stated that one of the early requirements of the finite element approxi-mation was the choice of shape functions which did no lead to infinite strains on elementinterfaces and which therefore preserved a necessary degree of continuity. Satisfaction ofthis requirement guarantees convergence and in the case of energy minimisation processes,an energy bound.

While in the case of simple elasticity and other self adjoint problems governed by secondorder equations such continuity is easy to satisfy, this requirement in case of thin platebending using the Kirchhoff–Germain postulates, in which fourth order equations arise ismuch more difficult to achieve (name of Germain is added here to honour the French ladywho, many years before Kirchhoff, formulated a nearly correct plate theory). Now C1

continuity has to be introduced and the continuity of both the function and of its normal


gradient assured. This requirement was well known in the early days of the sixties decade,and the development of plate bending elements followed two lines.

(i) the search for elements satisfying rigorously the continuityor

(ii) designing elements where continuity of slopes was imposed only at the nodes withthe hope that the interface work would tend to zero as the refinement proceeded, andthat convergence would be still achieved.

The first approach was, as mentioned before, difficult. The problem lies in the factthat if only the value of the displacement function w and of its slopes (∇∇∇∇∇∇∇∇∇∇∇∇∇∇w) were used asparameters at the nodes it is impossible to determine unique polynomial shape functions inthe element. The proof of this fact was established by Irons and Draper (1965) (viz p. 12–13of Zienkiewicz and Taylor, Vol. II 1991) who show that for continuous polynomial shapefunctions it is necessary to specify some of the second displacement derivative. In generalspecification of such nodally continuous second derivatives, though used successfully by someauthors (viz here the simultaneous development of quintic triangular elements with 21 d.o.f.and their reduction to an 18 d.o.f. form by Argyris et al. 1968, Cowper et al. 1968, Bosshard1968, Visser 1968, Bell 1969 and Irons 1969), does not permit abrupt thickness changes inthe plate (or moment discontinuity) such as may occur at irregularities and should in ouropinion not be used as it is a case of excessive continuity imposition. The single exceptionto this is the continuity of the cross derivative, ∂2w/∂x∂y, on an orthogonal mesh – andan element of rectangular shape using this derivative as a nodal variable was presented byBogner et al. (1965) at the first Wright–Patterson Conference in Dayton, Ohio.

At the same meeting, however, an alternative derivation of fully conforming triangleswas presented by Clough and Tocher (1965) and by the authors group – Bazeley et al.(1965). In both of these elements the shape functions are so formed that non unique secondderivatives arise at the nodes. In the first element the shape functions are derived by usingthree separate polynomials in different parts of the triangular element. In the second a nonpolynomial expression is used to achieve the desired end.

With much exercise of ingenuity the various ‘conforming’ elements described aboveproved, as expected, convergent but unfortunately were poor performers.

Much more fortune was however experienced by those who flouted the rules and ignoredthe interelement continuity requirements. Here, the first element, of rectangular shape wasproduced independently by Adini and Clough (1961) in an unpublished report and indepen-dently by Zienkiewicz and Cheung (1964). This element performed well and experimentsshowed it to be convergent.

Similar nonconforming triangular elements were described in Bazeley et al. (1965) andmuch later modified by Specht (1988). In Figure 8 we show a typical example comparingthe convergence of the conforming and non–conforming triangular elements discussed above.However, the assurance of convergence was not automatic and needed investigation. Theanswer to this was proposed by Irons and published with the author in Bazeley et al. (1965).This answer was the Patch Test.

6.2 The Patch Test

The original idea of the patch test stemmed from very physical, intuitive, considerations. Itsimply stated that a “patch” or any assembly of elements of the form shown in Figure 9 foran elastic continuum must be able to reproduce exactly all constant stress and strain stateswhen subject to appropriate (linear) variation of displacements.

The patch test was first applied to the non conforming plate bending elements in whichconstant curvature/moment states were modelled. It was easily shown that the non con-forming rectangle was fully convergent but the triangle derived by Bazeley et al. (1965)

16 O.C. Zienkiewicz

Figure 8 Performance comparison of some conforming and non–conforming thin plate bend-ing triangles with 9 D.O.F. Error in central displacement for a uniformly loaded,simply supported square plate

Figure 9 Patch test of (a) multiple element and (b) single element type. Boundary conditionsand internal forces imposed on the patch correspond (at least) to any arbitrarylowest order solution necessary for convergence (e.g. constant stress conditions).Higher order solution can be imposed to check convergence order

was only convergent for regular meshes formed by three sets of intersecting parallel lines –(though its performance was roughly acceptable for other meshes!).


Though the application of the patch test was first envisaged for testing non conformingelements its application, by the original arguments, was universal. It soon became obviousthat it should be used for all elements, even those based on fully convergent assumptions,as a test of the correctness of programming (viz Irons and Razzaque 1972, de Veubeke 1974and Oliveira 1977). For such purposes, it even got the blessing of some mathematicians, (vizStrang and Fix 1972) as a necessary condition for convergence and in due course became astandard test required for the verification of all codes.

However, in the form originally given it was only a necessary condition of convergencebut clearly not sufficient. Considerable confusion was engendered in more recent dayswhen a mathematician denied even its necessity (Stummel 1980) and some updating of itsvalidity was needed. This provided some of us with a stimulus for further research. In 1984the author with his colleages published a paper which extended the test to provide bothnecessary and sufficient conditions for convergence of all finite element forms (viz Taylor etal. 1984).

The essence of the extension was that of testing the stability of patches by ensuring thatno zero eigenvalues (singularity) were present under any acceptable boundary condition ona patch. Indeed this part of the test extended the usual structural ideas in which non–singularity of individual elements was generally imposed, but now the question was posedin a more mathematical form valid for all applications.

A further extension was that permitting the assessment of the convergence order by theimposition of higher order test solutions.

A very useful corrolary of the test was that made later for mixed formulations whereit provided a guide for the “design” and testing of various possible approximations,(Zienkiewicz et al. 1986). Here it gave a simpler alternative to the well known Babuska(1971,73)–Brezzi (1974) conditions which are recognised to be necessary and sufficient forconvergence. While generally the patch test has to be applied numerically with randomlyselected low order exact solutions, in the case of mixed forms which in two field situationsgive frequently rise to algebraic equations of the type containing a zero on the diagonal

[A QQT 0

]up

=

f1f2

(6.1)

it is possible to determine the existence of singularities ‘a priori’. Here u and p are thevariables describing the two fields and some indication of behaviour can be obtained byinspection. It is simple to show that to obtain the solution and avoid zero eigenvalues it isnecessary that

nu ≥ np (6.2)

where nu and np stand for the number of variables in the u and p sets. The verificationthat this condition is satisfied for all patches of elements (from a single element onwards)helps to eliminate most of the elements which were known to be either unusable or non–robust by virtue of failing the Babuska–Brezzi conditions. Further it indicates immediatelyas “possible” candidates other elements which have hitherto been derived by more elaboratemeans. (However, it must be stressed that the simple “count” provides a necesary conditiononly, and that in general further numerical tests on zero eigenvalues are needed).

If u and p of Eq. (6.1) stand for instance for displacements and pressures in typicalincompressible elasticity equations (Herrmann 1964) then it is clear from Figure 10a thatthe triangular element with linear continuous interpolation of u and p fails in all patchassemblies on which u is prescribed on boundaries. (Note that one value of p in any patchneeds to be always prescribed for physical reasons).

In Figure 10b a simple ‘bubble’ function with u displacement parameters has been addedand inmediately the count is satisfied for all patches. This indeed provides now a convergent

18 O.C. Zienkiewicz

Figure 10 Patch test count for incompressible elasticity (or Stokes flow) with C0 continuousinterpolation of displacements u and pressures p. (a) Simple linear interpolationof both variables fails count on all patches. (b) Addition of bubble function ondisplacement u ensures count is satisfed on all patches


element (as can be verified by eigenvalue tests), but of course merely reproduces a well knownelement of Fortin (1982). It is of course difficult to add to the study of such a well knownproblem (and the count here is strictly educational) but in the case of some later elements forplate bending problems where theree independent fields interact, the patch count has led tonovel forms giving amongst others, the first fully robust element based on Reissner–Mindlinassumptions (Zienkiewicz and Lefebvre 1987).

It should be mentioned here that the robustness of mixed forms of limiting kind such asthose given by Eq. (6.1) is of importance even if a non zero, but small, diagonal term existspermitting the elimination of the variables p. Now the element is usually identical to theirreducible form – and if robust in the limit will behave well generally. Thus for instance itis well known that the simple linear triangle behaves very erratically as incompressibility isapproached.

The patch test so far discussed is an essentially numerical verification procedure. How-ever it is often possible to derive from its requirements analytical conditions required forconvergence. Here a very useful criterion for the design of incompatible elements has beenpostulated by Taylor et al. (1986) which can lead to the development of incompatible ele-ment displacement functions fully satisfying convergence criteria. Specht (1988) develops onthis basis an incompatible element superceding that of Bazeley et al. (1965). This elementis convergent for all types of meshes an performs well as was shown in Figure 8.

Indeed if we accept that the satisfaction of the patch test is the only convergence criterionthis could be used as the sole means of deriving element stiffness matrices without specifyingdisplacement shape functions, or indeed the background theory, directly. Such elements havebeen successfully derived by Bergan et al. (1977,84).

Clearly, the establishment of the patch test is a major landmark in providing a basis forfinite elements.

6.3 “Diffuse” Element Approximation

In the previous sections we discussed the use and limitations of ‘non conforming’, discontin-uous, shape functions and showed under what conditions such conformity was restored inthe limit h → 0. However there is a very simple way of ensuring such limiting conformity byuse of overlapping shape functions. In Figure 7 we showed the genesis of such overlappingfunctions for the purpose of using a finite difference (or finite volume) approximations. Inthat figure quadratics generated by a nodal overlap were used and it is physically evidentthat in the limit the approximation will ensure the continuity of both the function u and ofthe first derivative. A formal proof of above is available in Neyroles et al. (1991, 92).

The idea was first put to practical use by Nay and Utku (1973) who derived shapefunctions for plate bending elements by using quadratic polynomial approximations obtainedby least square fitting of a number of nodes in the vicinity of a particular node consideredin the manner shown in Figure 11. Here the seven nodes together with the node i are usedto define the quadratic (six parameters) for the displacement variable and a tributary areaΩi defines the element.

Clearly now we have elements with external nodes but the identical formulation to thatpreviously used can be followed, now indeed achieving limiting compatibility without useof gradients as nodal variables. This indeed has been of some recent practical interest. Ofcourse special treatment of boundaries is now necessary, which we shall not discuss here,and it is of interest to note that approximations are convergent and of a form similar to thegeneralised finite difference approximation. However standard finite element assembly etc.still apply.

The procedure was extended and widely used by Pavlin and Perrone (1979) and Liszkaand Orkisz (1980) and very recently adapted by Neyroles et al. (1991,92) for precisemodelling without defining elements as obviously only an integration field needs to be

20 O.C. Zienkiewicz

Figure 11 Diffuse approximation formulations of different types of local quadratic interpo-lation. (a) Based on underlying mesh structure (Ut Ku 1972, Liszka and Orkisz1980). (b) Based on nearest seven nodes to an integration module (Neyroles et al.1992)

defined. The ideas are basically simple and it is surprising that such processes had towait so long to see practical use. It may well be that in the future their merits will becomemore evident.

Of course there are many possible ways of achieving the interpolation needed or creatingthe tributary element volumes as shown in Figure 11 and the interested reader should consultthe literature. Here of interest is the recent work by Belytschko et al. (1994). Indeed it canbe observed that in such formulation similarites with the so called “wavelet” forms existsas shown by Liu et al. (1993).

Clearly, the patch test again has a dominant role for all these problems to assureconvergence.

7. HIGHER ORDER ELEMENTS AND ISOPARAMETERIC MAP-PING. THREE DIMENSIONAL ANALYSIS, PLATES, SHELLS ANDREDUCED INTEGRATION

7.1 The Need Governs Development – Isoparametric Mapping

In the early sixties only linear elements were generally used and the solution of realistic prob-lems was possible (with some difficulty) for relatively small 2D problems on the computersavailable in that era. However the need for solving three dimensional problems was pressingand for these the obvious linear, tetrahedral element rapidly overstretched the capability of


computation (Gallagher et al. 1962, Melosh 1963). It was clear that higher order elementswould berequired to achieve the accuracy desired with a reasonable number of degrees offreedom. Here other problems were immediatly encountered. If for instance the quadratictriangle originally suggested by de Veubeke (1965) were extended to its equivalent threedimensional, tetrahedral, form then purely geometrical difficulties would be encounteredin dealing with complex boundary shapes as obviously a smaller number of such elementswould now be used. Clearly some form of co–ordinate mapping would be necesary to dealwith this situation.

It was my good to encounter at the time Bruce Irons who currently worked at Rolls Royce,and whom I managed to persuade to join the academia where he would work with peoplewho had more interest in reading his reports and acting upon them. It was clear that hehad given the problem much thought and that he had already formulated a possible way ofmapping using the essential element shape functions with higher order polynomials. Furtherhe suggested that the complex transformation integrals could be evaluated numerically. Thiswork published in 1966 by Irons generalised an original sugestion of Taig (1961) who firstsucceeded in arbitrary mapping a rectangle into a general quadrilateral form. Together wesucceeded in elaborating and applying the idea to both two and three dimensional problems(viz Ergatoudis et al. 1968 a,b, Zienkiewicz et al. 1969).

Figure 12 Analysis of a nuclear reactor using linear tetrahedra (10,000 DOF) and quadratichexahedra (2,000 DOF)

One of the first practical problem at that time was the analysis of a nuclear pressure vesselshown in Figure 12 where, by use of quadratic brick elements, the problem was adequatelysolved using only 2121 DOF in 1966. The same figure shows a similar analysis published by

22 O.C. Zienkiewicz

Rashid and Rockhauser (1968), using some 18000 DOF with linear tetrahedra. Obviouslyboth analysis were done almost simultaneously – but commercial “secrecy” prevented earliercomparison of results!.

At the same time the writer was engaged in a civil engineering project attempting todevise a method for computing stresses in arch dams. Figure 13 shows some typical results ofthis study using full three dimensional analysis with quadratic and cubic elements. Clearlythe gain of accuracy (per degree of freedom) between quadratic and linear elements wasdramatic though at the time we could not appreciate this fully as the accuracy of differentsolutions could not be fully compared. However, in Figure 14 we show (using adaptivityconcepts) an identical problem solved with the same accuracy by linear and quadraticelements and the reduction of degrees of freedom is very substantial, (Zienkiewicz and Zhu1987). This of course does not necessarily mean that computational cost is proportionallyreduced though with conventional solvers this indeed can be the case. However I wouldclassify the step of isoparametric mapping as one of the major landmarks which, by permitingthe practical use of higher order elements, had a very substantial effect on the finite elementscene. Since the early days described above many other alternative mapping processes havebeen introduced (viz Zienkiewicz and Taylor 1989) but none have become so universallypopular.

7.2 Physics Govern Theory – the Essential Development for Plates and Shells

With the easy and readily accessible form of isoparametric mapping in the late sixties, itappeared to the author that perhaps the difficulties initially associated with plates andshells which were inherent in the Kirchhoff–Germain, “thin plate”, assumption could beovercome by simply treating such structure using thin isoparametric elements of the three–dimensional, solid kind. It was obvious that here a neglect of the stresses in the transversedirection should be made and that linear variation of other stresses in that direction wouldbe sufficient to model the relatively thin behaviour of plates and shells. Thus the ideaillustrated in Figure 15 was introduced almost simultaneously with full three dimensionalanalysis. Here plates and shells could be treated by identical processes already developedfor 3D analysis.

The first work published on this by Ahmad et al. (1968, 1970) was a little disappoint-ing. It was found that the performance of the new approach was good only when fairlythick elements were used and deteriorated rapidly as this thickness was reduced (or as theKirchhoff–Germain assumptions were enforced). However as the direct approach obviatedcompletely the complex plate and shell theory – and did not require the enforcement ofslope continuity, methods of improving the poor performance had to be urgently found.

Here intuition and some rather heuristic arguments pointed to an answer which, at leastin part, was provided by using reduced (or selectively reduced) integration (Zienkiewicz etal. 1971, Pawsey and Clough 1971). It turned out that, simply by using a certain lowerintegration answers could be dramatically improved both for thin and thick plate and shellforms. From this day the formulation based on the three dimensional degeneration (oralternatively by re–introducing the Reissner 1945 and Mindlin 1951 assumptions) becamethe favourite approach to this class of problems with many high order elements being used,(viz Zienkiewicz and Taylor 1991).

From the practical point of view the problem was thus nearly solved with the variouselements performing reasonably in most situations though occasional malfunctions occurred.From the theoretical aspect however a difficulty remained; Why should a mere reduction ofthe integration labour result in an improvement? (this surely went against the “protestantwork ethic” as said by some at the time!).

The answers to this problem took a long time in coming. First important step was heremade by Malcus and Hughes (1978) who showed that reduced integration was (in many


Figure 13 A test analysis of an arch dam (1968). (a) And (b) two meshes for quadraticelements. (c) and (d) two meshes for cubic elements (e) Results giving deflectionson centre–line

24 O.C. Zienkiewicz

Figure 14 Performance of linear (a) and quadratic (b) triangular elements in an elastic solu-tion. Adaptive refinement of one stage is followed aiming at 5 % error in energynorm error. η – energy norm error; θ – effectivity of error estimator


Figure 15 Modelling of plates and shells by using thin, isoparametric element forms

cases) equivalent to the use of a mixed formulation in which, in addition to displacementinterpolation, the shear forces were independently interpolated from ‘nodes’ associated withthe integration points.

The second important step was, we believe, the introduction of the requirements of thepatch test which led to the development of the first two robust elements which could beguaranteed to work in all conditions, (Zienkiewicz and Lefebvre 1987 and Arnold and Falk1987). Other approaches were at the same time becoming popular and again could bejustified by the same arguments (Dvorkin and Bathe 1984, Hinton and Huang 1986).

Now at last the problem could be considered solved though there are many corrolariesyet subject to exploration as discussed in Zienkiewicz and Taylor (1991).

8. ADAPTIVITY AND ERROR ESTIMATION

In the preceding section I have dwelt at some length on development and understandingof matters initially introduced by the engineers rather than mathematicians (though thedifference of these two ‘species’ is by no means clear in the individuals – though perhapsis more evident in the language!). The serious problem of estimating errors of the finiteelement discretisation economically was however first addressed by mathematicians. Herethe work of Babuska and Rheinboldt (1978,79) was the major landmark which not onlyshowed the possibilities of economic error estimation but indicated how, by successive,subdivion of meshes a desired accuracy of the numerical solution could be reached. Thiswork was obviously important as the only practical procedures previously available to judgethe accuracy of the solution were.

1. Comparison with, occasionally available, exact solutions2. Full, uniform subdivision of the mesh and use of an extrapolation – which of course was

too costly for the majority of problems

26 O.C. Zienkiewicz

To facilitate the knowledge transfer Babuska and myself ‘teamed’ up to “translate” anddevelop the procedures for a wider audience. This resulted in a series of papers, (Kelly et al.1983) and an international meeting on the subject, (Babuska et al. 1986). However muchneeded to be done and the subject became a largely popular area of research in which theprocesses of h–refinement (element size adjustment), p–refinement (uniform or non uniformincrease of polynomial order) or even h–p combinations were investigated.

Nevertheless practical use of adaptive procedures remained quite small. What wasmissing, especially in the context of h refinement which could by used in general purposefinite element codes, centred around three important points.(a) A robust, simple and economical process of error estimation on an existing mesh.(b) A means of predicting directly the required mesh density satisfying economically

the specified accuracy and thus avoiding the previously used and costly progressiverefinement strategies.

(c) A mesh generator capable of deriving a mesh of specified density.An important step in this direction was taken by Zienkiewicz and Zhu (1987 and 1989)

who spell out above requirements provide the necessary answers making use of the firsttriangular mesh generator specifically designed for the adaptive process by Peraire et al.(1987).

The error estimator, postulated in this paper for self adjoint problems, is based on theapproximation that the error in fluxes (stresses) can be written as

eσ = σσσσσσσσσσσσσσ − σσσσσσσσσσσσσσh ≈ σσσσσσσσσσσσσσ∗ − σσσσσσσσσσσσσσh (8.1)

in which σσσσσσσσσσσσσσ are the exact values of the stresses (or other fluxes) σσσσσσσσσσσσσσh is the finite elementapproximation to these and σσσσσσσσσσσσσσ∗ are “recovered” values of σσσσσσσσσσσσσσh obtained by some post processingoperation which improves their accuracy.

Indeed it was shown by Zienkiewicz and Zhu (1992) that the effectivity of such an errorestimator, i.e.

θ =estimated erroractual error

(8.2)

where the errors are specified in any suitable norm is always bounded by

1− α ≤ θ ≤ 1 + α (8.3)

where

α =‖e∗σ‖‖ehσ‖

(8.4)

In above the numerator is the error of the recovered solution σσσσσσσσσσσσσσ∗ and the denominator thatof the finite element solution σσσσσσσσσσσσσσh. The need for a small value of α is obvious. Clearly theeffectivity of the estimation process will depend on the accuracy attainable by the recoveryprocess. In the 1987 paper the currently much used processes of “nodal averaging” or ofthe so called L2 projection (Brauchli and Oden 1971, Hinton and Campbell 1974) were usedwith some success though with these frequently the value of α was not always very small.

A major step forward was made here only very recently by Zienkiewicz and Zhu (1992a,b) where the well known properties of “superconvergence” of σσσσσσσσσσσσσσh at certain sampling pointswere used. This process, now given the name of SPR (superconvergent patch recovery), isso simple and self evident that in retrospect I found it hard to believe it was not used before!How did we all miss it?

The ‘SPR’ algorithm simply assumes that in patches of elements surrounding a typicalcorner node the value of σσσσσσσσσσσσσσ is approximated by a simple polynomial expansion one order


higher than that exactly reproduced by the element in question. This expansion which canalways be written as

σσσσσσσσσσσσσσ∗ = Pa (8.5)

where

P =

p 0p

. . .0 p

, a =

a1a2...aj

(8.6)

p = [1, x, y, ...]

with j being the number of components of σσσσσσσσσσσσσσ∗ which we assume to fit in a least square senseto superconvergent values of σσσσσσσσσσσσσσh

k where k is the appropriate sampling point.Minimisation with respect to a

∑k

∣∣σσσσσσσσσσσσσσhk − σσσσσσσσσσσσσσ∗∣∣T ∣∣σσσσσσσσσσσσσσh

k − σσσσσσσσσσσσσσ∗∣∣ (8.7)

givesPT

[σσσσσσσσσσσσσσh

k −Pa] = 0 (8.8)

ora =

[PTP

]−1PTσσσσσσσσσσσσσσh

k (8.9)

with summation with respect to k implied determines σσσσσσσσσσσσσσ∗ uniquely.The nodal values σσσσσσσσσσσσσσ∗ of the final ‘recovered’ approximation which is given below and in

which N are the ‘displacement’ shape functions

σσσσσσσσσσσσσσ∗ = Nσσσσσσσσσσσσσσ∗ (8.10)

can now be simply determined. Clearly in above each component of stress can be solved forseparately.

Figure 16 illustrates some typical two–dimensional patches which can be used and inFigure 17 we show the dramatic improvement in the value of recovered stresses for a simpleelastic problem.

Of course the process is equally applicable in three–dimensions or indeed in one. Forthe latter it is easy to see from Figure 18 why all the values recovered are superconvergent,with the linear u element being capable of modelling exactly a linear variation of σ, thequadratic u, a quadratic σ etc. In Figure 19 indeed we show how typical nodal values givesuperconvergence in a typical second order 1–D problem for various degrees of polynomialused in the original approximation.

Certainly error estimates based on σσσσσσσσσσσσσσ∗ are now extremely accurate (though a furtherimprovement can be added by modifying the functional of Eq. (8.8) to include overallequilibrium satisfaction – viz Wiberg and Abdulahab (1992).

The error estimator is now ready to be used with adaptivity and the process of predictingthe new mesh density will not be here described as it is much dependent on the objectivesof the analysis; reference to the original papers should be made for details. In Figure 20we show results of adaptive analysis applied to a simple heat conduction problem withdistributed sources for which the exact solution is known, (Zienkiewicz and Zhu 1992b).The fast convergence to the required solution accuracy and the efficiency of the estimatorsshould be noted.

28 O.C. Zienkiewicz

Figure 16 Superconvergent points in one dimension (a), two dimensional quadrilaterals (b),and triangles (c) in which a continuous polynomial approximates the stress dis-tribution. (Superconvergence in quadrilaterals is only true for simple Laplaceequation and for triangles their sampling points are not truly superconvergent)

Figure 16 (cont.) Computation of supercovergent nodal values for typical recovery patchesof different quadrilateral and triangular elements


Figure 17 Accuracy of the Superconvergent Patch Recovery (SPR) illustrated on an elasticperforated continuum for which exact solution is available (a) the analysis region,(b) three meshes using nine node, quadratic, quadrilaterals, (c) error of stressesobtained on various meshes using SPR, L2 recovery (sup. L) and local L2 recoverywith averaging (sup. HC)

30 O.C. Zienkiewicz

Figure 18 Recovery of exact solution by SPR in a 1D problem (a) Linear elements recover ex-act linear stresses (b) Quadratic elements recover exact quadratic stresses variationetc.

Figure 19 Convergence of stresses recovered by SPR in a 1D model problem (dotted line)and convergence of the original finite element stresses. Various orders of elementsp = 1 to p = 6 shown. (Note two order higher convergence rate for even orderelements–this occurs only for equal size elements)


Figure 20 Adaptive solution of a model 2D problem. Heat conduction with source termsleading to exact solution in (a) for ∂u/∂x and (b) for ∂u/∂y. Figure 20(c) showsthree stages of adaptive solution using quadratic triangles reducing the energynorm error below 1% from 55%. Figure 20(d) shows (1 − θ) for local effectivityindices during the adaptive refinement

32 O.C. Zienkiewicz

Figure 21 A return to dams! Practical automatic adaptive analysis of a dam (quadratictriangular elements)

Clearly in practical application the degree of effectivity of the estimator will not be testedbut as shown in a typical problem of Figure 21 optimal meshes are readily generated withconsiderable assurance of quality.

In parallel with the ‘h’ adaptivity procedures much progress has been made in recentyears on p and h− p processes (viz Demkowicz et al. 1992), however space does not permitto record here all the major steps achieved.

9. FLUID MECHANICS AND NON–SELF ADJOINT PROBLEMS

I have already refered to the wide application of finite element method beyond structural me-chanics which occurred after 1965 and in particular to fluid mechanics where its applicationseemed natural. The extension to potential problems of ideal, inviscid flow and indeed topredominantly viscous, Stokes, flow was obvious. The previously developed approaches (andindeed frequently the same computer codes) could be used directly without modification,(Martin 1968, Atkinson et al. 1970).

However the situation where convective accerations occurred which are typical of NavierStokes equations presented a difficulty. Here the non–self adjoint nature of the equationsprecluded the use of variational (extremum) principles and the only discretisation possibilitypresented itself via the use of weighted residual, Galerkin type, approaches. This possibilitywas first outlined by Oden (1969,73) and Zienkiewicz and Taylor (1973) with first realisticsolutions produced by Taylor and Hood (1973).

However simple application of the Galerkin weighting was soon found to be inapplicablein problems in which convective terms were dominant. Indeed here the same difficulties ofoscillatory (or divergent) solutions observed earlier by finite difference practitioners usingcentral differences were observed and it was natural to seek remedies from those working in


that field. Indeed a standard “medicine” for such problems in the finite difference fraternitywas the use of “upwind differences” (Spalding 1972) and the translation of this processto upwind weighting of the Petrov Galerkin kind in the finite element approach was firstsuggested by Zienkiewicz et al. (1975). The detailed elaboration of this by Christie et al.(1976) and Zienkiewicz et al. (1977) led to the adoption of such upwind Petrov Galerkinmethods for effective solutions of both simple convective problems and of the incompressibleNavier Stokes equation permitting in the later a larger range of Reynolds numbers to bedealt with effectively.

An important development of the techniques originally suggested was made by Hughesand Brooks (1979) and Kelly et al. (1980) and later Johnson et al. (1984) resulting inschemes which avoid the introduction of any “cross wind” diffusion.

But this first step allowing the treatment of steady state problems was not sufficient asthe use of the upwind Petrov Galerkin procedure was not justified when source or transientterm existed. Here the major step forward was taken by the introduction of a characteristicGalerkin process making use of the wave propagation features of these together with theoptimality of Galerkin methods for the self adjoint, diffusion part of the problem. Thisindeed led to a family of new algorithms, (Bercovier et al. 1982, Pironeau 1982, Zienkiewiczet al. 1984, Lohner et al. 1984) which permitted a wide and new range of problems to besolved. The simple algorithm produced in the last two references, in which the characteristic‘search’ is dealt with by a simple Taylor expansion could be reinterpreted as a finite elementversion of the Lax and Wendroff (1960) method (now ensuring a better accuracy due to theGalerkin approximation) and was identified with the so–called Taylor–Galerkin method,(Donea 1984). The latter, not based on a single characteristic velocity, can deal with anumber of characteristic speeds and as such allowed extension to problems of high speedcompressible gas flow to be made by the mid 1980’s.

This indeed was a landmark and within the last decade this procedure for the solutionwithin both compressible (and incompressible) flows was widely used and revolutionisedthis field of mechanics. The work of Morgan, Lohner, Peraire, the writer and others inthe years of 1985–1990 is reported in the bibliography. The development of algorithms forcompressible flow is still in progress despite the achievements recorded viz Zienkiewicz etal. (1995) and has opened the doors to finite element analysis of very complex aeronauticalproblems.

The rapid progress made in this context has thrown up many interesting points:1. Adaptive refinement in essential for realistic compressible flow solution if the capture

of such features as shock, boundary layers etc. is to be made. Further, the overallrefinement necessary in such complex, three dimensional problems as the analysis of thewhole aircraft (viz Peraire et al. 1988) requires very large number of elements and degreesof freedom (a milion is quite a common mean!) and only by the adaptive process caneconomy be achieved.

2. Lowest order elements have been universally used in all solutions for two reasons; firstbecause it is felt that these are optimal in modelling discontinuities, second because withthese explicit dynamic transient problems can be most efficiently computed.

3. Steady state solutions are always achieved by an iterative process (Gaussian eliminationbeing not applicable to such large systems). Here all the tools available to finite differencepractitioners, who invariably use such iterations, are availabe. Once again there havebeen re–interpreted in terms of unstructured finite element forms including such devicesas multigrid techniques etc.Figures 22–25 illustrate some typical compressible and incompressible analysis in both

steady and unsteady situations.

34 O.C. Zienkiewicz

Figure 22 Use of adaptive refinement for shock capture in compressible flow aerodynamics.Mach 2 flow of ideal gas round a cylinder. Note mesh elongation adopted foreconomy

Figure 23 Adaptive, three dimensional solution of Mach 2 flow around a “generic” fighteraircraft. Peraire et al. (1987)


Figure 24 Pressure contours for problem of Figure 23

36 O.C. Zienkiewicz

Figure 25 An early study of the European shuttle by Euler equations, (a) mesh on surfaceof shuttle, (b) pressure contours on surface


The procedures shown have been extended to viscous compressible flow and here problemsbecome even more complex as shown in Figures 26 and 27.

At this stage it is perhaps worth pondering whether finite element procedures have takenover from finite difference ones or whether simply a merging of the methods has occurredin the field of fluid mechanics. Certainly the unstructured nature of typical finite elementshas been duplicated recently by the, so called, finite volume techniques which can andindeed do on occasions use identical triangular forms (Jameson et al. 1986). Adaptivitycan therefore be used on a similar basis. However, the finite volumes here bear a closerresemblance to finite elements than they do to standard finite differences. Computationallyboth procedures are extremely similar and there is little that can be said to distinguishdifferences. However the clear cut finite element approximation is preferable in transientproblem where the correct ‘mass matrices’ generated add considerably to the accuracy viza viz the lumped forms traditionally used by the ‘difference’ proponents.

The algorithms used for the solution are, as already mentioned, similar but once againthe finite element forms allow in principle to use a more rational derivation – and generallyavoid rather arbitrary ‘artificial diffusion’ operators.

The biggest controversy still raging is that between the use of structured viz a vizunstructured meshes. The first of course can lead to simpler numerical algorithms buthere the difficulties associated with the mapping of complex shapes and the impossibility ofachieving optimal meshes by adaptivity, carry severe penalties of computational time.

It is my opinion that in fact the balance has turned today firmly in favour of finiteelement approximation in this field – particularly if these are given a liberal interpretationand make use of strictly numerical improvements made by ‘the other side’. I have alreadyremarked that simple elements are much in vogue in the fluid field, reversing the swing ofthe late sixties and seventies towards higher order approximation. The same phenomenonhas been observed in the development of solid mechanics codes based on explicit dynamicformulation in the early eighties, e.g. DYNA (1980).

Is it possible that the future will see a possible swing of the pendulum and a return tohigher order elements? Only time will tell.

10. THE EPILOGUE

In concluding this recital of the “landmarks” I am very conscious that omission has beenmade of many achievements which others would choose in this context. In particular I havenot talked about the important aspect of non–linear application in the fields of structuralplasticity and large deformation, metal forming and geomechanics (which inevitably occu-pied much time of my own and many others viz Zienkiewicz and Onate 1978, Zienkiewicz etal. 1990). Nor have I discussed the many problems associated with such non–linear compu-tations to which many have contributed. The reasons for this are manifold; what I intendedto present are in the main the various features of the generalised finite element formulationwhich can be widely applied and offers many possibilities. The view that “the finite elementmethod is simply a systematic technique for construction of Ritz–Galerkin approximationsfor irregular domains” is I find too restrictive and I hope that wider possibilities are impliedin the name.

However nothing in the field of F.E.M. activity is done in isolation. The technologytransfer is now rapid between one or other area of activity. In Figure 28 I show how thedevelopments for adaptive shock capture in fluid mechanics have influenced the currentlyfashionable problems of stress localization in plasticity.

Finally, let me stress that the process is a playground of many, including both engineersand mathematicians. The first, using intuition, frequently act before proof of correctness ismade available by the latter. The history shows that in general this has been the path ofprogress!.

38 O.C. Zienkiewicz

Figure 26 A solution for pressure distribution for a Boeing 747 in a landing configuration(incomplete geometry and Euler equations used here)

Figure 27 Shock and boundary layer interaction at Mach 3 near a surface ‘lump’


Figure 28 Technology transfer. Adaptive methods developed for compressible flow capture astrain localization in plastic flow

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Zienkiewicz, O.C. and Zhu, J.Z. (1989), “Error estimation and adaptive refinement for plate bendingproblems”, Int. J. Num. Meth. Eng., 28, pp. 2839–2853.

Zienkiewicz, O.C., Chan, A.H.C., Pastor, M., Paul, D.K. and Shiomi, T. (1990), “Static and dynamicbehaviour of geomaterials–A rational approach to quantitative solutions. Part I–Fully saturatedproblems”, Proc. Roy. Soc. London, A429, pp. 285–309.

Zienkiewicz, O.C., Xie, Y.M., Schrefler, B.A., Ledesma, A. and Bicanic, N. (1990), “Static anddynamic behaviour of geomaterials–a rational approach to quantitative solutions. Part II–Semisaturated problems”, Proc. Roy. Soc. London, A429, pp. 311–321.

Zienkiewicz, O.C. and Onate, E. (1991), “Finite volumes vs. finite elements. Is there really a choice”,Nonlinear Computational Mechanics, State of the Art, Eds. P. Wriggers and W. Wagner, Springer.

Zienkiewicz, O.C. and Taylor, R.L. (1991), “The finite element method”, II, (Solid and FluidMechanics, Dynamics and Nonlinearity), McGraw–Hill, p. 807.


Zienkiewicz, O.C. and Zhu, J.Z. (1992), “The superconvergent patch recovery (SPR) and adaptivefinite element refinement”, Comp. Meth. in Appl. Mech. Eng., 101, pp. 207–224.

Zienkiewicz, O.C. and Zhu, J.Z. (1992), “Superconvergent patch recovery and a posteriori errorestimation in the finite element method. Part I–A general superconvergent recovery technique”, Int.J. Num. Meth. Eng., 33, pp. 1331–1364.

Zienkiewicz, O.C. and Zhu, J.Z. (1992), “Superconvergent patch recovery and a posteriori errorestimation in the finite element method. Part II–The Zienkiewicz Zhu a posteriori error estimator”,Int. J. Num. Meth. Eng., 33, pp. 1365–1382.

Zienkiewicz, O.C. and Codina, R. (1995), “A general algorithm for compressible and incompressibleflow. Part I–The split chartacteristic based scheme”, Int. J. Num. Meth. in Fluids.

Zienkiewicz, O.C., Morgan, K., Satya Sai, B.V.K., Codina, R. and Vazquez, M. (1995), “A generalalgorithm for compressible and incompressible flow. Part II–Tests on the explicit form”, Int. J.Num. Meth. in Fluids.

APPENDIX†

The work which I associate with the beginning of the computerized FEM was doneduring summer 1953 when I was again employed by Boeing Airplane Company on theirsummer faculty program. Again, I was assigned to Mr. M.J. Turner’s Structural DynamicsUnit, to work on methods of evaluating the stiffness of a delta airplane wing for use influtter analysis. Because the bar assemblage approach tried during the previous summerhad been unsatifactory, Mr. Turner suggested that we should merely try dividing the wingskin into appropriate triangular segments. The stiffness properties fo these segments were tobe evaluated by assuming constant normal and shear stress states withing the triangles andapplying Castigliano’s’s theorem; then the stiffness of the complete wing system (consistingof skin segments, spars, stringers etc.) could be obtained by appropriate addition of thecomponent stiffness (the direct stiffness method). Thus, at the beginning of the summer,1953, Mr. Turner had completetly outlined the FEM concept and those of us working onthe project merely had to carry out the details and test the results by numerical experiment.

Our paper describing this initial effort was presented at the New York meeting of theInstitute of Aeronautical Sciences in January, 1954. I have never known why the decisionwas made not to submit the paper for publication until 1955, so the publication date ofSeptember 1956 (see Turner et al. 1956) was more than two years after the first presentationand over three years after the work was done. As was mentioned, this is graphic evidencethat the FEM did not attain instant recognition. Undoubtedly, a major factor which limitedits acceptability was that the original work was done in the Structural Dynamics Unit, wherethe objective was limited to stiffness and deflection analysis; it was several years before theconcept was accepted and put to use by the stress analysis groups at Boeing. Thus, itis possible that the orientation of this initial step toward a specific engineering applicationrended to obscure the general applicability of the FEM concept, even though the individualsworking with the development at Boeing were quite aware of its broader implications.

Although I maintained close contact with several of my Boeing colleagues for many yearsafter 1953, I did not work there again and I had no opportunity for further study of theFEM until 1956–57, when I spent my first sabbatical leave in Norway (with the SkipstekniskForsknings Institutt in Trondheim). This “Norwegian connection” also was a factor in mydecision to prepare a historical summary for this Conference; it was this period which madepossible my continued contact with the finite element concept. Lack of computer facilitiesin Norway limited the type of work I could do at this time, but I was intrigued by plane

† Extract form R.W. Clough (1979) p. 1.6–1.7

48 O.C. Zienkiewicz

stress aplications of the method and I carried out some very simple analyses of rectangularand triangular element assemblages using a desk calculator. Although this work was tootrivial to warrant publication, it convinced me of the potential of the FEM for the solutionof general continuum problems.

About the time I returned to Berkeley from my sabbatical leave, the Engineering Collegeacquired an IBM 701 Computer (replacing the old Card Programmed Calculator) andwe began to develop structural analysis capabilities with this machine. For educationalpurposes, a Matrix Interpretive Program of the type pioneered in England (Hunt 1956)offered the best means of making the computer capabilities accessible to the students, andmost of my early efforts went into developing such a program (Clough 1958). Then itwas possible to continue my work with the FEM which had been undergoing continuingdevelopment at Boeing but had attracted only little attention elsewhere.

Early FEM studies at Berkeley were greatly limited by the two thousand word centralprocessor capacity of the IBM 701, but by utilizing the seven 2000 word drum storage unitsit was possible to carry out some creditable analyses. Our first concentrated effort towardsplane stress analysis was in response to a challenge by one of my continuum mechanicscolleagues who was skeptical of the validity of the procedure and wanted to see a solutionof some classical problem. To me it seemed obvious that the method could solve any planestress problem to any desired accuracy – limited only by the time and energy one wishedto expend on the calculations. But in the hopes of attracting wider interest toward theFEM concept, I allocated part of a small NSF research grant to the solution of a few sampleplane stress problems. The principal problem that arose in writting the paper was choosinga suitable name for this analytical procedure and I decided finally on the Finite ElementMethod. This name first appeared in that paper (Clough 1960), and I can only concludefrom subsequent history that it was an apt choice.

In retrospect, the next red letter event in my personal FEM history occurred in December1960, when Professor O.C. Zienkiewicz invited me to Northwestern University to give aseminar lecture on the new procedure. We were friends from previous meetings, and I knewthat the had been brought up in the Southwell finite difference tradition, so it was apparentthat his invitation was prompted by skepticism and a desire to discuss the relative merits offinite elements vs. finite differences. Certainly, we did have such discussions during my visit,but Professor Zienkiewicz obviously is a very intelligent person and was quick to recognisethe advantages of the FEM. During that short visit an illustrious convert was won to thecause, and I think it is not coincidental that rapid worldwide acceptance of the FEM startedalmost from that moment.

Please address your comments or questions on this paper to:

International Center for Numerical Methods in EngineeringEdificio C-1, Campus Norte UPCGran Capitan s/n08034 Barcelona, SpainPhone: 34-3-4016035; Fax: 34-3-4016517E-mail: [email protected]

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