Olgierd (Olek) Cecil Zienkiewicz (1921-2009)
By Prof. D. Roger J. Owen
Olek Zienkiewicz was one of the greatest academics to have worked at Swansea University in its one-hundred-year history. His research was ground-breaking, and his legacy and reputation profound. Zienkiewicz is internationally recognised as one of the leading developers of the Finite Element Method, a computer-based technique which has, since the 1960s, revolutionised design and analysis procedures in civil, mechanical, aerospace and other branches of engineering. Initially, the formulation of the method followed a traditional structural engineering approach but as the underlying mathematical basis became understood, its application to other disciplines became possible. The methodology remains a flourishing research topic and its application has considerable potential in new scientific areas, including biomedical engineering and the life sciences.
Olek Zienkiewicz was born on 18th May, 1921, in Caterham, England, of a Polish father and English mother. He was the seventh son of a seventh son, which in folklore would have endowed him with magical and clairvoyant powers - and may, or may not, have been responsible for his remarkable career. His early life was greatly influenced by the turbulence of European history over the first half of the twentieth century. Towards the end of the First World War, Czar Nicholas of Russia was first overthrown by revolutionaries who formed a government under Alexander Kerensky, which was in turn overthrown by the Bolsheviks. In 1917 Zienkiewicz’s father, Casimir, held the post of Consul in Birmingham for the Kerensky government, which only lasted for a short period. The practice of law is tied to the legal system of one’s government, so employment in his chosen profession was effectively closed to him in England. Therefore, in 1922 the Zienkiewicz family returned to Poland, first to Warsaw and then to Lodz, before moving once more in 1926 to Katowice where his father held the post of district judge up to the outbreak of World War II.
Recognising his obvious academic talents his parents sent him to a boarding school near Poznan and his schooling went well for two years until he suffered a sporting injury which became serious due to a staphylococcus infection in his hip bone. Known as osteomyelitis, the disease required lengthy hospital stays and operations, which left him with a limp. The latter, it should be noted, never impeded him in a vigorous outdoor life which encompassed sailing, scuba diving, hiking and a general love of the outdoors and of nature. Extended hospitalisation did not deter the young Zienkiewicz from scholarly excellence and he completed his high school studies in June, 1939. He planned to compete for admission to the Warsaw Polytechnic in September and moved to Warsaw in preparation for the examination.
Fate, of course, intervened in the form of the outbreak of the Second World War on 1st September, 1939. Warsaw was soon under siege and the days that followed were among the most eventful of his entire life, which he spent building barricades, overturning trams and digging trenches. After several days, the order came to evacuate and Zienkiewicz made hasty preparations to leave, taking a new camera, drawing instruments, a pair of socks – and nothing else except the clothes he was wearing. He had arranged to meet his mother and sister at a certain bridge and found that his sister was even less prepared for the times that lay ahead, having only a pair of high-heeled shoes and two packets of cigarettes. Many exciting days were to follow, including wandering through the countryside for ten days, a return to Warsaw and, eventually, a reuniting of the family in Katowice.
After several weeks the family managed to obtain a visa to Italy, which had not yet entered the war, and then subsequently on to Angiers in France, where Casimir Zienkiewicz worked for the Polish government in exile. This stay proved to be short-lived, however, with the fall of France occurring in June 1940. The family was able to keep one step ahead of the invaders and reached the west coast of France at St. John de Luz, near Bayonne, where they sailed to England on the Polish ship Batory on 22nd June 1940, the very day the armistice between France and Germany was signed.
Once in England Zienkiewicz began to think of a university education. The British Council had available to Poles a special scholarship for study at Imperial College, London, where he opted for a course in Civil Engineering, as the university did not have a degree scheme in Naval Architecture which was his preferred choice. Placed first in his studies at the end of the first year, he was given two scholarships for the remaining time at Imperial. He graduated in 1943, one of two first-class honours recipients. Upon graduation, Zienkiewicz discussed the possibility of research work with A. J. S. Pippard who offered a scholarship to work jointly with him and R. V. Southwell on a dam analysis project. Southwell and Pippard were two of the principal figures in the development of structural mechanics in the 1930s and 1940s. Southwell’s relaxation method bridged the gap between the classical methods developed over the century prior to 1930 and the large-scale computational methods of today, which emerged in the late 1950s.
After the award of his Ph.D. in 1945, Zienkiewicz moved directly into engineering practice. With his appetite for dams having been whetted by his postgraduate work, he became attracted to the design and construction of these massive structures. He approached the consultancy firm of Sir William Halcrow and was offered a post leading a survey party on a dam project in Scotland. After salary negotiation culminating in the offer of £250 per annum, he was asked if he could drive a car. ‘Naturally’ came the reply and he promptly went out and took the test, in those days, for a provisional license. To the time of his death he did not take a test for a regular driving license. Olek spent four years with Halcrow, first with the survey task and then on the design and construction of the Glen Affric Hydroelectric System in northern Scotland (Figure 1).
In 1949, various opportunities for a change of direction came to Zienkiewicz’s attention and one which appealed to him, which he accepted, was a lectureship in the Department of Engineering at Edinburgh University. This was a period of ease in his academic career, steadily developing his teaching skills and honing his research techniques (Figure 2). To add to this idyllic scene, he met a young graduate student from Canada, Helen Fleming who was working towards a M.Sc. degree in chemistry. They were soon engaged and were married in 1952. Their first son, Andrew, arrived in 1953 and son David was born a year later.
Towards the end of 1956, offers materialised for posts elsewhere. He eventually decided on accepting a position as Associate Professor of Civil Engineering at Northwestern University, USA, facilitated by Pippard who had spent some time there as a visitor. During this period their daughter Krystyna was born in 1958. Sadly, his father passed away in this same year at age 86.
In 1961, Zienkiewicz began a new adventure that led to the activities for which he became world renowned. The chair of Professor of Civil Engineering became vacant at Swansea, due to the departure of B. G. Neal to Imperial College, and Zienkiewicz was successful in the ensuing competition. Although the salary was substantially less than he was receiving at Northwestern, he relished the attraction of a professorship in the UK system at that time, wherein it was possible to influence the work and professional development of many others. Moreover, it was a time of unprecedented expansion of higher education in the United Kingdom. Figure 3 depicts the Zienkiewicz family en route from Northwestern to Swansea. It was lucky for Swansea that he decided to come, because he helped the Department of Civil Engineering there achieve a reputation as one of the world’s foremost centres of finite element research.
Over his lifetime Zienkiewicz published close to six hundred papers. Clearly, not all of them can be summarised here and the reader is directed to the website address given in the Bibliography for a full list of his publications. Here, only some of the most significant contributions that led to advances to the field of finite elements are cited. From a chronological aspect, his published works are remarkable in terms of the rate at which they appeared. For a full nine years after the receipt of his Ph.D. he produced only one paper (1), that one derived from his thesis[1]! In the next five years (1954-59) he wrote ten papers and then eleven in the next two years (1960-61). Beyond 1970, and well past his formal retirement in 1988, he produced in the order of fifteen papers per year.
Plate bending and non-structural problems
It should be recognised that when Zienkiewicz and his colleagues began research into the finite element method at Swansea in 1961, the method was already a known technique. The mathematical foundations of the technique were laid in the early decades of the last century, with contributions by Ritz, Galerkin, Courant and others. Within the finite element community, the origins of the method are attributed to the contribution of Hrenikoff & McHenry (1941) who, using engineering intuition, approximated the response of a plate by an equivalent framework system and Courant (1943) who developed a more rigorous mathematical approach to the solution of a torsion problem by subdivision of the domain into triangular sub-regions. Progress from that time onward was slow, no doubt hampered by the lack of computing power, until the first recognisable finite element paper, by Turner, Clough, Martin & Topp appeared in 1956. By the end of that decade a substantial body of literature had been accumulated and the term ‘finite element’ was coined by R. W. Clough, University of California at Berkeley in 1960. It was against this background that Zienkiewicz entered the finite element research arena. It should also be mentioned that it was at this time that J. H. Argyris FRS, Imperial College and later the University of Stuttgart, embarked on finite element research and their careers ran in parallel for the next four decades, both contributing prominently to the field. Figure 4 shows the last meeting of the “holy trinity of finite elements” in Munich in 1999.
The Wright-Patterson Conference held in Dayton, Ohio in 1965 was a landmark in finite element analysis and a paper describing the first triangular plate bending elements from the Swansea group contained a remarkable number of major new ideas [2]. One was the use of area coordinates for integration over triangular regions. Another of the contributions in the paper, co-authored with Bruce Irons - then working at Rolls Royce and who was destined to become a colleague at Swansea - was the notion of the patch test. In a displacement finite element formulation displacement continuity is assured across element boundaries, but the interface element stresses are generally discontinuous. Elements of this type are of vital importance in a majority of engineering applications, but no one had previously attempted to establish criteria for their mathematical convergence. The paper devised a simple test to be applied to a collection, or ‘patch’, of elements to achieve this goal. The patch test has proven to be of fundamental importance in the finite element theory. It was also at this meeting that J. Tinsley Oden, another pioneer of the finite element method, made first contact with Zienkiewicz:
“I first met Olek at the Dayton finite element meetings in the mid 1960s, which is where some say the finite element method, being born in the mid 1950s, reached its adolescence. There were a number of original and important works that formed the foundations of the subject that were presented there by engineers and scientists working in this new and exciting field. Of course, Olek was already known to many there because of his first textbook on finite elements, co-authored with Y. K. Cheung. Olek’s intense interest and warmth was intriguing. We hit it off immediately and began a friendship that lasted until his passing some four decades later”. [3]
It was during this period that the first industrial application of the finite element method, in Europe at least, took place in 1963 when Zienkiewicz and co-workers undertook the stress analysis of the Clywedog dam in mid-Wales. As can be seen from Figure 5, which illustrates the dam and the finite element discretisation employed, the mesh was extremely coarse by present day standards; which reflected the limited computational power available at that time. Nevertheless, the analysis provided valuable insight to the behaviour of the dam and its foundation.
A paper that appeared in 1965 and which was to have profound impact in later years was ‘Finite Elements in the Solution of Field Problems’, co-authored with Y. K. Cheung[4]. Up to this time the method had been restricted to structural mechanics problems, by expanding techniques for the analysis of frameworks into a method for the analysis of elastic continua. As such, the methodology relied heavily of the theorem of total potential energy. Zienkiewicz was able to perceive it as a more general tool for the analysis of all types of problems in mathematical physics that could be described in terms of a differential equation with given boundary and initial conditions. By employing weighted residual procedures, and in particular a Galerkin approach, computational solutions could be obtained for classes of problem where a potential function cannot be easily derived. In particular, he identified the procedure as a scheme for solving Laplace’s equation which governs the behaviour of ideal fluids and the torsion of prismatic sections. Zienkiewicz and his colleagues soon amplified these ideas to deal with problems of heat transfer[5] and electromagnetics[6].
It is well established that isoparametric representation is fundamental to many modern aspects of finite element analysis. The essence of the idea originated in contributions by Ian Taig of the British Aircraft Corporation (a forerunner of British Aerospace) and Bruce Irons, then at Rolls Royce. A further contribution from Swansea brought together the ideas put forward in these earlier contributions and formalised the procedure for general application[7]. It was around this time that Thomas J. R. Hughes, another finite element pioneer, first saw Zienkiewicz in person at a seminar at the Massachusetts Institute of Technology:
“There was a lot of excitement in the room because Olek was about to present the ‘isoparametric concept’, which ultimately revolutionised the subject, but the image that remains in my mind was he began the lecture with the picture of a man made out of finite elements. It got a big laugh. At the time it was so incongruous to think of a man made out of finite elements, but now it is a serious business. Who could have thought at that time we would be eventually using medical imaging data to build anatomically and physiologically realistic models of real people for simulating medical interventions”.[8]
This early picture, intended to convey the message that the study of finite elements should be an enjoyable pursuit, is reproduced in Figure 6.
In 1966 Olek and his colleagues turned their attention to inelastic analysis. Time dependant problems were the first in the class of inelastic problems to draw Zienkiewicz’s attention. In 1966, with Margaret Watson, he formulated finite element procedures for the analysis of concrete structures undergoing creep deformation.[9]
A key publication represented a major contribution to the finite element analysis of time-independent inelastic problems[10]. To be sure, contributions by others had preceded this paper, but this work brought together the characterisation of material behaviour in terms of incremental theory and a new procedure for advancing along the load-displacement curve. Termed the ‘initial stress’ approach it represented, at that time, a fundamental advancement in the computational modelling of inelastic problems.
Collaborations with two Ph.D. students resulted in further developments in the field of inelastic analysis. Extensive studies in the modelling of plasticity problems were undertaken with G. C. Nayak[11]. Noteworthy contributions are the so-called ‘alpha-constant stiffness’ algorithm for iterative solution of the algebraic equations arising in nonlinear finite element analysis[12] and a convenient form of the third invariant of the stress tensor suitable for yield surface computations[13].
The other collaboration, with I. Cormeau, involved the concept of viscoplasticity. Zienkiewicz had for a long while envisioned a scheme whereby both time-dependent and time-independent inelastic phenomena could be handled by the same algorithmic structure. The first paper developing this approach appeared in 1972[14]. The viscoplastic approach later found extensive usage in the analysis of extrusion and forming processes[15] and a range of geotechnical problems [16],[17]. Many of the problems related to geotechnical foundations involve boundary conditions at infinity. Zienkiewicz and his colleagues, most notably Peter Bettess, created and applied families of ‘infinite elements’ (18, 19) to account for such conditions[18] [19].
Around this time the concept of an ‘overlay model’ was introduced for materials where characterisation of the constitutive response in terms of a single yield surface is inadequate. The overlay approach, in which a series of yield surfaces is introduced, was formulated by Zienkiewicz and co-workers[20].
The work in flow problems began in the 1965 paper with Y. K. Cheung, as discussed earlier and, continued on to the treatment of seepage, electromagnetics and heat transfer. This set the stage for consideration of the more sophisticated problems which involve convective terms and nonlinearities, such as viscous, incompressible flow described by the Navier-Stokes equations for which substantial problems remained for the solution of practical problems. One such difficulty, which was already well recognised in the finite difference literature, related to the fact that in the momentum equations, written in terms of velocities, the convection terms are proportional to the first derivative while the viscous term is proportional to the second derivative. The latter dominates at low values of the Reynolds number while the reverse is true at high Reynolds number. There is numerical instability as one increases from low to high speeds and to cope with this a technique known as ‘upwinding’ had been devised by finite difference researchers.
Adaptation of this process to upwind weighting of the Petrov-Galerkin kind in the context of the finite element approximation was at first unclear. After much effort, Zienkiewicz was able to identify the proper approach, and following numerical experimentation and theoretical refinement, a procedure was described in 1977 permitting the solution of both simple convective problems and of the incompressible Navier-Stokes equations over a large range of Reynolds numbers[21]. This development of upwinding in a finite element setting provided a general theoretical basis that did not previously exist.
Zienkiewicz remained at Swansea until his retirement at age 67 in 1988 and subsequently became Professor Emeritus of the University of Wales, as well as holding the UNESCO Chair of Numerical Methods in Engineering at the University of Technology of Catalunya, Barcelona for 15 years. Although formally retired, he remained active in finite element research.
The ‘holy grail’ in the field of finite elements had been a search for a method of estimating the error in the computational solution – a difficult task given that the exact solution to the problem is invariably unknown. This was of long-time concern to Zienkiewicz and he first turned his attention to addressing the problem in 1983. Working in collaboration with the mathematician I. Babuska, Zienkiewicz developed procedures for practical problems through construction of an appropriate error measure, culminating in the so-called ‘Zienkiewicz-Zhu (Z2)’ method (22, 23) that he principally developed with his Ph.D. student J. Z. Zhu[22][23]. This procedure represented the single most significant advancement in the control of error in finite element analysis and is widely used to the present day. The method relies upon using the difference between the smoothed and unsmoothed finite element solutions to construct a local error measure and then employing adaptive mesh refinement to equalise this error over the entire domain. Zienkiewicz’s work on the topic continued into the 1990s with publication of an improved error estimation procedure, termed the ‘Superconvergent Patch Recovery’ which is based on the superconvergence properties of certain points within the finite elements [24]. It is worth noting that Zienkiewicz was over 70 years old when this work was completed, which arguably is one of his most important contributions to finite element research.
In the years leading to his formal retirement, a substantial portion of Zienkiewicz research moved into the field of computational fluid dynamics. Commencing in 1983, with colleagues, Ken Morgan, Rainald Lőhner and Jaime Peraire, finite element approaches were extended to the field of high-speed compressible flow, where traditionally finite difference or finite volume approaches held sway. This allowed for the first time high speed flows, including supersonic behaviour, to be solved effectively using finite elements [25], [26]).
After retirement from Swansea in 1987, Zienkiewicz spent two months each year at the International Centre for Numerical Methods in Engineering (CIMNE) at Universitat Politecnica de Catalunya (UPC) in Barcelona, Spain, lecturing (Figure 7) and still undertaking research. This was brought about by his close connection with his former Ph.D. student Eugenio Oñate who obtained a position in UPC, where he subsequently founded the International Center for Numerical Methods in Engineering (CIMNE), a research centre specializing in the development and application of numerical methods in engineering.
In 1989 Zienkiewicz was appointed to the UNESCO Chair of Numerical Methods in Engineering at UPC. This was the very first UNESCO Chair in the world and arose from interactions with Geoffrey Holister who was working at UNESCO developing support to technology and engineering and who previously had been on the Civil Engineering staff at Swansea. The idea for such a position arose from the book “Small World” by David Lodge [27]. In the book professors of literature imagine a UNESCO Chair that will allow them to retire into a world of continuous travel with no lecturing obligations at an extravagant salary. The award to Zienkiewicz attracted interest worldwide and was taken as a model to create many other UNESCO Chairs in different fields in universities around the world. Today there exist some three hundred.
Zienkiewicz’s great passion over the remaining period of his life was revision and extension of his finite element books. It would not be out of place to refer to his text on finite element analysis as simply ‘The Book’. Surely, in no other field is there a reference which has had greater supremacy among competing texts. This dominance was established immediately on publication of the version published in 1967, which was written with the assistance of Y. K. Cheung. This text, numbering 272 pages and was translated into Japanese and Russian. The next version appeared four years later in 1971, was almost twice the length of the first version, and was translated into Japanese, Russian, Polish, French and German. Subsequent versions expanded in size and scope of topics addressed, culminating with the sixth edition which is a three-volume set[28]. This work is best described in the words of his principal co-author Bob Taylor:
“In the late 1990s we undertook a major revision of the FEM book with much of the revision completed while we both visited CIMNE for two months each year. The fifth edition was greatly expanded and divided into three volumes. Work on the revisions was carried out either at CIMNE or at the apartment of Olek and Helen at Sitges. The writing was hindered somewhat due to the deterioration of Olek’s eyesight from macular degeneration. However, his ability to recall and formulate complex technical information from memory made the writing possible. Working together, we discussed sections for each volume, the modifications were formulated verbally, dictated to tape by Olek and later transcribed and edited to computer files for printing”.
Over his long career in the field of computational modelling, Zienkiewicz made numerous, and often simultaneous, contributions to disparate fields that included: viscous flow analysis, geotechnical engineering, dynamics, forming processes, rock mechanics, inelastic material analysis, heat transfer, computational fluid dynamics, fluid-structure interaction and the mathematical fundamentals of the finite element method. His accomplishments did not go unrecognised. Over his career, he was awarded over 30 honorary degrees including ones from Portugal, Ireland, Belgium, Norway, Sweden, China, Poland, Scotland, Germany, France, England, Italy, Hong Kong, Hungary and the United States. He also received numerous special honours and medals including the Royal Medal of the Royal Society (1990) and the Prince Philip Medal of the Royal Academy of Engineering (2006).
He was elected to the Royal Society and the Royal Academy of Engineering in 1978 and was a Foreign Member of the United States Academy of Engineering, the Polish Academy of Science and the National Academies of Science in China and Italy. He was appointed CBE in 1989.
To those who are familiar with the work of Olek Zienkiewicz, he will surely be regarded as the major figure of the twentieth century in the numerical analysis of problems in engineering science. Hopefully, the above pages describe a person of even greater dimensions.
I first met Olek Zienkiewicz in 1961 when he took up his position as Professor of Civil Engineering at Swansea, where I was an undergraduate student at the time. After graduating, he encouraged me to go abroad for postgraduate study and arranged a Ph.D. position at Northwestern University. Upon completing this degree in 1967, Olek persuaded me to take a position as a Research Fellow at Swansea. I was rapidly introduced to the finite element method and was seduced by the exciting research prospects in the area and the enthusiastic and refreshing manner in which Olek ran the department. Administrative tasks were kept to an absolute minimum, teaching duties were efficiently dispatched and research was encouraged at every opportunity. Through creation of the world-renowned Institute for Numerical Methods in Engineering at Swansea, there was a continual exchange of research visitors and it was not unusual to find six to eight scholars, each of them an academic of substantial rank at his/her home institution, visiting the department at any given time. I subsequently spent the next forty years at Swansea.
Zienkiewicz’s stature in the field of finite elements is not only due to his scientific achievements but in no small measure a result of his extraordinary personality. As well as relishing academic debate with his peers, he was never happier than when discussing research issues with younger colleagues. One of his significant strengths was the ability to synthesise a particular research topic, extract the essential theoretical and computational features and to describe the resulting solution algorithms to students in a concise and transparent manner. In this respect, he provided countless engineers and researchers with the incentive and enthusiasm to participate in the exciting field of computational modelling. Over his research career, he supervised some seventy Ph.D. students, many of whom today hold leading positions in academia and industry.
Zienkiewicz’s decision to spend his entire academic career at Swansea was undoubtedly influenced by his love of sailing (Figure 8). Students and research visitors to Swansea were frequently roped in to act as crew, but all competitive instincts during a race would be quickly abandoned if a particularly interesting finite element discussion arose.
He was gastronomically adventurous, especially with regard to fungi where his Polish ancestry made him particularly bold, and many friends and colleagues can bear witness to his love of oysters which he consumed in large quantities at every conceivable opportunity. He had an extremely engaging nature and made countless friends during his extensive travels. A particular characteristic was his enquiring mind, which combined with his formidable intellect, ensured that late night discussions after dinner were far ranging and challenging. Olek Zienkiewicz is greatly missed by all who had the privilege to know and work with him over his long career, but his presence is still felt by all in the legacy of his research and in his impact on Swansea’s College of Engineering, which to this day is home to several locations in his name, and which hosts an annual lecture in his honour.
Bibliographical Note:
Much of the material on Zienkiewicz’s earlier life came from an unpublished autobiographical account, on which he was still working at the time of his death. His views on the first 25 years of finite element developments relies in part on: O. C. Zienkiewicz, ‘The first 25 years of the finite element method at Swansea’, in J. H. Purnell (ed.), The University of Wales Review, 2 (1987), pp. 5-19. The following ten years of progress is summarised in O. C. Zienkiewicz, ‘Origins, milestones and directions of the finite element method – A personal view’. Archives of Computational Methods in Engineering, 2, 1 (1995), pp. 1-48.
The paper has also drawn on the material in R. H. Gallagher, ‘Olgierd C. Zienkiewicz – The man and his work’ in R. W. Lewis, P. Bettess and E. Hinton (eds), Numerical Methods in Coupled Systems (John Wiley & Sons, 1984), which reviews Olek’s contributions to the field over the earlier year. A full bibliography is available as electronic supplementary material at http://rsbm.royalsocietypublishing.org.
[1] O.C. Zienkiewicz, ‘The stress distribution in gravity dams’. J. Inst. Civ. Eng., 27, (1947), pp. 244-271.
[2] G.P. Bazeley, Y.K. Cheung, B.M. Irons and O. C. Zienkiewicz, ‘Triangular elements in plate bending: Conforming and non-conforming solutions’, Proc. Conf. on Matrix Methods in Structural Mechanics, J. S. Przemieniecki, R.M. Bader, W.F., Bozich, J.R. Johnson, and W.J. Mykytow (eds), Oct.,1964, pp. 26-28, 547-576, Report No. AFFDL-TR-66-80, Wright-Patterson AFB, Ohio.
[3] J. T. Oden,. Private communication, 2009.
[4] O. C. Zienkiewicz and Y. K. Cheung, ‘Finite elements in the solution of field problems’ The Engineer, 220, 24 September 1958, pp. 507-510.
[5] O.C. Zienkiewicz and C. J. Parekh, ‘Transient field problems:2-dimensional and 3-dimensional analysis by isoparametric finite elements’,. Int. J. Num. Meth. Engng., 2, pp. 61-81.
[6] O.C. Zienkiewicz, J. F. Lyness, D. R. J. Owen, ‘Three dimensional magnetic field determination using a scalar potential – a finite element solution’, I.E.E.E. Transactions of Magnetics, MAG-13, No. 5 (1977), pp. 1649-1656.
[7] Ergatoudis, I., Irons, B. M. and Zienkiewicz, O. C. (1968) Curved isoparametric quadrilateral elements for finite element analysis. Int. J. Solids Struct., 4, 31-42.
[8] Private communication with T. J. R. Hughes, 2009.
[9] O.C. Zienkiewicz and M. Watson, ‘Some creep effects in stress analysis with particular reference to concrete pressure vessels’, J. Nuclear Engng. and Design, 4 (1966), pp. 406-412.
[10] O. C. Zienkiewicz, S. Valliappan and I. P. King, ‘Elasto-plastic solution of engineering problems; initial stress, finite element approach’, Int. J. Num. Meth. Engng, 1 (1969), pp. 75-100.
[11] O. C. Zienkiewicz and G. C. Nayak, ‘A general approach to problems of plasticity and large deformation’ in J. Whiteman (ed.), The Mathematics of Finite Elements and Application (Academic Press, 1973), pp. 1-37.
[12] G. C. Nayak and O. C. Zienkiewicz, ‘Note on the alpha-constant stiffness method for the analysis of non-linear problems’, Int. J. Num. Meth. Engng,4 (1972), pp. 579-582.
[13] G. C. Nayak and O. C. Zienkiewicz, ‘Convenient form of stress invariants for plasticity’, Proc. A.S.C.E., J. of Struct. Div., 98, ST4 (1972), pp. 949-953.
[14] O. C. Zienkiewicz and I. C. Cormeau, ‘Visco-plasticity; plasticity and creep in elastic solids A unified numerical solution approach’, Int. J. Num. Meth. Engng., 8 (1974), pp. 821-845.
[15] O. C. Zienkiewicz and P. N. Godbole, ‘Flow of plastic and visco-plastic solids with special reference to extrusion and forming processes’, Int. J. Num. Meth. Engng, 8 (1974), pp. 3-16.
[16] O. C. Zienkiewicz, C. Humpheson and R. W. Lewis, ‘Associated and non-associated visco-plasticity and plasticity in soil mechanics’, Geotechnique, 25 (1975), pp. 671-689.
[17] M. B. Kanchi, O. C. Zienkiewicz and D. R. J. Owen, ‘The visco-plastic approach to problems of plasticity and creep involving geometric nonlinear effects’, Int. J. Num. Meth. Engng., 12 (1979), pp. 169-181.
[18] O. C. Zienkiewicz. and P. Bettess, ‘Infinite elements in the study of fluid-structure interaction problems. Computing Methods in Applied Sciences, Lecture Notes in Physics’, Volume 58 (Berlin: Springer-Verlag, 1975), pp. 133-172.
[19] O. C. Zienkiewicz, D. W. Kelly, and P. Bettess, ‘The Sommerfeld (radiation) condition on infinite domains and its modelling in numerical procedures. Lecture Notes in Mathematics’, Volume 703 (Berlin: Springer-Verlag, 1977), pp. 169-203.
[20] O. C. Zienkiewicz, G. C. Nayak and D. R. J. Owen, ‘Composite and ‘overlay’ models in numerical analysis of elasto-plastic continua’, in A Sawczuk, ed., Foundations of Plasticity. (Noordhoff Press, 1972), pp. 107-122.
[21] J. C. Heinrich, P. Huyakorn, A. R. Mitchell. and O. C. Zienkiewicz. ‘An ‘upwind’ finite element scheme for two-dimensional convective transport equation’, Int. J. Num. Meth. Engng., 11 (1977), pp. 131-143.
[22] J. Z. Zhu and O. C. Zienkiewicz, ‘Adaptive techniques in the finite element method’, Communications in Applied Numerical Methods, 4 (1988), pp. 197-204.
[23] M. Ainsworth, J. Z. Zhu, A. Craig, and O. C. Zienkiewicz, ‘Analysis of the Zienkiewicz-Zhu a-posteriori error estimator in the finite element method’, Int. J. Num. Meth. Engng., 28 (1989), pp. 2161-2174.
[24] O. C. Zienkiewicz and J. Z. Zhu, ‘Superconvergent patch recovery and a-posteriori error estimation in the finite element method, Parts I and II’, Int. J. Num. Meth. Engng., 33 (1992), pp. 1331-1382.
[25] R. Lőhner, K. Morgan and O. C. Zienkiewicz, ‘Domain splitting for an explicit hyperbolic solver’, Comp. Meth. Appl. Mech. Eng., 45 (1984), pp. 313-329.
[26] O. C. Zienkiewicz, R Lőhner, K. Morgan and J. Peraire, ‘High speed compressible flow and other advection dominated problems of fluid mechanics;. Vol. 6, Ch. 2’ in R. H. Gallagher et al. (eds), Finite Elements in Fluids (John Wiley and Sons, 1986), pp. 41-88.
[27] D. Lodge, Small World, (Penguin Books, 1984).
[28] O. C. Zienkiewicz, R. L. Taylor, J. Z. Zhu and P. Nithiarasu, The Finite Element Method (Butterworth-Heinemann, 2005).
