**1997 John von Neumann Award Winner**

for pioneering work*"in recognition of contributions
to the field of Computational Mechanics"*

** **Also in USACM Bulletin Volume 10, Number 2, November 1997

**"Recollections of My Education in the Finite Element Method"**

by T.J.R. Hughes

As I reflected on this award, I thought about the field of computational mechanics, my initial exposure to it, and how I was influenced by the old, and not so old, masters. I view myself as a second generation computational mechanician, having been first exposed to the field in 1967. I learned finite elements from the generation of individuals who created the field. I never took a course in the subject.

It is difficult to say when and where the field started, although it is clear that important papers were published in the 1940's. By the mid-1950's the first landmark papers on structural mechanics had appeared. In particular, I may mention the famous series of articles by Argyris and Kelsey in Aircraft Engineering during the period 1954-55, which was later republished as a monograph entitled Energy Theorems and Structural Mechanics, and the classic paper "Stiffness and Deflection Analysis of Complex Structures," by Turner, Clough, Martin and Topp.

In 1967 I was working at the Electric Boat Division of General Dynamics where Hrennikoff framework analogies where being employed to analyze shell structures. I was expressing my feelings of discomfort to my colleagues about the ad hoc way inertial properties were being derived and I recall another young engineer, Hugh Davidson, mentioned that there was something called the "finite element method" in which a "consistent mass matrix " could be derived. (Did those words ever change my life!) My curiosity was piqued, but not a whole lot was known about the finite element method by my colleagues, so I set out to educate myself. It seemed that the only possible way to do this was to read the literature as no courses were offered that I was aware of. Tackling the literature proved very difficult. I studied Argyris and took away from his works the underlying sense of geometrical beauty that guided his thinking, but I really did not understand the essentials. I had similar difficulties with almost all the papers I initially encountered.

During this time, it came to my attention that at an AIAA conference in New York City, a single session of papers was devoted to matrix methods of structural mechanics. I attended that session and listened to papers that I did not understand. I recall one of the speakers was talking about energy methods and elicited discussion from another individual in attendance, Dick Gallagher. The comments amounted to an impromptu but remarkably precise and authoritative discourse on energy methods. I felt I learned something and all in attendance seemed to agree that these remarks were incisive. Even more importantly, it was clear that at least one individual really understood what to me at that time was a very mysterious subject, and it gave me hope that I too could eventually master it.

Despite the initial frustrations, later on I encountered two works that were remarkably clear: The first was a paper by Ray Clough which appeared in the book Stress Analysis, edited by Zienkiewicz and Holister. After reading this paper I really felt I had developed a rudimentary understanding of the essentials. It was the first paper that I read that I felt I fully understood.

The second work was the first text on finite elements by Olek Zienkiewicz. When this little book appeared in 1967, I immediately ordered it and, upon receiving it, read it thoroughly. When I was done, I felt, at last, I had achieved some level of understanding and I was on my way! Nevertheless, from the current vantage point, I would suggest that these works represented perhaps an over simplification, but one that was certainly beneficial to me at the time, given my limited background. (It has been said that a little inaccuracy saves tons of explanation.)

After two years of working on, and continuing my study of, finite elements, I decided to pursue a Ph.D. During this period, in which I published my first research papers with Henno Allik, I had become profoundly influenced by Tinsley Oden's work wherein he pioneered the view that continuum mechanics and mathematics were the essential foundations of finite element research. Thus I enrolled at what I perceived to be the mecca of finite elements, the University of California at Berkeley, where I set out to study mechanics and mathematics with a view to come back to finite elements after amassing a toolbox of new technical skills. I studied and worked with many outstanding faculty members in the process and had a particularly fruitful mathematical collaboration with Jerry Marsden. The torch bearers of finite elements at Berkeley were Ed Wilson and Bob Taylor. I learned from Ed and Bob how software architecture and algorithms were inextricably interlinked with fundamental theory, and I enjoyed a particularly intense and productive period of research with Bob. Tinsley cited my work in stabilized methods in his introduction. The seeds of these techniques were planted during the time I worked with Bob. (Another antecedent of stabilized methods is the seminal work of Von Neumann and Richtmyer on the computation of shock waves in gas dynamics, so I feel some direct empathy with the individual whose name adorns this award.) During my Berkeley years I also studied the delightful book of Strang and Fix, which provided me with an understanding of the mathematical basis of the finite element method.

Today, in way of contrast, most university students in mechanical and civil engineering take courses in finite elements at the graduate level. There is a current trend to introduce finite elements even earlier at the undergraduate level. At Chamlers University in Gothenburg, Claes Johnson and colleagues have taught a course covering the rudiments of calculus and finite elements simultaneously at the freshman level. Apparently, this experiment has proven successful. I would not be surprised if, in the not too distant future, the method is introduced in high school curricula!

It is unlikely that current and future generations will learn finite elements the way I did. It was not an efficient way to learn, but it did have the advantage that one got to know the minds of the subject's originators. On becoming active in the field, I met and interacted with all of them in one way or another, and continue to do so. That has been perhaps the best part of my education. The quality of those lessons has provided continual inspiration to my work in this fascinating field, if "work" is the right word. This field is fun. In fact, it is more fun than fun!

Thank you very much for your kind attention.