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.
top
