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Early numerical computations in engineering

2001; American Society of Mechanical Engineers; Volume: 54; Issue: 6 Linguagem: Inglês

10.1115/1.1390522

2379-0407

Frithiof I. Niordson,

History and Theory of Mathematics

Frithiof I NiordsonFrithiof Niordson was born in Johannesburg, South Africa in 1922 of Swedish-Russian parentage. Most of his formative years were spent in Sweden, and he graduated from the Royal Institute of Technology (KTH) with a MSc in 1945. As he describes in his article, he was invited to serve as a replacement for Folke Odqvist, during a period of absence from KTH of his teacher and internationally known professor. This was the beginning of his own prominent career in Scandinavia.Shortly after Odqvist’s return to Stockholm, Niordson departed for Brown University to obtain a PhD in solid mechanics under William Prager. This was the Golden Era of mechanics at Brown, as Niordson and many of his fellow graduate students from that period are fond of claiming. Indeed, during his period at Brown, Niordson established friendships with many colleagues who would later spend sabbatical periods in Copenhagen, where Niordson was to assume a professorship at the Technical University of Denmark (DTU) in 1958. In relatively short order, Niordson established an active research group in Denmark specializing in dynamics and vibrations, shells, and optimization of structures. He identified a number of outstanding young Danes amongst the students at DTU, and it was his effectiveness in fostering their academic careers that led to the position Denmark enjoys in the mechanics community today. Niordson’s leadership role extended far beyond his solid mechanics department and the mechanical engineering faculty at DTU which he led for more than 20 years. He served for many years on the executive committee of IUTAM, serving as its president from 1976-80. He organized the highly successful International Congress in 1984 held in Copenhagen. In recognition of his contributions to mechanics in Scandinavia and to his leadership role in the international community, the American Society of Mechanical Engineers bestowed Honorary Membership on Niordson. Although retired from his professorship at DTU, he remains an active member of the department pursuing his current research interests. He recently published a paper on accurate equations for computing shell vibrations with his youngest son, Christian, who is in the process of finishing his PhD in mechanics.Powerful computers have completely changed the world today, not in the least in engineering. Analysis by finite element programs is routine for almost every engineer graduating now. Experimental methods like photoelasticity and brittle coating are all but forgotten. Analytical methods are of course still the top preference, but they are too difficult to use and applicable only in very simple cases. How did it all come about? My early career started in Sweden. I graduated from the Royal Institute of Technology in 1945 and after a compulsory service in the army, which lasted two years, on the advice of my tutor Folke Odqvist went to his old friend and colleague Bill Prager at Brown University to obtain a PhD in applied mathematics. Back home, I had the good fortune to be asked by the Royal Institute to replace Folke Odqvist temporarily as professor for a year, while he spent this time in the USA as a guest lecturer. The duties (and the salary) were for a full professor. I was very young at the time and the position carried much prestige. It also carried the responsibility for a group of young people whose job was to find solutions to a statically indeterminate system of a new fighter airplane for the Swedish Air Force. For each small change in the construction of the wing, they had to solve numerically a set of about 20 linear equations with the same number of unknowns. At that time, we had only mechanical adding machines at hand, and it took about three weeks for a group of six men to solve the problem! After a year as a temporary professor, I started my own consulting company. There were any number of jobs available at the time. Just then the first computer was established at the old premises of the Royal Institute, BESK—the Binary Electronic Sequence Calculator. It could store 4096 numbers or orders (your choice) of 80 bits (not bytes!), could handle all numbers between –1 and +1, had no software and punched paper tape as input and output. BESK had no transistors and special cooling equipment was needed to remove the heat produced by all vacuum tubes. For anyone today, it would be an unbelievable monster, but for me it was like a gift from heaven; a tool for solving engineering problems. I attended the very first programing course and learned the hexadecimal machine language, the only existing one at the time. With the help of BESK, I could solve efficiently some of the problems urgently needed by the Swedish industry. To do so, elementary software had to be developed. First of all I had to make a program for floating numbers, so that the tedious scaling to numbers between –1 and +1 became unnecessary. Secondly, trigonometric functions were necessary and primitive subroutines had to be be developed. An effective program for computing critical speeds of rotors was the first to appear. A fellow student of mine, Curt Nicolin, was at the time head of the newly established gas turbine department of STAL, later ABB, Asea Brown Boverie, and on his advice the company hired me as a consultant. At the time critical speeds of rotors were evaluated graphically, a very primitive and inaccurate method, that was also limited to simply supported rotors. Although not employed by STAL, I became a member of a team of engineers, whose first duty was to develop a jet engine for the Swedish Air Force. Curt Nicolin was the man to do it, and in a short time he and his team succeeded to construct an engine in world class with a drag force of about 40000 N. The air force immediately ordered the design of a much stronger jet engine and the team at STAL produced the first Swedish double rotor engine, the Glan. In spite of the success of our construction and much to the regret of the whole team, the Air Force made a deal with the Rolls Royce Company in UK, and the Swedish production of jet engines was abandoned. The investment in the design was appreciable, and STAL quickly redesigned the Glan to a stationary power plant, a product that was a bestseller for STAL for more than 40 years! While working for STAL, I developed a number of computational programs which were used for analyzing rotors, turbine blades, turbine disks, and pipe systems. But my main interest in analytical solutions was always there. Programing has never been an easy task, and certainly not with BESK in the beginning. A real number was represented by 80 bits, the first one giving the sign and the remaining 79 the “decimals” in binary representation. Adding two positive numbers could give an overflow with a meaningless result. Multiplying numbers repeatedly resulted in smaller and smaller numbers the accuracy always being less than given by the smallest number. Division was even trickier. Do- loops were constructed with a jump condition but care had to be taken not to make them too short, since that could corrupt the memory! The reason being that the fast memory was a so called Williams-memory consisting of an appreciable amount of cathode ray tubes, in which each bit was represented by a very small dot or ring, which were “worn down” if not given sufficient time to regenerate between readings. The 80 bits could also represent an operation of 16 bits (such as adding to the accumulator) plus an address of 64 bits to the memory. Due to the very limited size of the memory the smart programmer used such computer orders as constants whenever possible! Automatic computing was ideal for iterations. Problems regarding natural frequencies of beams and critical speed of rotors were then usually solved (if not graphically!) by a series of successive iterations, the so-called Stodola method, which was easily translated into a computer program. Among the first problems as a consulting engineer, I was asked to investigate the vibration of tubes with flowing water for a company responsible for the construction of the Aswan Dam in Egypt. A summary of the paper was presented at the IUTAM Congress in Istanbul in 1952. But questions like “how can we change the shape of this rotor to increase its critical speed, without changing its weight” led to investigations of optimal design. By 1958, I was appointed full professor at the Technical University of Denmark, but I continued my consulting work for STAL, and my first paper related to this subject was “On the Optimal Design of a Vibrating Beam” in 1965. Optimal design became a rich and rewarding field of research. A long standing problem of the optimal shape of a column, subjected to a buckling load, was attacked by JL de Lagrange in 1770, but his solution was wrong, and JB Keller presented the correct solution in 1960. Professor Clifford Truesdell of Johns Hopkins University, who at that time was a well-known critical scientist, praised Keller’s work very appropriately, although it appeared afterwards that the problem had already been solved correctly by the German scientist, T Clausen, in 1853. Joseph Keller visited my department in Denmark several times, and we even wrote a paper together on “The Tallest Column.” Of course, one-dimensional problems concerning beams, columns and rotors with different boundary conditions and under different loading and response dominated the entire field in the beginning. Much attention was given to the singular behavior of the solutions at the endpoints. Also optimal trusses were given their attention. A very early paper by a student of mine, Peter Fleron, who investigated the optimal design of power line transmission towers, was heavily criticized by colleagues being used to design such trusses, who failed to understand that a lower bound is valuable, even if the optimal solution is unacceptable for esthetical or practical reasons. Pauli Pedersen extended the method to shape optimization for two- and three-dimensional trusses, also taking buckling into consideration, and Niels Olhoff to rectangular plates. But the smooth solution for the plate could only be a local optimum since any small amount of material could be remodeled into stringers lowering the compliance or raising the frequency. Deeper investigations showed that when formulated as a variational problem, there was no global optimum for plates. Keng-Tung Cheng and Olhoff wrote several papers on this subject with the somewhat unorthodox procedure of using the Kirchhoff plate theory for a plate “where the plate thickness is allowed to have an infinite number of discontinuities.” I always believed that this was not a well-posed problem and that by introducing a condition, which maximized the slope of the plate thickness, one would obtain a well-posed problem in which Kirchhoff’s plate theory was applicable. For circular plates I could show, that if such a maximum slope was prescribed, it was effective everywhere. George Rozvany has used this condition extensively in his work. The next step in optimization is to optimize the topology of a structure. Already much work has been done there, and the field is quickly expanding. Closely connected with optimal design problems is the inverse eigenvalue problem, the problem to determine a potential function from a given spectrum of eigenvalues. In engineering it appeared when, for instance, a rotor must have prescribed critical speeds, as was the case with Glan mentioned above. I published “A Method for Solving Inverse Eigenvalue Problem,” and this method was used by STAL in the design process. Of course, physicists had studied this problem long ago but in a quite different context. The extension of the method to a rectangular plate was rather trivial, but it could also be extended to find the boundary of a membrane with a given frequency spectrum. As a rather academic exercise John Hutchinson and I wrote a paper together on “The Harmonic Drum.” Early computational work was hampered by the severe limitations of the early computers, but as time went on, they became more efficient with fewer limitations. Today computational mechanics is a field in its own right with journals and congresses outnumbering its counterparts in Theoretical and Applied Mechanics by far. When I was a student at the Royal Institute, the world war was going on, and Sweden was a neutral country in a very dangerous position. The much neglected Swedish defense was now boosted, and the building of submarines became suddenly very important. Folke Odqvist helped the Swedish Navy to analyze the stability of the load carrying hulls, which consisted of cylindrical and truncated conical shells. Buckling of cylindrical shells was at that time well understood, but there was nothing about conical shells. Under his supervision I wrote my thesis on “Buckling of Conical Shells” in 1945, a paper I would hardly accept from my students today, but which in spite of its rather primitive analysis, was used by the Navy, nothing better being available. In 1943, a family friend of ours, a major in the reserve, was doing military service in Stockholm far away from his home and family. He felt lonely and suggested that we meet in the evening in his hotel room to study together AS Eddington’s book, The Mathematical Theory of Relativity. We met regularly and read the book together line by line, making sure that we understood every equation. I learned a lot from this and got a familiarity with tensor notations and non-Euclidean geometry, from which I have benefited all my life. In shell theory it became very useful a few years later. The courses I offered on shell theory at the University were based on the mathematics I learned from Eddington’s book. My interest in shell theory was awakened, and I attended the first IUTAM symposium on shell theory, organized by Warner Koiter 1959 in Delft and was asked to organize the Second IUTAM Symposium on the Theory of Thin Shells in Denmark in 1967. Both meetings were very successful, almost completely dominated by Warner Koiter. Research in shell theory was at that time a top subject and many excellent scientists contributed to it. Computers had not yet reached the stage when three-dimensional problems in elasticity could be attacked successfully, and two-dimensional finite element analysis of shell structures rather failed. The central problem was, of course, that the Kirchhoff plate theory and the Love-Kirchhoff shell theory were in contradiction to the well established theory of elasticity and any amount of ingenuity used to patch up the discrepancies by including effects of shear, rotational inertia, etc, to improve the accuracy was of no avail. Bernard Budiansky and Lyell Sanders wrote an excellent paper on the “best” shell theory, implying the sad fact that there was not a “correct” shell theory. This bothered me very much and I participated like many of my colleagues in trying to find a remedy. At the very end of the century, I succeeded in deriving a theory for cylindrical shells based on first principles only, but at that time interest in shell theory had faded considerably. Ironically, to convince the remaining established community of shell experts of the correctness of a mathematically derived shell theory, it was now necessary to “prove” its validity with examples solved by three-dimensional finite elements! Of course, in the sixties this possibility did not exist. On the other hand, this may not be interpreted in such a way that three-dimensional finite element solutions are on the verge of being substituted for plate or shell theory in engineering analysis. Like beam theory, plate and shell theories are—and will probably remain—necessary tools for engineers.