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The Finite E lement Method
The finite element method is the dominant computational tool of contemporary structural
and solid mechanics. It permits difficult problems of complicated geometry to be solved
with relative ease. A number of finite element-based general purpose computer programs
for analysis and design are available to the engineer. These programs can be used to solve
an array of difficult structural engineering problems. An introduction to finite element tech-
nology from the viewpoint of the structural mechanics philosophy of this topic is presented
in this chapter.
The finite element method has its origins in the matrix methods of structural analysis
(Chapter 5). Some four decades ago it was recognized to be an emerging viable com-
putational method for solid mechanics with applications to many problems of structural
analysis. Since then, a mathematical basis for finite elements has been developed and
applications have been expanded to include fluid mechanics, heat transfer, biomechan-
ics, geomechanics, and acoustics. This ever-widening applicability of the finite element
method is due in part to its common formulation based on the variational principles. Also,
one-, two-, and three-dimensional problems are handled in like fashion with a uniform
notation.
Technical histories of the finite element method are provided in Clough (1990), Felippa
(1994), and Gupta and Meek (1996). Biographies of some of the pioneers in the development
of finite element technology are available in Robinson (1985). The method can be considered
as a natural evolution of the standard methods of structural mechanics for frames modeled
as discrete elements, or as an approximate solution technique for continuum mechanics pro-
blems utilizing a regionally discretized model with assumed strain patterns for the regions.
The concept of regional discretization can be traced to the much earlier work of Courant 1
1 Richard Courant (1888-1972) was a German-born mathematician who studied with Hilbert at G ottingen. Begin-
ning with his doctoral thesis, he made several significant contributions to the calculus of variations. He authored
several topics, including Methods of Mathematical Physics with the first volume published in Germany and the
second when Courant was working in the United States. Courant suffered through difficult economic times in
Germany. His Prussian army service in World War I lasted more than 4 years. He was wounded early in the war,
but remained in the army until 1918. After a brief stint as a politician, he took a position at M unster and then
returned in 1920 to the University of G ottingen. With the support of the Prussian government and the Rockefeller
Foundation, he established the Mathematics Institute at G ottingen. The interference of the National Socialist gov-
ernment led to a breakup of the mathematics “club” in G ottingen, with Courant moving to Cambridge University
in 1933 and to New York University in 1934. In New York he collaborated with K.O. Friedrichs and J.J. Stoker, and
established the Institute of Mathematical Sciences, which now bears his name.
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