  This year marks the eightieth anniversary of the invention of the finite
element method (FEM). FEM has become the computational workhorse for
engineering design analysis and scientific modeling of a wide range of physical
processes, including material and structural mechanics, fluid flow and heat
conduction, various biological processes for medical diagnosis and surgery
planning, electromagnetics and semi-conductor circuit and chip design and
analysis, additive manufacturing, i.e. virtually every conceivable problem that
can be described by partial differential equations (PDEs). FEM has
fundamentally revolutionized the way we do scientific modeling and engineering
design, ranging from automobiles, aircraft, marine structures, bridges,
highways, and high-rise buildings. Associated with the development of finite
element methods has been the concurrent development of an engineering science
discipline called computational mechanics, or computational science and
engineering.
  In this paper, we present a historical perspective on the developments of
finite element methods mainly focusing on its applications and related
developments in solid and structural mechanics, with limited discussions to
other fields in which it has made significant impact, such as fluid mechanics,
heat transfer, and fluid-structure interaction. To have a complete storyline,
we divide the development of the finite element method into four time periods:
I. (1941-1965) Early years of FEM; II. (1966-1991) Golden age of FEM; III.
(1992-2017) Large scale, industrial applications of FEM and development of
material modeling, and IV (2018-) the state-of-the-art FEM technology for the
current and future eras of FEM research. Note that this paper may not strictly
follow the chronological order of FEM developments, because often time these
developments were interwoven across different time periods.
