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Eighty Years of the Finite Element Method (2022)
This document presents comprehensive historical accounts on the developments of finite element methods (FEM) since 1941, with a specific emphasis on developments related to solid mechanics. We present a historical overview beginning with the theoretical formulations and origins of the FEM, while discussing important developments that have enabled the FEM to become the numerical method of choice for so many problems rooted in solid mechanics.
On May 3rd, 1941, the same year that Hrennikoff published his paper, R. Courant of New York University delivered an invited lecture at a meeting of the American Mathematical Society held in Washington D.C. on his numerical treatment using a variational method to solve a second order PDE, which arises from Saint–Venant’s torsion problem of a cylinder. A significant breakthrough in computational fracture mechanics and FEM refinement technology came in the late 1990s, when Belytschko and his co-workers, including Black, Moes, and Dolbow, developed the eXtended finite element (X-FEM) (see [ 160, 161], which uses various enriched discontinuous shape functions to accurately capture the morphology of a cracked body without remeshing. To further reduce the FEM computational burden in multiscale analysis, Liu et al. [ 190] applied the unsupervised machine learning techniques, such as the k-mean clustering method to group the material points during the offline stage and obtain the final solutions by solving the reduced-order Lippmann–Schwinger micromechanics equations.
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