An historical note on the finite element method

Abstract

SUMMARYSome comparisons between the finite element method and two early methods in the calculus of variations are presented.

Some historic comments on finite elements

“…In 1851, SCHELLBACH [18Sl] proposed a finite-element-like solution to Plateau's problem of determining the surface S of minimum area enclosed by a given closed curve. SCHELLBACH used an approximation St, of S by a mesh of triangles over which the surface was represented by piecewise linear functions, and he then obtained an approximation of the solution to Plateau's problem by minimizing S, with respect to the coordinates of hexagons formed by six elements (see WILLIAMSON [1980]). Not quite the conventional finite element approach, !~;;;tii as much a finite element technique as that of Some'say that there is even an earlier work that uses some of the ideas underlying finite element methods: LEIBNIZ himself employed a piecewise linear approximation of the Brachistochrone problem proposed by BERNOULLI in 1696 (see the historical volume, LEIBNIZ [1962]).…”

Historical corner column: A prelude to finite elements: The fruitful problem of the brachistochrone

“…Schellbach used an approximation, S h , of S by a mesh of triangles over which the surface was represented by piecewiselinear functions. He then obtained an approximation of the solution to Plateau's problem by minimizing S h with respect to the coordinates of hexagons formed by six elements (see [4]). This was not quite the conventional fi nite-element approach, but certainly as much a fi nite-element technique as that of Courant.…”

Analysis and prediction of residual stress in deep cold rolling of Ti-6Al-4V

“…A precursor of finite element analysis can be traced back to when Schellbach in 1851 broke down a surface into right triangles and used finite difference to determine the minimum area of a given closed curve [106]. FE in its present form was developed by Courant, when he calculated the torsional rigidity of a hollow shaft by splitting the cross section into triangles and determining a linear stress function φ for each triangle from a matrix of net-points (now known as nodes) [107].…”