History of the Finite Element Method - Mathematics Meets Mechanics - Part I: Engineering Developments

摘要

The birth of variational calculus and the principle of virtual work goes back to the 17th and 18th century, and the first draft of a discrete variational method with "elementwise" triangular shape functions was given by Leibniz (1697). First analytical studies were made by Schellbach (1851) and then, already with numerical results, by Rayleigh (1877). The mathematician Ritz (1909) marks the first discrete (direct) variational method for the linear elastic Kirchhoff plate, and the engineer Galerkin (1915) published his seminal article on FEM for linear elastic continua, postulating the orthogonality of the residua of equilibrium with respect to the test functions, but both, Ritz and Galerkin, used test and trial functions within the whole domain as supports. Courant (1943) was the first to introduce triangular and rectangular "finite elements" for the 2D-St.-Venant torsion problem of a prismatic bar (Poisson equ.), and Clough and his team (1956) published the first modern 2D-FEM for arrowed aircraft wings. Also Wilson, Melosh and Taylor with their important schools, e.g. Bathe and Simo, promoted in Berkeley the new discipline of "Computational Mechanics". Argyris (since 1959 in Stuttgart) and Zienkiewicz (since 1965 in Swansea), together with Irons e.g., developed primal FEM in a systematic way: hierarchical classes of FEs with different topologies and ansatz techniques for solid and fluid mechanics. Mixed finite elements, advocated for non-robust problems, are usually based on dual mixed variational functionals (yielding saddle point problems) by Hellinger (1914), Prange (1916) and Reissner (1950). Important elements came, e.g., from Cruceix, Raviart (1973), Raviart, Thomas (1977) and Brezzi, Douglas, Marini (1987). For numerical stability of saddle point problems, the Brezzi-Babuska global (infsup) stability conditions have to be fulfilled. Rather new extensions and variants of FEM are XFEM by Belytschko (1996) and SFM (Singular Function Method) by Fix, Strang (1973) and Grisvard (1992), as well as Isogeometric Analysis by Hughes (2005). This chapter and also chapter 23 are restricted to linear elastic solid and structural mechanics. The topics are designed not only in a descriptive way with verbal comments on the origins, main features and the importance of publications on FEM, but moreover, essential steps of continuous and discrete variational theory, finite element algorithms and error estimators as a basis for adaptivity are communicated as concentrated as possible. This seems to be advantageous for conceiving the subjects.