Milestones of Direct Variational Calculus and its Analysis from the 17th Century until today and beyond- Mathematics meets Mechanics- with restriction to linear elasticity

This treatise collects and re?ects the major developments of direct (discrete) variational calculus since the end of the 17th century until about 1990, with restriction to classical linear elastomechanics, such as 1D-beam theory, 2D-plane stress analysis and 3D-problems, governed by the 2nd order elliptic Lam´e-Navier partial di?erential equations. The extension of the historical review to non-linear elasticity, or even more, to inelastic deformations would need an equal number of pages and, therefore, should be published separately. A comprehensive treatment of modern computational methods in mechanics can be found in the Encyclopedia of Computational Mechanics [83]. The purpose of the treatise is to derive the essential variants of numerical methods and algorithms for discretized weak forms or functionals in a systematic and comparable way, predominantly using matrix calculus, because partial integrations and transforming volume into boundary integrals with Gauss’s theorem yields simple and vivid representations. The matrix D of 1st partial derivatives is replaced by the matrix N of direction cosines at the boundary with the same order of non-zero entries in the matrix; ?/?xi corresponds to cos(n, ei),x = xiei , n = cos(n, ei)ei, i = 1, 2, 3 for ? ? R3. A main goal is to present the interaction of mechanics and mathematics for getting consistent discrete variational methods and from there – in a comparative way – the properties of primal, dual and dual-mixed ?nite element methods in their historical development. Of course the progress of proper mathematical analysis since the 1970s and 1980s is outlined, concerning consistency, convergence and numerical stability as well as a priori and a posteriori error estimates in the frame of Sobolev spaces for the mostly C 0 -continuous test and trial ansatz functions at element interfaces. This mathematical research followed after the more intuitive engineering developments since the 1950s, using the principle of virtual work and the principle of minimum of total potential energy. A proper ?nite element method has to regard the motto of this treatise: “mathematics meets mechanics”. Comparative numerical results are not included because numerous new calculations would have been necessary for getting usable comparisons for the various cited articles.