Milestones of Direct Variational Calculus and its Analysis from the 17th Century until today and beyond – Mathematics meets Mechanics – with restriction to linear elasticity - Stein, Erwin (Computer Assisted Methods in Engineering and Science, 2018)
This treatise collects and reﬂects 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 Lamé-Navier partial diﬀerential 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 algorithm 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 Ω⊂R3.