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Milestones of Direct Variational Calculus and its Analysisfrom 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

Vol. 25, no. 4

141--225

This treatise collects and reflects the major developments of direct (discrete) variational calculussince the end of the 17th century until about 1990, with restriction to classical linear elastome-chanics, 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 inthe Encyclopedia of Computational Mechanics.The purpose of the treatise is to derive the essential variants of numerical methods and algorithmsfor discretized weak forms or functionals in a systematic and comparable way, predominantly usingmatrix 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 derivativesis replaced by the matrix N of direction cosines at the boundary with the same order of non-zero entries in the matrix.