Mathematical and Historical Reflections on the Lowest-Order Finite Element Models for Thin Structures;Research rept

摘要

We discuss the mathematical theory and history of the lowest-order linear and bilinear finite element models for beams, arches, plates and shells. The finite element formulations considered are based on the non-asymptotic Timoshenko beam and Reissner-Mindlin plate models and the analogies of these models for arches and shells. We follow some of the historical roots of the successful linear and bilinear elements, to find various physical justifications for formulations that now may be understood as purely numerical modifications within the usual energy principle. The simplified mathematical theory of such formulations is outlined, first in cases of the beam, arch and plate. We finally focus on the still challenging and largely open problems arising in the modeling of shell deformations.