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History of the Finite Element Method – Mathematics Meets Mechanics – Part II: Mathematical Foundation of Primal FEM for Elastic Deformations, Error Analysis and Adaptivity

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The History of Theoretical, Material and Computational Mechanics - Mathematics Meets Mechanics and Engineering

Part of the book series: Lecture Notes in Applied Mathematics and Mechanics ((LAMM,volume 1))

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Abstract

This chapter treats the history of mathematical foundation of primal FEM, especially a posteriori error estimates and adaptivity, based on functional analysis in Sobolev spaces. This is of equal importance as the creation of multifarious computational methods and techniques in engineering and computer sciences. BVPs for linear elliptic PDEs, mainly the Lamè equations for linear static elasticity are treated.

Bounded residual explicit and various implicit error estimators of primal FEM were mainly developed by Babuška and Rheinboldt (1978), Bank and Weiser (1985), Babuška and Miller (1987) and Aubin (1967) and Nietsche (1977).

Mechanically motivated explicit and implicit error estimators were created by Zienkiewicz and Zhu (1987), using gradient smoothing of the C 0- continuous displacements and stress recovery for which convergence and upper bound property were proven by Carstensen and Funken (2001).

A variant of implicit a posteriori error estimators is the error of consitutive equations by Ladevèze et al. (1998). Equilibrated test stresses on element and patch levels are required, Ladevèze, Pelle (2005). Gradient-free formulations, e.g. by Cottereau, Díez and Huerta (2009), are also competitive. Generalizations of a priori and a posteriori error estimates, using the three-functional theorem by Prager and Synge (1947), are very useful.

Goal-oriented error estimators for quantities of interest (as linear or nonlinear functionals, defined of closed finite supports) are of practical importance, Eriksson et al. (1995), Rannacher and Suttmeier (1997), Cirac and Ramm (1998), Ohnimus et al. (2001), Stein and Rüter (2004) and others. Textbooks by Verfürth (1996, 1999, 2013), Ainsworth and Oden (2000), Babuška and Strouboulis (2001), are available.

Verification with prescribed error tolerances is realized with the above cited bounded error estimators and related discretization adaptivity, provided that the solution exists in the used test space.

Moreover, model validation requires model adaptivity of the adequate physical and mathematical modeling which additionally needs experimental verification, requiring a posteriori model error estimators combined with discretization error estimators.

Model reductions, e.g. for reinforced laminates, were treated by Oden (2002), and model expansions, e.g. for 3D boundary layers of 2D plate and shell theories by Stein and Ohnimus (1997), and Stein, Rüter and Ohnimus (2011).

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Stein, E. (2014). History of the Finite Element Method – Mathematics Meets Mechanics – Part II: Mathematical Foundation of Primal FEM for Elastic Deformations, Error Analysis and Adaptivity. In: Stein, E. (eds) The History of Theoretical, Material and Computational Mechanics - Mathematics Meets Mechanics and Engineering. Lecture Notes in Applied Mathematics and Mechanics, vol 1. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-39905-3_23

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