Finite Element Methods for Elasticity with Error‐Controlled Discretization and Model Adaptivity

Abstract

The essential topics of the finite element method for linear and finite elastic deformations of solids are presented in this chapter from both the mechanical and mathematical point of view. As a starting point, the nonlinear, and linearized theory of elasticity are derived in a rigorous way, followed by the classical variational principles of elasticity, which are the basis for the finite element method in its various forms. More precisely, the discrete variational approach of the (one‐field) Dirichlet minimization principle of the total potential energy is presented, followed by concise representations of the (two‐field) Hellinger–Reissner stationary dual‐mixed principle and the (three‐field) Hu–Washizu stationary mixed principle, including the main features of the associated finite element methods. The main objective of this chapter is the systematic treatment of error estimation procedures and adaptivity for the linearized and finite elasticity problem covering both global and goal‐oriented a posteriori error estimators. The three basic classes, that is, residual‐, hierarchical‐, and averaging‐type error estimators are presented and applied to a fracture mechanics problem as an example. A further challenging topic is the combination of error‐controlled adaptive finite element solutions with hierarchical model and dimension adaptivity of the underlying mathematical model, especially for thin‐walled structures where model expansion is necessary in subdomains with boundary layers and other disturbances.

Computational Phase‐Field Modeling

“…It is also possible to employ goal-oriented error estimates to control the accuracy in output quantities of interest (van der Zee et al, 2011;Mahnken, 2013). These are based on general frameworks as described in, e.g., (Becker and Rannacher, 2001;Stein and Rüter, 2004).…”

A fast, certified and “tuning free” two-field reduced basis method for the metamodelling of affinely-parametrised elasticity problems

“…Now, substituting equations (40) and 14into (42) and noting (31), we obtain the variational form that we will use to compute the stress field σ r,0 (µ):…”

“…We first show that the errors in quantities of interest can be written in terms of errors measured in energy norms, using standard adjoint techniques [39,40,41,42,22,43,44]. In turn, computable bounds can be obtained by substituting these exact error measures by the CRE.…”

A mixed finite element method for non‐linear and nearly incompressible elasticity based on biorthogonal systems

“…Although the formulation can easily be generalized to any hyperelastic material law, we present the formulation specifically for Mooney-Rivlin material law, which includes neoHookean material law as a special case [28]. We use standard linear finite elements to discretize the displacement variable whereas a Petrov-Galerkin discretization is employed to discretize the pressure variable.…”