Error-controlled adaptive goal-oriented modeling and finite element approximations in elasticity

In our earlier work [E. Stein, S. Ohnimus, Coupled model- and solution-adaptivity in the finite-element method, Comput. Methods Appl. Mech. Engrg. 150 (1997) 327–350; E. Stein, S. Ohnimus, Anisotropic discretization- and model-error estimation in solid mechanics by local Neumann problems, Comput. Methods Appl. Mech. Engrg. 176 (1999) 363–385; E. Stein, M. Rüter, S. Ohnimus, Adaptive finite element analysis and modelling of solids and structures. Findings, problems and trends, Int. J. Numer. Methods Engrg. 60 (2004) 103–138.], residual-based combined model and finite element discretization error estimators as a basis for adaptivity were derived that rely on the solutions of Neumann problems on the element level. In particular, these estimators control the errors obtained while expanding the model hierarchically from 1D- to 2D- and from 2D- to 3D-elastic theories in the (global) energy norm. In this paper, we extend both types of estimates to goal-oriented a posteriori error estimation within the framework of nonlinear structural mechanics. Structural model adaptivity is characterized by the fact that the dimensions, in general the constitutive equations and especially the related subdomains—and thus the test and solution spaces of the process-dependent models considered—are different. This has the important consequence that consistent prolongations of kinematic and static nodal quantities of finite elements of the coarser model “m” to those of the refined model “m + 1” are necessary. Therefore, such prolongation operators are discussed in detail. Furthermore, duality techniques are used for the model and discretization error estimates, using system and load-dependent quantities of interest. In this paper, we also consider different finite element meshes for the primal and the dual problem. Finally, some numerical examples demonstrate the expected effects, realizing verification and even validation in structural mechanics, i.e. approaching a new paradigm in Computational Mechanics for achieving reliability and efficiency.