Goal-oriented h-type adaptive finite elements for micromorphic elastoplasticity

Abstract

Generalized continuum theories become necessary for many scenarios like strain localization phenomena or size effects. In this work, we consider a class of higher order continua, namely micromorphic continua, where the kinematics is enhanced by means of a microstructure undergoing an affine micro deformation. The increasing number of the degrees of freedom in such a theory clearly motivates an application of adaptive methods. For linear elastic micromorphic problems, we have shown the consistency of the resulted dual problem, ensuring an optimal convergence order, in Ju and Mahnken (2017). In this work, we extend the framework of goal-oriented adaptivity to time-dependent problems, i.e. plasticity problems, where a backwards-in-time dual problem is crucial to account for the error accumulation over time caused by both error generation and error transport. Our focus is limited to an adaptive control of spatial discretization errors of the finite element method (FEM), while the temporal discretization errors are not considered for simplicity. Based on duality techniques, we derive exact error representations. For practice, four computable error estimators are proposed, where two different ways to obtain enhanced solutions are considered, and where additionally an approximate forwards-in-time dual problem neglecting error transport is introduced. By means of certain localization techniques, these estimators are used to drive an adaptive mesh refinement algorithm. Their effectiveness is shown by several numerical examples based on a prototype model.