Goal-oriented error estimation and h-adaptive finite elements for hyperelastic micromorphic continua

摘要

We develop a goal-oriented finite element method for a class of micromorphic hyperelasticity problems, where size effects are taken into account by an enriched kinematics. Upon introducing a notion of generalized solutions, an abstract weak formulation that is convenient to error estimation is established along with its finite element discretization. Based on duality techniques, exact error representations aiming at a user-defined quantity of interest are derived. The dual problem is first introduced in a secant form, and subsequently linearized and discretized. The resulting discretizations of both primal and dual problem are shown to be consistent, thus theoretically ensuring an optimal convergence order. Some relevant numerical aspects are discussed in detail. In combination with a patch recovery technique avoiding nonlinear computations, an efficient error estimator is developed to guide a greedy adaptive mesh refinement algorithm. The effectiveness of the resulting algorithm is investigated by several numerical examples.