Abstract
Accurate simulations of lattice-based metamaterials is often limited less by constitutive modelling than by discretisation errors caused by singular stress fields. These localised errors can substantially affect relevant quantities of interest, including stresses, displacements, and reaction forces. This paper presents applicable goal-oriented adaptive finite element methods for the computationally efficient simulation of lattice structures in the framework of linear elasticity, implemented on top of the open-source software FEniCSx. The novelty of the contribution lies in three main aspects: first, the formulation of guaranteed goal-oriented error estimates based on equilibrated, H(div) conforming stresses without strong symmetry; second, the comparison of equilibration based error estimators forming either a guaranteed upper bound or are just asymptotically exact; and third a comparative study of adaptive finite elements methods and the underlying error estimates, specifically for lattice structures. Therefore, three estimators are derived, analysed, and implemented. Their performance is assessed through a series of representative benchmarks and by comparing uniform and adaptive mesh refinement strategies for lattice structures of increasing complexity, ranging from a small patch of unit cells to a meso-structured aircraft wing rib. In addition, the predictive capability of the proposed framework is validated against experimentally determined force–displacement data for two different auxetic lattice structures. The results demonstrate that the proposed adaptive schemes provide improved computational efficiency over uniform refinement while maintaining high accuracy, thereby supporting reliable numerical prediction and simulation-based design of architected structures in structural engineering.
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