Keywords: adaptive analysis, damage, localization, nonlocal continuum.

This paper describes a mesh-adaptive technique developed for regularized softening continuum models of the integral type. The approach is based on an error indicator for the inelastic part of the domain of interest, combined with an error estimator for the elastic part. A truly adaptive procedure is developed, with mapping of displacements and internal variables, which allows to continue the analysis from the currently reached state, instead of restarting the analysis from the very beginning after each mesh refinement. The application of the proposed algorithms will be illustrated by several examples.

The inelastic behavior of quasibrittle materials (such as concrete, rock, tough ceramics or ice) under tension is caused by diffuse microcracking that later localizes in relatively narrow zones, referred to as the fracture process zones. The localization of strain and damage leads to a gradual development of macroscopic stress-free cracks. Despite a considerable progress in the past two decades, theoretical modeling and computational resolution of the localization process up to structural failure is still a challenging issue.

Continuum-based modeling of the progressive growth of microcracks and their coalescence exploits constitutive laws with strain softening. In the context of standard continuum mechanics, softening leads to serious mathematical and numerical difficulties. The boundary value problem becomes ill-posed, and the numerical solution exhibits a pathological sensitivity to the computational grid. The use of regularization techniques enforcing a mesh-independent profile of localized strain is needed. A wide class of localization limiters is based on the concept of a nonlocal continuum.

An accurate resolution of bands of highly localized strain typically requires very fine computational grids. The efficiency of the analysis can be greatly increased by using adaptive techniques that adjust the mesh during the simulation, depending on the intermediate localization pattern and its evolution. The proposed approach is based on -adaptivity, i.e., on the adjustment of the element size, keeping the order of the elements constant (and usually low).

The basic blocks of the general adaptive procedure include an error estimator or indicator, a corresponding remeshing criterion, transfer algorithms for primary unknowns and internal variables, and a mesh generator interface. After reaching the equilibrium state after each load increment and updating the solution state, the "a posteriori" error estimation is performed, in order to identify the regions that need a mesh refinement. The proposed approach is based on a damage indicator defined as the maximum principal value of the damage tensor. The indicator is initially equal to zero for a virgin material and grows to one as the material is subjected to increasing tensile cracking. This heuristic criterion is combined with the Zienkiewicz-Zhu error estimator, which provides an estimate of the error distribution in the elastic part of the domain. Based on the damage indicator and the error estimator, a remeshing criterion determines the further strategy. If the mesh in the damaged regions is sufficiently fine, the analysis continues with the next load increment on the current mesh. If the required mesh density is higher than the current one, a new spatial discretization is generated.

After generating a new discretization, the corresponding data structure corresponding to the new mesh is created and the transfer of displacements and internal variables from the old mesh to the new one is performed. The mapping of primary unknowns (displacements) is done first, usually using the shape function projection, and this is followed by the transfer of certain internal variables. The type and meaning of these internal variables depend on the material model. The internal variables transfer algorithm is one of the crucial parts. Its stability and diffusion properties must be carefully checked, since they significantly influence the overall solution stability. For constitutive models based on the continuum damage theory, a suitable and computationally simple transfer algorithm is proposed.

After the internal history has been mapped, it is used together with the strain vector computed from the mapped displacements to update the internal state of each integration point on the new mesh (to achieve local consistency). When the transfer is finished, the old discretization is deleted and an equilibrium iteration process is invoked in order to bring the mapped configuration into global equilibrium. Afterwards, the solution continues with the next load increment.

All the aspects of the proposed methodology are illustrated by several examples of adaptive computations in two and three dimensions. These examples demonstrate the ability of the proposed methodology to accurately describe the nonlinear fracture processes in quasibrittle materials.

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