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The finite element method has become the most powerful tool for structural analysis. During the last decades, the method has matured to such a state that it is massively used in practical engineering for the solution of broad range of problems starting from linear elasticity up to highly non-linear transient simulations of the behaviour of real materials and complex structures. However, it is often applied without good understanding of the method's background, leading to incorrect results and inadequate design, possibly causing damage or failure of the structure. A good way to prevent these undesirable effects is to check the quality of the obtained solution. If the results do not meet the prescribed level of accuracy, the discretization of the problem must be adequately adjusted and the problem recalculated. This process of solution enhancement is called the adaptive analysis.

Quasi-brittle materials, such as concrete, rock, tough ceramics, or ice, are characterized by the development of large nonlinear fracture process zones. Modeling progressive growth of microcracks and their gradual coalescence leads to constitutive laws with softening, i.e., with a descending branch of the stress-strain diagram. In the context of standard continuum mechanics, softening leads to serious mathematical and numerical difficulties. The boundary value problem becomes ill-posed, and numerical solutions exhibit a pathological sensitivity to the finite element mesh.

The most advanced and potentially most efficient techniques ensure objectivity of the numerical results by enriching the standard continuum by supplying additional information about the internal structure of the material. Such regularization techniques can enforce a realistic and mesh-independent size of the region of localized strain. Therefore they are called localization limiters. A wide class of localization limiters is based on the concept of a nonlocal continuum [2,5,15,21], which was introduced into the continuum mechanics in the sixties and applied as a localization limiter in the eighties. A differential form of the nonlocal concept was exploited by various gradient models [9,20]. Nonlocal formulations were elaborated for a wide spectrum of models, including softening plasticity [3], damage models [5,15,17,21], smeared crack models [2], or microplane models [4].

A very natural goal of the adaptive finite element analysis is to calculate solution of the governing partial differential equation with uniformly distributed error not exceeding a prescribed threshold in the most economical manner. This is achieved by improving the discretization in areas where the finite element solution is not adequate. It is therefore essential to have a quantitative assessment of the quality of the approximate solution and a capability of discretization enrichment. The most common error estimators can be classified into two basic groups: (i) the projection-based estimators [24,25] and (ii) the residual-based estimators [1,6,10,16]. These error estimation strategies are well established for linear problems and many of them have been more or less successfully generalized for application in nonlinear problems. Nevertheless, most of them lose the sound theoretical basis when applied to nonlinear problems because the rely on properties that are valid only for linear problems.

There are three main directions of the adaptive discretization enrichment. The first one, a natural way for most engineers, is the h-version [7,8,11], which refines the computational finite element mesh while preserving the approximation order of the elements. The p-version [23] keeps the mesh fixed but increases hierarchically the order of the approximation being used. The hp-version [8,18,26] is a proper combination of h- and p-versions and exhibits an exponential convergence rate independently of the smoothness of the solution. However, its implementation is not trivial. Similarly, the treatment of higher order elements in the p-version is rather complicated, especially when nonlinear analysis is considered.

The aim of this paper is to show the integration of individual components of the h-adaptive analysis into a single framework that can be used for adaptive simulation of behaviour of materials with softening. The remainder of the paper is structured as follows. Section 2 describes the regularization of the adopted material model using the nonlocal continuum concept. In Section 3 the philosophy of the residual-based error estimation is recalled. The employed refinement strategy is described in Section 4. Section 5 then outlines the individual components of the h-adaptive analysis and their integration into a functional unit. The application of the presented adaptive framework is demonstrated on an example in Section 6. And finally, some concluding remarks are made in Section 7.

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Daniel Rypl