Keywords: adaptive FEM, quick solvers, crack growth.

Today, fracture mechanics forms an autonomous area of research in solid mechanics in order to explain phenomena of fracture, fatigue, and strength of materials. Although all fracture mechanics approaches cannot altogether be ascribed to crack type problems, the consideration and examination of the essential conditions that lead to crack propagation, crack deflection and crack arrest are of highly practical and theoretical interest. Generally, analytical solutions for most of these crack problems in finite body domains are not attainable. Thus, it is necessary to develop numerical techniques for strength analysis of cracked structures subjected to various kinds of loads. The finite element method (FEM) has been used to solve crack and crack propagation problems for over thirty years. However, crack propagation modeling still represents a complicated problem, and different specific definitions of crack propagation are used in the literature. The most difficult part is the modeling of the changeover from a continuous solid medium that has strong continuity requirements on the displacements and their derivatives, to final displacement jumps with new free surfaces inside the solid which should lead to well-posed problems. In addition, the numerical solution procedure for the corresponding boundary value problem with changing boundaries has to be as efficient as possible. Thus, it is necessary to have excellent solutions surrounding the tips, including the asymptotic behavior, where the fracture process occurs. Away from the cracks, the numerical solution does not require very high resolution. In fact, one needs adaptivity of the solution based on a posteriori error estimation together with effective capable solvers for the discrete solution system at each step of crack propagation. This matter is the field of our intentions within the given paper. In connection with adaptive mesh refinement and mesh coarsening, the hierarchical iterative solution technique represents an optimal solution algorithm. The necessary accuracy will be achieved automatically. This way, the benefits of adaptive hierarchical iterative solvers lie in the optimum effort with respect to a given accuracy for the specific problems considered, e.g., for crack propagation because an effective preconditioner can be constructed for this problem as well. This is the essential message of the given paper emphasizing the efficiency of the given approach.

Thus, with the help of modern adaptive finite element techniques we are able to predict the path of a slow moving crack quite well. The definition of the crack propagation conditions are based on classical approaches by means of the stress intensity factors and . In general, it is impossible to extract these factors from the usual numerical near tip solution without taking into consideration the special asymptotic behavior in the solution procedure and in the postprocessing stage. This is, in fact, the case, even though a very fine finite element net is reached after appropriate adaptive mesh refinement at the crack tip. Therefore, we use the J-integral concept for the numerical determination of the K-factors exploiting the path independence in order to execute the numerical calculations away from the tip. This is possible only for a generally curved crack propagation, when the crack propagation length scales may be assumed as straight lines, whereby the so-called two-dimensional interaction integral technique can be used.

Applying adaptive iterative hierarchical solvers, we are able to reverse each of the (up to 1000) intermediate linear systems (arising from the finite element discretization of an actual crack situation characterized by the K - factors) quickly enough. We present a hierarchical-basis-like preconditioner that fulfills this demand very naturally in connection with the adaptive procedure.

Based on the paper [1] we consider the numerical simulation of crack growth with the help of a series of adaptive finite element computations. Due to the use of hierarchical preconditioners in the preconditioned conjugate gradient solvers we are able to solve each of the resulting linear systems on a standing crack very efficiently (about 10 to 15 iterations on each mesh are typically, see [1]). The problem of crack growth requires a new technique whenever we try to avoid re-meshing after the propagation of the crack line. Re-meshing would destroy the hierarchical nodal information from adaptivity and lead to more inefficient solvers without hierarchical techniques.

The way out of this problem uses "double nodes" along the crack line combined with a domain decomposition-like preconditioner based on the transformation of the basis of the ansatz functions along the crack line.

We demonstrate the (near) constant number of iterations for longer cracks to reach about 30 for each consecutive mesh.

1

A. Meyer, F. Rabold, M. Scherzer, "Efficient Finite Element Simulation of Crack Propagation Using Adaptive Iterative Solvers", Communications in Numerical Methods in Engineering, 22, pp.93-108, 2006. doi:10.1002/cnm.799

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