Exponential finite element shape functions for a phase field model of brittle fracture
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In phase ﬁeld models for fracture a continuous scalar ﬁeld variable is used to indicate cracks, i.e. the value 1 of the phase ﬁeld variable is assigned to sound material, while the value 0 indicates fully broken material. The width of the transition zone where the phase ﬁeld parameter changes between 1 and 0 is controlled by a regularization parameter. As a ﬁnite element discretization of the model needs to be ﬁne enough to resolve the crack ﬁeld and its gradient, the numerical results are sensitive to the choice of the regularization parameter in conjunction with the mesh size. This is the main challenge and the computational limit of the ﬁnite element implementation of phase ﬁeld fracture models. To overcome this limitation a ﬁnite element technique using special shape functions is introduced. These special shape functions take into account the exponential character of the crack ﬁeld as well as its dependence on the regularization length. Numerical examples show that the exponential shape functions allow a coarser discretization than standard linear shape functions without compromise on the accuracy of the results. This is due to the fact, that using exponential shape functions, the approximation of the surface energy of the phase ﬁeld cracks is impressively precise, even if the regularization length is rather small compared to the mesh size. Thus, these shape functions provide an alternative to a numerically expensive mesh reﬁnement.
CitationKuhn, C.; Müller, R. Exponential finite element shape functions for a phase field model of brittle fracture. A: COMPLAS XI. "COMPLAS XI : proceedings of the XI International Conference on Computational Plasticity : fundamentals and applications". CIMNE, 2011, p. 478-489.
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