Anisotropic Mesh Adaptation for Crack Detection In Brittle Materials

Abstract

The quasistatic brittle fracture model proposed by G. Francfort and J.-J. Marigo can be Γ-approximated at each time evolution step by the Ambrosio-Tortorelli functional. In this paper, we focus on a modification of this functional which includes additional constraints via penalty terms to enforce the irreversibility of the fracture as well as the applied displacement field. Secondly, we build on this variational model an adapted discretization to numerically compute the time-evolving minimizing solution. We present the derivation of a novel a posteriori error estimator driving the anisotropic adaptive procedure. The main properties of these automatically generated meshes are to be very fine and strongly anisotropic in a very thin neighborhood of the crack, but only far away from the crack tip, while they show a highly isotropic behavior in a neighborhood of the crack tip instead. As a consequence of these properties, the resulting discretizations follow very closely the propagation of the fracture, which is not significantly influenced by the discretization itself, delivering a physically sound prediction of the crack path, with a reasonable computational effort. In fact, we provide numerical tests which assess the balance between accuracy and complexity of the algorithm. We compare our results with isotropic mesh adaptation and we highlight the remarkable improvements both in terms of accuracy and computational cost with respect to simulations in the pertinent most recent literature.

The Benefits of Anisotropic Mesh Adaptation for Brittle Fractures Under Plane-Strain Conditions

“…The numerical experiments in Sect. 4 show that the proposed method is very stable and it allows us to reproduce all the previously obtained predictions on fracture development, in particular its directionality features. Additionally, we expect that our method, based on an extremely careful tuning of the anisotropic adaptation, outperforms significantly the ones used to achieve similar degrees of accuracy in previous studies.…”

“…Following Proposition 2.2 in [4], we can prove that condition 0 ≤ v ≤ 1 is automatically guaranteed for any critical point.…”

“…The following proposition states the main result of the paper and provides a variant on the anti-plane case addressed in [4].…”

“…Following [4], we relax the constraint in (2) with two penalization terms which lead us to rewrite the Plane-strain Ambrosio-Tortorelli elasticity functional as…”

“…In the approach of Süli et al [8], the adaptivity is driven exclusively by an a posteriori first order estimator, but only isotropic mesh refinement was considered. In our recent work [4], we tried to combine these two previous approaches, designing an appropriate a posteriori anisotropic first order estimator, leading to mesh coarsening far from the fracture and fine mesh elements exclusively very close to the crack path. Again this new method resulted being very efficient and effective, producing stable and realistic results for some test cases where the force applied to the domain is orthogonal to the domain itself.…”