We demonstrate the accuracy and the robustness of our approach for two and three dimensional examples related to mussel shell like shape and faceted surfaces of fracture and show that our approach can accurately capture the specific details of cracked surfaces, such as the rippled breakages of conchoidal fracture. Here, we propose to use multigrid methods as solution technique, which leads to a solution method of optimal complexity. We moreover present and discuss efficient methods for the numerical simulation of the arising large scale non-linear systems. We present phase-field approaches for conchoidal fracture for both, the linear-elastic case as well as the case of finite deformations. The main characteristic of conchoidal fracture is that the point of crack initiation is typically located inside of the body. In this paper, we consider phase-field models and their numerical simulation for conchoidal fracture. Phase-field methods have already been proven to predict complex fracture patterns in two and three dimensional numerical simulations for brittle fracture. Moreover, they allow for the representation of complex crack patterns without changing the computational mesh via the damage parameter-which however comes at the cost of larger spatial systems to be solved. Phase-field models address both challenges in an elegant way, as they are able to represent arbitrary crack paths by means of a damage parameter. A first challenge for modeling and simulation of crack propagation is to identify the location of the crack initiation accurately, a second challenge is to follow the crack paths accurately. Crack propagation involves the creation of new internal surfaces of a priori unknown paths.
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