This work considers a Stokes flow in a deformable fracture interacting with a linear elastic medium. To this end, we employ a phase-field model to approximate the crack dynamics. Phase-field methods belong to interface-capturing approaches in which the interface is only given by a smeared zone. For multi-domain problems, the accuracy of the coupling conditions is, however, of utmost importance. Here, interface-tracking methods are preferred, since the interface is resolved on mesh edges up to discretization errors, but it does not depend on the length scale parameter of some smeared zone. The key objective of this work is to construct a robust framework that computes first a crack path via the phase-field method (interface-capturing) and then does an interface-tracking reconstruction. We then discuss several approaches to reconstruct the Eulerian description of the open crack domain. This includes unfitted approaches where a level-set of the crack interface is constructed and an approach where the geometry is re-meshed. Using this reconstructed domain, we can compute the fluid-structure interaction problem between the fluid in the crack and the interacting solid. With the explicit mesh reconstruction of the two domains, we can then use an interface-tracking Arbitrary-Lagrangian-Eulerian (ALE) discretisation approach for the resulting fluid-structure interaction (FSI) problem. Our algorithmic procedure is realised in one final numerical algorithm and one implementation. We substantiate our approach using several numerical examples based on Sneddon's benchmark and corresponding extensions to Stokes fluid-filled regimes.

翻译：本研究考虑可变形裂缝中流体流动与线弹性介质相互作用的Stokes流问题。为此，我们采用相场模型近似刻画裂缝动力学。相场方法属于界面捕获方法，其界面仅由弥散区域表示。但在多域问题中，耦合条件的精度至关重要。此时界面追踪方法更具优势，因为其界面在网格边上的分辨率仅受离散误差限制，而不依赖于弥散区域长度尺度参数。本研究核心目标是构建一个鲁棒性框架：首先通过相场方法（界面捕获）计算裂缝路径，随后进行界面追踪重构。我们讨论了几种重构开放裂缝域欧拉描述的方法，包括构建裂缝界面水平集的无适配方法以及几何重网格方法。利用重构后的域，可计算裂缝内流体与相互作用固体间的流固耦合问题。通过对两个域进行显式网格重构，我们采用界面追踪的任意拉格朗日-欧拉（ALE）离散方法求解所得的流固耦合（FSI）问题。该算法流程最终整合为一个数值算法与单一实现。基于Sneddon基准问题及其向Stokes流体充填工况的扩展，通过多个数值算例验证了方法的有效性。