论文标题
基于固定压力概念的脆性断裂相位模型的快速交错方案
Fast staggered schemes for the phase-field model of brittle fracture based on the fixed-stress concept
论文作者
论文摘要
相位场模型有望解决各种断裂问题，在这些问题中引入了扩散裂纹，并使用相变量进行了建模。由于能量功能的非跨性别性，衍生的部分微分方程通常以交错的方式解决。但是，这种方法遭受低收敛速率，需要大量交错的迭代，尤其是在断裂成核和传播时。在这项研究中，我们提出了新颖的交错方案，该方案的灵感来自Poromechanic中的固定压力分裂方案。通过解决损伤演变时的应力，位移增量以相变量的增量表示。这两个增量之间的关系可以基于相变的增量来预测位移和活动能量。因此，减少了交错的最大迭代次数，并提高了计算效率。我们通过固定第一个不变的，第二个不变的或两个不变性的压力，以S1，S2和S3方案表示，我们提出了三个交错的方案。然后，通过与标准交错方案通过三个基准示例（即拉伸，剪切和L形面板测试）进行比较来验证这些方案的性能。结果表明，使用快速方案计算出的力置换关系和裂纹模式与基于标准交错方案的方案一致。此外，在所有基准测试中，提出的S1和S2方案可以很大程度上减少交错迭代的最大数量和总CPU时间。 S2方案的性能比在剪切测试中表现相当，在剪切测试中，靠近裂缝的区域违反了基本假设。
Phase field models are promising to tackle various fracture problems where a diffusive crack is introduced and modelled using the phase variable. Owing to the non-convexity of the energy functional, the derived partial differential equations are usually solved in a staggered manner. However, this method suffers from a low convergence rate, and a large number of staggered iterations are needed, especially at the fracture nucleation and propagation. In this study, we propose novel staggered schemes, which are inspired by the fixed-stress split scheme in poromechanics. By fixing the stress when solving the damage evolution, the displacement increment is expressed in terms of the increment of the phase variable. The relation between these two increments enables a prediction of the displacement and the active energy based on the increment of the phase variable. Thus, the maximum number of staggered iterations is reduced, and the computational efficiency is improved. We present three staggered schemes by fixing the first invariant, second invariant, or both invariants of the stress, denoted by S1, S2, and S3 schemes. The performance of the schemes is then verified by comparing with the standard staggered scheme through three benchmark examples, i.e., tensile, shear, and L-shape panel tests. The results exhibit that the force-displacement relations and the crack patterns computed using the fast schemes are consistent with the ones based on the standard staggered scheme. Moreover, the proposed S1 and S2 schemes can largely reduce the maximum number of staggered iterations and total CPU time in all benchmark tests. The S2 scheme performs comparably except in the shear test, where the underlying assumption is violated in the region close to the crack.
