In this work, preconditioning strategies are developed in the context of generalized continuum formulations used to regularize multifield models for simulating localized failure of quasi-brittle materials. Specifically, a micropolar continuum extended by a nonlocal damage formulation is considered for regularizing both, shear dominated failure and tensile cracking. For such models, additional microrotation and nonlocal damage fields, and their interactions, increase the complexity and size of the arising linear systems. This increases the demand for specialized preconditioning strategies when iterative solvers are adopted. Herein, a block preconditioning strategy, employing algebraic multigrid methods (AMG) for approximating the application of sub-block inverses, is developed and tested in three steps. First, a block preconditioner is introduced for linear systems resulting from micropolar models. For this case, a simple sparse Schur complement approximation, which is practical to compute, is proposed and analyzed. It is tested for three different discretizations. Second, the developed preconditioner is extended to reflect the additional nonlocal damage field. This extended three-field preconditioner is tested on the simulation of a compression test on a sandstone sample. All numerical tests show an improved performance of the block preconditioning approach in comparison to a black-box monolithic AMG approach. Finally, a problem-adapted preconditioner setup strategy is proposed, which involves a reuse of the multigrid hierarchy during nonlinear iterations, and additionally accounts for the different stages occurring in the simulation of localized failure. The problem-adapted preconditioning strategy has the potential to further reduce the total computation time.

中文翻译：

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具有微极性和非局部损伤公式的广义连续介质模型的块预处理策略

在这项工作中，预处理策略是在广义连续介质公式的背景下开发的，用于规范多场模型以模拟准脆性材料的局部失效。具体来说，考虑通过非局部损伤公式扩展的微极性连续体来规范剪切主导的失效和拉伸裂纹。对于此类模型，额外的微旋转和非局部损伤场及其相互作用会增加所产生的线性系统的复杂性和尺寸。当采用迭代求解器时，这增加了对专门预处理策略的需求。在此，分三个步骤开发和测试了块预处理策略，采用代数多重网格方法（AMG）来近似子块逆的应用。首先，为由微极性模型产生的线性系统引入了块预处理器。针对这种情况，提出并分析了一种计算实用的简单稀疏Schur补近似。它针对三种不同的离散化进行了测试。其次，扩展了所开发的预处理器以反映额外的非局部损伤场。这种扩展的三场预处理器通过模拟砂岩样品的压缩测试进行了测试。所有数值测试均表明，与黑盒单片 AMG 方法相比，块预处理方法的性能有所提高。最后，提出了一种适应问题的预处理器设置策略，其中涉及在非线性迭代期间重用多重网格层次结构，并另外考虑了局部故障模拟中发生的不同阶段。适应问题的预处理策略有可能进一步减少总计算时间。