A phase-field monolithic scheme based on the gradient projection method is developed to model crack propagation in brittle materials under cyclic loading. As a type of active set method, the gradient projection method is particularly attractive to enforce the irreversibility condition imposed on the phase-field variables as bound constraints, or box constraints. This method has the advantages of allowing the rapid change of active constraints during iterations and computing the projected gradient with a negligible cost. The gradient projection method is further combined with the limited-memory BFGS (L-BFGS) method to overcome the convergence difficulties arising from the non-convex energy functional. A compact representation of the BFGS matrix is adopted as the limited-memory feature to avoid the storage of fully dense matrices, making this method practical for large-scale finite element simulations. By locating the generalized Cauchy point on the piecewise linear path formed by the projected gradient, the active set of box constraints can be determined. The variables in the active set, which are at the boundary of the box constraints, are kept fixed to form a subspace minimization problem. A primal approach and a dual approach are presented to solve this subspace minimization problem for the remaining free variables at the generalized Cauchy point. Several two-dimensional (2D) and three-dimensional (3D) examples are provided to demonstrate the capabilities of the proposed monolithic scheme, particularly in enforcing the phase-field irreversibility during crack propagation under cyclic loading. In these numerical examples, the proposed monolithic scheme is combined with an adaptive mesh refinement technique to alleviate the heavy computational cost incurred by the fine mesh resolution required around the crack region. The proposed method is further compared with two other phase-field solving techniques regarding the convergence behavior. To ensure a fair comparison, the same problem settings and implementation techniques are adopted. The proposed monolithic scheme provides a unified framework to overcome the numerical difficulties associated with the non-convex energy functional, effectively enforce the phase-field irreversibility to ensure the thermodynamic consistency, and alleviate the heavy computational cost through adaptive mesh refinement in 2D and 3D phase-field crack simulations.

中文翻译：

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在鲁棒整体相场方案中将裂纹不可逆性强制为箱约束的梯度投影方法


开发了一种基于梯度投影法的相场整体方案，用于模拟循环载荷下脆性材料中的裂纹扩展。作为一种主动集合方法，梯度投影方法特别有吸引力，可以强制执行施加在相场变量上的不可逆条件作为边界约束或框约束。这种方法的优点是允许在迭代期间快速更改活动约束，并以可忽略不计的成本计算投影梯度。梯度投影方法进一步与有限内存 BFGS （L-BFGS） 方法相结合，以克服非凸能量泛函引起的收敛困难。采用 BFGS 矩阵的紧凑表示作为有限内存特征，以避免存储完全密集的矩阵，使该方法可用于大规模有限元仿真。通过在投影梯度形成的分段线性路径上定位广义柯西点，可以确定活动的箱约束集。活动集中位于框约束边界的变量保持固定，以形成子空间最小化问题。提出了一种原始方法和一种对偶方法来解决广义柯西点处剩余自由变量的子空间最小化问题。提供了几个二维 （2D） 和三维 （3D） 示例来演示所提出的整体方案的能力，特别是在循环载荷下裂纹扩展过程中增强相场不可逆性。 在这些数值示例中，所提出的整体方案与自适应网格细化技术相结合，以减轻裂纹区域周围所需的精细网格分辨率所产生的沉重计算成本。在收敛行为方面，将所提出的方法与其他两种相场求解技术进行了进一步比较。为了确保公平的比较，采用了相同的问题设置和实现技术。所提出的整体方案提供了一个统一的框架来克服与非凸能量泛函相关的数值困难，有效地加强了相场不可逆性以确保热力学的一致性，并通过二维和三维相场裂纹模拟中的自适应网格细化减轻了沉重的计算成本。