The phase-field fracture free-energy functional is non-convex with respect to the displacement and the phase field. This results in a poor performance of the conventional monolithic solvers like the Newton-Raphson method. In order to circumvent this issue, researchers opt for the alternate minimization (staggered) solvers. Staggered solvers are robust for the phase-field based fracture simulations as the displacement and the phase-field sub-problems are convex in nature. Nevertheless, the staggered solver requires very large number of iterations (of the order of thousands) to converge. In this work, a robust monolithic solver is presented for the phase-field fracture problem. The solver adopts a fracture energy-based arc-length method and an adaptive under-relaxation scheme. The arc-length method enables the simulation to overcome critical points (snap-back, snap-through instabilities) during the loading of a specimen. The use of an under-relaxation scheme stabilizes the solver by preventing the divergence due to an ill-behaving stiffness matrix. The efficiency of the proposed solver is further amplified with an adaptive mesh refinement scheme based on PHT-splines within the framework of isogeometric analysis. The numerical examples presented in the manuscript demonstrates the efficacy of the solver. All the codes and data-sets accompanying this work will be made available on GitHub (https://github.com/rbharali/IGAFrac).
翻译:相位断裂自由能源功能在迁移和阶段字段方面是非隐形的。 这导致像牛顿- 拉夫森方法这样的常规单流解决器的性能不佳。 为了绕过这个问题, 研究人员选择了替代最小化( 错开) 解答器。 相位断裂解器在基于阶段的断裂模拟中是强大的, 因为迁移和阶段字段子问题在性质上具有共性。 然而, 错开解器需要大量迭代( 千分之和) 才能汇合。 在这项工作中, 显示一个强大的单流解解器, 用于处理相位断裂问题。 为了避免这个问题, 研究人员选择了一种以断裂能量为基础的弧线解解解( 错开) 解脱裂解( 错开) 解裂解( 平流后, 快速解析( 快速解析) 方案将稳定解决方案, 防止因磁场断断断裂而导致的分解( 错误的分解) 。 在轨/ 平流/ 平流/ 平流/ 平流/ 平流分析中, 的平流分析中, 以 放大分析 放大图图图中, 的平流/ 效率 的平流/ 将进一步的平流/ 演示图图图图图图图图图解法 。