The numerical solution of continuum damage mechanics (CDM) problems suffers from convergence-related challenges during the material softening stage, and consequently existing iterative solvers are subject to a trade-off between computational expense and solution accuracy. In this work, we present a novel unified arc-length (UAL) method, and we derive the formulation of the analytical tangent matrix and governing system of equations for both local and non-local gradient damage problems. Unlike existing versions of arc-length solvers that monolithically scale the external force vector, the proposed method treats the latter as an independent variable and determines the position of the system on the equilibrium path based on all the nodal variations of the external force vector. This approach renders the proposed solver substantially more efficient and robust than existing solvers used in CDM problems. We demonstrate the considerable advantages of the proposed algorithm through several benchmark 1D problems with sharp snap-backs and 2D examples under various boundary conditions and loading scenarios. The proposed UAL approach exhibits a superior ability of overcoming critical increments along the equilibrium path. Moreover, in the presented examples, the proposed UAL method is 1-2 orders of magnitude faster than force-controlled arc-length and monolithic Newton-Raphson solvers.

翻译：连续介质损伤力学问题的数值求解在材料软化阶段面临收敛性挑战，因此现有迭代求解器需要在计算成本与求解精度之间进行权衡。本文提出了一种新颖的统一弧长方法，推导了适用于局部和非局部梯度损伤问题的解析切线矩阵及控制方程组的公式。与现有将外力向量整体缩放的弧长求解器不同，本方法将外力向量视为独立变量，并基于外力向量的所有节点变分确定系统在平衡路径上的位置。这种处理方式使所提出的求解器在损伤力学问题中显著优于现有求解器，具有更高效率和鲁棒性。通过若干个具有明显回弹行为的基准一维问题以及多种边界条件和加载场景下的二维算例，我们展示了所提算法的显著优势。该统一弧长方法展现出沿平衡路径克服关键增量步的卓越能力。此外，在示例算例中，所提统一弧长方法比力控制弧长法和牛顿-拉夫逊整体求解器快1-2个数量级。