评估用于信赖域相场断裂的非线性消去预处理
Assessing Nonlinear Elimination Preconditioning for Trust-Region Phase-Field Fracture
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中文总结 AI 辅助
针对相场断裂，提出按需非线性消去预处理的信赖域牛顿法，在保持鲁棒性的同时减少迭代次数，但总功无优势。
中文摘要 AI 辅助
相场断裂的每个准静态载荷步都是在损伤不可逆性约束下，对非凸的位移-损伤耦合能量进行有界约束最小化。当非线性在扩展裂纹前沿局部化时，整体牛顿法停滞，而交错（交替最小化）格式在该处收敛缓慢。我们提出一种按需非线性消去预处理的信赖域牛顿法：采用能量Steihaug-Toint信赖域、用于不可逆性的原始-对偶活动集，以及有界约束场分裂扫描，该扫描在每一步之前消去一个代数识别的跨越两个场的“硬集”。消去按需应用——由耦合牛顿法自身的停滞触发，否则跳过——因此该方法在步长健康时无额外开销地退化为整体牛顿法。我们发现鲁棒性来自能量信赖域：在该全局化固定后，整体牛顿法无需削减即可完成所有加载历史，而残差度量牛顿法死亡螺旋，交替最小化停滞，全场扫描失去鲁棒性。相对于该良好全局化的基线，按需消去将外部非线性迭代减少19-25%（脆性）和17%（延性），而始终开启的消去达到26-28%，在延性成核步约为39%。以机器无关方式测量，作为全网格等效组装功代理而非墙钟时间，它与整体牛顿法相比具有竞争力——但不更快（约10%以内），而始终开启的扫描增加多达30%。因此，非线性消去是一种迭代减少机制，其开销由按需门控限制，相对于良好全局化的整体牛顿法，没有显示出总功优势。
英文摘要
Each quasi-static load step of phase-field fracture is a bound-constrained minimization of a nonconvex, coupled displacement-damage energy under an irreversibility bound on the damage. Monolithic Newton stalls once the nonlinearity localizes at the advancing crack front, and staggered (alternate-minimization) schemes converge slowly there. We present an on-demand nonlinear-elimination preconditioned trust-region Newton method: an energy Steihaug-Toint trust region, a primal-dual active set for irreversibility, and a bound-constrained field-split sweep that eliminates an algebraically-identified "hard set" spanning both fields before each step. The elimination is applied on demand -- triggered by the coupled Newton's own stalling and otherwise skipped -- so the method reduces to monolithic Newton at no surcharge where the step is already healthy. We find the robustness to come from the energy trust region: with that globalization fixed, monolithic Newton already completes every loading history without cutbacks, where residual-merit Newton death-spirals, alternate minimization stalls, and the full-field sweep loses robustness. Against that well-globalized baseline, the on-demand elimination cuts outer nonlinear iterations by 19-25% (brittle) and 17% (ductile), with always-on elimination reaching 26-28% and about $39\%$ at the ductile nucleation step. Measured machine-independently, as a full-mesh-equivalent assembly-work proxy rather than wall-clock, it is competitive with -- not faster than -- monolithic Newton (within about 10%), whereas an always-on sweep adds up to 30%. Nonlinear elimination is thus an iteration-reduction mechanism whose overhead the on-demand gate bounds, with no demonstrated total-work advantage over well-globalized monolithic Newton.
