Assessing Nonlinear Elimination Preconditioning for Trust-Region Phase-Field Fracture

RESUMO

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.