Search arXiv⌕ Search
Home/Archive/arXiv

arXiv · 2609.05458

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

Tianchen Hu
View original source ↗Read source document ↗Export metadata ↓

Abstract

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.

Explore related subjects

cs.CEcs.NAmath.APmath.NA

Keep this discovery

BibTeXRIS

Tianchen Hu. 2026-08-09. Assessing Nonlinear Elimination Preconditioning for Trust-Region Phase-Field Fracture. https://arxiv.org/abs/2609.05458

Cite the original work for its findings. Save a collection to share your selection of sources.

Discover connections

cs.CE subjectcs.NA subjectmath.AP subjectmath.NA subject

Connections use source metadata and explicit phrase matches, not verified experimental comparisons.

KEEP EXPLORING

Related papers

arXiv2026

Regularity of Second-Order Elliptic PDEs in Spectral Barron Spaces

We establish a regularity theorem for second-order elliptic PDEs on $\mathbb{R}^{d}$ in spectral Barron spaces. Under mild ellipticity and smallness assumptions, the solution gains two additional orders of Barron regularity. As a corollary, we identify a class of PDEs whose solutions can be approximated by two-layer neural networks with cosine activation functions, where the width of the neural network is independent of the spatial dimension.

math.AP↗
arXiv2026

Two Adjoint Perspectives on Fokker-Planck Optimization: A Microscopic-Macroscopic Correspondence

The Fokker-Planck equation admits both a macroscopic Eulerian description through probability densities and a microscopic Lagrangian description through stochastic trajectories. Consequently, optimization problems constrained by the Fokker-Planck equation can be formulated from either perspective. Surprisingly, the corresponding adjoint equations appear to be fundamentally different: the macroscopic adjoint is governed by the backward Kolmogorov equation, whereas the microscopic adjoint evolves pathwise along stochastic trajectories. In this note, we reconcile these two formulations by establishing their correspondence in the continuum setting. We further show that, although their discrete gradients no longer coincide after discretization, both provide consistent numerical approximations of the continuum gradient. Explicit convergence rates are established for both discretization strategies.

math.NA↗
arXiv2024

The Ramshaw-Mesina Hybrid Algorithm applied to the Navier Stokes Equations

In 1991, Ramshaw and Mesina proposed a novel synthesis of penalty methods and artificial compression methods. When the two were balanced they found the combination was 3-4 orders more accurate than either alone. This report begins the study of their interesting method applied to the Navier-Stokes equations. We perform stability analysis, semi-discrete error analysis, and tests of the algorithm. Although most of the results for implicit time discretizations of our numerical tests comply with theirs for explicit time discretizations, the behavior in damping pressure oscillations and violations of incompressibility are different from their findings and our heuristic analysis.

math.NA↗