Allen--Cahn Residual-Constrained Topology Optimization via an Augmented Lagrangian Method
Abstract
This paper presents a residual-constrained topology optimization framework that controls interface transition bandwidth without requiring external density filters. The steady conserved Allen--Cahn (AC) equation is formulated as an equality residual constraint and solved via a mass-weighted Augmented Lagrangian Method (ALM) coupled with the Method of Moving Asymptotes (MMA). A closed-form chemical potential projection eliminates the constant residual mode, ensuring algebraic compatibility with external volume targets. Variational sensitivity analysis reveals a symmetric-plus-rank-one Jacobian that yields matrix-free sensitivities without additional adjoint PDE solves, reducing residual sensitivity overhead to only 0.11% of the state elasticity solve time in 3D. Systematic 2D benchmarks confirm that interface bandwidth is governed intrinsically by reaction--diffusion parameters across mesh refinements. While diffuse interfaces incur a continuous compliance elevation (1.4%--12.3%), a volume-preserving 0/1 post-processing audit across 48 benchmark cases demonstrates that this gap stems primarily from intermediate-density penalization: binarized AC-ALM designs achieve equal or superior stiffness in 45 configurations (averaging 1.2% stiffer than SIMP). Simultaneously, single-node hinges and strut fragmentation are substantially suppressed compared to unregularized baselines. Three-dimensional benchmarks further demonstrate practical spatial viability for complex structures.
Related
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BibTeX
@misc{du2026allencahn,
title = {{Allen--Cahn Residual-Constrained Topology Optimization via an Augmented Lagrangian Method}},
author = {Run Du and Xuan Xiao and Wenming Cheng and Zeqiang Zhang},
howpublished = {arXiv},
year = {2026},
month = nov,
doi = {10.48550/arxiv.2611.02375},
eprint = {2611.02375},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2611.02375},
}
RIS
TY - UNPB
TI - Allen--Cahn Residual-Constrained Topology Optimization via an Augmented Lagrangian Method
AU - Run Du
AU - Xuan Xiao
AU - Wenming Cheng
AU - Zeqiang Zhang
PY - 2026
DA - 2026/11/02
PB - arXiv
DO - 10.48550/arxiv.2611.02375
UR - https://arxiv.org/abs/2611.02375
AB - This paper presents a residual-constrained topology optimization framework that controls interface transition bandwidth without requiring external density filters. The steady conserved Allen--Cahn (AC) equation is formulated as an equality residual constraint and solved via a mass-weighted Augmented Lagrangian Method (ALM) coupled with the Method of Moving Asymptotes (MMA). A closed-form chemical potential projection eliminates the constant residual mode, ensuring algebraic compatibility with external volume targets. Variational sensitivity analysis reveals a symmetric-plus-rank-one Jacobian that yields matrix-free sensitivities without additional adjoint PDE solves, reducing residual sensitivity overhead to only 0.11% of the state elasticity solve time in 3D. Systematic 2D benchmarks confirm that interface bandwidth is governed intrinsically by reaction--diffusion parameters across mesh refinements. While diffuse interfaces incur a continuous compliance elevation (1.4%--12.3%), a volume-preserving 0/1 post-processing audit across 48 benchmark cases demonstrates that this gap stems primarily from intermediate-density penalization: binarized AC-ALM designs achieve equal or superior stiffness in 45 configurations (averaging 1.2% stiffer than SIMP). Simultaneously, single-node hinges and strut fragmentation are substantially suppressed compared to unregularized baselines. Three-dimensional benchmarks further demonstrate practical spatial viability for complex structures.
ER -