Topology optimization controls interface thickness with improved structural designs
Allen--Cahn Residual-Constrained Topology Optimization via an Augmented Lagrangian Method
Computational Engineering, Finance, and Science
Summary
Designing strong and efficient structures often requires balancing material shapes and transitions inside the design. This paper shows a new way to control the thickness of boundaries between solid and empty spaces in 3D printed or modeled objects without extra smoothing steps. The method uses a special mathematical equation called the Allen–Cahn equation combined with advanced optimization techniques to get designs that are both stiff and smooth. The authors find that after turning the designs into solid shapes, their approach gives designs that can be even stiffer than traditional methods and avoids tiny weak points.
What this means in practice
- •For structural engineers: Create optimized 3D structural designs with controlled boundary widths to improve stiffness and reduce weak connection points.
- •For additive manufacturing teams: Develop printable components with smoother interfaces and fewer design artifacts by integrating the proposed topology optimization framework directly into design workflows.
Authors
Run Du, Xuan Xiao, Wenming Cheng, Zeqiang Zhang
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.