A fully variational numerical method for structural topology optimization based on a Cahn-Hilliard model

TL;DR

The paper addresses topology optimization in solid mechanics by recasting the problem as a fully variational evolution driven by a Cahn–Hilliard gradient flow. It introduces VTORCH, a single incremental functional whose stationarity yields the coupled updates for displacement, density reparameterization, and mass transport, ensuring exact mass conservation and a symmetric tangent without Lagrange multipliers. A reparameterization of density via a logistic map and a Modica–Mortola regularization promote binary phases while permitting efficient numerical treatment, and a continuation strategy accelerates convergence. Numerical experiments on MBB and Michell cantilevers demonstrate competitive stiffness gains, sharp interfaces, and robustness across mesh refinements, with VTORCH often outperforming SIMP and Allen–Cahn baselines. The approach is versatile and can be extended to other elliptic problems, offering a mass-preserving, variationally consistent framework for topology optimization.

Abstract

We formulate a novel numerical method suitable for the solution of topology optimization problems in solid mechanics. The most salient feature of the new approach is that the space and time discrete equations of the numerical method can be obtained as the optimality conditions of a single incremental potential. The governing equations define a gradient flow of the mass in the domain that maximizes the stiffness of the proposed solid, while exactly preserving the mass of the allocated material. Moreover, we propose a change of variables in the model equations that constrains the value of the density within admissible bounds and a continuation strategy that speeds up the evolution of the flow. The proposed strategy results in a robust and efficient topology optimization method that is exactly mass-preserving, does not employ Lagrange multipliers, and is fully variational.

Figures (18)

• Figure 1: Continuation method for the double-well potential. Potential at times $t/T=0$ (blue), $t/T=0.8$ (orange), and $t/T=1$ (gray).
• Figure 2: Geometry, supports, and loading of the MBB beam benchmark.
• Figure 3: Relative density in the solution of the MBB problem obtained with the VTORCH method. From top to bottom, snapshots correspond to $t=0.01$ s, $t=0.16$ s, $t=0.36$ s, $t=0.45$ s, $t=0.54$ s, and $t=0.7$ s.
• Figure 4: Time step evolution for the VTORCH method with the original double-well potential vs. the proposed continuation technique.
• Figure 5: MBB problem. Evolution of the potential energy and Modica-Mortola functional in the VTORCH solution with mesh size $h = {1/64}$ mm.