Differentiating through binarized topology changes: Second-order subpixel-smoothed projection

A key challenge in topology optimization (TopOpt) is that manufacturable structures, being inherently binary, are non-differentiable, creating a fundamental tension with gradient-based optimization. The subpixel-smoothed projection (SSP) method addresses this issue by smoothing sharp interfaces at the subpixel level through a first-order expansion of the filtered field. However, SSP does not guarantee differentiability under topology changes, such as the merging of two interfaces, and therefore violates the convergence guarantees of many popular gradient-based optimization algorithms. We overcome this limitation by regularizing SSP with the Hessian of the filtered field, resulting in a twice-differentiable projected density during such transitions, while still guaranteeing an almost-everywhere binary structure. We demonstrate the effectiveness of our second-order SSP (SSP2) methodology on both thermal and photonic problems, showing that SSP2 has faster convergence than SSP for connectivity-dominant cases -- where frequent topology changes occur -- while exhibiting comparable performance otherwise. Beyond improving convergence guarantees for CCSA optimizers, SSP2 enables the use of a broader class of optimization algorithms with stronger theoretical guarantees, such as interior-point methods. Since SSP2 adds minimal complexity relative to SSP or traditional projection schemes, it can be used as a drop-in replacement in existing TopOpt codes.