Unifying and accelerating level-set and density-based topology optimization by subpixel-smoothed projection

TL;DR

The paper tackles the ill-conditioning and vanishing gradients that plague traditional density-based topology optimization as binarization is pursued via the projection parameter $\beta$. It introduces subpixel-smoothed projection (SSP), a gradient-aware, local operator that computes $\hat{\tilde{\rho}}(\tilde{\rho}, \|\nabla \tilde{\rho}\|)$ using a distance to the interface $d=(\eta-\tilde{\rho})/\|\nabla \tilde{\rho}\|$ and a monotone fill-factor $F(d)$ (e.g., a polynomial $F_2(d)$). SSP yields an almost-everywhere binary design as $\Delta x \to 0$, while maintaining a well-conditioned, differentiable relationship to the design variables for all finite $\beta$ and enabling rapid progression toward $\beta=\infty$. The authors demonstrate SSP across photonics inverse-design problems using three Maxwell solvers (FDTD, FMM, FEM), showing faster convergence, compatibility with existing topology-optimization pipelines, and final designs that rival traditional TO results. This approach promises broader applicability and simpler optimization workflows by removing the strict coupling between topology changes and gradient conditioning, while enabling rapid exploration of binary, manufacturable designs.

Abstract

We introduce a new "subpixel-smoothed projection" (SSP) formulation for differentiable binarization in topology optimization (TopOpt) as a drop-in replacement for previous projection schemes, which suffer from near-non-differentiability and slow convergence as binarization improves. Our new algorithm overcomes these limitations by depending on both the underlying filtered design field and its spatial gradient, instead of the filtered design field alone. We can now smoothly transition between density-based TopOpt (in which topology can easily change during optimization) and a level-set method (in which shapes evolve in an almost-everywhere binarized structure). We demonstrate the effectiveness of our method on several photonics inverse-design problems and for a variety of computational methods (finite difference, Fourier-modal, and finite-element methods). SSP exhibits both faster convergence and greater simplicity.

Figures (8)

• Figure 1: Schematic of "three-field" approach to density-based topology optimization: (a) an initial density field $\rho$ (here for the waveguide-crossing problem of Fig. \ref{['fig:waveguide_TO']}) is (b) low-pass filtered to a smoothed/grayscale density $\tilde{\rho}$ and then (c) projected to a binarized density $\hat{\rho}$ with steepness hyper-parameter $\beta$, which represents the physical material arrangement. Step (c) is not differentiable for a step-function ($\beta = \infty$) projection, but could theoretically be made differentiable by smoothing the edges at infinite spatial resolution to obtain (d) $\hat{\tilde{\rho}}$, indicated by the dashed "not practical" arrow. Our new algorithm obtains (d) $\hat{\tilde{\rho}}$ directly from (b) $\tilde{\rho}$. The result $\hat{\tilde{\rho}}$ is almost-everywhere binary except for a thin "subpixel" boundary layer (e) within the grid resolution $\sim \Delta x$ of the interface. (f) This layer allows an objective function $f$ (here from Sec. \ref{['sec:crossing']}) to have a finite gradient (nonzero only near boundaries) with respect to the density $\rho$ (a); here, the positive gradients (red) are $\approx 20\times$ larger than the negative gradients (blue).
• Figure 2: The number of CCSA Svanberg2002 optimization steps required to achieve 99% transmission for a waveguide-crossing example (Sec. \ref{['sec:convergence']}), using standard projection (orange) and our SSP method (blue) as a function of $\beta$. As expected, standard projection requires a diverging number of optimization iterations to achieve the specified tolerance as $\beta$ increases. In contrast, SSP converges to a finite number of optimization steps as $\beta\to\infty$. Inset: The optimized structure for $\beta=\infty$, along with an enlarged section of the structure illustrating the 1-pixel-thick smoothed interface.
• Figure 3: Schematic of proposed subpixel smoothing routine, shown here in the $\beta \to \infty$ limit (step-function projection $\hat{\rho}$). A simulation algorithm discretizes the geometry on some grid or mesh with spacing $\sim \Delta x$ (a). At a particular grid point $\mathbf{x}$, a smoothing sphere is chosen with diameter $2\hat{R} \gtrsim \Delta x$ (b). In a small region, the interface is approximately planar ($\perp \nabla \tilde{\rho}$), at a distance $d$ from $\mathbf{x}$. Conceptually, a subpixel-smoothed $\tilde{\rho}$ is constructed by convolving $\hat{\rho}$ with some smoothing kernel of radius $\hat{R}$, resulting in an almost-every binary discretized structure (c) ($\hat{\tilde{\rho}} \in \{0,1\}$ for $d \ge \hat{R}$). To make this practical and differentiable, we compute $\hat{\tilde{\rho}}$ from $\tilde{\rho}$ instead of from $\hat{\rho}$.
• Figure 4: Subpixel-smoothed projections $\hat{\tilde{\rho}}(x)$, Eq. \ref{['eq:DOFsmooth_beta']}, applied to an example $\tilde{\rho}(x) = \frac{1}{2} \tanh(x/\tilde{R}) + \frac{1}{2}$ (dashed black) for $\tilde{R} = 10\hat{R}$, with a threshold $\eta = 0.5$, for various steepness parameters $\beta$. ($\tilde{\rho}$ is nearly linear for $-4 < x/\hat{R} < 4$, so Eq. \ref{['eq:d']} gives $d \approx -x$.) The shaded region indicates the smoothing radius $\hat{R}$ around the level set $\tilde{\rho} = \eta$. For $\beta=0^+$ (dashed black), our formula is equivalent to no projection ($\hat{\tilde{\rho}} = \tilde{\rho}$), while for $\beta = \infty$ (solid green) we obtain our chosen "fill-fraction" curve $F(d)$; for reference, the step-function projection $\hat{\rho}$ for $\beta = \infty$ is also shown (dashed green). Here, we use $F = F_2$ from Eq. \ref{['eq:Fpoly']}, so the $\hat{\tilde{\rho}}$ curves are all twice differentiable by construction, despite the fact that the piecewise definitions change at $d/\hat{R}=\pm 1$.
• Figure 5: Comparison of the new subpixel-smoothed projection $\hat{\tilde{\rho}}$ (orange) to standard projection $\hat{\rho}$ (blue), for optimizing transmission through a waveguide crossing (inset, Sec. \ref{['sec:crossing']}). (a) Projected densities vs. position in the vicinity of an interface (small red circle, inset): for $\beta =\infty$, standard projection $\hat{\rho}$ is a discontinuous step function, whereas the new $\hat{\tilde{\rho}}$ rises continuously over a 1-pixel ($\sim \Delta x$) distance. (b) Gradient norm $\Vert \nabla_\rho \text{FOM}\Vert$ of the optimization figure of merit (FOM) as a function of projection steepness $\beta$: for $\hat{\rho}$, the gradient is (almost everywhere) zero as $\beta \to \infty$, preventing optimization from making progress; for the new $\hat{\tilde{\rho}}$, the gradient converges to a nonzero value. (c) Optimization progress (FOM vs. iteration) for the waveguide crossing (Sec. \ref{['sec:crossing']}) with $\beta = 64$: the standard projection converges much more slowly than smoothed projection, because the former is "stiff" (ill-conditioned second derivatives) for large $\beta$. (d) Optimization progress for $\beta = \infty$: standard projection makes no progress (due to zero gradients), whereas smoothed projection converges at almost the same rate as (c).