Quantifying Non-linearity in Topology Optimization with similarity based Visualization

Abstract

Topology optimization (TO) can be viewed as seeking an optimal solution in the design space of a given TO problem. For weakly non-linear TO problems, e.g., compliance minimization, sensitivity-based methods typically converge well, whereas for strongly non-linear problems, e.g., maximum stress minimization, stabilization strategies such as stabilization terms and projection functions are often required to enhance convergence. Especially in scenarios with massive design variables, it is difficult to intuitively demonstrate the non-linear complexity of different TO problems and to elucidate the mechanisms by which stabilization strategies affect convergence. To address this challenge, we propose a visualization framework and a quantitative non-linearity index for objectives with varying complexity. We employ a multi-start fixed-gradient sampling tailored to similarity-based dimensionality reduction while keeping the computational cost under control. The samples are then parameterized via cosine similarity to obtain a low-dimensional visualization surface of the objective function. Based on this visualization, we construct a dimensionless complexity index with a clear geometric interpretation by measuring the gap between the visualization surface and the discrete approximation of its convex envelope, which enables quantitative comparisons of non-linearity across TO tasks, parameter choices, and stabilization strategies. Extensive comparative experiments show that the proposed approach is both adaptable and discriminative on a variety of representative TO problems, and it provides intuitive and measurable guidance for parameter selection.

Figures (22)

• Figure 1: Problem setting and optimal material distribution. (a) Design domain (dark gray) discretized by $30\times60$ elements; the left boundary is fixed and a downward distributed load of $1\,\mathrm{N}$ is applied at the lower-right corner. (b) Optimal material distribution for compliance minimization. (c) Optimal material distribution for the $p$-norm approximation of the von Mises stress objective with $p=10$. Optimization settings: optimizer, Method of Moving Asymptotes (MMA); initial element density, $0.5$; maximum allowable volume fraction, $0.5$; maximum iterations, $100$; density-filter radius, $2.5$ (element size). Grayscale shows density (white $=0$, black $=1$).
• Figure 2: Weakly vs. strongly non-linear objectives. The problem setting follows Fig. \ref{['setup']}(a) and is discretized by $30\times60$ elements. Optimizer: MMA; maximum iterations: 100; density-filter radius: 2.5 (elements); $p$-norm stress with $p=10$. (a) Three uniform initial designs with element density $\rho_e \in \{0.1,0.3,0.5\}$. (b) Compliance minimization, started from (a), converges to similar local optima. (c) Maximum von Mises stress minimization, due to higher non-linearity, converges to distinct local optima.
• Figure 3: Comparison of visualization surfaces. The problem setting follows Fig. \ref{['setup']}(a), except that the move limit control parameter $\eta_{\max}$ is reduced to $0.01$ after the fifth iteration, which leads to minor differences in the optimal material distribution and this modification is discussed in Section \ref{['sec:ablationaaa']}. To preserve the cosine distances between initial points, we use the unfiltered design variables as sampling points. Left, compliance as the objective yields a relatively simple surface with few peaks and valleys. Right, the $p$-norm approximation of the maximum von Mises stress objective ($p=10$) is visibly more complex. Blue squares indicate the starting points for each sampling group, with the red dot indicating the best performing design.
• Figure 4: Schematic of the proposed framework. (a) Multi-start fixed-gradient sampling under a fixed gradient from different initial points; within each group, the best result is highlighted with a red box. In (b), (c), and the left of (d), the left column shows the compliance minimization problem, and the right column shows the minimization of the maximum von Mises stress. (b) Cosine-based MDS maps the high-dimensional designs to 2D coordinates. (c) Adding normalized objective values on the embedding yields the visualization surface. (d) Left: visualization surface with sample points on its convex envelope. Right: a 2D toy example illustrating the distances between the visualization surface and its convex envelope.
• Figure 5: Multi-start fixed-gradient sampling. Starting from different initial points, we iterate with a fixed gradient for 100 steps and save the iterate every 20 steps. In both the top and bottom panels, each row corresponds to one group, with the red box marking the best structure in that group. We observe that, for (a), the per-group best structures are more similar to the final designs obtained by standard optimization without the fixed-gradient constraint (bottom strip), whereas for (b), the best structures differ markedly from those final designs in shape. These differences also motivate our use of a similarity-based dimensionality reduction method in the subsequent stage. The problem setting is identical to Fig. \ref{['visualization']}.