A Hierarchical Condensation Finite Element Framework for Large-Scale Topology Optimization
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Topology optimization (TO) is dominated by the repeated solution of large-scale finite element (FE) systems, which becomes prohibitive at industry-scale resolutions. This work introduces HiCon-TO, a hierarchical condensation finite element framework that accelerates density-based TO by embedding model reduction directly into a standard SIMP loop. The key idea is to project the fine-scale displacement field within each macroelement onto a compact hierarchical basis and then apply static condensation prior to global assembly. This yields a reduced global system involving only macroelement boundary unknowns, while preserving the variational structure of classical FE formulations and enabling consistent recovery of fine-scale displacements for objective evaluation and sensitivity analysis. Unlike multiscale or homogenization-based strategies, HiCon-TO does not require scale separation assumptions and can be used as a drop-in replacement for the conventional FE solve without modifying the underlying optimization algorithm. The method is assessed on the classical MBB beam benchmark over multiple resolutions, including ultra-high-resolution problems up to 3000× 2000 finite elements. Across all cases, HiCon-TO achieves end-to-end runtime reductions exceeding 97%, corresponding to total speedups of up to 40× relative to a Julia-optimized classical SIMP implementation. Per-iteration comparisons indicate intrinsic accelerations above 20×, confirming that gains arise from the reduced-order structural analysis rather than differences in convergence history. Timing breakdowns show that the dominant savings stem from the substantially smaller global solve enabled by hierarchical condensation, while maintaining accurate fine-scale displacement recovery. Robust performance is observed under different regularization strategies, including classical sensitivity filtering and Gaussian convolution-based filtering, with favorable memory scaling for very large design domains where conventional filtering matrices become impractical.
