Fixed point formulation of optimality criteria for efficient topology optimization
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The traditional Optimality Criteria (OC) update in topology optimization suffers from slow convergence, thereby requiring a large number of iterations to result in only a small improvement in the performance and design. To address this problem, we propose to use a novel fixed-point formulation of the OC update to accelerate the convergence. Such strategies can achieve higher convergence rates without overly complexifying the update process. In this thesis, we first provide some mathematical background on fixed-point iteration methods. Then, based on theoretical analysis and numerical experiments, we analyze these methods' respective advantages and drawbacks in the context of topology optimization. The analysis focuses on the methods' design update stability, effectiveness in reducing the design cycles, computational cost, and robustness. Through numerical studies, we found one of the methods, called Periodic Anderson Extrapolation (PAE), is the most stable, effective, economic, and robust approach to speed up OC's convergence. The overall update is named Periodically Anderson Extrapolated Optimality Criteria (PAE-OC). Via several 2D and 3D benchmarks, we demonstrate that the PAE-OC can effectively reduce both the number of iterations and computation time. In addition, this scheme shows good robustness with respect to the change of boundary conditions, problem sizes, and parameters. Finally, we show the scalability of the PAE-OC through a 3D problem consisting of more than 3 million degrees of freedom.