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Despite a solid theoretical foundation and straightforward application to structural design problems, 3D topology optimization still suffers from a prohibitively high computational effort that hinders its widespread use in industrial design. One major contributor to this problem is the cost of solving the finite element equations during each iteration of the optimization loop. To alleviate this cost in large-scale topology optimization, the authors propose a projection-based reducedorder modeling approach using proper orthogonal decomposition for the construction of a reduced basis for the FE solution during the optimization, using a small number of previously obtained and stored solutions. This basis is then adaptively enriched and updated on-the-fly according to an error residual, until convergence of the main optimization loop. The method of moving asymptotes is used for the optimization. The techniques are validated using established 3D benchmark problems. The numerical results demonstrate the advantages and the improved performance of our proposed approach.

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Most of the methods used today for handling local stress constraints in topology optimization, fail to directly address the non-self-adjointness of the stress-constrained topology optimization problem. This in turn could drastically raise the computational cost for an already large-scale problem. These problems involve both the equilibrium equations resulting from finite element analysis (FEA) in each iteration, as well as the adjoint equations from the sensitivity analysis of the stress constraints. In this work, we present a paradigm for large-scale stress-constrained topology optimization problems, where we build a multi-grid approach using an on-the-fly Reduced Order Model (ROM) and the p-norm aggregation function, in which the discrete reduced-order basis functions (modes) are adaptively constructed for adjoint problems. In addition to reducing the computational savings due to the ROM, we also address the computational cost of the ROM learning and updating phases. Both reduced-order bases are enriched according to the residual threshold of the corresponding linear systems, and the grid resolution is adaptively selected based on the relative error in approximating the objective function and constraint values during the iteration. The tests on 2D and 3D benchmark problems demonstrate improved performance with acceptable objective and constraint violation errors. Finally, we thoroughly investigate the influence of relevant stress constraint parameters such as the p norm factor, stress penalty factor, and the allowable stress value.

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