### Rapid Topology Optimization using Path-Dependent Reduced-Order Models

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Matthew Zahr, Stanford University
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Topology optimization of structural systems is an important tool in engineering to find the most efficient design of a structure for its intended use. The high-dimensional nature of practical computational mechanics (CM) simulations make structural topology optimization a very large-scale nonlinear program with non-convex constraints defined by the discretized, nonlinear continuum equations. It is well-known that such large-scale, non-convex nonlinear programs are difficult and computationally prohibitive to solve. Despite the high-dimensional nature of many CM models, the solution trajectory is typically confined to a low-dimensional affine subspace. This is the inherent assumption in model order reduction (MOR), whereby a reduced order model (ROM) is constructed with many fewer degrees of freedom than the original high-dimensional model (HDM) without significant loss of fidelity. To address the computational cost of topology optimization, the HDM constraint is replaced with the ROM constraint. The result is a potentially dramatic reduction in the number of optimization variables and constraints. The contribution of this work is a greedy, physics-based method for simultaneously constructing a ROM and solving a PDE-constrained nonlinear program. The method is tailored to the case of topology optimization of geometrically nonlinear, St. Venant-Kirchoff material. The proposed method is a sequence of small nonlinear programs, constrained by the ROM, whose objective function is a weighting between the importance of ROM construction and solving the topology optimization problem. In regions of parameter space where the ROM is known to be accurate and robust, more weight is given to the topology optimization objective. In regions where the ROM lacks accuracy, weight is given to the ROM construction objective. Upon termination of the optimization algorithm, the HDM is sampled at the optimal point, the ROM is updated, and the optimization algorithm is restarted until the optimal topology is found.

**Abstract Author(s): **Matthew J. Zahr, Charbel Farhat