While harmful vibration is prevalent in many engineering systems, the relationship between a structure's form and its vibration properties can be complex and unintuitive. Topology optimization (TO) techniques – which can determine optimal structural form to minimize or control dynamic response – have particular use then in the design process. However, this class of design problems faces several mathematical and computational challenges, including the treatment of resonance phenomena and high-dimensional design variables, that can prevent successful optimization.
I present two strategies, inspired by methods to solve related parameter-inversion problems, that improve performance of TO for dynamic response. I first present a modified error-in-constitutive-equations (MECE) strategy as a new approach for constructing the PDE-constrained topology optimization problem. The main idea of the MECE strategy is to relax enforcement of the governing elastodynamic PDE constraint and instead include a model-error term in the objective function that is minimized along with the original objective measuring response magnitude. The resulting optimization problem features an objective more amenable to minimization, as resonant responses are suppressed in the relaxed elastodynamic model. I will show results of successful design using the MECE method for high- and multi-frequency vibration control, leveraging a preconditioned parallelized iterative solver for large-scale 3-D problems. Next, I will present the adaptive eigenspace basis (AEB) method, a reduced-dimension approach for parameterizing designs in TO problems. Here, the design is represented by a sum of eigenfunction bases, computed for an elliptic operator defined over the design domain. Restriction to the low-dimensional eigenspace significantly reduces the number of design variables, imposes length-scale control upon the solution, and provides computational cost savings. A scheme for adaptive construction of the basis enables further reduction in dimensions throughout the optimization process. I will demonstrate the performance of the AEB method, when paired with the MECE strategy, for improved dynamic-response design.
