Adaptive methods for linear dynamic systems in the frequency domain with application to global optimization
Wixom, Andrew S.
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Designers often seek to improve their designs by considering several discrete modifications. These modifications may require changes in materials and geometry, as well as the addition or removal of individual components. In general, if the modifications are applied one at a time, none of them may sufficiently improve the performance. Also, the total number of modifications that may be included in the final design is often limited due to cost or other constraints. The designer must therefore determine the optimal combination of modifications in order to complete the design. While this design challenge arises fairly commonly in practice, very little research has studied it in its full generality. This work assumes that the mathematical description of the design and its modifications are frequency dependent matrices. Such matrices typically arise due to finite element analysis as well as other modeling techniques. Computing performance metrics related to steady-state forced response, also known as performing a frequency sweep, involves factorizing these matrices many times. Additionally, determining the globally optimum design in this case involves an exhaustive search of the combinations of modifications. These factors lead to prohibitively long run times particularly as the size of the system grows. The research presented here seeks to reduce these costs, making such a search feasible. Several innovative techniques have been developed and tested over the course of the research, focused in two primary areas: adaptive frequency sweeps and efficient combinatorial optimization. The frequency sweep methods rely on an adaptive bisection of the frequency range and either a subspace approximation based on implicit interpolatory model order reduction or an elementwise approximation using piecewise multi-point Padé interpolants. Additionally, a strategy for augmenting the adaptive methods with the system's modal information is presented. For combinatorial optimization, an approximation algorithm is developed that capitalizes on any presence of dynamic uncoupling between modifications. The net effect of this work is to allow designers and researchers to develop new dynamic systems and perform analyses faster and more efficiently than ever before.