Application of the Ritz Method to the Optimization of Vibrating Structures

This work presents an application of the Ritz Method to the optimization of vibrating structures. The optimization problems considered here involve local design choices made in various regions of the structure in hopes of improving the vibration characteristics of the structure. In order to find the global optimum, one must perform an exhaustive search over all combinations of such choices. Even a modest number of design choices may give rise to a large number of combinations, so that an exhaustive search becomes computationally intensive. In the present work, the Ritz Method is employed to efficiently compute cost functions related to the vibration characteristics of the structure. Since the Ritz Method is based on integral expressions of the potential and kinetic energies of the structure, one may naturally divide these integrals over regions of the structure. In doing so, the concept of substructuring appears naturally in the formulation without explicitly considering boundary conditions between regions. This advantage, combined with the well-known convergence properties of the Ritz Method, provide for a computationally efficient approach for optimization problems. Numerical examples related to the optimization of a vibrating plate illustrate the approach.