Abstract Summary
Rank deficiency of the dynamic stiffness matrix at a given frequency is an indicator of resonance. In order to achieve resonance at several eigenfrequencies, the sum of ranks of the dynamic stiffness matrix at these frequencies can be used as a heuristic optimization objective. This objective avoids the explicit computation of natural frequencies and eigenmodes. The rank of the matrices is not minimized directly. Instead, a surrogate function for matrix rank is used, e.g. log-det function. This surrogate function is known to work well in the case of affine dependency of the matrix on parameters, like the dynamic stiffness matrix of truss structures depending on cross-sectional areas. Reducing the rank of the dynamic stiffness matrix for higher frequencies implies that the matrix is not semi-positive definite. For this case, the log-det function is valid with a combination of interior-point methods and Fazel's semi-definite embedding via linear matrix inequalities. Further constraints on the fundamental frequency and compliance can be easily added within the framework as additional linear matrix inequalities. This describes the heuristic design procedure. However, an obtained design is not guaranteed to show resonance at each frequency. Thus, any design should be checked after the optimization procedure. Several successful numerical examples illustrate the performance of the approach.
