Design of truss structures with multiple eigenfrequency constraints via rank minimization

Abstract

Rank deficiency of the dynamic stiffness matrix is an indicator for resonance of a structure at a given frequency. This indicator can be exploited as a heuristic optimization objective to achieve resonance at several frequencies. Log-det heuristic provides a tractable surrogate function for matrix rank in the case of affine dependency of stiffness and mass matrices on design parameters, which applies to truss structures. 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 heuristic 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 linear matrix inequalities. Several successful numerical examples illustrate the performance of the approach.