We consider a new class of performance functions for dispersion design of 1D periodic discrete systems using matrix rang and its regularization using log-det heuristics [1]. As an input, the desired dispersion dependency of a branch is used. Ideally, the representative dynamic stiffness matrix (RDSM) is singular at every point of the desired branch. Instead, the sum of ranks for RDSM evaluated at several discrete points is minimized using a surrogate log-det objective [2]. An example of a periodic system with a side branch is given in the Figure. The system has four free stiffness and three free mass parameters with admissible ranges given in Figure. The desired dispersion relation should have a constant frequency branch at 1.58. Thank of RDSM is evaluated at 12 discrete points. The obtained design satisfies dispersion requirements. This approach avoids ordering or tracking of eigenfrequencies and reduces the problem to a sequence of quadratic programming problems. The considered periodic discrete systems are simplified objects for method development with a further dispersion design goal for acoustic metamaterials.

[1] FAZEL, M.; HINDI, H.; BOYD, S.P. Log-det heuristic for matrix rank minimization with applications to Hankel and Euclidean distance matrices. In: Proc. of the ACC2003. IEEE, 2003. PP 2156-2162.[2] TKACHUK, A. Customization of reciprocal mass matrices via log‐det heuristic. IJNME, 2020, 121., PP.690-711.