Designing Phononic Crystal with Convex Optimization

Designing phononic crystals by creating frequency bandgaps is of particular interest in elastic and acoustic metamaterial engineering. Mathematically, the problem of optimizing the frequency bandgaps is often nonconvex, as it requires the maximization of the higher indexed eigenfrequency and the minimization of the lower indexed eigenfrequency. A novel algorithm [1] has been previously developed to reformulate the original nonlinear, nonconvex optimization problem to an iteration-specific semidefinite program (SDP). This algorithm separates two consecutive eigenvalues – effectively, maximizing bandgap (or bandwidth) – by separating the gap between two orthogonal subspaces, which are comprised columnwise of “important” eigenvectors associated with the eigenvalues being bounded. By doing so, the computation of eigenvalue gradient can be efficiently approximated by the computation of gradient of affine matrices with respect to the decision variables. In this work, we propose an even more efficient algorithm based on linear program (LP). The new formulation is obtained via approximation of the semidefinite cones by judiciously chosen linear bases, coupled with “delayed constraint generation”. We apply the two convex conic formulations, namely, the semidefinite program and the linear program, to solve the bandgap optimization problems. By comparing the two methods, we demonstrate the efficacy and efficiency of the LP-based algorithm in solving the category of eigenvalue bandgap optimization problems. ∗Address all correspondence to this author.