In this paper we present comparative results regarding the computational efficiency, speed, and versatility of three fundamental variational schemes for phononic band-structure calculation: a. traditional Rayleigh quotient where the displacement field is varied, b. inverse Rayleigh quotient where the stress field is varied, and c. mixed-method where both the displacement and the stress fields are varied. We show that the variational principles can be used for the fast evaluation of various phononic properties such as the dispersion relation, equi-frequency contours, and the density of states.

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