The aim of this work is to present a finite element methodology, based upon a 3-D extension of the classical 'Hermite interpolation' and of the 'Coons Patch', for the evaluation of the natural modes of vibration of the air inside cavities (interior acoustics) and of elastic structures (structural dynamics), with the distinguishing feature of high efficiency and high flexibility and, therefore, extremely useful in Multidisciplinary Design Optimization (MDO). External and internal nodes configurations have been developed and investigated. The validation of the methodology is based upon the evaluation of the natural eigenvalues and modes of the air vibrating inside hexahedral cavities as well as of those of elastic thin plates. Applications to more complicated structures, such as curved domains or simplified wing-boxes are included. Results are compared with Ansys and with the literature.