RUS-resonant ultrasound spectroscopy is a recent experimental-numerical method for the determination of moduli of elastic materials. Generally, all 21 elastic components of the elastic tensor can be determined by the numerical procedure based on the knowledge of a mechanical spectrum of a specimen. This involves the solution of a demanding inverse problem. Currently, the RUS technology allows the determination of material parameters of composite materials consisting of several layers with different material properties. In the present work, the fixed point iteration method in connection with the finite element method developed earlier is extended to optimize elastic moduli of layered materials. Properties of the fixed point iteration method are tested on a bicrystal specimen.