Modeling of noise reduction metamaterials using a mixed FEM-VTRC method

To improve acoustic confort in railroad cars, companies such as SNCF focus on the use of metamaterials to design more absorbant while lighter panels. However, predicting their performances remains very costly because of their complex geometries and the necessity of accounting for a wide frequency bandwidth. Therefore we propose to use a mixed FEM-VTRC method to get the best of these two methods. On the one hand, the Variational Theory of Complex Rays (VTCR) is a Trefftz method, whose principle is to search for the pressure field as a linear combination of plane waves, which are solutions of the Helmholtz equation inside the domain, and which satisfy the boundary conditions in a weak sense. As such, this method does not need any mesh, but the mathematical requirement of considering star-shaped domains only implies introducing numerous subdomains in the case of complex geometries, hence reducing the VTCR efficiency. On the other hand, the Finite Element Method (FEM) is an extremely widespread and compliant method, its main limitation in the context of this study being its highly increasing cost when dealing with medium frequency problems to compensate for the pollution effect. As a result, we propose to use FEM to model the vibroacoustic coupling inside metamaterials complex geometries and VTCR to calculate acoustic propagation around them. The methods are coupled by expressing the acoustic pressure and velocity continuities at the interface between the two methods subdomains. Academic examples as well as industrial cases will be presented to assess the resulting performances.