Nullspace technique for imposing constraints in the Rayleigh–Ritz method

The key issue to overcome when applying the Rayleigh–Ritz method (RRM) to structural analysis is to find a set of admissible functions that satisfy all system constraints. These constraints may involve from boundary conditions to continuity requirements at the junction between different substructures in a built-up system, among others. Unfortunately, admissible functions are difficult to find, assuming they exist, which strongly limits the applicability of the RRM. Although some methods have been proposed in literature to address this situation, like the penalty function method (PFM) and the Lagrange multiplier method (LMM), they still present significant limitations. In this rapid communication we introduce a new approach for dealing with constraints in the RRM. The core idea is to compute a set of nullspace fundamental solutions starting from the problem boundary conditions and/or from other restraints. It is then assumed that the response of the mechanical system consists of a linear superposition of these fundamental solutions, so that its final response will simultaneously satisfy the equations of motion and the constraints. In this communication, the essentials of the proposed nullspace method (NSM) are illustrated by applying the method to a system composed of an acoustic black hole plate and a uniform plate coupled at right angle. The accuracy of the NSM is validated against finite element method (FEM) simulations and the results are compared to those provided by the PFM and LMM approaches. It is shown how the NSM can overcome the difficulties of the latter at a smaller computational cost. It is expected that the NSM will strongly facilitate the application of the RRM to complex built-up mechanical systems.