Approximating the inertia forces in the floating frame of reference formulation using the consistent finite element mass matrix

In the floating frame of reference formulation, the exact form of the inertia forces is derived using a continuum-based approach. This yields a mass matrix and quadratic velocity terms containing inertia shape integrals. To avoid these integrals, many implementations of the floating frame formulation approximate the inertia forces by defining the kinetic energy using the lumped finite element mass matrix. This work proposes an alternative approximation of the inertia forces based on the consistent finite element mass matrix for structural elements, addressing cases where the exact solutions available in literature for most solid elements are not applicable. The inertia forces are derived by defining the kinetic energy using the consistent finite element mass matrix or by using the inertia forces from the equation of motion of the corresponding linear finite element model. In this way, the inertia shape integrals are replaced by a readily available mass matrix. In comparison with the lumped approach, the proposed definition yields more accurate results for coarser meshes since a more realistic representation of the mass and inertia properties of the body is used. Furthermore, the proposed approach yields inertia forces similar to the exact continuum-based approach under the assumption of small deformations. If the influence of deformation on the mass matrix is significant or the quadratic velocity terms are important, mesh refinement is required to accurately represent the inertia forces. The accuracy of the proposed definition of the inertia forces is compared to the exact and lumped mass approaches through simulation of flexible systems.