A theoretical study on the effect of mass lumping on the discrete frequencies in immersogeometric analysis

In structural dynamics, mass lumping techniques are commonly employed for improving the efficiency of explicit time integration schemes and increasing their critical time step constrained by the largest discrete frequency of the system. For immersogeometric methods, Leidinger first showed in 2020 that under mild conditions on the spline's order and smoothness, the largest frequency was not affected by small trimmed elements if the mass matrix was lumped. This finding was later supported by independent numerical studies. This article is a first attempt at unraveling this property from a theoretical perspective. By combining linear algebra with functional analysis, we derive analytical estimates capturing its behavior for various trimming configurations. Our estimates are then verified numerically for 1D and 2D problems.