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

TL;DR

This paper provides a rigorous theoretical analysis of how mass lumping affects the largest and smallest discrete frequencies in immersogeometric analysis, confirming previous numerical observations and offering new analytical estimates.

Contribution

It introduces a mathematical framework combining linear algebra and functional analysis to derive sharp estimates for discrete frequencies under mass lumping in immersogeometric methods.

Findings

Mass lumping does not affect the largest frequency for smooth spline discretizations.

Theoretical estimates for the smallest discrete frequency are provided.

Numerical validation confirms the analytical results in 1D and 2D problems.

Abstract

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 \cite{leidinger2020explicit} first showed in 2020 that for sufficiently smooth spline discretizations, the largest frequency was not affected by small trimmed elements if the mass matrix was lumped, a finding later supported by several independent numerical studies. This article provides a rigorous theoretical analysis aimed at unraveling this property. By combining linear algebra with functional analysis, we derive sharp analytical estimates capturing the behavior of the largest discrete frequency for lumped mass approximations and various trimming configurations. Additionally, we also provide estimates for the smallest discrete frequency, which has lately drawn closer scrutiny. Our estimates are then confirmed numerically for 1D and 2D problems.