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.

翻译：在结构动力学中，质量集中技术常被用于提高显式时间积分方案的效率，并增加受系统最大离散频率约束的临界时间步长。对于浸入等几何方法，Leidinger \cite{leidinger2020explicit} 于2020年首次证明，对于足够光滑的样条离散化，如果质量矩阵被集中，则最大频率不受小裁剪单元的影响，这一发现后来得到了多项独立数值研究的支持。本文旨在通过严格的理论分析来揭示这一特性。通过结合线性代数与泛函分析，我们推导了捕捉集中质量近似及各种裁剪配置下最大离散频率行为的精确解析估计。此外，我们还提供了对最小离散频率的估计，该频率近来受到了更密切的关注。我们的估计随后通过一维和二维问题的数值计算得到了验证。