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. This finding was later supported by 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. 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，并增大受系统最大离散频率限制的临界时间步长。对于浸入式几何分析方法，Leidinger \cite{leidinger2020explicit} 于2020年首次证明，对于足够光滑的样条离散化，若质量矩阵采用集中质量法，最大频率不会受到微小裁剪单元的影响。这一发现后来得到了独立数值研究的支持。本文旨在通过严格的理论分析揭示这一特性。通过结合线性代数与泛函分析，我们推导出精确的解析估计，以捕捉采用集中质量近似及不同裁剪配置下最大离散频率的行为规律。随后，我们通过一维和二维问题的数值计算验证了这些估计。