Explicit time integration schemes coupled with Galerkin discretizations of time-dependent partial differential equations require solving a linear system with the mass matrix at each time step. For applications in structural dynamics, the solution of the linear system is frequently approximated through so-called mass lumping, which consists in replacing the mass matrix by some diagonal approximation. Mass lumping has been widely used in engineering practice for decades already and has a sound mathematical theory supporting it for finite element methods using the classical Lagrange basis. However, the theory for more general basis functions is still missing. Our paper partly addresses this shortcoming. Some special and practically relevant properties of lumped mass matrices are proved and we discuss how these properties naturally extend to banded and Kronecker product matrices whose structure allows to solve linear systems very efficiently. Our theoretical results are applied to isogeometric discretizations but are not restricted to them.

翻译：显式时间积分方案与时间相关偏微分方程的伽辽金离散化相结合时，需要在每个时间步求解包含质量矩阵的线性系统。在结构动力学应用中，该线性系统的解常通过所谓的质量集中法进行近似，即用某种对角近似替代质量矩阵。质量集中法已在工程实践中广泛应用数十年，并针对使用经典拉格朗日基的有限元方法建立了完善的理论支撑。然而，更一般基函数的理论仍然缺失。本文部分填补了这一空白。我们证明了集中质量矩阵的某些特殊且具有实际重要性的性质，并探讨了这些性质如何自然推广至带状矩阵和克罗内克积矩阵——其结构可高效求解线性系统。理论成果应用于等几何离散化，但并不仅限于此。