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含时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质量矩阵构成的线性系统。在结构动力学应用中，通常通过所谓的质量凝聚方法来近似求解该线性系统，即用对角近似替代质量矩阵。质量凝聚已在工程实践中广泛应用数十年，对于采用经典拉格朗日基的有限元方法，已有完善的数学理论支撑。然而，针对更一般基函数的理论仍存在空白。本文部分弥补了这一不足。我们证明了凝聚质量矩阵的若干特殊且具有实际意义的性质，并讨论了如何将这些性质自然推广至带状矩阵和克罗内克积矩阵——这类矩阵的结构特性能够高效求解线性系统。我们的理论结果虽以等几何离散化为应用背景，但适用范围并不局限于此。