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:明确的时间整合计划,加上基于时间的局部差异方程式的Galerkin离散化,要求每个时间步骤用质量矩阵解决线性系统。对于结构动态的应用,线性系统的解决方案经常通过所谓的质量整块法近似于以某种对角近似法取代质量矩阵。大规模整块法在工程实践中已广泛使用数十年了,并且有一个可靠的数学理论支持它使用传统的拉格朗基底基底的有限元素方法。然而,关于更一般性基本功能的理论仍然缺乏。我们的文件部分地讨论了这一缺陷。一些包块质量矩阵的特殊和实际相关特性得到了证明,我们讨论了这些特性如何自然延伸至带状和克伦克尔产品矩阵,其结构允许非常高效地解决线性系统。我们的理论结果被应用到等分解法分解法中,但并不局限于它们。