论文标题
用于质量大块的数学理论及其在同几觉分析中应用的概括
A mathematical theory for mass lumping and its generalization with applications to isogeometric analysis
论文作者
论文摘要
显式时间集成方案，以及时间依赖性偏微分方程的盖尔金离散，需要在每个时间步骤求解具有质量矩阵的线性系统。对于在结构动力学中的应用，通常通过所谓的质量大块近似线性系统的溶液，这包括用一些对角线近似代替质量基质。数十年来，质量集团已经在工程实践中广泛使用，并且具有合理的数学理论，该理论使用经典的拉格朗日为有限的元素方法支持它。但是，对于更一般的基础函数的理论仍然缺少。我们的论文部分解决了这一缺点。证明了总质量矩阵的一些特殊且实际上相关的特性，我们讨论了这些属性如何自然扩展到带状和kronecker产品矩阵，其结构允许非常有效地求解线性系统。我们的理论结果应用于等几年离散化，但不限于它们。
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.
