Direct and sparse construction of consistent inverse mass matrices: general variational formulation and application to selective mass scaling
Direct and sparse construction of consistent inverse mass matrices: general variational formulation and application to selective mass scaling
复制标题
一致逆质量矩阵的直接和稀疏构造：一般变分公式及其在选择性质量缩放中的应用
DOI：
10.1002/nme.4805
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发表时间：
2015
影响因子：
2.9
通讯作者：
M. Bischoff
中科院分区：
文献类型：
--
作者：
A. Tkachuk;M. Bischoff
经典的显式有限元公式依赖于集总质量矩阵。对角化的质量矩阵使从力矢量计算加速度矢量成为可能。近年来，由于选择性质量标度（SMS）技术的发展，用于显式有限元分析（FEA）的非对角质量矩阵受到了人们的关注。SMS允许更大的时间步长而不会严重损失精度。然而，每个时间步都需要一个昂贵的加速解决方案。在本研究中，通过直接构造逆质量矩阵来解决这一问题。首先，利用改进的哈密顿原理建立了具有独立位移和动量变量的一致稀疏反质量矩阵；使用双正交基的动量可以消除动量未知数而不需要矩阵反转，并直接产生逆质量矩阵，这里表示为互反质量矩阵（RMM）。其次，将变分质量标度技术应用于RMM。它是基于附加速度变量和自由参数的惩罚汉密尔顿原理。使用单元基的速度和局部消去产生可变比例的RMM。第三，给出并讨论了该方法求解单纯形单元的有效性。版权所有©2014 John Wiley & Sons， Ltd。
Classical explicit finite element formulations rely on lumped mass matrices. A diagonalized mass matrix enables a trivial computation of the acceleration vector from the force vector. Recently, non‐diagonal mass matrices for explicit finite element analysis (FEA) have received attention due to the selective mass scaling (SMS) technique. SMS allows larger time step sizes without substantial loss of accuracy. However, an expensive solution for accelerations is required at each time step. In the present study, this problem is solved by directly constructing the inverse mass matrix. First, a consistent and sparse inverse mass matrix is built from the modified Hamiltons principle with independent displacement and momentum variables. Usage of biorthogonal bases for momentum allows elimination of momentum unknowns without matrix inversions and directly yields the inverse mass matrix denoted here as reciprocal mass matrix (RMM). Secondly, a variational mass scaling technique is applied to the RMM. It is based on the penalized Hamiltons principle with an additional velocity variable and a free parameter. Using element‐wise bases for velocity and a local elimination yields variationally scaled RMM. Thirdly, examples illustrating the efficiency of the proposed method for simplex elements are presented and discussed. Copyright © 2014 John Wiley & Sons, Ltd.