A lumped mass finite element formulation with consistent nodal quadrature for improved frequency analysis of wave equations  

波动方程频率分析的一致节点积分集中质量高精度有限元方法

作　　者：Xiwei Li Hanjie Zhang Dongdong Wang 李希伟;张汉杰;王东东(Department of Civil Engineering,Xiamen University,Xiamen 361005,China;College of Civil and Architectural Engineering,North China University of Science and Technology,Tangshan 063210,China)

机构地区：[1]Department of Civil Engineering,Xiamen University,Xiamen 361005,China [2]College of Civil and Architectural Engineering,North China University of Science and Technology,Tangshan 063210,China

出　　处：《Acta Mechanica Sinica》2022年第5期97-112,I0003,共17页力学学报（英文版）

基　　金：supported by the National Natural Science Foundation of China(Grant Nos.12072302 and 11772280);the Natural Science Foundation of Fujian Province of China(Grant No.2021J02003);the Natural Science Foundation of Hebei Province of China(Grant No.A2018209319).

摘　　要：A lumped mass finite element formulation with consistent nodal quadrature is presented for improved frequency analysis of wave equations with particular reference to the Lagrangian elements with Lobatto nodes.For the Lagrangian Lobatto elements,a lumped mass matrix can be conveniently constructed by employing the nodal quadrature rule that takes the Lobatto nodes as integration points.In the conventional finite element analysis,this nodal quadrature-based lumped mass matrix is usually accompanied by the stiffness matrix computed via the Gauss quadrature.In this work,it is shown that this combination is not optimal regarding the frequency accuracy of finite element analysis of wave equations.To elevate the frequency accuracy,in addition to the lumped mass matrix formulated by the nodal quadrature,a frequency accuracy measure is established as a function of the quadrature rule used in the stiffness matrix integration.This accuracy measure discloses that the frequency accuracy can be optimized if both lumped mass and stiffness matrices are simultaneously computed by the same nodal quadrature rule.These theoretical results are well demonstrated by two-and three-dimensional numerical examples,which clearly show that the proposed consistent nodal quadrature formulation yields much higher frequency accuracy than the conventional finite element analysis with nodal quadrature-based lumped mass and Gauss quadrature-based stiffness matrices for wave equations.针对波动方程的频率计算,提出了一种基于Lobatto单元的一致节点积分集中质量高精度有限元分析方法.Lobatto单元的集中质量矩阵通常可方便地采用节点积分进行构造.而在传统的有限元方法中,与集中质量矩阵一起参与频率计算的刚度矩阵通常采用高斯积分方案.本文通过研究发现,这种传统的集中质量和刚度矩阵组合方式在求解波动方程频率问题时并未达到最优精度.为了提高波动方程的频率计算精度,本文建立了用刚度矩阵数值积分点和权重表示的频率误差理论表达式.理论分析结果表明,当集中质量矩阵与刚度矩阵同时采用相同的节点积分方案时,能够有效优化波动方程的频率计算精度.文中通过系列二维、三维数值算例验证了所提方法的有效性.与传统的集中质量矩阵采用节点积分方案、刚度矩阵采用高斯积分方案的组合计算方式相比,本文所提的一致节点积分集中质量有限元分析方法能够显著提升波动方程的频率计算精度.

关 键 词：Lobatto element Wave equation Frequency accuracy Lumped mass matrix Consistent nodal quadrature 

分 类 号：O15[理学—数学]