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 ...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.展开更多
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).
文摘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.
