NDLI: A unified quadrature-based superconvergent finite element formulation for eigenvalue computation of wave equations

Content Provider	Springer Nature Link
Author	Wang, Dongdong Li, Xiwei Pan, Feixu
Copyright Year	2016
Abstract	A simple and unified finite element formulation is presented for superconvergent eigenvalue computation of wave equations ranging from 1D to 3D. In this framework, a general method based upon the so called $$\alpha $$ mass matrix formulation is first proposed to effectively construct 1D higher order mass matrices for arbitrary order elements. The finite elements discussed herein refer to the Lagrangian type of Lobatto elements that take the Lobatto points as nodes. Subsequently a set of quadrature rules that exactly integrate the 1D higher order mass matrices are rationally derived, which are termed as the superconvergent quadrature rules. More importantly, in 2D and 3D cases, it is found that the employment of these quadrature rules via tensor product simultaneously for the mass and stiffness matrix integrations of Lobatto elements produces a unified superconvergent formulation for the eigenvalue or frequency computation without wave propagation direction dependence, which usually is a critical issue for the multidimensional higher order mass matrix formulation. Consequently the proposed approach is capable of computing arbitrary frequencies in a superconvergent fashion. Meanwhile, numerical implementation of the proposed method for multidimensional problems is trivial. The effectiveness of the proposed methodology is systematically demonstrated by a series of numerical examples. Numerical results revealed that a superconvergence with $$2(p+1)\hbox {th}$$ order of frequency accuracy is achieved by the present unified formulation for the $$p\hbox {th}$$ order Lobatto element.
Starting Page	37
Ending Page	72
Page Count	36
File Format	PDF
ISSN	01787675
Journal	Computational Mechanics
Volume Number	59
Issue Number	1
e-ISSN	14320924
Language	English
Publisher	Springer Berlin Heidelberg
Publisher Date	2016-11-03
Publisher Place	Berlin, Heidelberg
Access Restriction	One Nation One Subscription (ONOS)
Subject Keyword	Eigenvalue Wave equation Superconvergence $$\alpha $$ Mass matrix Higher order mass matrix Superconvergent quadrature rule Lobatto element Theoretical and Applied Mechanics Computational Science and Engineering Classical and Continuum Physics
Content Type	Text
Resource Type	Article
Subject	Applied Mathematics Ocean Engineering Computational Theory and Mathematics Mechanical Engineering Computational Mechanics Computational Mathematics

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