A unified quadrature-based superconvergent finite element formulation for eigenvalue computation of wave equations
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- 作者：Dongdong Wang ; Xiwei Li ; Feixu Pan
- 关键词：Eigenvalue ; Wave equation ; Superconvergence ; \(\alpha \) ; Mass matrix ; Higher order mass matrix ; Superconvergent quadrature rule ; Lobatto element
- 刊名：Computational Mechanics
- 出版年：2017
- 出版时间：January 2017
- 年：2017
- 卷：59
- 期：1
- 页码：37-72
- 全文大小：
- 刊物类别：Engineering
- 刊物主题：Theoretical and Applied Mechanics; Computational Science and Engineering; Classical and Continuum Physics;
- 出版者：Springer Berlin Heidelberg
- ISSN：1432-0924
- 卷排序：59
文摘
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.