A mixed Legendre-Galerkin spectral method for the buckling problem of simply supported Kirchhoff plates

Junying Cao,Ziqiang Wang,Waixiang Cao,Lizhen Chen

doi:10.1186/s13661-017-0767-z

Abstract

In this paper, we develop a mixed Legendre-Galerkin spectral method to approximate the buckling problem of simply supported Kirchhoff plates subjected to general plane stress tensor. By the spectral theory of compact operators, the rigorous error estimates for the approximate eigenvalues and eigenfunctions are provided. Finally, we present some numerical experiments which support our theoretical results.

Highlights

Buckling problem has attracted lots of interest since it is frequently encountered in engineering applications such as bridge, ship, and aircraft design
Many numerical methods for the buckling problem have been studied, for example, finite element schemes [, – ]. They are based on the well-known mixed methods to deal with the source problem of thin plates modeled by the biharmonic equation which was introduced by Ciarlet and Raviart [ ]
The main purpose of this paper is to propose a mixed Legendre-Galerkin spectral method to approximate the buckling problem of supported Kirchhoff plates subjected to general plane stress tensor

Summary

Introduction

Buckling problem has attracted lots of interest since it is frequently encountered in engineering applications such as bridge, ship, and aircraft design. Many numerical methods for the buckling problem have been studied, for example, finite element schemes [ , – ]. They are based on the well-known mixed methods to deal with the source problem of thin plates modeled by the biharmonic equation which was introduced by Ciarlet and Raviart [ ]. The main idea is to introduce an auxiliary variable ω := Δψ (with ψ being the transverse displacement of the mean surface of the plate) to write a variational formulation of the spectral problem. This mixed trick has been widely used. The author presented a spline collocation method for two different integral equations which were split by Fredholm-Hammerstein integral equations of the second kind over a rectangular region in a plane [ ]

Objectives

Conclusion