A mixed LegendreGalerkin spectral method for the buckling problem of simply supported Kirchhoff plates
 Junying Cao^{1, 2},
 Ziqiang Wang^{1, 2},
 Waixiang Cao^{3} and
 Lizhen Chen^{2}Email author
Received: 6 December 2016
Accepted: 24 February 2017
Published: 14 March 2017
Abstract
In this paper, we develop a mixed LegendreGalerkin 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.
Keywords
buckling problem LegendreGalerkin spectral method simply supported Kirchhoff plates1 Introduction
Buckling problem has attracted lots of interest since it is frequently encountered in engineering applications such as bridge, ship, and aircraft design. The buckling problem has been studied for years by many researchers (see [1–7] and the references therein). Furthermore, many numerical methods for the buckling problem have been studied, for example, finite element schemes [3, 8–12]. They are based on the wellknown mixed methods to deal with the source problem of thin plates modeled by the biharmonic equation which was introduced by Ciarlet and Raviart [13]. The main idea is to introduce an auxiliary variable \(\omega:=\varDelta \psi \) (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 now has been widely used. Marin and Lupu solved the unknowns of the displacement and microrotation on harmonic vibrations in thermoelasticity of micropolar bodies [14]. Pop et al. proposed a novel algorithm for the condition detection in which the solution breaks down [15]. The author presented a spline collocation method for two different integral equations which were split by FredholmHammerstein integral equations of the second kind over a rectangular region in a plane [16].
The main purpose of this paper is to propose a mixed LegendreGalerkin spectral method to approximate the buckling problem of simply supported Kirchhoff plates subjected to general plane stress tensor. We introduce a compact operator to analyze the continuous problem. The basis functions are constructed by combining the Legendre polynomials which satisfy the boundary condition automatically. Finally, we prove the optimal order error estimate for the eigenfunctions and a double order for the eigenvalues.
The remainder of this paper is organized as follows. Section 2 describes simply supported Kirchhoff plates subjected to general plane stress tensor. The LegendreGalerkin spectral method and the error estimate are proposed in Section 3. The details of implementation and the expression of a linear algebra system corresponding to the discrete variational formulation are given in Section 4. Section 5 presents the main numerical results of this work which demonstrate the efficiency and accuracy of this method. Finally, a conclusion of this paper is made in Section 6.
2 The spectral problem

Find \(\omega\in H_{0}^{1}(\varOmega )\) such that$$ \int_{\varOmega } \nabla\omega\cdot\nabla v \,d\varOmega = \int_{\varOmega }(\boldsymbol{\sigma}\nabla f)\cdot\nabla v\,d\varOmega , \quad \forall v \in H_{0}^{1}(\varOmega ); $$(9)

Find \(\psi\in H_{0}^{1}(\varOmega )\) such that$$ \int_{\varOmega } \nabla\psi\cdot\nabla\eta \,d\varOmega = \int_{\varOmega }\omega \eta \,d\varOmega ,\quad \forall\eta\in H_{0}^{1}(\varOmega ). $$(10)
Problems (9) and (10) are equivalent to the source problem of equations (1) as follows:
In order to imply the spectral method on problem (8), we introduce the discrete space as follows.
Lemma 1
Proof
For details of the proof, one can refer to Theorem 7.3 in [18]. □
3 Error estimate for eigenvalues

Find \(\lambda_{N}, \omega_{N} \in V_{N}\) such that$$ \int_{\varOmega } \nabla\omega_{N} \cdot\nabla v_{N} \,d\varOmega =\lambda_{N} \int _{\varOmega }(\boldsymbol{\sigma}\nabla\psi_{N})\cdot \nabla v_{N} \,d\varOmega ,\quad \forall v_{N} \in V_{N}; $$(13)

Find \(\psi_{N} \in V_{N}\) such that$$ \int_{\varOmega } \nabla\psi_{N} \cdot\nabla \eta_{N} \,d\varOmega = \int_{\varOmega }\omega_{N} \eta_{N} \,d\varOmega , \quad \forall\eta_{N} \in V_{N}. $$(14)

Find \(\omega_{N} \in V_{N}\) such that$$ \int_{\varOmega } \nabla\omega_{N} \cdot\nabla v_{N} \,d\varOmega = \int_{\varOmega }(\boldsymbol{\sigma}\nabla f)\cdot\nabla v_{N} \,d\varOmega ,\quad \forall v_{N} \in V_{N}; $$(16)

Find \(\psi_{N} \in V_{N}\) such that$$ \int_{\varOmega } \nabla\psi_{N} \cdot\nabla \eta_{N} \,d\varOmega = \int_{\varOmega }\omega_{N} \eta_{N} \,d\varOmega , \quad \forall\eta_{N} \in V_{N}. $$(17)
Theorem 1
Proof
Let \(E(\lambda)\) be the eigenfunctions space of (3) corresponding to the eigenvalue λ, and let C stand for a generic positive constant independent of any functions and of any discretization parameters. Then we give theoretical analysis of the eigenvalue problem by assuming here that all eigenvalues have ascent.
Theorem 2
Proof
4 Implementation
In this subsection, we start with some implementation details in the basis function construction.
The nonzero entries of B and C can be easily determined from the properties of Legendre polynomials as follows.
Lemma 2
5 Numerical results
In this section, we will show some numerical results which demonstrate the accuracy and efficiency of the LegendreGalerkin spectral method for the buckling problem of simply supported Kirchhoff plates on the reference square. We have taken the unit square \(\varOmega =(0,1)\times(0,1)\) as an example of a convex domain.
Example 1
Uniformly compressed square plate
The lowest four buckling coefficients of a uniformly compressed simply supported square plate
N = 10  Ref. [ 12 ] N = 40  Ref. [ 12 ] N = 80  Exact  

\(\lambda_{N}^{1}\)  19.7392088021788  19.7614  19.7448  19.739208802178716 
\(\lambda_{N}^{2}\)  49.3480220530284  49.5155  49.3899  49.348022005446794 
\(\lambda_{N}^{3}\)  49.3480220530284  49.5155  49.3899  49.348022005446794 
\(\lambda_{N}^{4}\)  78.9568353038783  79.4444  79.0786  78.956835208714864 
Example 2
Square plate under combined bending and compression in one direction
The lowest four buckling coefficients of a uniformly compressed simply supported square plate
α  N = 10  N = 20  N = 30  Ref. [ 12 ] N = 80 

2.0  25.5283500058494  25.5283479481006  25.5283479481008  25.5619 
4/3  11.0117810710886  11.0117810716443  11.0117810716443  11.0152 
1  7.81195727517932  7.81195727522176  7.81195727522178  7.8142 
4/5  6.59506010731459  6.59506010732159  6.59506010732162  6.6026 
2/3  5.96338451327805  5.96338451327973  5.96338451327975  5.9701 
Example 3
Shear loaded square plate
The lowest four buckling coefficients of a uniformly compressed simply supported square plate with ten convergence digits
N = 10  N = 20  N = 30  N = 40  Ref. [ 12 ]  

\(\kappa_{1}\)  9.3245119409  9.3245202591  9.3245202616  9.3245202616  9.3236 
6 Conclusion
We have proposed an efficient mixed LegendreGalerkin spectral method for the solution of the buckling problem of simply supported Kirchhoff plates. The optimal error estimates for eigenvalues and eigenfunctions are also provided. Finally, the efficiency and accuracy of the spectral method for the buckling problem of simply supported Kirchhoff plates have been illustrated by numerical results. In the future, we will consider dealing with more complicated domains by the mixed LegendreGalerkin spectral element method. Firstly, we will divide the domain into lots of subdomains by domain decomposition, then we will construct the same basis functions in each subdomain similar to the square plate examples and construct the hat basis functions at each interface between the two domains. Here we want to mention that if the subdomain is irregular, the expressions of the mass matrix and the stiffness matrix should be computed by numerical integration.
Declarations
Acknowledgements
JYC and ZQW would like to acknowledge the support from the National Natural Science Foundation of China (NSFC) under Grant 11501140, the Foundation of Guizhou Science and Technology Department under Grants [2014]2098 and Innovation Group Major Program of Guizhou Province KY[2016]029. WXC would like to acknowledge the support from China Postdoctoral Science Foundation under Grant 2016T90027. LZC would like to acknowledge the support from NSFC through Grants 11671166 and U1530401, Postdoctoral SFC through Grant 2015M580038.
Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
Authors’ Affiliations
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