A piecewise linear finite element method for the buckling and the vibration problems of thin plates

Mora, David; Rodrı́guez, Rodolfo

doi:10.1090/s0025-5718-09-02228-5

Cited by 18 publications

(27 citation statements)

References 19 publications

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“…There has been a long history on developing the finite element methods of the biharmonic eigenvalue problem, and many schemes have been proposed for discretization [9,11,25,36], computation of guaranteed upper and lower bounds [10,22,23,43], and adaptive method and its convergence analysis [17]. This paper is devoted to studying the multilevel efficient method of the biharmonic eigenvalue problem.…”

Section: Introductionmentioning

confidence: 99%

A Multi-Level Mixed Element Method for the Eigenvalue Problem of Biharmonic Equation

Zhang

2017

J Sci Comput

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Abstract. In this paper, we discuss approximating the eigenvalue problem of biharmonic equation. We first present an equivalent mixed formulation which admits amiable nested discretization.Then, we construct multi-level finite element schemes by implementing the algorithm as in [33] to the nested discretizations on series of nested grids. The multi-level mixed scheme for biharmonic eigenvalue problem possesses optimal convergence rate and optimal computational cost. Both theoretical analysis and numerical verifications are presented.

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Section: Introductionmentioning

confidence: 99%

A Multi-Level Mixed Element Method for the Eigenvalue Problem of Biharmonic Equation

Zhang

2017

J Sci Comput

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“…The table includes orders of convergence, as well as accurate values extrapolated by means of a least-squares fitting. The last column shows the values obtained by extrapolating those computed with method in [35] on the same uniform triangular meshes.…”

Section: L-shaped Platementioning

confidence: 99%

A virtual element method for the vibration problem of Kirchhoff plates

2018

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The aim of this paper is to develop a virtual element method (VEM) for the vibration problem of thin plates on polygonal meshes. We consider a variational formulation relying only on the transverse displacement of the plate and propose an H 2 (Ω) conforming discretization by means of the VEM which is simple in terms of degrees of freedom and coding aspects. Under standard assumptions on the computational domain, we establish that the resulting scheme provides a correct approximation of the spectrum and prove optimal order error estimates for the eigenfunctions and a double order for the eigenvalues. The analysis restricts to simply connected polygonal clamped plates, not necessarily convex. Finally, we report several numerical experiments illustrating the behaviour of the proposed scheme and confirming our theoretical results on different families of meshes. Additional examples of cases not covered by our theory are also presented.

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“…In this case, we have used a Poisson ratio ν = 0.25 and a correction factor k = 5/6. Additionally, we have also computed the lowest buckling intensity of a Kirchhoff-Love plate by using the finite element method analyzed in [41].…”

Section: Uniformly Compressed Platementioning

confidence: 99%