Boundary element method for buckling eigenvalue problem and its convergence analysis

Ding, Rui; Fang-yun, Ding; Zhang, Ying

doi:10.1007/bf02436557

Cited by 4 publications

(1 citation statement)

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“…Numerical methods are often used by engineers and researchers for plate buckling, bending and vibration analysis, because they can usually provide numerical solutions that meet engineering requirements with acceptable errors. Such methods include, but are not limited to, the finite element method EC 40,6 1330 (Kanaka and Venkateswar, 1976;Tianyu and Kapania, 2018), finite difference method (Karamooz Ravari et al, 2014;Karimi and Shahidi, 2017), boundary element method (Ding et al, 2002;Paiva, 2018), discrete singular convolution method (Civalek, 2007;Omer et al, 2010;Xinwei and Zhangxian, 2017) and Rayleigh-Ritz method (Bhat, 1985;Monterrubio, 2012), etc.…”

Section: Introductionmentioning

confidence: 99%

Unified solution of some problems of rectangular plates with four free edges based on symplectic superposition method

Su,

Bai,

Hai

2023

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PurposeA unified framework for solving the bending, buckling and vibration problems of rectangular thin plates (RTPs) with four free edges (FFFF), including isotropic RTPs, orthotropic rectangular thin plates (ORTPs) and nano-rectangular plates, is established by using the symplectic superposition method (SSM).Design/methodology/approachThe original fourth-order partial differential equation is first rewritten into Hamiltonian system. The class of boundary value problems of the original equation is decomposed into three subproblems, and each subproblem is given the corresponding symplectic eigenvalues and symplectic eigenvectors by using the separation variable method in Hamiltonian system. The symplectic orthogonality and completeness of symplectic eigen-vectors are proved. Then, the symplectic eigenvector expansion method is applied to solve the each subproblem. Then, the symplectic superposition solution of the boundary value problem of the original fourth-order partial differential equation is given through superposing analytical solutions of three foundation plates.FindingsThe bending, vibration and buckling problems of the rectangular nano-plate/isotropic rectangular thin plate/orthotropic rectangular thin plate with FFFF can be solved by the unified symplectic superposition solution respectively.Originality/valueThe symplectic superposition solution obtained is a reference solution to verify the feasibility of other methods. At the same time, it can be used for parameter analysis to deeply understand the mechanical behavior of related RTPs. The advantages of this method are as follows: (1) It provides a systematic framework for solving the boundary value problem of a class of fourth-order partial differential equations. It is expected to solve more complicated boundary value problems of partial differential equations. (2) SSM uses series expansion of symplectic eigenvectors to accurately describe the solution. Moreover, symplectic eigenvectors are orthogonal and directly reflect the orthogonal relationship of vibration modes. (3) The SSM can be carried to bending, buckling and free vibration problems of the same plate with other boundary conditions.

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Section: Introductionmentioning

confidence: 99%

Unified solution of some problems of rectangular plates with four free edges based on symplectic superposition method

Su,

Bai,

Hai

2023

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Multiple reciprocity method with two series of sequences of high-order fundamental solution for thin plate bending

Fang-yun

Ding²,

2003

Appl Math Mech

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The boundary value problem of plate bending problem on two-parameter foundation was discussed. Using two series of the high-order fundamental solution sequences, namely, the fundamental solution sequences for the multi-harmonic operator and Laplace operator, applying the multiple reciprocity method (MRM), the MRM boundary integral equation for plate bending problem was constructed. It proves that the boundary integral equation derived from MRM is essentially identical to the conventional boundary integral equation. Hence the convergence analysis of MRM for plate bending problem can be obtained by the error estimation for the conventional boundary integral equation. In addition, this method can extend to the case of more series of the high-order fundamental solution sequences.

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An analytical spectral stiffness method for buckling of rectangular plates on Winkler foundation subject to general boundary conditions

Xiang

Xiao

Zhou

2020

Applied Mathematical Modelling

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Boundary element method for buckling eigenvalue problem and its convergence analysis

Cited by 4 publications

References 1 publication

Unified solution of some problems of rectangular plates with four free edges based on symplectic superposition method

Unified solution of some problems of rectangular plates with four free edges based on symplectic superposition method

Multiple reciprocity method with two series of sequences of high-order fundamental solution for thin plate bending

An analytical spectral stiffness method for buckling of rectangular plates on Winkler foundation subject to general boundary conditions

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