Analytic approximations of Von Kármán plate under arbitrary uniform pressure—equations in integral form

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SCIENCE CHINA Physics, Mechanics & Astronomy
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SCIENCE CHINA Physics, Mechanics & Astronomy, Volume 61, Issue 1: 014611(2018) https://doi.org/10.1007/s11433-017-9096-1

Analytic approximations of Von Kármán plate under arbitrary uniform pressure—equations in integral form

XiaoXu Zhong1, ShiJun Liao1,2,3,4,*
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  • ReceivedJun 19, 2017
  • AcceptedAug 25, 2017
  • PublishedOct 30, 2017
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Abstract

Analytic approximations of the Von Kármán's plate equations in integral form for a circular plate under external uniform pressure to arbitrary magnitude are successfully obtained by means of the homotopy analysis method (HAM), an analytic approximation technique for highly nonlinear problems. Two HAM-based approaches are proposed for either a given external uniform pressure $Q$ or a given central deflection, respectively. Both of them are valid for uniform pressure to arbitrary magnitude by choosing proper values of the so-called convergence-control parameters $c_1$ and $c_2$ in the frame of the HAM. Besides, it is found that the HAM-based iteration approaches generally converge much faster than the interpolation iterative method. Furthermore, we prove that the interpolation iterative method is a special case of the first-order HAM iteration approach for a given external uniform pressure $Q$ when $c_{1}=-\theta$ and $c_{2}=-1$, where $\theta$ denotes the interpolation iterative parameter. Therefore, according to the convergence theorem of Zheng and Zhou about the interpolation iterative method, the HAM-based approaches are valid for uniform pressure to arbitrary magnitude at least in the special case $c_{1}=-\theta$ and $c_{2}=-1$. In addition, we prove that the HAM approach for the Von Kármán's plate equations in differential form is just a special case of the HAM for the Von Kármán's plate equations in integral form mentioned in this paper. All of these illustrate the validity and great potential of the HAM for highly nonlinear problems, and its superiority over perturbation techniques.


Acknowledgment

This work was supported by the National Natural Science Foundation of China (Grant Nos. 11272209, and 11432009), and the State Key Laboratory of Ocean Engineering (Grant No. GKZD010063).


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    Figure 1

    (Color online) The squared residual error $\cal~E$ versus the times of iteration (a) and the CPU times (b) in the case of $Q=1000$, given by the HAM-based iteration approach using the convergence-control parameter $c_{0}=-0.02$. Solid line: first-order; long-dashed line: second-order; dashed line: third-order; dash-dotted line: fourth-order; dash-double-dotted line: fifth-order.

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    Figure 2

    (Color online) The squared residual error $\cal~E$ versus the CPU time in the case of $Q=1000$ (corresponding to $w(0)/h=6.1$), given by the interpolation iterative method [3], and the HAM-based approach for given external uniform pressure $Q$ and central deflection $a$, respectively. Dash-double-dotted line: results given by the interpolation iterative method [3]using the interpolation parameter $\theta=0.02$; dashed line: results given by the first-order HAM iteration approach for the Von Karman's plate equations in integral form with a given external uniform pressure $Q$ using $c_{0}=-0.05$; dash-dotted line: results given by the first-order HAM iteration approach [5]for the Von Kármán's plate equations in differential form with a given central deflection $a$ using $c_{0}=-0.25$; solid line: results given by the first-order HAM iteration approach for the Von Kármán's plate equations in integral form with a given central deflection $a$ using $c_{0}=-0.25$.

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    Figure 3

    (Color online) The convergence-control parameter $c_{0}$ and the interpolation parameter $\theta$ versus the central deflection $a$ for a circular plate with clamped boundary, given by the first-order HAM iteration approach for given central deflection $a$ and the interpolation iterative method [2]for given external uniform pressure $Q$. Dashed line: $-\theta$; solid line: $c_{0}$ given by eq. (sect. 3.3).

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