Analytic Solutions of Von Karman Plate under Arbitrary Uniform Pressure (II): Equations in Integral Form

TL;DR

This paper applies the homotopy analysis method (HAM) to solve Von Karman plate equations in integral form for circular plates under arbitrary uniform pressure, demonstrating faster convergence and broader applicability than traditional methods.

Contribution

The paper introduces two HAM-based approaches for solving Von Karman's equations under arbitrary pressure, showing their validity and improved convergence over existing iterative methods.

Findings

HAM-based methods are valid for arbitrary uniform pressure

HAM approaches converge faster than interpolation iterative method

HAM demonstrates superiority over perturbation methods

Abstract

In this paper, the homotopy analysis method (HAM) is successfully applied to solve the Von Karman's plate equations in the integral form for a circular plate with the clamped boundary under an arbitrary uniform external pressure. Two HAM-based approaches are proposed. One is for a given external load Q, the other for a given central deflection. Both of them are valid for an arbitrary uniform external pressure by means of choosing a proper value of the so-called convergence-control parameters c_1 and c_2 in the frame of the HAM. Besides, it is found that iteration can greatly accelerate the convergence of solution series. In addition, we prove that the interpolation iterative method is a special case of the HAM-based 1st-order iteration approach for a given external load Q when c_1=-theta and c_2=-1, where theta denotes the interpolation parameter of the interpolation iterative method. Therefore, like Zheng and Zhou}, one can similarly prove that the HAM-based approaches are valid for an arbitrary uniform external pressure, at least in some special cases such as c_1=-theta and c_2=-1. Furthermore, it is found that the HAM-based iteration approaches converge much faster than the interpolation iterative method. All of these illustrate the validity and potential of the HAM for the famous Von Karman's plate equations, and show the superiority of the HAM over perturbation methods.