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

Abstract

In this paper, the homotopy analysis method (HAM) is successfully applied to solve the Von Kármán'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 [1, 2] is a special case of the HAM-based 1st-order iteration approach for a given external load Q when c 1 = −θ and c 2 = −1, where θ denotes the interpolation parameter of the interpolation iterative method. Therefore, like Zheng and Zhou [3], 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 = −θ and c 2 = −1. Furthermore, it is found that the HAM-based iteration approaches converge much faster than the interpolation iterative method [1,2]. All of these illustrate the validity and potential of the HAM for the famous Von Kármán's plate equations, and show the superiority of the HAM over perturbation methods.

On the limiting Stokes wave of extreme height in arbitrary water depth

“…Thus, the HAM can indeed greatly enlarge the convergence interval of solution series by means of properly choosing the so-called "convergence-control parameter" . Note that perturbation methods for many problems have been found to be a special case of the HAM when = −1, as illustrated by Zhong & Liao (2017, 2018, and this well explains why perturbation results are often invalid in case of high nonlinearity. In addition, the HAM provides us great freedom to choose initial approximation so that iteration can be easily used to accelerate convergence in the frame of the HAM.…”

“…In this paper, the limiting Stokes' wave in arbitrary water depth is successfully solved by an analytic approximation method, namely the homotopy analysis method (HAM) (Liao 1992(Liao , 1999(Liao , 2003(Liao , 2009(Liao , 2010(Liao , 2012Van Gorder & Vajravelu 2008;Mastroberardino 2011;Kimiaeifar et al 2011;Vajravelu & Van Gorder 2012;Sardanyés et al 2015;Liao et al 2016;Zhong & Liao 2017, 2018Liu et al 2018a,b). Unlike perturbation methods (Schwartz 1974;Cokelet 1977), the HAM is independent of any small/large physical parameter.…”

“…For example, perturbation techniques are invalid for large deformation of Von Kármán plate, a famous classic problem in solid mechanics. However, Zhong & Liao (2017, 2018 successfully applied the HAM to gain convergent series solution even for extremely large deformation of Von Kármán plate. It is worthwhile mentioning that some mathematical theorems of convergence have been rigorously proved in the frame of the HAM (Liao 2012).…”

On the generalized wavelet-Galerkin method

“…Besides, different from all other approximation techniques, the HAM provides us a convenient way to guarantee the convergence of solution series. As illustrated by Zhong and Liao [38], perturbation methods are often only special cases of the HAM, but the HAM still works well even if perturbation methods fail. More importantly, as a new analytic method, the HAM can provide us something completely new/different [39][40][41].…”

An analytical approximate technique to investigate a finite extensibility nonlinear oscillator

“…Furthermore, homotopy analysis method (HAM) (Xu et al, 2015;Nawaz et al, 2015;Zhong and Liao, 2016;Ellahi et al, 2015a,b;Rashidi et al, 2015;Zeeshan et al, 2014) has been developed for solving strongly nonlinear systems. The HAM is based on the homotopy in topology, which is, however, different from perturbation method.…”