The nonlinear hydroelastic response of very large floating structures (VLFSs) or an ice sheet floating on the surface of deep water, idealized as a semiinfinite thin elastic plate, is investigated analytically in the case of nonlinear incident waves. Assuming that the fluid is inviscid and incompressible and the motion is irrotational, we consider incident progressive waves with a given angular frequency within the framework of potential flow theory. With the aid of the homotopy analysis method (HAM), the convergent analytical series solutions are derived by solving the simultaneous equations in which we apply a convergencecontrol parameter to obtain convergent solutions with relatively few terms. The clear calculation results are represented to show nonlinear waveplate interaction. The effects of different physical parameters, including incident wave amplitude, Young’s modulus, the thickness and density of the plate on the wave scattering, and the hydroelastic response of the floating plate, are considered. We find that the variations of the plate stiffness, thickness, and density greatly change amount of wave energy which is reflected into the open water region and is transmitted into the platecovered region. Further, the hydroelastic response of the plate also can be affected by the amplitude of incident wave.
In recent decades, there have been intensive researches on hydroelastic interaction between water waves and elastic plates in the area of ocean engineering and polar engineering in view of their academic research and applications. The very large floating structures (VLFSs), which are expected to serve as floating airports, artificial floating islands, ultralarge ships, or even mobile offshore bases, are mathematically assumed as elastic plates floating on a fluid for their strong flexibility and huge horizontal scales compared to vertical scales and characteristic wavelength. Accordingly, the hydroelastic deformation in the response of the structure to water waves is a predominant factor to design and maintain a safe and economic VLFS. Early studies in this field were mainly focused on hydroelastic interaction between the water waves and the large ice sheets which were also idealized as floating elastic plates [
More importantly, it is inevitable that largeamplitude waves will occur more often in the future with the growing frequency of extreme weather events on a warmer and warmer Earth. Thus, some scholars investigated nonlinear hydroelastic interaction between semiinfinite elastic plates and water waves by using the wellknown perturbation method. Forbes [
It is well known that the traditional perturbation and asymptotic techniques depend on the small physical parameters and approximations of nonlinear problems usually break down when the nonlinearity becomes strong. Therefore, they are only valid for weakly nonlinear problems. In this paper, we apply the homotopy analysis method (HAM) developed by Liao [
We consider the nonlinear hydroelastic interaction between incident progressive waves and a semiinfinite elastic plate floating on infinitely deep water for the twodimensional case, as shown in Figure
Schematic diagram for the interaction between incident progressive waves and a semiinfinite elastic plate floating on a fluid of infinite depth.
We consider the nonlinear hydroelastic interaction between incident progressive waves and a semiinfinite elastic plate floating on infinitely deep water for the twodimensional case, as shown in Figure
In the platecovered region, under the assumption that any fluid particle which is once between the thin elastic plate and the water surface will remain on it, the dynamic boundary condition on the fluidplate interface can be written as
Further, considering the continuities of the pressure and the velocity on the boundary between the platecovered region and the open water region, we express the matching conditions as
For the elastic plate freely floating on the water surface, it is reasonable that the motion of the plate satisfies the free edge conditions as follows:
The homotopy analysis method (HAM) is an analysis approximation method developed by Liao [
Let
Similarly, in the platecovered region, the socalled zerothorder deformation equations are the governing equation (
In order to obtain the series solutions for the velocity potential
Assuming that
As shown later in Section
Considering the progressive gravity wave propagating periodically in
According to the linear wave theory, the governing equation (
In the platecovered region, if there is no gap between the bottom surface of the thin elastic plate and the top surface of the fluid layer, it is reasonable that the plate or the surface fluid still travels periodically in the
Subsequently, we consider the expression of the potential function as
Equations (
As mentioned by Liao [
It should be noted that the HAM can provide extremely large freedom in the choice of auxiliary linear operators; then for simplifying the computation procedure we can only choose linear operators of
The linear PDEs for the unknown
The elaborated expressions for
At the boundary between the platecovered region and the open water region, the matching conditions of the velocity and pressure of the free edge conditions are
Now we see that the unknown terms
If we fix all model physical parameters in our nonlinear boundaryvalue problem, there will still be an unknown convergencecontrol parameter
To verify the convergence of our approximate analytic solutions to the nonlinear hydroelasticity problem considered here by means of the HAM, we employ the dimensional case of
The total residual square error



1 

3 

5 

7 

Residual squares of
Here we only change one physical parameter in the above case to exclude the effects of other ones on the nonlinear hydroelastic interaction between the incident progressive waves and the elastic plate. Firstly, we compare the HAMbased nonlinear solutions of hydroelastic response of a semiinfinite elastic plate floating on a fluid due to incident progressive waves with corresponding linear results also obtained by HAM. As shown in Figure
Comparison of my nonlinear wave profiles with those of corresponding linear theory. Solid line: linear solution; dashed line: nonlinear solution.
Variation of incident progressive waves and hydroelastic deflection of the plate versus
Variation of incident progressive waves and hydroelastic deflection of the plate versus
Variation of incident progressive waves and hydroelastic deflection of the plate versus
Practically speaking, in Figure
Variation of incident progressive waves and hydroelastic deflection of the plate versus
By the homotopy analysis method, convergent series solutions are derived for the free wave surface elevation and the deflection of a semiinfinite elastic plate floating on a fluid of infinite depth, respectively, to analytically study the nonlinear hydroelastic response of the plate with a free edge due to incident progressive waves. For the whole flow domain mathematically including an open water region and a platecovered region, we obtain solutions with fairly low residual error after computing only a small number of terms by introducing a convergencecontrol parameter. Further, in the platecovered region, it is noted that all of two nonlinear operators
The influences of several physical parameters on the hydroelastic responses of the plate are considered in detail. The amplitude of incident wave, Young’s modulus
Let
There is no conflict of interests in this paper. The author obtained the approximate analytical solutions of the PDEs considered here by means of computer algebra system like Mathematica.
This research was sponsored by the Natural Science Foundation of Shanghai under Grant no. 14ZR1416200, the National Natural Science Foundation of China under Grant no. 11472166, and the Natural Science Foundation of Shandong Province of China under Grant no. ZR2013AL012.