黏弹性板水弹性响应的非线性分析

通过将超大型浮式结构物(a very large floating structure,简称VLFS)模拟为黏弹性薄板,本工作对VLFS的非线性水弹性响应进行了解析研究。运用同伦分析方法(the homotopy analysis method,简称HAM),计算出速度势和板挠度的近似迭代解,并根据计算结果着重探究了几个重要的物理参数对黏弹性板形变的影响。结果发现:黏弹性板的挠度随着黏弹性时间、杨氏模量和板厚度增加而减少,而板挠度随着入射波波幅的增加而增加。最后,还对非线性色散关系和波幅之间的联系进行了探讨。

多外载联合作用下圆板的非线性弯曲

分析了多外载联合作用下圆板的轴对称非线性弯曲问题。分析从极坐标系下圆板弯曲的Ｖｏｎ－Ｋａｒｍａｎ方程出发，运用微分求积方法（ＤＱ法）导出了控制方程的ＤＱ形式；边缘径向位移和边缘力矩由两个统一的方程来表示，通过改变方程中的约束刚度和边缘载荷系数，实现了对任意边界条件的模拟；对最终得到的非线性方程组，用Ｎｅｗｔｏｎ－Ｒａｐｈｓｏｎ方法进行了迭代求解。文中给出了圆板受横向均布力、板心横向集中力、边缘均布径向力、边缘均布弯矩等四种载荷两两联合作用下的计算结果曲线，讨论了不同联合载荷对回板非线性弯曲的影响。与文献结果比较表明，该方法能满足各种边界条件，具有较高的求解精度。

摄动迭代法应用于正交异性圆板大挠度问题

在这篇文章里,我们介绍摄动-迭代法求正交异性圆板非线性弯曲的渐近解。对正交圆板的非线性控制方程构造了一个迭代格式,对其具有某些特殊边界条件(含固支、铰支等齐次边界条件)的解,构造了积分渐近式,并对它的收敛性作了严格的证明。

A Coiflet Wavelet Homotopy Technique for Nonlinear PDEs:Application to the Extreme Bending of Orthotropic Plate with Forced Boundary Constraints

A generalized homotopy-based Coiflet-type wavelet method for solving strongly nonlinear PDEs with nonhomogeneous edges is proposed.Based on the improvement of boundary difference order by Taylor expansion,the accuracy in wavelet approximation is largely improved and the accumulated error on boundary is successfully suppressed in application.A unified high-precision wavelet approximation scheme is formulated for inhomogeneous boundaries involved in generalized Neumann,Robin and Cauchy types,which overcomes the shortcomings of accuracy loss in homogenizing process by variable substitution.Large deflection bending analysis of orthotropic plate with forced boundary moments and rotations on nonlinear foundation is used as an example to illustrate the wavelet approach,while the obtained solutions for lateral deflection at both smally and largely deformed stage have been validated compared to the published results in good accuracy.Compared to the other homotopy-based approach,the wavelet scheme possesses good efficiency in transforming the differential operations into algebraic ones by converting the differential operators into iterative matrices,while nonhomogeneous boundary is directly approached dispensing with homogenization.The auxiliary linear operator determined by linear component of original governing equation demonstrates excellent approaching precision and the convergence can be ensured by iterative approach.

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

Analytic approximations of the Von Karman＇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 c1 and c2 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 c1= -0 and c2 = -1, where 0 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 c1 = -0 and c2= -1. In addition, we prove that the HAM approach for the Von karman＇s plate equations in differential form is just a special case of the HAM for the Von karman＇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.

Analytic approximation to nonlinear hydroelastic waves traveling in a thin elastic plate floating on a fluid

An analytic approximation method known as the homotopy analysis method(HAM)is applied to study the nonlinear hydroelastic progressive waves traveling in an infinite elastic plate such as an ice sheet or a very large floating structure(VLFS)on the surface of deep water.A convergent analytical series solution for the plate deflection is derived by choosing the optimal convergencecontrol parameter.Based on the analytical solution the efects of diferent parameters are considered.We find that the plate deflection becomes lower with an increasing Young’s modulus of the plate.The displacement tends to be flattened at the crest and be sharpened at the trough as the thickness of the plate increases,and the larger density of the plate also causes analogous results.Furthermore,it is shown that the hydroelastic response of the plate is greatly afected by the high-amplitude incident wave.The results obtained can help enrich our understanding of the nonlinear hydroelastic response of an ice sheet or a VLFS on the water surface.

Winkler基础薄圆板非线性弯曲的半解析法

该文应用半解析法,分析了 Winkler 基础上轴对称薄圆板的非线性弯曲问题。文中根据 Hertz 理论,将板面荷载假设为半球面形分布。文末通过具体算例,给出了圆板的荷载——挠度曲线和薄膜力曲线,并且分析了线性理论和非线性理论的误差。

外加直流电场作用下高阶弱非线性复合介质的电势分布

利用同伦分析方法,研究了一类由柱形杂质随机嵌入基质所形成的、电场和电流密度满足J=σE+χ|E|2E+η|E|4E形式本构关系的高阶弱非线性复合介质在外加直流电场作用下的电势分布问题.首先利用模函数展开法,将本构方程及边界条件化成了一系列非线性常微分方程的边值问题;再利用同伦分析方法进行求解,给出了电势在基质和杂质区域的渐近解析解.