A HAM-based Wavelet Approach And Its Applications In Nonlinear Mechanics And Ocean Engineering

Nonlinear problems are widely existed in mechanics of ocean engineering.Based on the comprehensive analysis of homotopy analysis method(HAM)and wavelet,we introduce Generalized Orthogonal Coiflets into the framework of HAM and propose a ham-based wavelet approach for nonhomogeneous boundary value problem.By selecting proper convergence control parameters,initial guesses and auxiliary linear operators,nonlinear differential equations are transformed into a set of linear ones.Variables are expanded by the Coiflets and substituted into high order deformation equation.Finally,high accurate Coiflets solutions are reconstituted by iterating equations formulated to obtain coefficients of wavelet series with Galerkin method.The approach has been applied to solve typical problems in mechanics of ocean engineering,described briefly as follows:1.The framework of the ham-based wavelet approach for nonhomogeneous problem and solution procedure are detailedly illustrated and mathematical feasibility analysis are conducted.Two examples of large geometric deformation of uniform cantilever beam and small deflection deformation of nonlinear elastic foundation plate are given in order to validate the efficiency of our approach.2.Linear differential equation governed by biharmonic operator and coupled F(?)ppl-von K(?)rm(?)n equations are selected as comparative calculation example.The above coupled equations subjected to circled simply supported edge,clamped edge,and mixed one,respectively,nonlinearity of which is only related dimensionless loads,ratio of length to width and poisson ratio of materials.The deflection results are in agreement with numerical or exact ones.Our approach is able to obtain high accurate Coiflets solution for high nonlinearity and performs good efficiency while linear theory is only valid for weak nonlinear case.3.Large deflection of plates and bending of non-uniform square plate resting on different elastic foundations subjected to non-homogeneous boundary are studied with nonlinear analysis of ultimate loads,including linear and nonlinear Winkler basis,Pasternak basis,Winkler-Pasternak basis.High accurate Coiflets solution for deflection and stresses in middle point of plate are given in good accordance with previous studies,which is also effective for ultimate deformation conditions.The wavelet Homotopy Analysis Method is extended to solve partial differential equations with variable coefficients governed by bending plate on non-uniform elastic foundation.4.Classical lid-driven cavity flow is investigated.For 1D case,Coiflets expansion performs good accuracy without introducing optimal homogenizing function.For 2D case subjected to high order Neumann-type boundary,the high accurate Coiflets solution is also given since the homogenizing function doesn't exist.For the typical cavity flow,given a few wavelet basis(64 × 64),wavelet approximation overcoming the boundary singularities is proposed to obtain the results in good agreement with numerical ones by FVM,FEM,FDM,LBM,Spectral,Wavelet BEM-FEM.5.Classical mixed cavity flow with heat transfer is investigated.Compared to uniform,linear and exponential distributions of temperature,sinusoidal type performs better properties of heat transfer,which greatly changed the fields of flow and heat.As ratio of temperature increases from 0 to 1,heat transfer rate is improved on the upper lid but stay the same in the bottom side,where the direction inverse point of heat transfer is fixed.To add inclined angle contributes to weaken buoyancy effect and reduces heat transfer rate but is prevalent to variation of energy rate absorption in fluid.The periodic change of temperature amplitude ratio caused by phase difference brings about approximate periodic variation of the heat transfer property on boundary.6.Nanofluid flow with heat and mass transfer in an inclined cavity is investigated.It is found that Grashof number,lid-driven types,coefficients related to nanofluid,temperature ratio and phase deviation play an important role in complex physical field.Parametric analysis of complex flow fields,temperature and nanoparticle concentration are conducted to validate the efficiency of the nanofluid model.