Numerical Solutions Of Extended Mild Slope Equation By Local RBF-DQM

The mild slope equation(MSE)is the basic wave model for studying the wave propagation deformation law of nearshore.Due to its simple form and strong applicability,the mild slope equation has attracted wide attention since it was put forward.In order to further expand the application range of the mild slope equation,many scholars have revised and improved the original mild slope equation by considering the influence of the steep slope topography and the energy dissipation caused by wave breaking.The numerical method for solving the extended mild slope equation usually belongs to the grid method.Although these traditional methods can solve the problem,they all have some defects.In order to find an efficient,accurate and stable numerical method for solving the extended mild slope equation,this paper adopts an emerging meshless method—Local RBF-based Differential Quadrature Method(Local RBF-DQM)to solve the extended mild slope equation,and then to simulate the complex wave motion in the coastal region.The research contents and related achievements of this paper are mainly in the following aspects:(1)The extended mild slope equation model,which considering the influence of the steep slope terrain and rapidly varying bathymetry: In the present study,we attempt to adopt the variable shape parameter strategy proposed by Golbabai to determine the shape parameters in the Local RBF-DQM,and then use this numerical method to discretize the partial differential items in the extended mild slope equation.Therefore,the numerical model of the extended mild slope equation combined with meshless method is established.Next,this model was applied to simulate wave deformation phenomena in three complex terrains and corresponding analysis was also made.By comparing the numerical solution with the previous research results,the present model is tested to have good applicability for wave propagation in steep slope terrain and rapidly varying bathymetry.In the case of sinusoidally varying topography,the present model even produces better results than other numerical methods.In addition,the stability and convergence of the numerical model are further tested by setting different total points and comparing the simulation results.(2)The extended mild slope equation model,which combined energy dissipation factor for wave breaking and the influence of the steep slope of the seabed: Determined the solving steps of the nonlinear equation after considering the wave breaking and established the Local RBF-DQM extended mild slope equation model.Then used this model to simulate wave deformation phenomena in three numerical case: slope topography,shore-parallel breakwater and shore-normal breakwater experimental.The numerical solutions were compared with previous research results to test the feasibility of applying the present model to capture the wave breaking phenomenon.Moreover,the effects of different incident wave heights and water depths on the relative wave height are analyzed.In addition,the stability and convergence of the numerical model are further tested.In order to accurately describe the wave diffraction phenomenon near the breakwater,the shape of the local support domain in the numerical method is improved.Moreover,it is found that the numerical oscillation problem occurs when used the conventional wave breaking solution procedure.To solve this problem,we proposes a processing technique: when the i(i?2)iteration is performed,the wave energy dissipation term is estimated based on the wave height average of the previous i-1 iteration.This technique successfully smoothes numerical oscillations and accelerates convergence.(3)The extended mild slope equation model,which combined energy dissipation factor for wave breaking and the influence of the steep slope terrain and rapidly varying bathymetry.Two cases with complex boundary conditions,topographic conditions and wave conditions are selected for numerical simulation: Pan Jun-ning physical model experiment and large-scale reef topography on Mokuleia Beach in Hawaii.Meanwhile,we studied the wave height distribution law around the harbor and the beach,tested the effects of different incident wave heights and slope gradients on the wave deformation.The differences of wave breaking points under different wave conditions were also analyzed.The simulation results show that the present model has good adaptability to complex wave propagation problems,which provides a reference for practical engineering applications.