Coupling Algorithm Of SBFEM And SPH For Water Entry Of Infinite Fluid Domain

In the ocean engineering,it often involves the problems of water exit and water entry,such as underwater missile launching,torpedo water entry,ship water entry and so on.These problems usually involve the breaking and overturning of free surface and the infinite fluid domain.Smoothed Particle Hydrodynamics(SPH)is a kind of particle method developed in recent years.It can track the free surface well and is suitable for solving large deformation problems.Therefore,it is often used to simulate water entry problems.However,when the simulated domain is infinite,the SPH method will lead to low efficiency due to the rapid increase of the number of particles.In order to solve this problem,we combine the Scaled Boundary Finite Element Method(SBFEM)and SPH method and propose a new non-reflecting boundary called SBFE virtual particle boundary.After that,we apply the SBFE virtual particle boundary to the numerical simulation of infinite water entry.The results show that the SBFE virtual particle boundary can not only simulate the transmission of pressure wave accurately,but also has the advantages of high computational efficiency.Firstly,the basic principles of the scaled boundary finite element method(SBFEM)and the derivation process of the infinite domain dynamic stiffness matrix in the scaled boundary coordinate system are described.Furthermore,the high-frequency approximation method that solves the infinite domain dynamic stiffness matrix and its significance are described.On this basis,the stability of the high-frequency approximation method that solves the infinite domain dynamic stiffness matrix is mainly solved,so as to apply this method to the pressure calculation in the SBFE virtual particle boundary.When the present high-frequency approximation method is used to solve the infinite domain dynamic stiffness matrix,due to the need of solving the Lyapunov equation and inverse matrix for many times in the calculation process,the spurious mode will inevitably be introduced,resulting in the divergence of the calculation results of the boundary pressure value.In this chapter,we analyze the reason for the divergence through the spectral analysis method,and improve the stability of the method by modifying the stiffness matrix of the infinite domain dynamic stiffness matrix.At last,the modified high-frequency approximation method is used to analyze the transient response of underwater structures,and the stability and accuracy of the modified high frequency approximation method is verified by comparing with the analytical solution.Secondly,this paper introduces the governing equations and approximation theory of SPH method.After that,the improvement of stability and accuracy of SPH algorithm in recent years are also introduced.On this basis,a new non-reflecting boundary called SBFE virtual particle boundary is proposed by combining the scaled boundary finite element method(SBFEM)and smoothed particle method(SPH).The numerical simulation results show that the new non-reflecting boundary can accurately and efficiently simulate the non-reflecting characteristics of the boundary.In this nonreflecting boundary,the pressure value of SBFE virtual particles is obtained by the modified high-frequency approximation method,while the acceleration value is obtained by the kernel approximation of SPH particles nearby.Thirdly,the SBFE virtual particle boundary is used to simulate the problem of water entry in infinite water.In this chapter,we use SBFE virtual particle boundary to simulate the non-reflecting characteristics of infinite water.The simulation results show that the results obtained by the non-reflecting boundary are more accurate,especially for the high-speed water entry problems.In addition,we investigate the influence of the artificial sound velocity that we choose in SPH simulations on the accuracy of water entry.The results show that the selection of the real sound velocity of the fluid is conducive to improve the accuracy of the simulation.Although an appropriate artificial sound velocity can improve the simulation efficiency,it also affects the accuracy of the simulations.when the artificial sound velocity is lower than real sound velocity,the compressibility of the fluid is overestimated,while when the artificial sound velocity is higher than real sound velocity,the compressibility of the fluid is underestimated.Generally speaking,this paper mainly improves the stability of the high frequency approximation of the scaled boundary finite element method,and applies the improved method to the coupling with the smoothed particle method,so as to simulate the nonreflecting characteristics of the boundary.Finally,the SBFEM-SPH coupling algorithm is applied to the numerical simulations of the infinite water entry problems,which shows the importance of the coupling algorithm.