Time-Domain Higher-Order Boundary Element Method for Simulating High Forward-Speed Ship Motions in Waves

The hydrodynamic performance of a high forward-speed ship in obliquely propagating waves is numerically examined to assess both free motions and wave field in comparison with a low forward-speed ship.This numerical model is based on the time-domain potential flow theory and higher-order boundary element method,where an analytical expression is completely expanded to determine the base-unsteady coupling flow imposed on the moving condition of the ship.The ship in the numerical model may possess different advancing speeds,i.e.stationary,low speed,and high speed.The role of the water depth,wave height,wave period,and incident wave angle is analyzed by means of the accurate numerical model.It is found that the resonant motions of the high forward-speed ship are triggered by comparison with the stationary one.More specifically,a higher forward speed generates a V-shaped wave region with a larger elevation,which induces stronger resonant motions corresponding to larger wave periods.The shoaling effect is adverse to the motion of the low-speed ship,but is beneficial to the resonant motion of the high-speed ship.When waves obliquely propagate toward the ship,the V-shaped wave region would be broken due to the coupling effect between roll and pitch motions.It is also demonstrated that the maximum heave motion occurs in beam seas for stationary cases but occurs in head waves for high speeds.However,the variation of the pitch motion with period is hardly affected by wave incident angles.

A Wave Superposition-Finite Element Method for Calculating the Radiated Noise Generated by Volumetric Targets in Shallow Water

A combined method of wave superposition and finite element is proposed to solve the radiation noise of targets in shallow sea.Taking the sound propagation of spherical sound source in shallow sea as an example,the radiation sound field of the spherical sound source is equivalent to the linear superposition of the radiation sound field of several internal point sound sources,and then the radiated noise induced by spherical sound source can be predicted quickly.The accuracy and efficiency of the method are verified by comparing with the numerical results of finite element method,and the rapid prediction of underwater radiated noise of cylindrical shell is carried out based on the method.The results show that compared with the finite element method,the relative error of the calculation results under different simulation conditions does not exceed 0.1%,and the calculation time is about 1/10 of the finite element method,so this method can be used to solve the radiated noise of shallow underwater targets.

A weak formulation of heterogenous viscoacoustic wave propagation in infinite domain

The accurate simulation of wave propagation in real media requires properly taking the attenuation into account,which leads to wave dissipation together with its causal companion,wave dispersion.In this study,to obtain a weak formulation of heterogenous viscoacoustic wave propagation in an infinite domain,the viscoacoustic medium is first characterized by its frequency-dependent complex bulk compliance instead of the classically used complex bulk modulus.Then,a mechanical model using serially connected standard linear solids(SSLS)is built to obtain the rational approximation of the complex bulk compliance whose parameters are calculated via an adapted nonlinear optimization method.Utilizing the obtained bulk compliance-based constitutive relation,a novel second-order viscoacoustic wave equation in the frequency domain is derived,of which the weak formulation can be physically explained as the virtual work equation and can thus be discretized using a continuous spectral element method in space.Additionally,a new method is introduced to address the convolution terms involved in the inverse Fourier transform,whose accurate time integration can then be achieved using an explicit time scheme,which avoids the transient growth that exists in the classical method.The resulting full time-space decoupling scheme can handle wave propagation in arbitrary heterogeneous media.Moreover,to treat the wave propagation in an infinite domain,a perfectly matched layer in weak formulation is derived for the truncation of the infinite domain via complex coordinate stretching of the virtual work equation.With only minor modification,the resulting perfectly matched layer can be implemented using the same time scheme as for the wave equation inside the truncated domain.The accuracy,numerical stability,and versatility of the new proposed scheme are demonstrated with numerical examples.

NUMERICAL INVESTIGATION OF TOROIDAL SHOCK WAVES FOCUSING USING DISCONTINUOUS GALERKIN FINITE ELEMENT METHOD

A numerical simulation of the toroidal shock wave focusing in a co-axial cylindrical shock tube is inves- tigated by using discontinuous Galerkin （DG） finite element method to solve the axisymmetric Euler equations. For validating the numerical method, the shock-tube problem with exact solution is computed, and the computed results agree well with the exact cases. Then, several cases with higher incident Mach numbers varying from 2.0 to 5.0 are simulated. Simulation results show that complicated flow-field structures of toroidal shock wave diffraction, reflection, and focusing in a co-axial cylindrical shock tube can be obtained at different incident Mach numbers and the numerical solutions appear steep gradients near the focusing point, which illustrates the DG method has higher accuracy and better resolution near the discontinuous point. Moreover, the focusing peak pres- sure with different grid scales is compared.

PARTITION OF UNITY FINITE ELEMENT METHOD FOR SHORT WAVE PROPAGATION IN SOLIDS

A partition of unity finite element method for numerical simulation of short wave propagation in solids is presented. The finite element spaces were constructed by multiplying the standard isoparametric finite element shape functions, which form a partition of unity, with the local subspaces defined on the corresponding shape functions, which include a priori knowledge about the wave motion equation in trial spaces and approximately reproduce the highly oscillatory properties within a single element. Numerical examples demonstrate the performance of the proposed partition of unity finite element in both computational accuracy and efficiency.

Three-Dimensional Boundary Element Method Applied to Nonlinear Wave Transformation

For higher accuracy in simulating the transformation of three dimensional waves, in consideration of the advantages of constant panels and linear elements, a combined boundary elements is applied in this research. The method can be used to remove the transverse vibration due to the accumulation of computational errors. A combined boundary condition of sponge layer and Sommerfeld radiation condition is used to remove the reflected waves from the computing domain. By following the water particle on the water surface, the third order Stokes wave transform is simulated by the numerical wave flume technique. The computed results are in good agreement with theoretical ones.

IN-PLANE WAVE MOTION IN FINITE ELEMENT MODEL

The analysis method of lattice dynamics in classical physics is extended to study the properties of in-plane wave motion in the hybrid-mass finite element model in this paper. The dispersion equations of P and SV waves in the discrete model are first obtained by means of separating the characteristic equation of the motion equation, and then used to analyse the properties of P-and SV-homogeneous, inhomogeneous waves and other types of motion in the model. The dispersion characters, cut-off frequencies of P and SV waves, the polarization drift and appendent anisotropic property of wave motion caused by the discretization are finally discussed.

Propagations of Rayleigh and Love waves in ZnO films/glass substrates analyzed by three-dimensional finite element method

Propagation characteristics of surface acoustic waves（SAWs） in ZnO films/glass substrates are theoretically investigated by the three-dimensional（3D） finite element method. At first, for（11ˉ20） ZnO films/glass substrates, the simulation results confirm that the Rayleigh waves along the [0001] direction and Love waves along the [1ˉ100] direction are successfully excited in the multilayered structures. Next, the crystal orientations of the ZnO films are rotated, and the influences of ZnO films with different crystal orientations on SAW characterizations, including the phase velocity, electromechanical coupling coefficient, and temperature coefficient of frequency, are investigated. The results show that at appropriate h/λ, Rayleigh wave has a maximum k^2 of 2.4% in（90°, 56.5°, 0°） ZnO film/glass substrate structure; Love wave has a maximum k^2 of 3.81% in（56°, 90°, 0°） ZnO film/glass substrate structure. Meantime, for Rayleigh wave and Love wave devices, zero temperature coefficient of frequency（TCF） can be achieved at appropriate ratio of film thickness to SAW wavelength. These results show that SAW devices with higher k^2 or lower TCF can be fabricated by flexibly selecting the crystal orientations of ZnO films on glass substrates.

A Finite Element Solution of Wave Forces on Submerged Horizontal Circular Cylinders

In this paper, a numerical model is established for estimating the wave forces on a submerged horizontal circular cylinder. For predicting the wave motion, a set of two dimensional Navier Stokes equations is solved numerically with a finite element method. In order to track the moving non linear wave surface boundary, the Navier Stokes equations are discretized in a moving mesh system. After each computational time step, the mesh is modified according to the changed wave surface boundary. In order to stabilize the numerical procedure, a three step finite element method is applied in the time integration. The water sloshing in a tank and wave propagation over a submerged bar are simulated for the first time to validate the present model. The computational results agree well with the analytical solution and the experimental data. Finally, the model is applied to the simulation of interaction between waves and a submerged horizontal circular cylinder. The effects of the KC number and the cylinder depth on the wave forces are studied.

Scaled Boundary Finite Element Analysis of Wave Passing A Submerged Breakwater

The scaled boundary finite element method （SBFEM） is a novel semi-analytical technique combining the advantage of the finite element method （FEM） and the boundary element method （BEM） with its unique properties. In this paper, the SBFEM is used for computing wave passing submerged breakwaters, and the reflection coeffcient and transmission coefficient are given for the case of wave passing by a rectangular submerged breakwater, a rigid submerged barrier breakwater and a trapezium submerged breakwater in a constant water depth. The results are compared with the analytical solution and experimental results. Good agreement is obtained. Through comparison with the results using the dual boundary element method （DBEM）, it is found that the SBFEM can obtain higher accuracy with fewer elements. Many submerged breakwaters with different dimensions are computed by the SBFEM, and the changing character of the reflection coeffcient and the transmission coefficient are given in the current study.

Nonlinear finite element analysis of effect of seismic waves on dynamic response of Shiziping dam

Many high earth-rockfill dams are constructed in the west of China. The seismic intensity at the dam site is usually very high, thus it is of great importance to ensure the safety of the dam in meizoseismal area. A 3D FEM model is established to analyze the seismic responses of Shiziping earth-rockfill dam. The nonlinear elastic Duncan-Chang constitutive model and the equivalent viscoelastic constitutive model are used to simulate the static and dynamic stress strain relationships of the dam materials, respectively. Four groups of seismic waves are inputted from the top of the bedrock to analyze the dynamic responses of the dam. The numerical results show that the calculated dynamic magnification factors display a good consistency with the specification values. The site spectrum results in larger acceleration response than the specification spectrum. The analysis of relative dynamic displacement indicates that the displacement at the downstream side of the dam is larger than that at the upstream side. The displacement response reduces from the center of river valley to two banks. The displacement responses corresponding to the specification spectrum are a little smaller than those corresponding to the site spectrum. The analysis of shear stress indicates that a large shear stress area appears in the upstream overburden layer, where the shear stress caused by site waves is larger than that caused by specification waves. The analysis of dynamic principal stress indicates that the minimum dynamic stresses in corridor caused by specification and site waves have little difference. The maximum and minimum dynamic stresses are relatively large at two sides. The largest tensile stress occurs at two sides of the floor of grouting corridor, which may result in the crack near the corridor side. The numerical results present good consistency with the observation data of the grouting corridor in Wenchuan earthquake.

A Multi-Domain Boundary Element Method to Analyze the Reflection and Transmission of Oblique Waves From Double Porous Thin Walls

In the present paper, we examine the performance of an efficient type of wave-absorbing porous marine structure under the attack of regular oblique waves by using a Multi-Domain Boundary Element Method(MDBEM). The structure consists of two perforated vertical thin barriers creating what can be called a wave absorbing chamber system. The barriers are surface piercing, thereby eliminating wave overtopping. The problem of the interaction of obliquely incident linear waves upon a pair of perforated barriers is first formulated in the context of linear diffraction theory. The resulting boundary integral equation, which is matched with far-field solutions presented in terms of analytical series with unknown coefficients, as well as the appropriate boundary conditions at the free surface, seabed, and barriers, is then solved numerically using MDBEM. Dissipation of the wave energy due to the presence of the perforated barriers is represented by a simple yet effective relation in terms of the porosity parameter appropriate for thin perforated walls. The results are presented in terms of reflection and transmission coefficients. The effects of the incident wave angles, relative water depths, porosities, depths of the walls, and other major parameters of interest are explored.

Dynamic Adaptive Finite Element Analysis of Acoustic Wave Propagation Due to Underwater Explosion for Fluid-structure Interaction Problems

In this paper, an investigation into the propagation of far field explosion waves in water and their effects on nearby structures are carried out. For the far field structure, the motion of the fluid surrounding the structure may be assumed small, allowing linearization of the governing fluid equations. A complete analysis of the problem must involve simultaneous solution of the dynamic response of the structure and the propagation of explosion wave in the surrounding fluid. In this study, a dynamic adaptive finite element procedure is proposed. Its application to the solution of a 2D fluid-structure interaction is investigated in the time domain. The research includes：a） calculation of the far-field scatter wave due to underwater explosion including solution of the time-depended acoustic wave equation, b） fluid-structure interaction analysis using coupled Euler-Lagrangian approach, and c） adaptive finite element procedures employing error estimates, and re-meshing. The temporal mesh adaptation is achieved by local regeneration of the grid using a time-dependent error indicator based on curvature of pressure function. As a result, the overall response is better predicted by a moving mesh than an equivalent uniform mesh. In addition, the cost of computation for large problems is reduced while the accuracy is improved.

PARTITION OF UNITY FINITE ELEMENT METHOD FOR SHORT WAVE PROPAGATION IN SOLIDS

A partition of unity finite element method for numerical simulation of short wave propagation in solids is presented. The finite element spaces were constructed by multiplying the standard isoparametric finite element shape functions, which form a partition of unity, with the local subspaces defined on the corresponding shape functions, which include a priori knowledge about the wave motion equation in trial spaces and approximately reproduce the highly oscillatory properties within a single element. Numerical examples demonstrate the performance of the proposed partition of unity finite element in both computational accuracy and efficiency.

Modeling of Fully Nonlinear Wave Radiation by Submerged Moving Structures Using the Higher Order Boundary Element Method

The higher-order boundary element method is applied to the numerical simulation of nonlinear waves radiated by a forced oscillating fully submerged vertical circular cylinder. In this time-domain approach, the mixed boundary value problem based on an Eulerian description at each time step is solved using the higher order boundary element method. The 4th-order Runge–Kutta scheme is adopted to update the free water surface boundary conditions expressed in a Lagrangian formulation. Following completion of the numerical model, the problems of radiation(heave) of water waves by a submerged sphere in finite depth are simulated and the computed results are verified against the published numerical results in order to ensure the effectiveness of the model. The validated numerical model is then applied to simulate the nonlinear wave radiation by a fully submerged vertical circular cylinder undergoing various forced sinusoidal motion in otherwise still conditions. The numerical results are obtained for a series of wave radiation problems; the completely submerged cylinder is placed in surging, heaving and combined heave-pitching motions with different drafts, amplitudes and frequencies. The corresponding numerical results of the cylinder motions, wave profiles, and hydrodynamic forces are then compared and explained for all the cases.

Seismic wave modeling in viscoelastic VTI media using spectral element method

Spectral element method（SEM） for elastic media is well known for its great flexibility and high accuracy in solving problems with complex geometries.It is an advanced choice for wave simulations.Due to anelasticity of earth media,SEM for elastic media is no longer appropriate.On fundamental of the second-order elastic SEM,this work takes the viscoelastic wave equations and the vertical transversely isotropic（VTI） media into consideration,and establishes the second-order SEM for wave modeling in viscoelastic VTI media.The second-order perfectly matched layer for viscoelastic VTI media is also introduced.The problem of handling the overlapped absorbed corners is solved.A comparison with the analytical solution in a twodimensional viscoelastic homogeneous medium shows that the method is accurate in the wave-field modeling.Furtherly,numerical validation also presents its great flexibility in solving wave propagation problems in complex heterogeneous media.This second-order SEM with perfectly matched layer for viscoelastic VTI media can be easily applied in wave modeling in a limited region.

FINITE ELEMENT ANALYSIS OF WAVE PROPAGATION IN FLUID-SATURATED POROUS MEDIA

With the porous media model based on mixture theory, a finite element formulation for dynamic transient analysis of fluid_saturated two_phase porous media is presented. Time integration of the equation, deduced with penalty method, can be performed by using implicit or explicit method. One_dimensional wave propagation in column under step loading and impulsive loading are analyzed with the developed finite element program. The obtained curves of displacements, velocities, effective stresses and pore pressures against time demonstrate the existence of wave propagation phenomena, which coincide with the theoretical results.

Surface-mounted bender elements for measuring horizontal shear wave velocity of soils

The bender element testing features its in-plane directivity, which allows using bender elements to measure the shear wave velocities in a wider range of in-plane configurations besides the standard tip-to-tip alignment. This paper proposed a novel bender element testing technique for measuring the horizontal shear wave velocity of soils, where the bender elements are surface- mounted and the axes of the source and receiver elements are parallel to each other. The preliminary tests performed on model ground of silica sand showed that, by properly determining the travel distance and time of the shear waves, the surface-mounted bender elements can perform as accurately as the conventional "tip-to-tip" configuration. Potentially, the present system provides a promising nondestructive tool for characterizing geomaterials and site conditions both in laboratory and in the fields.

Finite Elements Based on Deslauriers-Dubuc Wavelets for Wave Propagation Problems

This paper presents the formulation of finite elements based on Deslauriers-Dubuc interpolating scaling functions, also known as Interpolets, for their use in wave propagation modeling. Unlike other wavelet families like Daubechies, Interpolets possess rational filter coefficients, are smooth, symmetric and therefore more suitable for use in numerical methods. Expressions for stiffness and mass matrices are developed based on connection coefficients, which are inner products of basis functions and their derivatives. An example in 1-D was formulated using Central Difference and Newmark schemes for time differentiation. Encouraging results were obtained even for large time steps. Results obtained in 2-D are compared with the standard Finite Difference Method for validation.

Second-generation wavelet finite element based on the lifting scheme for GPR simulation

Ground-penetrating radar(GPR)is a highly efficient,fast and non-destructive exploration method for shallow surfaces.High-precision numerical simulation method is employed to improve the interpretation precision of detection.Second-generation wavelet finite element is introduced into the forward modeling of the GPR.As the finite element basis function,the second-generation wavelet scaling function constructed by the scheme is characterized as having multiple scales and resolutions.The function can change the analytical scale arbitrarily according to actual needs.We can adopt a small analysis scale at a large gradient to improve the precision of analysis while adopting a large analytical scale at a small gradient to improve the efficiency of analysis.This approach is beneficial to capture the local mutation characteristics of the solution and improve the resolution without changing mesh subdivision to realize the efficient solution of the forward GPR problem.The algorithm is applied to the numerical simulation of line current radiation source and tunnel non-dense lining model with analytical solutions.Result show that the solution results of the secondgeneration wavelet finite element are in agreement with the analytical solutions and the conventional finite element solutions,thereby verifying the accuracy of the second-generation wavelet finite element algorithm.Furthermore,the second-generation wavelet finite element algorithm can change the analysis scale arbitrarily according to the actual problem without subdividing grids again.The adaptive algorithm is superior to traditional scheme in grid refinement and basis function order increase,which makes this algorithm suitable for solving complex GPR forward-modeling problems with large gradient and singularity.