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.

Using a Time-domain Higher-order Boundary Element Method to Simulate Wave and Current Diffraction from a 3-D Body

To study wave-current actions on 3-D bodies a time-domain numerical model was established using a higher-order boundary element method(HOBEM).By assuming small flow velocities,the velocity potential could be expressed for linear and higher order components by perturbation expansion.A 4th-order Runge-Kutta method was applied for time marching.An artificial damping layer was adopted at the outer zone of the free surface mesh to dissipate scattering waves.Validation of the numerical method was carried out on run-up,wave exciting forces,and mean drift forces for wave-currents acting on a bottom-mounted vertical cylinder.The results were in close agreement with the results of a frequency-domain method and a published time-domain method.The model was then applied to compute wave-current forces and run-up on a Seastar mini tension-leg platform.

Second-Order Wave Diffraction Around 3-D Bodies by A Time-Domain Method

A time-domain method is applied to simulate nonlinear wave diffraction around a surface piercing 3-D arbitrary body. The method involves the application of Taylor series expansions and the use of perturbation procedure to establish the corresponding boundary value problems with respect to a time-independent fluid domain. A boundary element method based on B-spline expansion is used to calculate the wave field at each time step, and the free surface boundary condition is satisfied to the second order of wave steepness by a numerical integration in time. An artificial damping layer is adopted on the free surface for the removal of wave reflection from the outer boundary. As an illustration, the method is used to compute the second-order wave forces and run-up on a surface-piercing circular cylinder. The present method is found to be accurate, computationally efficient, and numerically stable.

Time-Domain Analysis of Underground Station-Layered Soil Interaction Based on High-Order Doubly Asymptotic Transmitting Boundary

Based on the modified scale boundary finite element method and continued fraction solution,a high-order doubly asymptotic transmitting boundary(DATB)is derived and extended to the simulation of vector wave propagation in complex layered soils.The high-order DATB converges rapidly to the exact solution throughout the entire frequency range and its formulation is local in the time domain,possessing high accuracy and good efficiency.Combining with finite element method,a coupled model is constructed for time-domain analysis of underground station-layered soil interaction.The coupled model is divided into the near and far field by the truncated boundary,of which the near field is modelled by FEM while the far field is modelled by the high-order DATB.The coupled model is implemented in an open source finite element software,OpenSees,in which the DATB is employed as a super element.Numerical examples demonstrate that results of the coupled model are stable,accurate and efficient compared with those of the extended mesh model and the viscous-spring boundary model.Besides,it has also shown the fitness for long-time seismic response analysis of underground station-layered soil interaction.Therefore,it is believed that the coupled model could provide a new approach for seismic analysis of underground station-layered soil interaction and could be further developed for engineering.

DYNAMIC INTERACTION OF PLANE WAVES WITH A UNILATERALLY FRICTIONALLY CONSTRAINED INCLUSION-TIME DOMAIN BOUNDARY ELEMENT ANALYSIS

A 2D time domain boundary element method(BEM)is developed to solve the transient scattering of plane waves by a unilaterally frictionally constrained inclusion.Coulomb friction is assumed along the contact interface.The incident wave is assumed strong enough so that localized slip and separation take place along the interface.The present problem is in effect a nonlinear boundary value problem since the mixed boundary conditions involve unknown intervals (slip,separation and stick regions).In order to determine the unknown intervals,an iterative technique is developed.As an example,we consider the scattering of a circular cylinder embedded in an infinite solid.

A Method for Calculation of Low-Frequency Slow Drift Motions Based on NURBS for Floating Bodies

Through a higher-order boundary element method based on NURBS （Non-uniform Rational B-splines）, the calculation of second-order low-frequency forces and slow drift motions is conducted for floating bodies. In the floating body＇s inner domain, an auxiliary equation is obtained by applying a Green function which satisfies the solid surface condition. Then, the auxiliary equation and the velocity potential equation are combined in the fluid domain to remove the solid angle coefficient and the singularity of the double layer potentials in the integral equation. Thus, a new velocity potential integral equation is obtained. The new equation is extended to the inner domain to reheve the irregular frequency effects; on the basis of the order analysis, the comparison is made about the contribution of all integral terms with the result in the second-order tow-frequency problem; the higher-order boundary element method based on NURBS is apphed to calculate the geometric position and velocity potentials; the slow drift motions are calculated by the spectrum analysis method. Removing the solid angle coefficient can apply NURBS technology to the hydrodynamic calculation of floating bodies with complex surfaces, and the extended boundary integral method can reduce the irregular frequency effects. Order analysis shows that free surface integral can be neglected, and the numerical results can also prove the correctness of order analysis. The results of second-order low-frequency forces and slow drift motions and the comparison with the results from references show that the application of the NURBS technology to the second-order low-frequency problem is of high efficiency and credible results.

Time-domain Finite Element Method for Transient Electric Field and Transient Charge Density on Dielectric Interface

This paper is devoted to solving the transient electric field and transient charge density on the dielectric interface under the electroquasistatic(EQS)field conditions with high accuracy.The proposed method is suitable for both 2-D and 3-D applications.Firstly,the governing equations represented by scalar electric potential are discretized by the nodal finite element method(FEM)in space and the finite difference method in time.Secondly,the transient constrained electric field equation on the boundary(TCEFEB)is derived to calculate the normal component of the transient electric field intensities on the Dirichlet boundary and dielectric interface as well as the transient charge density on the dielectric interface.Finally,a 2-D numerical example is employed to demonstrate the validity of the proposed method.Furthermore,the comparisons of the numerical accuracy of the proposed method in this paper with the existing FEMs for electric field intensity and charge density on the dielectric interface are conducted.The results show that the numerical accuracy of the proposed method for calculating the normal component of transient electric field intensities on the Dirichlet boundary and dielectric interface as well as the transient charge density on the dielectric interface is close to that of nodal electric potential and an order of magnitude higher than those of existing FEMs.

An iterative Rankine boundary element method for wave diffraction of a ship with forward speed

A 3-D time-domain seakeeping analysis tool has been newly developed by using a higher-order boundary element method with the Rankine source as the kernel function. An iterative time-marching scheme for updating both kinematic and dynamic free-surface boundary conditions is adopted for achieving numerical accuracy and stability. A rectangular computational domain moving with the mean speed of ship is introduced. A damping beach at the outer portion of the truncated free surface is installed for satisfying the radiation condition. After numerical convergence checked, the diffraction unsteady problem of a Wigley hull traveling with a constant forward speed in waves is studied. Extensive results including wave exciting forces, wave patterns and pressure distributions on the hull are presented to validate the efficiency and accuracy of the proposed 3-D time-domain iterative Rankine BEM approach. Computed results are compared to be in good agreement with the corresponding experimental data and other published numerical solutions.

TIME-DOMAIN RESPONSES OF IMMERSING TUNNEL ELEMENT UNDER WAVE ACTIONS

The time domain responses of the tunnel element under wave actions during its immersion are investigated based on the linear wave diffraction theory. The integral equation is derived by using the time-domain Green function that satisfies the free water surface condition in the finite water depth, and is solved by the boundary element method. The motion equations of the tunnel element are solved by the fourth order Runge-Kutta method. A comparison between the computed and measured results reveals that the numerical model can effectively simulate the motion responses of the tunnel element and the cable tensions when the motions of the tunnel element are within some limit. Taking the tunnel element of 100 m in length, 15 m in width and 10 m in height as an example, the computational results of the motion responses of the tunnel element and the cable tensions in different immersing depths are obtained under different incident wave conditions.

Numerical investigation of solitary wave action on tworectangular boxes with a narrow gap

Based on the time-domain higher-order boundary element method(HOBEM), a two-dimensional numerical wave flume is developed to investigate solitary wave interaction with two rectangular boxes with a narrow gap.In the numerical model, the fully nonlinear boundary conditions are satisfied on the free surface, the mixed Eulerian-Lagrangian method is adopted to track the transient water surface and the fourth-order Runga-Kutta method is used to predict the velocity potential and wave elevation on the free surface. The acceleration potential technique is used to compute the transient wave forces along the wetted object surface. A piston-type wavemaker is used to generate solitary waves. The proposed model is validated by comparing the simulated wave run-up and the wave loads with the published experimental and numerical results of the reflection of a solitary wave from a vertical wall. Then, numerical experiments are performed to study the effects of the narrow gap and the size of each box on the wave run-ups at the two sides of the two-box system and in the narrow gap between two boxes, and the wave loads on the two boxes. The interaction between double solitary waves with a time interval between them with a two-box system is also investigated.

Wave Slamming on An OWSC Wave Energy Converter in Coupled Wave-Current Conditions with Variable-Depth Seabed

Coastal wave energy resources have enormous exploitation potential due to shorter weather window,closer installation distance and lower maintenance cost.However,impact loads generated by depth variation from offshore to nearshore and wave-current interaction,may lead to a catastrophic damage or complete destruction to wave energy converters(WECs).This objective of this paper is to investigate slamming response of a coastal oscillating wave surge converter(OWSC)entering or leaving water freely.Based on fully nonlinear potential flow theory,a time-domain wave-current-structure interaction model combined with higher-order boundary element method(HOBEM),is developed to analyze the coupled hydrodynamic problem.The variable-depth seabed is considered in the model to illustrate the shallow water effect on impact loads and free surface profiles in coastal zone.A domain decomposition approach is utilized to simulate the overlapping phenomenon generated by a jet falling into water under gravity effect.Through a series of Lagrangian interpolation methods,the meshes on boundaries are rearranged to avoid the mismatch between element size on free surface and body surface.The present model is validated against the existing experimental and numerical results.Simulations are also provided for the effects of wave-current interaction and uneven local seabed on the slamming responses.It is found that the length of the splash jet increases for a following current and decreases for an opposing current,and that the slamming response of the OWSC device is sensitive to the geometric features of the uneven seabed.

Numerical analysis of added mass and damping of floating production,storage and offloading system

An integral equation approach is utilized to in- vestigate the added mass and damping of floating produc- tion, storage and offloading system （FPSO system）. Finite water depth Green function and higher-order boundary ele- ment method are used to solve integral equation. Numeri- cal results about added mass and damping are presented for odd and even mode motions of FPSO. The results show ro- bust convergence in high frequency range and can be used in wave load analysis for FPSO designing and operation.

Nonlinear numerical simulation on extreme-wave kine-matics

A fully nonlinear numerical model based on a time-domain higher-order boundary element method （HOBEM） is founded to simulate the kinematics of extreme waves. In the model, the fully nonlinear free surface boundary conditions are satisfied and a semi-mixed Euler-Lagrange method is used to track free surface; a fourth-order Runga-Kutta technique is adopted to refresh the wave elevation and velocity potential on the free surface at each time step; an image Green function is used in the numerical wave tank so that the integrations on the lateral surfaces and bottom are excluded. The extreme waves are generated by the method of wave focusing. The physical experiments are carried out in a wave flume. On the horizontal velocity of the measured point, numerical solutions agree well with experimental results. The characteristics of the nonlinear extreme-wave kinematics and the velocity distribution are studied here.

An Efficient Model for Transient Surface Waves in Both Finite and Infinite Water Depths

A numerical model is developed to simulate fully nonlinear extreme waves in finite and infinite water-depth wave tanks. A semi-mixed Enlerian-Lagrangian formulation is adopted and a higher-order boundary element method in conjunction with an image Green function is used for the fluid domain. The botmdary values on the free surface are updated at each time step by a fourth-order Runga-Kutta time-marching scheme at each time step. Input wave characteristics are specified at the upstream boundary by an appropriate wave theory. At the downstream boundary, an artificial damping zone is used to prevent wave reflection back into the computational domain. Using the image Green function in the whole fluid domain, the integrations on the two lateral walls and bottom are excluded. The simulation results on extreme wave elevations in finite and infinite water-depths are compared with experimental results and second-order analytical solutions respectively. The wave kinematics is also discussed in the present study.

Development of TD-BEM Formulation for Dynamic Analysis for Twin-Parallel Circular Tunnels in an Elastic Semi-Innite Medium

In order to simulate the propagation process of subway vibration of parallel tunnels in semi-infinite rocks or soils,time domain boundary element method(TD-BEM)formulation for analyzing the dynamic response of twin-parallel circular tunnels in an elastic semi-infinite medium is developed in this paper.The time domain boundary integral equations of displacement and stress for the elastodynamic problem are presented based on Betti’s reciprocal work theorem,ignoring contributions from initial conditions and body forces.In the process of establishing time domain boundary integral equations,some virtual boundaries are constructed between finite boundaries and the free boundary to form a boundary to refer to the time domain boundary integral equations for a single circular tunnel under dynamic loads.The numerical treatment and solving process of time domain boundary integral equations are given in detail,including temporal discretization,spatial discretization and the assembly of the influencing coefficients.In the process of the numerical implementation,infinite boundary elements are incorporated in time domain boundary element method formulation to satisfy stress free conditions on the ground surface,in addition,to reduce the discretization of the boundary of the ground surface.The applicability and efficiency of the presented time domain boundary element formulation are verified by a deliberately designed example.

WAVE-CURRENT INTERACTIONS WITH THREE-DIMENSIONAL FLOATING BODIES

A Time-domain Higher-Order Boundary Element Method（THOBEM） is developed for simulating wave-current interactions with 3-D floating bodies.Through a Taylor series expansion and a perturbation procedure,the model is formulated to the first-order in the wave steepness and in the current velocity,respectively.The boundary value problem is decomposed into a steady double-body flow problem and an unsteady wave problem.Higher-order boundary integral equation methods are then used to solve the proposed problems with a fourth-order Runge-Kutta method for the time marching.An artificial damping layer is adopted to dissipate the scattering waves.Different from the other time-domain numerical models,which are often focused on the wave-current interaction with restrained bodies,the present model deals with a floating hemisphere.The numerical results of wave forces,wave run-up and body response are all in a close agreement with those obtained by frequency-domain methods.The proposed numerical model is further applied to investigate wave-current interactions with a floating body of complicated geometry.In this work,the regular and focused wave combined with current interacting with a truss-spar platform is investigated.

An iterative Rankine BEM for wave-making analysis of submerged and surface-piercing bodies in finite water depth

A 3-D iterative Rankine Boundary Element Method （BEM） for seakeeping problem in time domain is developed in the framework of linear potential theory. Waves generated by both submerged and surface-piercing bodies moving at a constant forward speed in otherwise calm water, and the resultant steady wave pattern, wave profile and resistance are computed to validate this newly-developed code. A rectangular computational domain moving with the same forward speed as the body is introduced, in which an artificial damping beach is installed at an outer portion of the free surface except the downstream side for satisfying the radiation condition. The velocity potential on the ship hull and the normal velocity on the free surface are obtained directly by solving the boundary integral equation, with the Rankine source used as the kernel function. An iterative time-marching scheme is employed for updating both kinematic and dynamic free surface boundary conditions to stabilize the calculation. Extensive results including the wave patterns, wave profiles and wave resistances for a submerged spheroid and a Wigley hull with forward speed are presented to validate the efficiency of the proposed 3-D time-domain higher-order approach. Finally, the sensitivity of ship-generated waves to the water depth is investigated. Computed results show satisfactory agreement with the corresponding experimental data and other numerical solutions.

Iterative Rankine HOBEM analysis of hull-form effects in forward-speed diffraction problem

A time-domain numerical algorithm based on the higher-order boundary element method and the iterative time-marching scheme is proposed for seakeeping analysis. The ship waves generated by a hull advancing at a constant forward speed in incident waves and the resultant diffraction forces acting on the hull are computed to investigate the hull-form effects on the hydrodynamic forces. A rectangular computational domain travelling at ship＇s speed is considered. An artificial damping beach for satisfying the radiation condition is installed at the outer portion of the free surface except the downstream side. An iterative time-marching scheme is employed for updating both kinematic and dynamic free-surface boundary conditions for numerical accuracy and stability. The boundary integral equation is solved by distributing higher-order boundary elements over the wetted body surface and the free surface. The hull-form effects on the naval hydrodynamics are investigated by comparing three different Wigley models. Finally, the corresponding unsteady wave patterns and the wave profiles around the hulls are illustrated and discussed.

Application of the generalized body-fixed coordinate system for the wave-body interaction problem of a small-depth elastic structure in head seas

The body-fixed coordinate system is applied to the wave-body interaction problem of a small-depth elastic structure which has both rigid and elastic body motions in head waves.In the weakly non-linear assumption,the perturbation scheme is used and the expansion is conducted up to second-order to consider several non-linear quantities.To solve the boundary value problem,linearization is carried out based not on inertial coordinate but on body-fixed coordinate which could be accelerated by a motion of a body.At first,the main feature of the application of body-fixed coordinate system for a seakeeping problem is briefly described.After that the transformation of a coordinate system is extended to consider an elastic body motion and several physical variables are re-described in the generalized mode.It has been found that the deformation gradient could be used for the transformation of a coordinate system if several conditions are satisfied.Provided there are only vertical bending in elastic modes and the structure has relatively small depth,these conditions are generally satisfied.To calculate an elastic motion of a body,the generalized mode method is adopted and the mode shape is obtained by solving eigen-value problem of dynamic beam equation.In the boundary condition of the body-fixed coordinate system,the motion effect reflected to free-surface boundary is considered by extrapolating each mode shape to the horizontal direction from a body.At last,simple numerical tests are implemented as a validation process.The second-order hydrodynamic force of a freely floating hemisphere is first calculated in zero forward speed condition.Next,motion and added resistance of a ship with forward speed are considered at different flexibility to confirm the effect of an elastic body motion in body-fixed coordinate system.