Time Domain Simulation of Transient Responses of Very Large Floating Structures Under Unsteady External Loads

A time domain finite element method （FEM） for the analysis of transient elastic response of a very large floating structure （VLFS） subjected to arbitrary time-dependent external loads is presented. This method is developed directly in time domain and the hydrodynamic problem is formulated based on linear, inviscid and slightly compressible fluid theory and the structural response is analyzed on the thin plate assumption. The time domain finite element procedure herein is validated by comparing numerical results with available experimental data. Finally, the transient elastic response of a pontoon-type VLFS under the landing of an airplane is computed by the proposed time domain FEM. The time histories of the applied force and the position and velocity of an airplane during landing are modeled with data from a Boeing 747-400 jumbo jet.     

Fully nonlinear simulations of wave resonance by an array of cylinders in vertical motions

The finite element method(FEM) is employed to analyze the resonant oscillations of the liquid confined within multiple or an array of floating bodies with fully nonlinear boundary conditions on the free surface and the body surface in two dimensions.The velocity potentials at each time step are obtained through the FEM with 8-node quadratic shape functions.The finite element linear system is solved by the conjugate gradient(CG) method with a symmetric successive overelaxlation(SSOR) preconditioner.The waves at the open boundary are absorbed by the combination of the damping zone method and the Sommerfeld-Orlanski equation.Numerical examples are given by an array of floating wedgeshaped cylinders and rectangular cylinders.Results are provided for heave motions including wave elevations,profiles and hydrodynamic forces.Comparisons are made in several cases with the results obtained from the second order solution in the time domain.It is found that the wave amplitude in the middle region of the array is larger than those in other places,and the hydrodynamic force on a cylinder increases with the cylinder closing to the middle of the array.     

Fully Nonlinear Simulations of Wave Resonance by An Array of Cylinders in Vertical Motions

The finite element method (FEM) is employed to analyze the resonant oscillations of the liquid confined within multiple or an array of floating bodies with fully nonlinear boundary conditions on the free surface and the body surface in two dimensions. The velocity potentials at each time step are obtained through the FEM with 8-node quadratic shape functions. The finite element linear system is solved by the conjugate gradient (CG) method with a symmetric successive overelaxlation (SSOR) preconditioner. The waves at the open boundary are absorbed by the combination of the damping zone method and the Sommerfeld-Orlanski equation. Numerical examples are given by an array of floating wedge- shaped cylinders and rectangular cylinders. Results are provided for heave motions including wave elevations, profiles and hydrodynamic forces. Comparisons are made in several cases with the results obtained from the second order solution in the time domain. It is found that the wave amplitude in the middle region of the array is larger than those in other places, and the hydrodynamic force on a cylinder increases with the cylinder closing to the middle of the array.     

Short-crested waves interaction with a concentric porous cylinder system with partially porous outer cylinder

In this paper, based on the linear wave theory, the interaction of short-crested waves with a concentric dual cylindrical system with a partially porous outer cylinder is studied by using the scaled boundary finite element method (SBFEM), which is a novel semi-analytical method with the advantages of combining the finite element method (FEM) with the boundary element method (BEM). The whole solution domain is divided into one unbounded sub-domain and one bounded sub-domain by the exterior cylinder. By weakening the governing differential equation in the circumferential direction, the SBFEM equations for both domains can be solved analytically in the radial direction. Only the boundary on the circumference of the exterior porous cylinder is discretized with curved surface finite elements. Meanwhile, by introducing a variable porous-effect parameter G, non-homogeneous materials caused by the complex configuration of the exterior cylinder are modeled without additional efforts. Comparisons clearly demonstrate the excellent accuracy and computational efficiency associated with the present SBFEM. The effects of the wide range wave parameters and the structure configuration are examined. This parametric study will help determine the various hydrodynamic effects of the concentric porous cylindrical structure.     

Wave radiation and diffraction by a two-dimensional floating body with an opening near a side wall

The radiation and diffraction problem of a two-dimensional rectangular body with an opening floating on a semi-infinite fluid domain of finite water depth is analysed based on the linearized velocity potential theory through an analytical solution procedure. The expressions for potentials are obtained by the method of variation separation, in which the unknown coefficients are determined by the boundary condition and matching requirement on the interface. The effects of the position of the hole and the gap between the body and side wall on hydrodynamic characteristics are investigated. Some resonance is observed like piston motion in a moon pool and sloshing in a closed tank because of the existence of restricted fluid domains.     

Interactions between nonlinear water waves and non-wall-sided 3D structures

Interactions between water waves and non-wall-sided cylinders are analyzed based on velocity potential theory with fully nonlinear boundary conditions on the free surface and the body surface. The finite element method (FEM) is adopted together with a 3D mesh generated through an extension of a 2D Delaunay grid on a horizontal plane along the depth. The linear matrix equation for the velocity potential is constructed by imposing the governing equation and boundary conditions through the Galerkin method and is solved through an iterative method. By imposing the gradient of the potential equal to the velocity, the Galerkin method is used again to obtain the velocity field in the fluid domain. Simulations are made for bottom mounted and truncated cylinders with flare in a numerical tank. Periodic waves and wave groups are generated by a piston type wave maker mounted on one end of the tank. Results are obtained for forces, wave profiles and wave runups. Further simulations are made for a cylinder with flare subjected to forced motion in otherwise still open water. Results are provided for surge and heave motion in different amplitudes, and for a body moving in a circular path in the horizontal plane. Comparisons are made in several cases with the results obtained from the second order solution in the time domain.     

Hydroelastic analysis of floating structures with liquid tanks and comparison with experimental tests

In this study, we develop a numerical method for a 3D linear hydroelastic analysis of floating structures with liquid tanks subjected to surface regular water waves and compare the numerical results with experimental tests. Considering direct couplings among structural motion, sloshing, and water waves, a mathematical formulation and a numerical method extended from a recent work [1] are developed. The finite element method is employed for the floating structure and internal fluid in tanks, and the boundary element method is used for the external fluid. The resulting formulation completely incorporates all the interaction terms including hydrostatic stiffness and the irregular frequency effect is removed by introducing the extended boundary integral equations. Through various numerical tests, we verify the proposed numerical method. We also performed 3D hydroelastic experimental tests of a floating production unit (FPU) model in an ocean basin. The measured dynamic motions are compared with the numerical results obtained using the proposed method.     

Nonlinear response of membranes to ocean waves using boundary and finite elements

The behavior of a highly deformable membrane to ocean waves was studied by coupling a nonlinear boundary element model of the fluid domain to a nonlinear finite element model of the membrane. The hydrodynamic loadings induced by water waves are computed assuming large body hydrodynamics and ideal fluid flow and then solving the transient diffraction/radiation problem. Either linear waves or finite amplitude waves can be assumed in the model and thus the nonlinear kinematic and dynamic free surface boundary conditions are solved iteratively. The nonlinear nature of the boundary condition requires a time domain solution. To implicitly include time in the governing field equation, Volterra's method was used. The approach is the same as the typical boundary element method for a fluid domain where the governing field equation is the starting point. The difference is that in Volterra's method the time derivative of the governing field equation becomes the starting point.The boundary element model was then coupled through an iterative process to a finite element model of membrane structures. The coupled model predicts the nonlinear interaction of nonlinear water waves with highly deformable bodies. To verify the coupled model a large scale test was conducted in the OH Hinsdale wave Research Laboratory at Oregon State University on a 3-ft-diameter fabric cylinder submerged in the wave tank. The model data verified the numerical prediction of the structure displacements and of the changes in the wave field.The boundary element model is an ideal modeling technique for modeling the fluid domain when the governing field equations is the Laplace equation. In this case the nonlinear boundary element model was coupled with a finite element model of membrane structures, but the model could have been coupled with other finite element models of more rigid structures, such as a pontoon floating breakwater.     

波浪作用下方箱-水平板浮式防波堤时域水动力分析

在线性化势流理论范围内求解方箱-水平板浮式防波堤的波浪绕射和辐射问题,从时域角度分析了浮式防波堤的水动力特性.采用格林函数法将速度势定解问题的控制微分方程变换成边界上的积分方程进行数值求解,浮式防波堤的运动方程采用四阶Runge-Kutta方法求解.对不同层数水平板的浮式防波堤的波浪透射系数、运动响应和锚链受力进行了计算分析,结果表明方箱相对宽度对方箱-水平板浮式防波堤的波浪透射作用有重要的影响,透射系数随着方箱相对宽度的增加而减小.对于方箱加二层水平板的浮式防波堤,在本研究的计算条件下,当方箱相对宽度从0.110增加至0.295时,透射系数从0.88减小至0.30.水平板有利于增加浮式防波堤对波浪的衰减作用,但随着水平板层数从0增加至2,这种波浪衰减作用增加的程度趋弱.方箱-水平板的浮式防波堤的运动量小于单一方箱防波堤的运动量.与此对应,方箱-水平板防波堤的锚链受力小于单一方箱防波堤的锚链受力.     

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

A numerical model is developed to simulate fully nonlinear extreme waves in finite and infinite water-depth wave tanks. A semi-mixed Eulerian-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 boundary 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.     

Water entry of a finite width wedge near a floating body

The problem of a two-dimensional finite-width wedge entering water near a freely floating body is considered through the velocity potential theory for the incompressible liquid with the fully nonlinear boundary conditions on the free surface. The problem is solved by using the boundary element method in the time domain. The numerical process is divided into two phases based on whether the interaction between the wedge and floating body is significant. In the first phase, when the single wedge enters water at initial stage, only a small part near its tip is in the fluid, the problem is studied in a stretched coordinate system and the presence of the floating body has no major effect. In the second phase, the disturbance by water entry of the wedge has reached the floating body, and both are considered together in the physical system. The auxiliary function method is adopted to decouple the nonlinear mutual dependence between the motions of the wedge and floating body, both in three degrees of freedom, and the fluid flow, as well as the interaction effects between them. Case studies are undertaken for a wedge entering water in forced or free fall motion, vertically or obliquely. Results are provided for the accelerations, velocities, pressure distribution and free surface deformation, and the interaction effects are discussed.     

Parametric Analysis of A Submerged Cylindrical Shell Subjected to Shock Waves

In this study,an FEM-SBFEM(scaled boundary finite element method)coupling procedure proposed by Fan et al.(2005)is adopted to obtain the dynamic responses of a submerged cylindrical shell subjected to plane step or exponential acoustic shock waves.The coupling procedure can readily be applied to three-dimensional problem,however for clarity,the problems to be presented are limited to two-dimensional domain.In the analyses,the cylindrical shell is modeled by simple beam elements(using FEM),while the effects of the surrounding infinite fluid is modeled by the SBFEM.In it,no free surface and seabed are involved.Compared with Fan and his co-authors' works,the FEM-SBFEM coupling procedure is further verified to be feasible for shock waves by benchmark examples.Furthermore,parametric studies are performed and presented to gain insight into effects of the geometric and material properties of the cylindrical shell on its dynamic responses.     

Coupled dynamic analysis for wave interaction with a truss spar and its mooring line/riser system in time domain

A full time-domain analysis program is developed for the coupled dynamic analysis of offshore structures. For the hydrodynamic loads, a time domain second order method is developed. In this approach, Taylor series expansions are applied to the body surface and free-surface boundary conditions, and the Stokes perturbation procedure is then used to establish the corresponding boundary value problems with time-independent boundaries. A higher-order boundary element method (HOBEM) is developed to calculate the velocity potential of the resulting flow field at each time step. The free-surface boundary condition is satisfied to the second order by fourth order Adams–Bashforth–Moultn method. An artificial damping layer is adopted on the free surface to avoid the wave reflection. The mooring-line/tendon/riser dynamics are based on the rod theory and the finite element method (FEM), with the governing equations described in a global coordinate system. In the coupled dynamic analysis, the motion equation for the hull and dynamic equations for mooring-lines/tendons/risers are solved simultaneously using the Newmark method. The coupled analysis program is applied for a truss Spar motion response simulation. Numerical results including motions and tensions at the top of mooring-lines/risers are presented, and some significant conclusions are derived.     

用时域法求解二维二阶非线性水波

本文用二阶理论在时域范围内计算二维二阶非线性水波,一、二阶问题分别满足各自的自由表面条件和物面条件,流场内的速度势通过求解有限元方程得到,计算采用八结点四边形等参单元,采用人工阻尼来吸收反射波,对楔形体在水面上的振荡问题进行了计算,计算结果与有关文献相比符合较好。     

An integro-variational method for interior and exterior free surface flow problems

An integro-variational method is used to solve free surface problems of linear potential flow. Results obtained by the proposed method are compared with solution of the finite element formulation and the boundary integral equation. The I.V. method uses isoparametric element distributed on the contour of the fluid domain.     

Numerical simulation of unsteady free-surface motions using a boundary integral formulation

A numerical time simulation method is described to solve fluid flow problems including unsteady free surface motion. The method is based on potential flow theory. At every time step, the problem is solved using a boundary integral formulation of the fluid domain. The linearized free surface conditions are integrated in time and the solution is marched forward. Computational results simulating the free surface motion for the cases of a linear progressive wave, wave propagating into a region of calm water and the wave maker motion are presented. Comparison with theoretical results demonstrate the feasibility of the proposed simulation scheme.     

完全非线性波浪与二维固定结构物作用的IGN-BEM耦合分析模型

为高效准确地对完全非线性波浪与二维固定结构物的相互作用进行模拟分析,建立了二维完全非线性时域耦合模型。耦合模型将计算域划分为靠近结构物的内域和远离结构物的外域,每个区域均采用满足完全非线性自由水面边界条件的波浪模型进行求解。在内域使用Laplace方程描述流体运动并采用高阶边界元法（BEM）对其进行求解;而在没有结构物的外域,波浪运动的控制方程为Irrotational Green-Naghdi（IGN）方程并采用有限元法（FEM）对其进行求解。内域和外域通过一段重叠区域进行耦合,从而实现模型间变量的传递。首先利用耦合模型分别对规则波的传播、直墙前立波的生成以及相关物理模型试验进行模拟,数值结果与精确解和试验结果的良好吻合验证了耦合模型耦合方式的合理性以及处理非线性问题的准确性;然后使用耦合模型模拟分析了波浪与固定结构物间的相互作用,并将结果与线性解析解以及完全非线性BEM模型的结果进行了对比分析,进一步证明了耦合模型的正确性与高效性。     

A novel decomposition of the quadratic transfer function (QTF) for the time-domain simulation of non-linear wave forces on floating bodies

In the present study, a novel method is proposed for the separation of the second-order sum- and difference-frequency wave forces—that is, quadratic transfer functions (QTFs)—on a floating body into three components due to wave–wave, wave–motion, and motion–motion action. By applying the new QTF components, the second-order wave forces on a floating body can be strictly computed in the time domain. In this work, the boundary value problems (BVPs) corresponding to the three kinds of QTF components were derived, and non-homogeneous boundary conditions on the free surface and the body surface were obtained. The second-order diffraction potentials were determined using the boundary integral equation method. In the solution procedure, the highly oscillatory and slowly converging integral on the free surface was evaluated in an accurate and effective manner. Furthermore, the application of the QTF components in the time domain was demonstrated. The second-order exciting forces in the time domain were divided into three parts. Each part of these forces was computed via a two-term Volterra series model based on the incident waves, the first-order motion response, and the QTF components. This method was applied to several numerical examples. The results demonstrated that this decomposition yields satisfactory results.     

三维二阶水波绕射问题的有限元时域计算

本文用有限元法配合时步处理来求解三维非线性水波的绕射问题,自由表面条件和物面条件都满足到二阶,采用人工阻尼区来吸收反射波,流场内的速度势通过求解有限元方程得到。对垂直圆柱体的绕射问题进行了计算,得到了自由表面波高时间历程和圆柱所受到的波浪力,计算结果和有关文献的理论计算结果进行了比较。     

Time-domain simulation of berthing problem between FPSO and shuttle tanker in waves

A time-domain simulation method based on potential flow model has been developed to investigate the berthing problem between two floating bodies in wave. The boundary value problem is formulated with respect to an earth-fixed coordinate system because the relative positions of the two vessels continuously change during the berthing operation. The classical finite element method is used to solve the Laplace equation in the fluid domain with moving boundary. The linearized free-surface boundary conditions are integrated in time by applying 4th-order Adams–Bashforth–Moulton method. A simple re-mesh algorithm with local and global mesh systems is introduced to update mesh by considering large horizontal movement of the berthing vessel. The developed numerical method is used to investigate the berthing problem between a FPSO and shuttle tanker in waves. The focus is on the wave-induced motion response during the berthing process. The characteristics of the motion responses in berthing operation are examined with various wave frequencies, berthing speeds and wave headings.