A modified scaled boundary finite-element method for problems with parallel side-faces. Part I. Theoretical developments

A modified scaled boundary finite-element method (SBFEM) for problems with parallel side-faces is presented in this study. To overcome the inherent difficulty of the original SBFEM for domains with parallel side-faces, a new type of local co-ordinate system is proposed. The new local co-ordinate system allows the so-called scaling centre of the SBFEM to move freely along an arbitrary curve and thus eliminates the non-parallel side-face restriction in the original SBFEM. The modified SBFEM equations are derived based on a weighted residual approach. It is found that the modified SBFEM solution retains the analytical feature in the direction parallel to the side-faces and satisfies the boundary conditions at infinity exactly, as in the original SBFEM. This paper develops a complete scaled boundary finite-element solution to a two-dimensional Laplace's equation with Neumann and Robin boundary conditions in a semi-infinite domain with parallel boundaries.     

The hydrodynamic coefficients for an oscillating rectangular structure on a free surface with sidewall

Based on a two dimensional linear water wave theory, the boundary element method (BEM) is developed and applied to study the heave and the sway problem of a floating rectangular structure in water to finite depth with one side of the boundary is a vertical sidewall and the other boundary is an open boundary. Numerical results for the added mass and radiation damping coefficients are presented. These coefficients are not only depend on the submergence and the width of the structure, but also depend on the clearance between structure and sidewall. Negative added mass and sharp peaks in the damping and added mass coefficients have been found when the clearance with a value close to integral times of half wave length of wave generated by oscillation structure. The important effect of the clearance on the added mass and radiation damping coefficients are discussed in detail. An analytical solution method is also presented. The BEM solution is compared with the analytical solution, and the comparison shows good agreement.     

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.     

Hydrodynamic coefficients of a two-dimensional,truncated rectangular floating structure in shallow water

The paper seeks to examine hydrodynamic coefficients of a rectangular structure in shallow water and to establish analytical formulae for fast computations. A two-dimensional rectangular profile is considered with the under-bottom clearance assumed to be small compared with structure dimensions and the water depth. Following the method of matched asymptotic expansions, the radiation problem is solved under assumptions of the linear wave theory, by matching two ‘outer’ flows with the ‘inner’ flow near the structure edge. Closed asymptotic formulae are obtained for all hydrodynamic coefficients for heave, sway and roll motions. The zero and infinite frequency values of the added mass are examined and formulae are derived intended for quick engineering estimations. Numerical results compare well with those published in literature, and the approach is shown to be consistent with known fundamental relations in the body–wave interaction theory.     

Wave radiation due to oscillation of two parallel spaced cylinders

The radiation problem for two parallel-spaced cylinders is studied. The solution is expressed explicitly in terms of well-behaved convergent series with elementary functions, which are convenient for numerical computation and readily applicable for two-dimensional two-body potential problems. The added mass and damping coefficients together with the phase angles of radiated wave potentials for the forced heave and sway motions of two identical submerged cylinders are presented. The results are useful for determination of the hydrodynamic properties of multi-hull semi-submersibles. In view of the close relationship between a radiation and a scattering problem, the application of the results to the problem of energy extraction from water waves is also noted.     

两层流体中矩形箱浮体的附加质量和阻尼系数

研究了两层流体中矩形箱浮体的辐射问题。基于特征函数匹配理论,针对矩形箱浮体的三种振荡运动模式(横荡、垂荡和横摇),建立两层流体中矩形箱浮体辐射势的求解方法,导出矩形箱浮体附加质量和阻尼系数的计算公式。对所建立的求解模型进行了数值计算分析,结果表明在矩形箱浮体的某个振荡频率范围内,流体的分层效应对其附加质量和阻尼系数有显著影响的。     

应用比例边界有限元法求解狭缝对双箱水动力的影响

比例边界有限元法(SBFEM)是一种半解析数值分析的新方法,既融合了有限元法和边界元法的优点,又有其特有的优点。用该方法可求解有限水深下狭缝对双箱水动力作用的影响,为波浪与多浮体超大型结构的相互作用探索一些规律。整个计算域划分成2个无限子域和4个有限子域,并利用加权余量法在各个子域上推导了SBFEM的积分方程;计算了4个数值算例并与边界元等其它数值方法进行了比较,验证了该方法是一种用很少单元便能得到精确结果的高效方法。应用SBFEM对不同箱体宽度、不同狭缝宽度、不同吃水深度条件的双箱作了计算,得出了狭缝对双箱水动力干涉影响的一些规律,对超大型浮体水动力分析和结构设计具有一定的参考价值。     

Nonlinear heave radiation forces on two-dimensional single and twin hulls

A study of nonlinear heave radiation of two-dimensional single and double hulls has been carried out in the time domain. The problem is analyzed by means of a fully nonlinear mathematical model, referred to as the mixed Eulerian–Lagrangian (MEL) model, which is based on an integral relation formulation coupled with time-integration of the nonlinear free-surface boundary conditions. The integral equation solver is based on a cubic-spline boundary-element scheme in which both potential and velocity continuity conditions can be enforced through the intersection points. The body undergoes periodic forced heave oscillation. By implementing effective wave-absorbing beaches at the two ends of the rectangular numerical tank, long-term steady-state force-histories could be achieved consistently in all computations.Results in terms of radiation forces for rectangular and triangular single- and twin-hull geometries are presented and discussed. Linear hydrodynamic forces in terms of added-mass and damping are validated for the rectangular hull. The Fourier-analyzed results reveal the extent of nonlinear (higher-order) components in the force-signals over different parameters which include the amplitudes of oscillation, hull-spacing for the twin-hulls and water depth.     

Diffraction–radiation of multiple floating structures in directional waves

The dynamics of multiple floating structures have been studied using the finite element method. The emphasis is on the hydrodynamic behaviour of multiple bodies under a multi-directional wave field. A two-dimensional numerical model has been adopted to evaluate hydrodynamic coefficients and forces in an oblique wave field. The responses in sway, heave and roll modes are reported. The linear filter technique is then used to extrapolate the responses under directional waves. The effect of mean wave direction and directional homogeneity on the hydrodynamic behaviour of the structure is studied. Based on the present study, it is found that the two-dimensional model is applicable to investigate the wave-structure interaction problems of the type herein considered.     

The prediction of the dynamic and structural motions of a floating-pier system in waves

In this study, a two-dimensional floating pier consists of single rectangular impermeable pontoon with side supporting pile-columns is studied. The purpose of this study is to present a theoretical solution for the linearized problem of incident waves exerting on a floating pier with pile-restrained. All boundary conditions are linearized in the problem, which is incorporated into a scattering problem and radiation problem with unit displacement. The method of separation of variables is used to solve for velocity potentials. For the radiation problem with unit heave and pitch amplitude, the boundary value problem with non-homogeneous boundary condition beneath the structure is solved by using a solution scheme. By calculating the wave force from velocity potential and solving the equation of motion of the floating structure simultaneously a close form theoretical solution for the problem is developed. The finite element method was also applied to calculate the dynamic responses on the supporting piles subjected to the pontoon motions and incident waves.     

应用比例边界有限元法求解多种二维浮体水动力特性

比例边界有限元法(SBFEM)是一种半解析的数值方法,比完全数值方法具有更高的精度,该方法结合了有限元和边界元的优点,采用相对少的剖分单元就可以得到较高精度的模拟结果。通过改变有限子域内部比例中心的位置,使这种方法可以应用到多种形式浮体在波浪作用下的水动力特性的计算中。同时还给出了各种形式浮体的波浪力及反射、透射系数的数值结果,并与边界元方法(BEM)计算结果和特征函数展开方法得到解析解进行了比较,均吻合良好。研究表明比例边界有限元不仅可以计算矩形的浮体结构,而且对于多种结构形式的浮体都可以计算,这为多种结构形式浮体的水动力分析提供了一个可行的方法。     

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.     

Time-domain floating body dynamics by rational approximation of the radiation impedance and diffraction mapping

The problem of approximating the dynamics of a floating structure in a transient wave environment with a set of constant-coefficient differential equations is explored. It is assumed that the solutions of the corresponding steady-state time-harmonic radiation and diffraction problems are available. It is proposed to fit the frequency responses associated with the ‘radiation impedance' and wave-exciting forces with appropriate analytic functions. In the case of the radiation problem, these possess certain properties corresponding to the passivity of the radiation mapping. By choosing rational approximations, the transformation from the frequency to the time domain is facilitated. The method is illustrated for both two-dimensional and three-dimensional problems using a floating cylinder, sphere, and a model of Salter's Duck which exhibits hydrodynamic coupling between sway, heave, and pitch motions.     

The dynamics of wave induced ship motions in shallow water

The analytical method developed by Svendsen (1968) for a forced heave motion is extended to the general problem of wave induced heave, roll and sway motions of a long ship at a depth of water which is only slightly larger than the draught of the ship. This corresponds, for example, to the situation of a fully loaded ship in a harbour area.After linearization of the problem, the water motion is considered for each of the three individual motions and for the wave reflection-transmission problem for a fixed ship. The ensuing results for the forces on the ship are then synthesized to form the equations of motion, which are presented with all coefficients given, including mooring forces.Analytical and numerical results are given for the three components of motion, for the associated resonance frequencies, and for the hydrodynamic masses and moments of inertia. Finally, the assumptions used are analyzed and evaluated by comparison with measurements and with other results for a special case.     

Floating pontoon breakwaters

The hydrodynamic properties of a pair of long floating pontoon breakwaters of rectangular section are investigated theoretically. The structures are partially restrained by linear symmetric moorings fore and aft. The fluid motion is idealized as linearized, two-dimensional potential flow. The breakwater motions are assumed to be two-dimensional, in surge, heave and pitch. The solution for the fluid motion is obtained by the boundary integral equation method using an appropriate Green's function. Numerical results are presented that illustrate the effects of the various wave and structural parameters on the efficiency of the breakwaters as barriers to wave action. It is found that the wave reflection properties of the structures depend strongly on their width, draft and spacing and the mooring line stiffnesses, while their excess buoyancy is of lesser importance.     

Wave interaction with multiple horizontal cylinders

The interaction of waves with submerged two-dimensional circular cylinder groups is investigated. Linear wave theory is used in the analysis and the viscous effects are neglected. The boundary value problem for the wave potential is based on Green's theorem, and the resulting integral equation is solved numerically. The added mass and damping coefficients in sway, heave and roll of oscillating twin-cylinders and the total wave forces on a fixed cylinder in a group are examined. The effects of the free surface, and particularly, the spacing of the cylinders are shown. The numerical results are tested against known solutions. Results for a two-cylinder configuration at different orientations are presented. It is found that the interaction between closely spaced cylinders is large. The spacings at which the interaction effect is important are shown graphically.     

Numerical modeling of dynamic responses and mooring forces of submerged floating breakwater

Using the volume of fluid (VOF) method, a numerical model is developed to estimate the nonlinear dynamics of a pontoon type moored submerged breakwater under wave action and the forces acting on the mooring lines, for both the vertical and inclined mooring alignments. The model is developed for a two-dimensional wave field in a vertical plane. The finite displacements of the breakwater such as sway, heave and roll in a very small time step are considered and the numerical grid cells intersected by the breakwater surfaces for changing its position due to wave action are treated using the concept of porous body model. Also, two-dimensional experimental studies are carried out to investigate the performance of the proposed model. The comparison of the computed and measured results reveals that the developed numerical model can reproduce well the dynamics of the floating body and the mooring line forces.     

Viscosity and nonlinearity effects on the forces and waves generated by a floating twin hull under heave oscillation

Nonlinear hydrodynamics of a twin rectangular hull under heave oscillation is analyzed using numerical methods. Two-dimensional nonlinear time-domain solutions to both inviscid and viscous problems are obtained and the results are compared with linear, inviscid frequency-domain results obtained in [26] to quantify nonlinear and viscous effects. Finite-difference methods based on boundary-fitted coordinates are used for solving the governing equations in the time domain [2]. A primitive-variables based projection method [6] is used for the viscous analysis and a mixed Eulerian–Lagrangian formulation [11] for inviscid analysis. The algorithms are validated and the order of accuracy determined by comparing the results obtained from the present algorithm with the experimental results of Vugt [22] for a heaving rectangle in the free surface. The present study on the twin-hull hydrodynamics shows that at large and non-resonant regular frequencies, and small amplitude of body oscillation, the fluid viscosity does not significantly affect the wave motion and the radiation forces. At low frequencies however the viscosity effect is found to be significant even for small amplitude of body oscillation. In particular, the hydrodynamic force obtained from the nonlinear viscous analysis is found to be closer to the linear inviscid force than the nonlinear inviscid force to the linear inviscid force, the reason for which is attributed to the wave dampening effect of viscosity. Since the wave lengths generated at smaller frequencies of oscillation are longer and therefore the waves could have a more significant effect on the dynamic pressure on the bottom of the hulls which contribute to the heave force, the correlation between the heave force and the wave elevation is found to be larger at smaller frequencies. Because of nonlinearity, the wave radiation and wave damping force remained nonzero even at and around the resonant frequencies – with the resonant frequencies as determined in [26] using linear potential flow theory. As to be expected, the nonlinear effect on the wave force is found to be significant at all frequencies for large amplitude of oscillation compared to the hull draft. The effect of viscosity on the force, by flow separation, is also found to be significant for large amplitude of body oscillation.     

Study on coupling effects of ship motion and sloshing

This study considers the coupling effects of ship motion and sloshing. The linear ship motion is solved using an impulse-response-function (IRF) method, while the nonlinear sloshing flow is simulated using a finite-difference method. The IRF method requires the frequency-domain solution prior to conversion to time domain, but the computational effort is much less than that of direct time-domain approaches. The developed scheme is verified by comparing the motion RAOs between the frequency-domain solution and the solution obtained by the IRF method. Furthermore, a soft-spring concept and linear roll damping are implemented to predict more realistic motions of surge, sway, yaw, and roll. For the simulation of sloshing flow in liquid tanks, a physics-based numerical approach adopted by Kim [2001. Numerical simulation of sloshing flows with impact load. Applied Ocean Research 23, 53–62] and Kim et al. [2004. Numerical study on slosh-induced impact pressures on three-dimensional prismatic tanks. Applied Ocean Research 26, 213–226] is applied. In particular, the present method focuses on the simulation of the global motion of sloshing flow, ignoring some local phenomena. The sloshing-induced forces and moments are added to wave-excitation forces and moments, and then the corresponding body motion is obtained. The developed schemes are applied for two problems: the sway motion of a box-type barge with rectangular tanks and the roll motion of a modified S175 hull with rectangular anti-rolling tank. Motion RAOs are compared with existing results, showing fair agreement. It is found that the nonlinearity of sloshing flow is very important in coupling analysis. Due to the nonlinearity of sloshing flow, ship motion shows a strong sensitivity to wave slope.     

Numerical and Experimental Investigation of Interactions Between Free-Surface Waves and A Floating Breakwater with Cylindrical-Dual/Rectangular-Single Pontoon

This paper investigates the hydrodynamic performance of a cylindrical-dual or rectangular-single pontoon floating breakwater using the numerical method and experimental study. The numerical simulation work is based on the multi-physics computational fluid dynamics (CFD) code and an innovative full-structured dynamic grid method applied to update the three-degree-of-freedom (3-DOF) rigid structure motions. As a time-marching scheme, the trapezoid analogue integral method is used to update the time integration combined with remeshing at each time step. The application of full-structured mesh elements can prevent grids distortion or deformation caused by large-scale movement and improve the stability of calculation. In movable regions, each moving zone is specified with particular motion modes (sway, heave and roll). A series of experimental studies are carried out to validate the performance of the floating body and verify the accuracy of the proposed numerical model. The results are systematically assessed in terms of wave coefficients, mooring line forces, velocity streamlines and the 3-DOF motions of the floating breakwater. When compared with the wave coefficient solutions, excellent agreements are achieved between the computed and experimental data, except in the vicinity of resonant frequency. The velocity streamlines and wave profile movement in the fluid field can also be reproduced using this numerical model.