柱形地形上单浮体的辐射和散射问题

在有限水深、同轴但半径大于或等于浮体半径的圆柱体障碍物地形条件下，基于特征函数展开法，推导了垂直放置的圆柱形浮体由于波的辐射和散射作用所表现的动力学和运动学特征表达式，涉及浮体做垂荡、横荡和横摇运动所产生的辐射势，以及在入射波的作用下，由于浮体固定不动而产生的散射势，并推导了激励力、附加质量和阻尼系数表达式。采用与同轴、同半径圆柱体障碍物地形上单浮体水动力学特性相比的方式和激励力计算两种方法验证了推导的表达式，最后分析了障碍物几何尺寸对浮体水动力学特性的特有影响。     

On the singularities of Green's formula and its normal derivative,with an application to surface‐wave–body interaction problems

The paper presents the non‐singular forms of Green's formula and its normal derivative of exterior problems for three‐dimensional Laplace's equation. The main advantage of these modified formulations is that they are amenable to solution by directly using quadrature formulas. Thus, the conventional boundary element approximation, which locally regularizes the singularities in each element, is not required. The weak singularities are treated by both the Gauss flux theorem and the property of the associated equipotential body. The hypersingularities are treated by further using the boundary formula for the associated interior problems. The efficacy of the modified formulations is examined by a sphere, in an infinite domain, subject to Neumann and Dirichlet conditions, respectively. The modified integral formulations are further applied to a practical problem, i.e. surface‐wave–body interactions. Using the conventional boundary integral equation formulation is known to break down at certain discrete frequencies for such a problem. Removing the ‘irregular’ frequencies is performed by linearly combining the standard integral equation with its normal derivative. Computations are presented of the added‐mass and damping coefficients and wave exciting forces on a floating hemisphere. Comparing the numerical results with that by other approaches demonstrates the effectiveness of the method. Copyright © 2000 John Wiley & Sons, Ltd.     

A rigid surface boundary element for soil-structure interaction analysis in the direct time domain

Many soil-structure interaction problems involve studies of single or multiple rigid bodies of arbitrary shape and soil media. The commonly used boundary element methods implement the equations of the rigid body in a form that depends on the particulars of the geometry and requires partitioning and condensation of the associated algebraic system of equations. The present work employs the direct time domain B-Spline BEM for 3D elastodynamic analysis and presents an efficient implementation of rigid bodies of arbitrary shape in contact with, or embedded in, elastic media. The formulation of a rigid surface boundary element introduced herein is suitable for direct superposition in the BEM system of algebraic equations. Consequently, solutions are computed in a single analysis step, eliminating, thus, the need for partitioning of the system of equations. Computational efficiency is also achieved due to the extremely sparse form of the associated coefficient matrices. The proposed element can be used for the modeling of single or multiple rigid bodies of arbitrary shape within the framework of the BEM method. The efficiency and general nature of the proposed element is demonstrated through applications related to the dynamic analysis of rigid surface and embedded foundations and their interaction with embedded rigid bodies of arbitrary shape. Received 17 December 1999     

Modelling and analysis of coupled nonlinear oscillations of a floating body in two degrees of freedom

Summary We describe a mathematical model to investigate the effect of coupled nonlinear oscillations of a floating body in time domain under the influence of sinusoidal waves. To account for hydrodynamic forces, a mathematical formulation for added mass moments of inertia, damping and restoring moments is presented for roll and yaw. Using perturbation technique, we obtain order wise solutions in the normalized domain wherein the assumption on small distortion holds. On applying Laplace transform, a zeroth-order solution is obtained in closed form whereas for higher order solutions we resort to the Runge-Kutta method with adaptive step size algorithm. For analyzing the model result we perform numerical experiments for a vessel of mass 19190 tons under the action of a beam wave of frequency 0.74 rad/sec and 1.0 m wave height. The validity of the numerical scheme is checked by comparing with the analytical solution for uncoupled zeroth-order roll and then we proceed to examine the effect of coupled behavior of roll and yaw for higher-order approximations. The inter-dependence between wave frequency (ω), system frequency (β) and damping factor (ζ) is obtained to reveal system stability. Model results indicate an artificial increase in amplitude for uncoupled roll, and also emphasize the contribution of viscous damping in roll in contrast to added mass in yaw.     

An analytical solution for the diffraction problem of a submerged vertical cylinder considering bottom-mounted barrier effect

Using the methods of variable separation and eigenfunction expansion for velocity potentials, an analytical expression for the composite structure of a submerged cylinder over a coaxial, bottom-mounted cylinder in finite depth is developed. This model is also applicable for the isolated submerged truncated cylinder by careful choice of bottom cylinder height. The exciting forces on the submerged cylinder obtained by the proposed analytical solution and the boundary element method (BEM) agree well with each other, which shows good reliability of the analytical expression. Based on the proposed model, the effects of wave force on submerged cylinder owing to the existence of topographical obstacles of different sizes are investigated. The results show that various geometrical parameters of the bottom cylinder alter wave force on the upper cylinder significantly. Considering the impact of barrier, horizontal wave force and vertical wave force on submerged cylinder arrive there are several peaks at different frequencies, while the isolated cylinder always reaches the peak at a certain frequency. The peak value of wave forces may increase or decrease, but that of overturning moment has a maximum value of two to four times amplitude.     

Estimation of wall effects on floating cylinders

Under the assumptions of classical linearised water-wave theory, the time-harmonic two-dimensional motion of a rigid body floating in the surface of a fluid may be characterised by various coefficients which express components of the hydrodynamic forces acting on that body. For a single body in isolation, these are relatively simple to calculate. For bodies placed next to a vertical wall, the corresponding calculations are often much more complicated. In this paper the well-known wide-spacing approximation is used to develop approximations to the hydrodynamic coefficients for a body having a vertical plane of symmetry, but otherwise of arbitrary cross-section, next to a wall solely in terms of the results for an isolated body. Exact results are compared with the wide-spacing approximations for semi-immersed circular cylinders and cylinders of rectangular cross-section. They show that the approximation works remarkably well over all frequency ranges and even when the cylinders are very close to the wall.     

A new 3-D finite element model to evaluate added mass for analysis of fluid-structure interaction problems

The paper presents the results of investigations conducted to evaluate the added mass to represent fluid-structure interaction effects in vibration/dynamic analysis of floating bodies such as ship hulls. While the structural plating is idealized by 9-noded plate/shell finite elements, the fluid domain is modelled by 20-noded/21-noded 3-D finite elements in the investigations conducted. A new 8-noded element has been developed to model the interface between the structure and the fluid. An efficient computational methodology has been used for computation of added mass. The finite element models are validated by comparing the results with those given by analytical solution for a submerged sphere. The efficacy of the finite element model is demonstrated through convergence of the results obtained for a floating barge problem. A better convergence rate and distribution of added mass in three orthogonal directions have been obtained.     

水中悬浮隧道在冲击载荷作用下的计算模型与数值模拟

建立了水中悬浮隧道在冲击载荷作用下的简化计算模型。用等效质量法将圆柱壳分布质量折算成冲击点处的集中质量,模型中考虑流体附加质量和系统阻尼的影响。根据碰撞过程中的动量守恒、变形过程中的能量守恒以及结构的位移与内力关系,得到问题的解析解。为验证解析解,在ANSYS/LS-DYNA中建立了动态冲击有限元分析模型。通过算例分别考察了在忽略和考虑流体附加质量两种情况下,冲击点位置和冲击速度对冲击点处最大径向位移的影响,将解析解与数值解进行对比,结果吻合较好。然后采用数值模拟方法得到了系统阻尼对计算结果的影响规律。数值模拟过程中还可以得到冲击点处的最大Mises应力。     

Scattering of acoustic waves by movable lightweight elastic screens

This paper computes the insertion loss provided by movable lightweight elastic screens, placed over an elastic half-space, when subjected to spatially sinusoidal harmonic line pressure sources. A gap between the acoustic screen and the elastic floor is allowed. The problem is formulated in the frequency domain via the boundary element method (BEM). The Green's functions used in the BEM formulation permit the solution to be obtained without the discretization of the flat solid–ground interface. Thus, only the boundary of the elastic screen is modeled, which allows the BEM to be efficient even for high frequencies of excitation. The formulation of the problem takes into account the full interaction between the fluid (air) and the solid elastic interfaces.The validation of the algorithm uses a BEM model, which incorporates the Green's functions for a full space, requiring the full discretization of the ground. The model developed is then used to simulate the wave propagation in the vicinity of lightweight elastic screens with different dimensions and geometries. Both frequency and insertion loss results are computed over a grid of receivers. These results are also compared with those obtained with a rigid barrier and an infinite elastic panel.     

A unified approach for the formulation of interaction problems by the boundary element method

A unified formulation is presented, based on boundary element method, in a form suitable for performing the interaction analyses by substructure method for solid–solid and soil–structure problems. The proposed formulation permits the evaluation of all the elements of impedance and input motion matrices simultaneously at a single step in terms of system matrices of the boundary element method without solving any special problem, such as, unit displacement or load problem, as required in conventional methods. It eliminates further the complicated procedure and the need for using scattering analysis in the evaluation of input motion functions. To explain the formulation, it is first given for an inclusion interacting with an infinite surrounding medium under the influence of a seismic input, where both the inclusion and surrounding medium are treated as viscoelastic. It is shown that the formulation for a rigid inclusion may be obtained from that for flexible inclusion as a special case through a transformation. Then, the formulation is extended to other types of interaction problems: a multi‐inclusion problem and an interaction problem involving a foundation embedded in a viscoelastic half‐space. It is found that the proposed formulation remains essentially the same for all kinds of interaction problems and it can be used not only in regular interaction analysis, but also in the analysis involving diffraction of waves in a medium containing holes. Copyright © 2005 John Wiley & Sons, Ltd.     

Modelling of radiation damping in fluids by finite elements

A very efficient technique is presented to model the effects of radiation damping in the computation of added mass for the dynamic analysis of submerged structures. The structure is assumed to be surrounded by an infinite, incompressible and inviscid fluid field and the effect of the free surface is neglected. The technique is implemented in the finite element analysis of two-dimensional problems, assuming pressure to be the nodal unknown. The implementation procedure is quite simple and the symmetrical and banded form of the matrix of coefficients remains unchanged. With the use of the proposed radiation condition, the fluid field may be truncated at a relatively very short distance from the solid—fluid interface. This results in great computational advantages. Furthermore, a guideline is suggested for the selection of the geometry and the location of the truncation boundary to enhance the computational efficiency. The effectiveness and efficiency of the technique is demonstrated by analysing several cases for different geometries of the solid—fluid interface and the truncation boundary.     

Wave forces on arrays of floating bodies

Summary Previous work on the scattering of an incident wave field by an array of fixed vertical cylinders is extended to calculate the added-mass and damping coefficients for an array of floating axisymmetric bodies. The method is based upon a large spacing approximation where diverging waves are replaced by plane waves. It is shown that, given the scattering and radiation properties of a single body, the interaction effects within an array can be calculated both simply and accurately.     

Frictional contact of flexible and rigid bodies

 This paper is about planar frictional contact problems of both flexible and rigid bodies. For the flexible case a nonlinear finite element formulation is presented, which is based on a modified Coulomb friction law. Stick-slip motion is incorporated into the formulation through a radial return mapping scheme. Linearly interpolating four node elements and three node contact elements are utilized for the finite element discretization. The corresponding tangent stiffness matrices and residual vectors of the equations of motion are presented. In the rigid body case the contact problem is divided into impact and continual contact, which are mathematically described by linear complementarity problems. The impact in normal direction is modeled by a modified Poisson hypothesis, which is adapted to allow multiple impacts. The formulation of the tangential impact is grounded on Coulombs law of friction. The normal contact forces of the continual contact are such that colliding bodies are prevented from penetration and the corresponding tangential forces are expressed by Coulombs law of friction. Examples and comparisions between the different methods are presented. Received: 10 January 2001     

Hydrodynamic coefficients for water-wave diffraction by spherical structures

Evaluation of hydrodynamic coefficients and loads on submerged or floating bodies is of great significance in designing these structures. Some special regular-shaped geometries such as those of cylindrical (circular, elliptic) and spherical (hemisphere, sphere, spheroid) structures are usually considered to obtain analytical solutions to wave diffraction and radiation problems. The work presented here is the result of water-wave interaction with submerged spheres. Analytical expressions for various hydrodynamic coefficients and loads due to the diffraction of water waves by a submerged sphere are obtained. The exciting force components due to surge and heave motions are derived by solving the diffraction problem. Theory of multipole expansions is used to express the velocity potentials in terms of an infinite series of associated Legendre polynomials with unknown coefficients and the orthogonality of the polynomials is utilized to simplify the expressions. Since the infinite series appearing in various expressions have excellent truncation properties, they are evaluated by considering only a finite number of terms. Gaussian quadrature is used to evaluate the integrals. Numerical estimates for the analytical expressions for the hydrodynamic coefficients and loads are presented for various depth to radius ratios. Consideration of more values for depth makes it easy to compare the results with those available. The results obtained match closely with those obtained earlier by Wang and Wu and their coworkers     

振荡浮子式波能转换装置动力输出系统特性研究

动力输出系统(Power Take-Off, PTO)作为波浪能转换装置(Wave Energy Converter, WEC)的主要构件之一,对系统运动及能量转换至关重要。首先基于势流理论,运用特征函数展开法得到圆柱形浮体所在流域的速度势函数级数表达式,进而通过边界匹配法得到作垂荡运动浮子的附加质量、阻尼系数及波浪激励力的解析表达式。针对阻尼器特性,分别研究线性和非线性PTO阻尼作用下,浮子的运动及波能转换特性,重点研究了线性PTO作用下的过阻尼问题。计算结果表明,低速度指数的PTO系统对装置运动的影响主要体现在PTO阻尼系数上,随着阻尼系数增大,波能装置的共振频率逐渐减小,但减小幅度很小;PTO系统的非线性特性并不能改变浮子的最优转换效率,但是较大的速度指数能有效改善PTO系统的阻尼容量;在较低频和较高频时,通过解析算法得到的最优PTO阻尼系数会使得装置处于过阻尼工作状态,且在低频部分需要进行最优PTO修正的最高频率和在高频部分需要进行修正的最低频率均随着半径和吃水的增大而逐渐减小。     

Analytical derivation and numerical experiments of degenerate scale for an ellipse in BEM

Degenerate scale of an ellipse is studied by using the dual boundary element method (BEM), degenerate kernel and unit logarithmic capacity. Degenerate scale stems from either the nonuniqueness of logarithmic kernel in the BIE or the conformal radius of unit logarithmic capacity in the complex variable. Numerical evidence of degenerate scale in BEM is given. Analytical formula for the degenerate scale can be derived not only from the conformal mapping in conjunction with unit logarithmic capacity, but also can be derived by using the degenerate kernel. Eigenvalues and eigenfunctions for the weakly singular integral operator in the elliptical domain are both derived by using the degenerate kernel. It is found that zero eigenvalue results in the degenerate scale. Based on the dual BEM, the rank-deficiency (mathematical) mode due to the degenerate scale is imbedded in the left unitary vector for weakly singular and strongly singular integral operators. On the other hand, we obtain the common right unitary vector of a rigid body (physical) mode in the influence matrices of strongly singular and hypersingular operators after using the singular value decomposition. Null field for the exterior domain and interior nonzero fields are analytically derived and numerically verified in case of the normal scale while the interior null field and nonzero exterior field are obtained for the homogeneous Dirichlet problem in case of the degenerate scale. No failure CHEEF point is confirmed in the nonzero exterior field to overcome the degenerate-scale problem. To deal with the nonuniqueness-solution problem, the constraint of boundary flux equilibrium instead of rigid body term, CHEEF and hypersingular BIE, is added to promote the rank of influence matrices to be full rank. Both analytical and numerical results agree well in the demonstrative example of an ellipse.     

A general exact transformation of body-force volume integral in BEM for 2D anisotropic elasticity

In a previous study (Zhang, Tan and Afagh, 1995), the present authors successfully transformed the body-force volume integrals in BEM for 2D anisotropic elasticity, to boundary ones. This restores the BEM as a truly boundary solution process for treating anisotropic bodies involving body forces. However, the formulation is valid only for problem domains which are geometrically convex and simply connected. This paper presents a general and exact transformation of the bodyforce volume integrals in BEM to line integrals for 2D anisotropic elasticity, in which the above-mentioned restriction on the geometry of the domain is eliminated. The successful implementation of the formulation is demonstrated by three practical examples.     

Formulation of consistent restoring stiffness in ship hydroelastic analysis

In this article, the role of the restoring stiffness, as one of the basic parameters in ship hydroelastic analysis, is brought out. It is formulated using the variational principle and the method of virtual displacements. It is shown that asymmetry of the restoring stiffness is a physical reality. Moreover, it is confirmed that modal variation, still disputed in the relevant literature, has to be taken into account to satisfy the ship’s stability. Consistent stiffness is formulated here by regarding stiffness definition as relation between forces and displacements. Hybrid stiffness known from the literature is wrong since some terms are specified as relation between forces and displacement gradient. Influence of the consistent and symmetrized stiffness matrix, and the hybrid one, on dynamic response is illustrated for a prismatic pontoon and a large container ship. It is found that the latter two matrices do not assure convergence of transfer functions of sectional forces to zero value as the wave frequency approaches zero. The rigid body and elastic responses are compared, and pertinent conclusions are drawn. It is also shown that it is not necessary to use the unified geometric and restoring stiffness for ordinary hydroelastic analysis of ship structures. The consistent formulation of the restoring stiffness matrix will be useful for extending linear potential theory hydrodynamic codes for rigid body analysis to deformable bodies.     

基于比例边界有限元法和有限元法的水下结构瞬态分析方法

针对冲击波作用下水下结构与无限声学水域的流固耦合问题,建立了基于比例边界有限元法和有限元法的瞬态分析方法。无限水域用比例边界有限元法离散,而水下结构等有限域用有限元法模拟。该方法利用声学近似法将无限水域施加给水下结构的载荷分解成冲击波载荷和散射波载荷。冲击波载荷由水下冲击波理论确定,而散射波载荷由比例边界有限元法估值。为改善比例边界有限元法动态质量矩阵的计算效率,发展了动态质量矩阵的时域递推公式。数值算例分析结果表明了所发展的瞬态分析方法和时域递推公式的正确性。