Transient Hydroelastic Response of VLFS by FEM with Impedance Boundary Conditions in Time Domain

A time-dependent finite element method (FEM) is developed to analyze the transient hydroelastie responses of very large floating structures (VLFS) subjected to dynamic loads. The hydrodynamic problem is formulated based on the linear theory of fluid and the structural response is analyzed based on the thin plate theory. The FEM truncates the unbounded fluid domain by introducing an artificial boundary surface, thus defining a finite computational domain. At this boundary surface an impedance boundary conditions are applied so that no wave reflections occur. In the proposed scheme, all of the procedures are processed directly in time domain, which is efficient for nonlinear analyses of structure floating on unbounded fluid. Numerical results indicate acceptable accuracy of the proposed method.     

有航速浮体时域分析的混合边界元法

给出了一种联合瞬态格林函数和Rankine源进行有航速浮体时域水动力分析的混合——边界元方法。在三维混合边界元方法中,通过一个匹配面将流体域划分为内域和外域,在内域中使用Rankine源以模拟直壁或非直壁船体及线性或非线性自由面条件,在外域中使用瞬态格林函数以满足自由面条件和远方辐射条件。使用该方法计算了一个有航速潜没圆球的波浪力,和解析结果的比较证明了该方法的正确性。进一步给出了一个有航速Wigley船的水动力结果,计算结果稳定,没有外传波向内反射的现象发生。     

An integral eigenvalue approach to three-dimensional field problems--A low-frequency variation of SEM

The three-dimensional (3-D) eddy-current transient field problem is formulated first using theu-vmethod. This method breaks the vector Helmholtz equation into two scalar Helmholtz equations. Null-field integral equations and the appropriate boundary conditions germane to the problem are used to set up an identification matrix which is independent of null-field point locations. Embedded in the identification matrix are the unknown eigenvalues of the problem representing its impulse response in time. These eigenvalues are found by equating the determinant of the identification matrix to zero. The eigenvalues, which can be equated with temporal response, are found to be intimately linked to the initial forcing function which triggers the transient in question. When this initial forcing function is Fourier decomposed into its respective spatial harmonics, it is possible to associate with each Fourier component a unique eigenvalue by this technique. The true transient solution comes through a convolution of the impulse response so obtained with the particular imposed external field governing the problem at hand. The technique is applied to the FELIX medium cylinder (a conducting cylinder placed in a collapsing external field) and compared to data. A pseudoanalytic confirmation of the eigenvalues so obtained is formulated to validate the procedure. The technique proposed is applied in the low-frequency regime where the near-field effects must be considered. Application of the technique to a high frequency follows directly if the Coulomb gauge is adopted to represent the vector potential.     

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.     

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.     

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.     

Pseudorandom realization of transient acoustic waves scattered fromrandomly rough patches

A simple numerical technique is developed for generating pseudorandom realizations of three-dimensional (3-D) transient acoustic waves that are scattered from two-dimensional (2-D) patches of randomly rough surfaces. The rough surface height of a patch is represented numerically in the 2-D horizontal wavenumber plane by choosing a scheme for interpolation between pseudorandom complex coefficients. Using this approach, the realizations of the patches can be generated from experimentally measured roughness power spectra, and phase information is generated in the frequency domain that leads to time spreads in the time domain. The acoustic scattering is modeled here with first-order perturbation theory. The boundary conditions considered here are pressure-release, rigid, and fluid-fluid. Three different spatial windows are considered for defining the patches. In the time domain, the time spreads of the scattered waveforms agree with predictions. In the frequency domain, the phase is seen as a random walk. The solutions developed here can be used with normal mode propagation models or ray propagation models     

Coupled SPHS–BEM method for transient fluid–structure interaction and applications in underwater impacts

Coupled SPHS–BEM method is proposed for transient fluid–structure interaction problems: SPH shell (SPHS) is selected to discretize shell structures, the second-order doubly asymptotic approximations (DAA₂) of boundary element method (BEM) is chosen to analyze flow-field. BEM can remedy the expensive costs for three-dimensional SPH (smoothed particle hydrodynamics), yet SPHS provides a structural solver for BEM. The coupled method is attractive, since only a layer of SPHS particles and a piece of flow-field boundary elements are needed to be modeled; the compatibility conditions of the coupled surface are performed with moving least square (MLS) function. The final two benchmarks on underwater impacts prove the feasibility, stability and accuracy of the proposed method.     

Moving shoreline boundary condition for nearshore models

The paper develops and analyzes two fully nonlinear boundary conditions that incorporate the motion of the shoreline in nonlinear time domain nearshore models. A moving shoreline essentially means the computational domain is changing with the solution of the flow. The problem is solved in two steps. The first is to establish an equation that determines the motion of the shoreline based on the local momentum balance. The second is to develop and implement into a shoreline model the capability of accommodating a changing computational domain. The two models represent two different ways of addressing this step: one is to track the position of the shoreline in a fixed grid by establishing a special shoreline point which generally is not a fixed grid point. The second is by a coordinate transformation that maps the changing domain onto a fixed domain and solves the basic equations in the mapped domain. The two shoreline conditions are tested against three known solution for nonlinear shoreline motion. Two are the 1-D solutions to the nonlinear shallow water (NSW) equations by Carrier and Greenspan [J. Fluid Mech. 4 (1958) 97], one representing the response to a transient change in the offshore water level, the other the motion due to a periodic standing wave, both on slopes steep enough to allow full reflection. The third is the 2-D horizontal (2DH) computational solution by Zelt [Coast. Eng. 15 (1991) 205] for the run-up of a solitary wave on a cusped beach. In all cases, both models are shown to behave well and give high accuracy results for suitably chosen grid and time spacings.     

Simulation of unsteady oceanic cable deployment by direct integration with suppression

An improved algorithm is developed for predicting the transient response of a system of serially connected cables and bodies during unsteady deployment from a surface vessel. The governing equations of a cable-body system are derived with dependent variables of cable velocities, direction cosines and tension magnitude to form a nonlinear combined initial-value and boundary-value problem. The problem is then solved by introducing a stable Newmark-like implicit integration scheme in time and by a direct integration method with suppression of extraneous erroneous solutions. Special boundary conditions simulating actively controlled payout and slack-cable/ocean-bottom contact boundary conditions are included in the present model.     

An analytical solution for wave-induced seabed response in a multi-layered poro-elastic seabed

In this study, a new analytical solution for the wave-induced seabed response in a multi-layered poro-elastic seabed is developed. The seabed is treated as a multi-layered porous medium and characterized by Biot’s theory. The displacements of the solid skeleton and the pore pressure are expressed in terms of two scalar potentials and one vector. Then, the Biot’s dynamic equation can be solved using Fourier transformation and reducing to Helmholtz equations. To obtain the general solutions for the multi-layered poro-elastic seabed in the frequency-wave-number domain, the transmission and reflection matrices (TRM) method is used to form the equivalent stiffness. Using the boundary conditions and continuous conditions, the frequency-wave-number domain solutions are obtained. Finally, the time-space domain solutions for the multi-layered poro-elastic seabed are obtained by means of the inverse Fourier transformation with respect to the horizontal coordinate. Based on the new solution, a parametric study is carried out to examine the effects of soil characteristics (number of layers, permeability and shear modulus) and wave characteristics (water depth and wave steepness) on seabed responses. The results indicate that the seabed response is affected significantly by permeability, shear modulus and relative water depth.     

一个近岸海流数值计算模式的开边界条件

为了计算的省时性,ＩＡＰ近岸海流数值模式采用了分解算法,即将控制方程分解为３个过程并用不同的时间步长进行积分。文章在此基础上对分解后的控制方程作局地线性化,得到了适合不同过程的开边界条件。对于适应过程,控制方程变换为等价的表示沿不同方向传播波动的特征方程。传出计算区域的波动用特征方程来描述,而传入波动则由无反射边界条件来消除。对于演变过程,求出了解析形式的通解。在流出点上,边界条件可以借助解析解由上一时刻区域内部或边界上的已知值求得;而在流入点上,边值保持定常。对于耗散过程,不需要边界条件。最后对所提出的开边界条件进行了数值检验,结果是令人满意的。     

Fast computation of the transient motions of moving vessels in irregular ocean waves

A fast time-domain method is developed in this paper for the real-time prediction of the six degree of freedom motions of a vessel traveling in an irregular seaway in infinitely deep water. The fully coupled unsteady ship motion problem is solved by time-stepping the linearized boundary conditions on both the free surface and body surface. A velocity-based boundary integral method is then used to solve the Laplace equation at every time step for the fluid kinematics, while a scalar integral equation is solved for the total fluid pressure. The boundary integral equations are applied to both the physical fluid domain outside the body and a fictitious fluid region inside the body, enabling use of the fast Fourier transform method to evaluate the free surface integrals. The computational efficiency of the scheme is further improved through use of the method of images to eliminate source singularities on the free surface while retaining vortex/dipole singularities that decay more rapidly in space. The resulting numerical algorithm runs 2–3 times faster than real time on a standard desktop computer. Numerical predictions are compared to prior published results for the transient motions of a hemisphere and laboratory measurements of the motions of a free running vessel in oblique waves with good agreement.     

Hydroelastic analysis of cantilever plate in time domain

Numerical solutions for the hydroelastic problems of bodies are studied directly in the time domain using Neumann–Kelvin formulation. In the hydrodynamic part of problem, the exact initial boundary value problem is linearized using the free stream as a basis flow, replaced by the boundary integral equation applying Green theorem over the transient free surface Green function. The resultant boundary integral equation is discretized using quadrilateral elements over which the value of the potential is assumed to be constant and solved using the trapezoidal rule to integrate the memory or convolution part in time. In the structure part of the problem, the finite element method is used to solve the hydroelastic problem. The Mindlin plate as a bending element, which includes transverse shear effect and rotary inertia effect are used. The present numerical results show acceptable agreement with experimental, analytical, and other published numerical results.     

Boundary element analysis of liquid sloshing characteristics in axisymmetric tanks with various porous baffles

This paper aims to investigate the effects of the porous baffles on the suppression of sloshing for the tanks with axisymmetric geometries under lateral excitation. Based on the assumptions of inviscid, irrotational, incompressible liquid and small amplitude sloshing, an axisymmetric boundary element method (BEM) for 3D Laplace equation is derived by using the Green's theorem together with the weighted residual method. And a zoning method is employed to model fluid domain in the tanks with complex porous baffles. Meanwhile, the porous baffles are treated motioning together with the tanks, and the velocity across the porous baffle is assumed to be linearly proportional to the pressure gradient between each side of the porous baffle. And the mechanism of suppressing the sloshing response is mainly the energy dissipation of the fluid passing through the porous baffle. Moreover, the linear free surface boundary conditions are also used to solve the governing equations. Compared with other numerical methods, the most prominent advantage of the BEM in solving axisymmetric potential problem is that only the boundaries of half the cross-section instead of the entire problem domain should be discretized, which can cut down large amount of memory and time costs. The present method is verified by comparing the numerical results with the existing literatures, and excellent agreements are obtained. Meanwhile, the proposed models are applied to investigate the effects of the porous baffles on sloshing response in circular cylindrical, annular cylindrical and conical tanks. The effects of the porous baffle length, porous-effect parameter, installation angle and baffle height on the sloshing force, natural frequency and surface elevation are studied. Additionally, some typical sloshing pressure distributions, velocity potential contours and velocity fields are plotted. The results show that swirls at the tips of the baffles can be observed in many cases, and the top-mounted porous baffle makes more significant suppression effects on sloshing response than that of bottom-mounted porous baffle, while increasing the number of ring porous baffles can achieve better restraint effects on sloshing response. And increasing the baffle length of the horizontal wall-mounted ring porous baffle can significantly decrease the sloshing frequencies, as well as the first non-dimensional natural frequency decreases with decrease in porous-effect parameter of the coaxial porous baffle. In addition, remarkable effects on sloshing can be obtained when reasonable designed by selecting the optimal porous-effect parameter, installation angle and baffle height. And this paper can be a useful guide for the seismic design and analysis of many actual liquid storage tanks (such as the Advanced Passive PWR, large water cooling tower, etc.).     

Numerical experiment on the circulation in the Japan Sea

By using a rectangular basin of uniform depth with inflow and outflow openings, the circulation in the Japan Sea is investigated numerically. Heat flux through the sea surface is determined from the annual mean atmospheric conditions for the Japan Sea, but no wind stress is considered.In the transient state, the warm water supplied through an inflow opening travels cyclonically along the coast as a density-driven boundary current in a rotating system. In the quasi-steady state, the warm water flows northward as a western boundary current which corresponds to the East Korean Warm Current and gradually separates from the coast as it flows northward. No strong boundary current corresponding to the nearshore branch of the Tsushima Current exists.Under annual mean atmospheric conditions, formation of the deep water characteristic of the Japan Sea and of the thermal front corresponding to the Polar Front do not take place.     

A pretty good sponge: Dealing with open boundaries in limited-area ocean models

The problem of computing within a limited domain surrounded by open boundaries is discussed within the context of the shallow-water wave equations by comparing three different treatments, all of which surround the domain by absorbing zones intended to prevent reflections of outgoing waves. The first, which has attracted a lot of attention for use in electromagnetic and aeroacoustic applications, is intended to prevent all reflections. However, it has not yet been developed to handle the second important requirement of open boundaries, namely the ability to pass information about external conditions into the domain of interest. The other two treatments, which absorb differences from a specified external solution, allow information to pass through the open boundary in both directions. One, based on the flow relaxation scheme of [Martinsen, E.A., Engedahl, H., 1987. Implementation and testing of a lateral boundary scheme as an open-boundary condition in a barotropic ocean model. Coastal Eng. 11, 603–627] and termed here the “simple sponge,” relaxes all fields toward their external counterparts. The other, a simplification and generalization of the perfectly matched layer, referred to here as the “pretty good sponge,” avoids absorbing the component of momentum parallel to the open boundary. Comparisons for a case that is dominated by outgoing waves shows the pretty good sponge to perform essentially as well as the perfectly matched layer and better than the simple sponge. In comparisons for a geostrophically balanced eddy passing through open boundaries, the pretty good sponge out-performed the simple sponge when the only external information available was about the advecting flow, but when information about the nature of the eddy in the sponge zones was also available, the simple sponge performed better. For the case of an equatorial soliton passing through the boundary and no information provided about its nature outside the open domain, again the pretty good sponge performed better. Proving useful for situations governed by nonlinear equations forced by external conditions and being easy to implement, the pretty good sponge should be considered for use with existing limited-area ocean models.     

波浪沿斜面传播过程的数值模拟

精确模拟非线性波沿斜面传播过程非常困难,为此论文从势函数的边界积分方程出发,建立了一种时域内二维波浪模拟的数值模型,主要用来模拟完全非线性波浪的传播变形过程。论文的数值模型使用高阶二维边界元方法,采用可调节时间步长的基于二阶显式泰勒展开的混合欧拉-拉格郎日时间步进来求解带自由表面的线性或完全非线性波浪传播问题。在计算区域一端造出线性或非线性的周期性波浪,另一端采用消除反射波的人工粘性吸收边界。通过与现有理论比较证明了论文数值方法所得结果是准确可靠的。     

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.     

Constant acceleration exit of two-dimensional free-surface-piercing bodies

The forced constant acceleration exit of two-dimensional bodies through a free-surface is computed for various 2D bodies (symmetric wedges, asymmetric wedges, truncated wedges and boxes). The calculations are based on the fully non-linear time-stepping complex-variable method of Vinje and Brevig (1981). The model was formulated as an initial boundary-value problem (IBVP) with boundary conditions specified on the boundaries (dynamic and kinematic free-surface boundary conditions) and initial conditions at time zero (initial velocity and position of the body and free-surface particles). The formulated problem was solved by means of a boundary-element method using collocation points on the boundary of the domain and stepped forward in time using Runge–Kutta and Hamming predictor–corrector methods. Numerical results for the deformed free-surface profile, pressure along the wetted region of the bodies and force experienced by the bodies are given for the exit. The analytical added-mass force is presented for the exit of symmetric wedges and boxes with constant acceleration using conformal mappings. To verify the numerical results, the added-mass force and the numerical force are compared and give good agreement for the exit of a symmetric wedge at a time zero (t = 0) as expected but only moderate agreement for the box.