Free-surface Green Function Research Articles

Enhanced Endo's approach for evaluating free-surface Green's function with application to wave-structure interactions

The Green function of the diffraction-radiation theory of time-harmonic waves under a moderate water depth is considered. The paper presents a newly developed enhanced algorithm based on Endo's approach, which employs a direct Gauss-Laguerre quadrature in the calculation of the principal value integrations. This methodology was firstly originated from Endo (1983), and had been subsequently improved by Liu et al. (2008) and Shan et al. (2019). Unfortunately, the latest version of Endo's approach still has two fatal problems with (1) incredibly large or small values at some “weird frequencies”, and (2) nonsense values in the high-frequency region under a large depth. The present algorithm proposes special techniques for the removal of the singularities at these “weird frequencies”, and provides special treatments to avoid exceeding hardware's limitation when the input parameter k0h is overlarge. The developed algorithm is thereafter tested against a variety of sea conditions, via a comparison with Newman's polynomial algorithm (Newman, 1985), certifying its good accuracy in various circumstances. By carrying out a benchmark test on the DeepCwind semisubmersible, through a comparison with Shan's algorithm (Shan et al., 2019) and the commericial software Hydrostar®, the cause of the occurrence of the “weird frequencies” is found, in association with other important findings. The computations demonstrate that the present enhanced algorithm is free from the preceding problems and is sufficiently accurate to compute wave loads in practice.

Prediction of Ship Motions Via a Three-Dimensional Time-Domain Method Following a Quad-Tree Adaptive Mesh Technique

Abstract A three-dimensional (3D) time-domain method is developed to predict ship motions in waves. To evaluate the Froude-Krylov (F-K) forces and hydrostatic forces under the instantaneous incident wave profile, an adaptive mesh technique based on a quad-tree subdivision is adopted to generate instantaneous wet meshes for ship. For quadrilateral panels under both mean free surface and instantaneous incident wave profiles, Froude-Krylov forces and hydrostatic forces are computed by analytical exact pressure integration expressions, allowing for considerably coarse meshes without loss of accuracy. And for quadrilateral panels interacting with the wave profile, F-K and hydrostatic forces are evaluated following a quad-tree subdivision. The transient free surface Green function (TFSGF) is essential to evaluate radiation and diffraction forces based on linear theory. To reduce the numerical error due to unclear partition, a precise integration method is applied to solve the TFSGF in the partition computation time domain. Computations are carried out for a Wigley hull form and S175 container ship, and the results show good agreement with both experimental results and published results.

Transient wave diffraction around cylinders by a novel boundary element method based on Fourier-Laguerre expansions

ABSTRACT A novel boundary element method based on series expansion is developed for wave diffraction by a vertical circular cylinder in the time domain. Unlike traditional potential flow boundary element methods in which velocity potentials are evaluated on each panel, both the velocity potential and its normal derivative are expanded by Laguerre functions in the vertical direction and Fourier series along the circumference of cylinder, respectively. A boundary integral equation is therefore established associated with expansion coefficients in the sense of Galerkin collocation. Free-surface Green function in the time domain is no longer explicitly computed but its integration involving Fourier-Laguerre basis functions on the whole cylindrical surface is evaluated. Incoming waves are transient with wavefronts, which are more general than steady-state plane progressive waves. Results obtained by using the present method are compared with those from existing time-domain analysing method.

Froude–Krylov nonlinear computations of three dimensional wave loads by a hybrid time domain boundary element method

Wave-induced loads on advancing ships are studied by using a three dimensional nonlinear hybrid high order boundary element method with the combination of the Rankine source Green function applied in the internal domain and the transient free surface Green function adopted in the external domain. A quadratic representation of both the boundary and the unknowns on the boundary is adopted. In the internal domain the forward speed problem is solved with the steady ship wave as the basis flow. In the external domain, an efficient algorithm combined with Chebyshev approximations and asymptotic expressions for the transient free surface Green function is adopted to the solution of radiation condition. The nonlinear Froude–Krylov and hydrostatic restoring forces resulted from large amplitude incident waves are taken into consideration over the instantaneous wetted hull surface. This paper presents an effective solution to the problems of excessive memory occupation and large time consuming in the long-duration simulation. A nonlinear numerical program is originally developed and the preliminary linear motions of Wigley-3 and S175 container hull at relatively high speed are analyzed. Furthermore, the nonlinear behavior of hydrodynamic forces, motions, and the wave loads of S175 hull are investigated in regular and irregular seas, especially the phenomenon of ‘super harmonics’ in severe regular waves are highlighted thoroughly. Comparisons with experiment results indicate the present method owns satisfactory numerical stability, accuracy, and efficiency for the practical application.

Environmental Aspects of Total Resistance of Container Ship in the North Atlantic

The total resistance of ship in waves is composed of calm water resistance and added resistance in waves. The added resistance in waves is one of the main causes of an involuntary speed reduction of a ship. It may cause a significant increase in ship resistance and consequently increased fuel consumption and CO2 emission, especially at heavier sea states. Thus, the added resistance is very important from both an economic and environmental point of view. In this paper, an increase in fuel consumption is calculated for container ship on a typical North Atlantic route. The calm water resistance is calculated utilizing Computational Fluid Dynamics (CFD) based on the viscous flow theory at model scale and the obtained results are extrapolated to full scale. The added resistance in waves is calculated using three-dimensional panel code based on the free surface Green function, for the sake of simplicity. It is calculated at regular waves and the mean value of the added resistance at certain sea states is obtained by means of spectral analysis. Two-parameter Bretschneider wave spectrum recommended for the North Atlantic, defined by significant wave height and zero crossing period, is used in the spectral analysis. The obtained results may provide valuable insight into important environmental issue of pollution reduction.

Comparative study on ship motions in waves based on two time domain boundary element methods

This paper presents a comparative study on the motions of a ship advancing in waves using two different three-dimensional time domain boundary element methods: a Transient Free surface Green's Function (TFGF) method and a Rankine Higher Order Boundary Element Method (HOBEM). Three models based on the HOBEM are also considered to identify the dominant factors affecting the accuracy of solutions: (i) the N–K model using Neumann–Kelvin (N–K) linearization, (ii) the N–K+DB model using the N–K linearization for the free surface boundary conditions and the double-body (DB) m-terms in the body boundary condition, (iii) the Rankine HOBEM based on DB linearization. The Wigley I and the Series 60 (CB = =0.7) are taken as study objects. The comparisons show that the Rankine HOBEM based on DB linearization is generally more accurate than the Rankine HOBEM with other two models and the TFGF method. The hydrodynamic coefficients are mainly affected by the m-terms, especially at low frequencies; while the influences of different linearizations of the free surface boundary conditions are negligible. Both the m-terms and the leading terms of steady velocity potential reserved in the free surface boundary conditions are important for motion responses, while the influence of the former is stronger than the latter.

A simplified derivation of the ordinary differential equations for the free-surface Green functions

Ordinary differential equations (ODEs) are derived for the free-surface Green functions and their gradients, in a fluid of infinite depth. The cases of harmonic time-dependence in the frequency domain and impulsive motion in the time domain are considered separately. These ODEs were derived originally by Clément, starting with fourth-oder equations in the time domain and using Fourier transforms to derive a second-order ODE in the frequency domain. In the present work a simpler procedure is followed independently in each domain, transforming the governing Laplace equation to an ordinary differential equation. The results are consistent with the ODEs derived using Clément’s method.

Consistent expressions for the free-surface Green function in finite water depth

Fast and accurate computation of the free-surface Green function is of key importance for the numerical solution of linear and second-order wave-structure interaction problems in three dimensions. Integral and series expressions for the Green function are derived for which the limiting values for zero and infinite frequency are consistent with the zero and infinite frequency Green function defined in terms of infinite series of Rankine image sources. The integral expressions presented here have the advantage that they are slowly varying with the non-dimensional wave frequency, making them more efficient to approximate compared with previous expressions.

Prediction of added resistance of a ship in waves at low speed

This paper presents predictions of the added resistance of a ship in waves at a low speed according to the IMO minimum propulsion power requirement by a hybrid Taylor expansion boundary element method (TEBEM). The flow domain is divided into two parts: the inner domain and the outer domain. The first-order TEBEM with a simple Green function is used for the solution in the inner domain and the zero order TEBEM with a transient free surface Green function is used for the solution in the outer domain. The TEBEM is applied in the numerical prediction of the motions and the added resistance in waves for three new designed commercial ships. The numerical results are compared with those obtained from the seakeeping model tests. It is shown that the prediction of the ship motions and the added resistance in waves are in good agreement with the experimental results. The comparison also indicates that the accuracy of the motion estimation is crucial for the prediction of the wave added resistance. In general, the TEBEM enjoys a satisfactory accuracy and efficiency to predict the added resistance in waves at a low speed according to the IMO minimum propulsion power requirement.

An efficient algorithm with new residual functions for the transient free-surface green function in infinite depth

Efficient evaluation of the transient free-surface Green function with sufficient accuracy is the key to employ time-domain panel method in solving hydrodynamic problems. In this paper, series and asymptotic expansions for the Green function and its two spatial derivatives are deduced in detail for the evaluation. New residual functions are derived straightforwardly from the resulted asymptotic formulas and it is concluded that the present residual functions decay rapidly than existing choices. The ultimate algorithm is the combination of Chebyshev approximations and asymptotic expressions, along with a special partitioning line introduced to speed up the evaluation. After comprehensive numerical validation, it is pointed out that the present algorithm owns at least 6 digits of accuracy and sufficient efficiency for practical application. Any efficient method for evaluating the Green function will use the asymptotic expansions. Furthermore, it should be pointed out that the present residual functions can also be conveniently applied to the scheme of interpolation in a precomputed table.

A new ordinary differential equation for the evaluation of the frequency-domain Green function

Clément (2013) derived a second order ordinary differential equation (ODE) satisfied by the free-surface Green function in the frequency domain. Since then, similar ODEs for the gradient of the Green function have been developed. Unfortunately, all these ODEs degenerate at zero frequency. Therefore, it is not possible to initialize the numerical solution of these ODEs from this zero frequency. Alternative methods based on the shifting of the initial condition to frequencies strictly greater than zero have then been developed.The present paper describes an alternative approach to address this issue. It involves a new function which is the solution of a modified ODE which can be solved from the zero frequency.Finally, comparisons with evaluations of the Green function using the classical direct integration method are provided. They show that the new ODE can provide accurate estimates of the Green function.

Straightforward integration for free surface Green function and body wave motions

An alternative manner is provided for solving the classical linearised problem of the radiation and diffraction of regular water waves caused by oscillation of a floating body in deep water. It is shown that the singular wave integrals of the three-dimensional free surface Green function $G$ and its gradient $\nabla G$ can be regarded as regular wave integrals and are integrated directly. The method is validated by comparing with benchmark data for a floating or submerged body undergoing oscillatory wave motions. The comparison shows that the evaluation is sufficiently accurate for practical purposes. As the significance of the method, the numerical approximation stability for the gradient $\nabla G$ is shown to be the same with that for $G$.

Comparison of existing methods for the calculation of the infinite water depth free-surface Green function for the wave–structure interaction problem

In this study, the mathematical expressions and numerical methods for the free-surface Green function of the linearized wave–structure problem in deep water and in the frequency domain are investigated. Twelve different expressions are reviewed and analyzed. All these expressions are exact mathematical solutions for the propagation of waves from a pulsating source located in the fluid domain. However, their numerical evaluation is challenging. Dedicated numerical methods have been developed. They include series expansions, polynomials, table interpolations, multipole expansions, approximations with elementary functions, etc. In this work, four methods were implemented: the Newman's method [1], the Delhommeau's method [2], the Telste-Noblesse's method [3] and the Wu et al.'s method [4]. Their CPU time and accuracy are compared. It is found that the average computational time for Newman's method is 5.745 × 10−7. It is 5.782 × 10−8 for the Delhommeau's method. For Telste-Noblesse's method and Wu et al.'s methods, they are 4.642 × 10−8 and 1.491 × 10−9, respectively. The accuracy is respectively 6D(6 decimals), 5D and 3D for the Newman's method, the Telste-Noblesse's method and the Wu et al.'s method. For the Delhommeau's method, it is 3D except when the vertical coordinate is close to 0. The accuracy of the Delhommeau's method can be increased significantly by refining the discretization of the space variables for the tabulated functions and by using higher interpolation methods, at cost of increased computational time.

Motion of a floating body in a harbour by domain decomposition method

A three-dimensional domain decomposition method is used to solve the problem of wave interaction with a ship floating inside a harbour with arbitrary shape. The linearized velocity potential theory is adopted. The total fluid domain is divided into two sub-ones: one for the harbour and the other for the external open sea. Boundary integral equations together with the free surface Green function are used in the both domains. Matching conditions are imposed on the interface of the two sub-domains to ensure the velocity and pressure continuity. The advantage of the domain decomposition method over the single domain method is that it removes the coastal surface from the boundary integral equation. This subsequently removes the need for elements on the coastal wall when the equation is discretized. The accuracy of the method is demonstrated through convergence study and through the comparison with the published data. Extensive results through the hydrodynamic coefficients, wave exciting forces and ship motions are provided. Highly oscillatory behaviour is observed and its mechanism is discussed. Finally, the effects of incident wave direction, ship location as well as the harbour topography are investigated in detail.

Efficient approximation of free-surface Green function and OpenMP parallelization in frequency-domain wave–body interactions

The evaluation of Green function without translation has always been the most time-consuming part of the hydrodynamic analysis based on potential flow theory. This research investigates the new implementations of efficient approximations of Green function and the code parallelization with OpenMP library. First, algorithms of infinite-depth case are developed with economized Chebyshev polynomials based on modified region and subdomain partitions. Second, an improved Gauss–Laguerre method for the Cauchy integral is presented to evaluate the finite-depth case with the convergence acceleration by the reduction of fraction and singularity elimination by the infinite-depth case together with the exponential integral. For the analyzed container vessel, on the premise of good numerical accuracy, we conclude that the present method has nearly obtain the same efficiency as Chen (Hydrodynamics in offshore and naval applications—Part I, Bureau Veritas, Paris, 2004) for the infinite-depth case and almost 30% higher efficiency for the finite-depth case. Moreover, we also point out that the numerical efficiency is significantly enhanced again due to the OpenMP parallelization of the serial code.

Fully coupled time domain solution for hydroelastic analysis of a floating body

A numerical method is developed to study the ship hydroelasticity problem in time domain. Boundary – Element Method (BEM) with three-dimensional transient free-surface Green function and Neumann – Kelvin approximation is used for the solution of the hydrodynamic part. The structural domain is modeled by finite element method (FEM) using 1D beam element in time domain. The coupling between BEM and FEM is made through body boundary condition in the hydrodynamic solution. Furthermore, direct integration scheme, i.e. Newmark –β method is used for obtaining the structural velocity and deformation. The efficiency and robustness of the proposed method are validated with available published results. Three different barge type structures and a large container ship are taken for the analysis. A good correspondence is observed between the proposed and reported numerical and experimental results. Also, it is observed that structural flexibility plays an important role while calculating structural deflection, shear force, bending moment etc. The developed numerical method seems to be robust, efficient and a very useful tool in predicting the hydrodynamic loads, structural deflections, shears force, bending moment etc. for flexible structure.

Coupled boundary element method and finite element method for hydroelastic analysis of floating plate

In this study, a numerical procedure has been proposed to analyze the equation of motion of the elastic plate which is elastic in nature and having shallow draft (small thickness) with arbitrary geometry subjected to linear wave force at a fixed frequency. Investigation on the convergence of maximum deflection of the floating plate has been carried out. A hybrid model has been developed (coupling between FEM and BEM) which contains same nodes, maintaining the same order and basis function in both the methods. To develop the relationship between the displacement of the plate and the velocity potential under the plate, two equations have been derived. The first equation is derived from the equation of motion for the plate and is solved by finite element method (FEM) to extract the displacement of the floating structure. The second equation is from water wave theory which is based on boundary integral equation that relates the displacement of the floating plate and velocity potential using free-surface Green's function. To get the displacement of floating elastic plate and velocity potential both the equations are solved simultaneously. Results are presented for modified Green's function which has been derived and validated with the results of Meylan (2004). The performance of the developed model is examined by the convergence rate, simulation time. It is learnt that the model works well in finite depth whereas its performance in infinite depth lags by an average of 20% in simulation time than the results obtained by Meylan (2004).

A new multi-domain method based on an analytical control surface for linear and second-order mean drift wave loads on floating bodies

A novel multi-domain method based on an analytical control surface is proposed by combining the use of free-surface Green function and Rankine source function. A cylindrical control surface is introduced to subdivide the fluid domain into external and internal domains. Unlike the traditional domain decomposition strategy or multi-block method, the control surface here is not panelized, on which the velocity potential and normal velocity components are analytically expressed as a series of base functions composed of Laguerre function in vertical coordinate and Fourier series in the circumference. Free-surface Green function is applied in the external domain, and the boundary integral equation is constructed on the control surface in the sense of Galerkin collocation via integrating test functions orthogonal to base functions over the control surface. The external solution gives rise to the so-called Dirichlet-to-Neumann[DN2] and Neumann-to-Dirichlet[ND2] relations on the control surface. Irregular frequencies, which are only dependent on the radius of the control surface, are present in the external solution, and they are removed by extending the boundary integral equation to the interior free surface (circular disc) on which the null normal derivative of potential is imposed, and the dipole distribution is expressed as Fourier–Bessel expansion on the disc. In the internal domain, where the Rankine source function is adopted, new boundary integral equations are formulated. The point collocation is imposed over the body surface and free surface, while the collocation of the Galerkin type is applied on the control surface. The present method is valid in the computation of both linear and second-order mean drift wave loads. Furthermore, the second-order mean drift force based on the middle-field formulation can be calculated analytically by using the coefficients of the Fourier–Laguerre expansion.

Time domain Rankine-Green panel method for offshore structures

To solve the numerical divergence problem of the direct time domain Green function method for the motion simulation of floating bodies with large flare, a time domain hybrid Rankine-Green boundary element method is proposed. In this numerical method, the fluid domain is decomposed by an imaginary control surface, at which the continuous condition should be satisfied. Then the Rankine Green function is adopted in the inner domain. The transient free surface Green function is applied in the outer domain, which is used to find the relationship between the velocity potential and its normal derivative for the inner domain. Besides, the velocity potential at the mean free surface between body surface and control surface is directly solved by the integration scheme. The wave exciting force is computed through the convolution integration with wave elevation, by introducing the impulse response function. Additionally, the nonlinear Froude-Krylov force and hydrostatic force, which is computed under the instantaneous incident wave free surface, are taken into account by the direct pressure integration scheme. The corresponding numerical computer code is developed and first used to compute the hydrodynamic coefficients of the hemisphere, as well as the time history of a ship with large flare; good agreement is obtained with the analytical solutions as well as the available numerical results. Then the hydrodynamic properties of a FPSO are studied. The hydrodynamic coefficients agree well with the results computed by the frequency method; the influence of the time interval and the truncated time is investigated in detail.

Control of Wave Energy Converters for Maximum Power Absorption with Time Domain Analysis

A discrete control of latching is used to increase the bandwidth of the efficiency of the Wave Energy Converters (WEC) in regular and irregular seas. When latching control applied to WEC it increases the amplitude of the motion as well as absorbed power. It is assumed that the exciting force is known in the close future and that body is hold in position during the latching time. A heaving vertical-cylinder as a point-absorber WEC is used for the numerical prediction of the different parameters. The absorbed maximum power from the sea is achieved with a three-dimensional panel method using Neumann-Kelvin approximation in which the exact initial-boundary-value problem is linearized about a uniform flow, and recast as an integral equation using the transient free-surface Green function. The calculated response amplitude operator, absorbed power, relative capture width, and efficiency of vertical-cylinder compared with analytical results.