Higher-order Boundary Element Research Articles

A Fully Nonlinear HOBEM with the Domain Decomposition Method for Simulation of Wave Propagation and Diffraction

A higher-order boundary element method (HOBEM) for simulating the fully nonlinear regular wave propagation and diffraction around a fixed vertical circular cylinder is investigated. The domain decomposition method with continuity conditions enforced on the interfaces between the adjacent sub-domains is implemented for reducing the computational cost. By adjusting the algorithm of iterative procedure on the interfaces, four types of coupling strategies are established, that is, Dirchlet/Dirchlet-Neumman/Neumman (D/D-N/N), Dirchlet-Neumman (D-N), Neumman-Dirchlet (N-D) and Mixed Dirchlet-Neumman/Neumman-Dirchlet (Mixed D-N/N-D). Numerical simulations indicate that the domain decomposition methods can provide accurate results compared with that of the single domain method. According to the comparisons of computational efficiency, the D/D-N/N coupling strategy is recommended for the wave propagation problem. As for the wave-body interaction problem, the Mixed D-N/N-D coupling strategy can obtain the highest computational efficiency.

Ringing of the roll motion of a two-dimensional barge in focused wave groups

The ringing of a floating barge’s roll motion under focused wave groups is studied using a fully nonlinear method in a numerical wave tank based on the potential flow theory. The higher order boundary element method (HOBEM) is used to solve the boundary value problem. The mixed Eulerian–Lagrangian (MEL) technique and the fourth-order Runge–Kutta time stepping scheme is used to simulate the interaction between the floating barge and focused wave groups. As the peak spectra frequency is set as one-third of the natural frequency of the barge’s roll motion, the influence of the peak wave amplitude of the incident focused wave groups on the roll response of the barge and wave loads is studied. The third-order components are found to be significant in both roll motion and moment as the incident peak wave amplitude increases. Relationship between the roll response of the barge and the peak spectrum frequency of the incident focused wave groups is also studied. The peak of the third-order roll response of the barge is found to decrease greatly as the triple peak spectral frequency deviates from the roll natural frequency.

Real-coded genetic algorithm optimization in reduction of wave drift forces on an array of truncated cylinders

To reduce wave drift forces acting on an array of truncated cylinders, the optimization of geometrical dimensions of the structure using a genetic algorithm is presented in this paper. Wave drift forces are always the major challenge for offshore floating structures facing to waves. Recently, a cloaking phenomenon has attracted the attention of researchers, which can reduce the scattered wave energy of offshore structures even to zero. In a diffraction problem of water waves, the term cloaking means that there are no scattered waves existing outside the structures. The configuration that a truncated inner cylinder surrounded by four truncated cylinders is considered as an effective way to realize this phenomenon. To accurately calculate the hydrodynamic characteristics of the truncated cylinder array, the Wave Interaction Theory combining with a higher-order boundary element method is adopted. Based on the above numerical methods, three parameters including the radius and the draft of the outer cylinders and the distance between the inner cylinder and outer cylinders are optimized to minimize the scattered wave energy and the wave drift force for a certain wavenumber. It is found that the effect of the draft of outer cylinders on the scattered wave energy is insignificant. Therefore, the optimization with two parameters is carried out additionally without consideration of the draft of outer cylinders. Moreover, the relationship between the wave drift force and the wave distribution around the inner cylinder is discussed.

Current effects on nonlinear wave slamming by an Oscillating Wave Surge Converter

A time-domain higher-order boundary element method (HOBEM) based on potential theory with fully nonlinear boundary is developed to simulate an Oscillating Wave Surge Converter (OWSC) rotating in waves in the presence of uniform currents. A domain decomposition method which neglects impact of jet flow to the main flow field is adopted. Auxiliary function is used to calculate the pressure distribution on body surface. The rotating coordinate system with origin fixed at the rotating center is established to avoid the water particle moving into or away from the body during the simulation. An iteration method is employed to assess the values of coupling wave characteristics affected by currents. Good agreement is found between present model and published fist-order boundary element method (FOBEM) model for the interaction of flap and current-free waves. Numerical simulations are then conducted in the presence of following current, opposing current and zero current, respectively. Several parametric studies, such as angle velocity and angular acceleration of the flap, initial phase, wave height and wave length, are performed to study the characteristics of the slamming problem with taking into account currents effects. The numerical results are presented through free surface profiles and pressure distributions.

Nonlinear Simulation of Focused Wave Group Action on a Truncated Surface-Piercing Structure

In this study, we numerically investigated the nonlinear focused wave group action on a truncated surface-piercing structure, and developed a two-dimensional fully nonlinear numerical tank using the higher-order boundary element method. We determined the amplitude of the wave components of the focused wave group based on the JONSWAP wave spectrum. We discuss the effects of the presence of a surface-piercing structure on the characteristics of the focused wave group and find that the location of the structure does not evidently change the focal location or focal time of the focused wave group. The largest amplitudes of the run-up and horizontal force on the structure occur when the front surface of the structure is at the focal location. The critical draught and breadth of the structure occur when the wave run-up reaches its maximum along the structure.

Numerical modelling of nonlinear extreme waves in presence of wind

A numerical wave flume with fully nonlinear free surface boundary conditions is adopted to investigate the temporal characteristics of extreme waves in the presence of wind at various speeds. Incident wave trains are numerically generated by a piston-type wave maker, and the wind-excited pressure is introduced into dynamic boundary conditions using a pressure distribution over steep crests, as defined by Jeffreys’ sheltering mechanism. A boundary value problem is solved by a higher-order boundary element method (HOBEM) and a mixed Eulerian-Lagrangian time marching scheme. The proposed model is validated through comparison with published experimental data from a focused wave group. The influence of wind on extreme wave properties, including maximum extreme wave crest, focal position shift, and spectrum evolution, is also studied. To consider the effects of the wind-driven currents on a wave evolution, the simulations assume a uniform current over varying water depth. The results show that wind causes weak increases in the extreme wave crest, and makes the nonlinear energy transfer non-reversible in the focusing and defocusing processes. The numerical results also provide a comparison to demonstrate the shifts at focal points, considering the combined effects of the winds and the wind-driven currents.

An investigation on HOBEM in evaluating ship wave of high speed displacement ship

A practical numerical tool is developed to evaluate ship waves of high speed displacement ships on the basis of potential flow theory, in which high order boundary element method (HOBEM) based on biquadratic shape functions is applied to solve the boundary value problem. Since the sinkage and trim of ship at high speeds are notable, influences of ship attitude on wave drag are investigated and three kinds of models are used to evaluate them. To make the numerical approach highly efficient, an incomplete LU factorization preconditioner is adopted and incorporated with the restarted generalized minimal residual method GMRES (m) to solve the boundary integral equation. A corresponding Fortran code is developed and applied to evaluate ship waves of the Wigley hull and 4a model, a transom stern ship. Computations are performed for both monohulls and catamarans over a wide range of Froude numbers (Fr = 0.10 - 1.00) . Numerical issues including mesh convergence and computational efficiency are investigated at first. Computed results of the wave drag, sinkage and trim show generally good agreement with experimental data. Reasonable wave patterns are obtained and physical phenomena that wake angle φmax, where the largest waves occur, would become narrow at high speeds is also captured by the present computations. Numerical results indicate the proposed method would be accurate and efficient to evaluate resistance for hull design of high speed displacement ship.

A 3D multi-domain high order boundary element method to evaluate time domain motions and added resistance of ship in waves

A newly developed three dimensional multi-domain high order boundary element method (MDHOBEM) is applied to solve time domain ship motions in waves for ships advancing with forward speed, as well as the corresponding added wave resistances. By the method, fluid domain is decomposed by a regular control surface into inner and outer parts, in which Rankine panel method and transient Green function method are adopted respectively. By the MDHOBEM, the problem of highly oscillatory nature of transient Green function near waterline can be avoided due to substitution of Rankine source in inner domain. Fewer discrete quantities than in Rankine panel method is needed as transient Green function is employed in outer domain, and the radiation condition can be analytically satisfied as well. Boundary value problem of velocity potential is solved in a translational and inertial frame without use of moving mesh, implying it is a highly efficient approach. A mixed form of ship wave linearized or double body flow linearized free surface condition, and the according m-terms are employed in inner domain to treat the steady flow effects more accurately. The yielding hybrid boundary integral equations are discretized with bi-quadratic isoparametric high order elements to minimize discretization error.A Fortran code is originally developed, and is used to simulate wave-body interactions of both the wall-sided and flared hulls including mathematical Wigley-3 hull, Series60, and more realistic hull forms of KCS and S175. Convergence of spatial and temporal discretization, and size of inner domain are detailedly investigated beforehand. Results indicate the present numerical scheme has quite good robustness, and the inner free surface can be reduced to a rather small region comparing to Rankine panel method. The computed hydrodynamic coefficients, added wave resistance and ship motions by MDHOBEM are compared with model test data and results of classic boundary element methods including Rankine panel method, transient Green function method, and translating and pulsating Green function method. It manifests the MDHOBEM has quite good accuracy and needs less computational cost.

Wave–current entry of an asymmetric wedge in 3DOF free motions

A time domain higher-order boundary element method (HOBEM) is developed to simulate the coupling process of a two-dimensional freely falling wedge entering obliquely waves in the presence of uniform currents. The wave–current-wedge slamming model is established in the stretched coordinate system, in which the velocity potential is decomposed as the potential induced by the current speed, the scattering potential and the incident potential at the initial stage. Some auxiliary functions are used to decouple the mutual dependence of structure motions with three degrees of freedom (3DOF) and fluid flow. The presented model is verified against the published first-order boundary element method (FOBEM) solutions for wedge entering calm water in free fall motion and wedge entering waves with the prescribed velocity in the absence of currents. Simulations and comparisons are then made for a wedge entering into waves in the following current, zero current and opposing current, respectively. Various parametric studies in the opposing current are performed for a wedge hitting wave crest vertically, obliquely and with initial rotational motion. Detailed results are provided through free surface elevation, pressure distribution, accelerations and velocities, and their physical implications are discussed.

π-BEM: A flexible parallel implementation for adaptive, geometry aware, and high order boundary element methods

Many physical phenomena can be modelled using boundary integral equations, and discretised using the boundary element method (BEM). Such models only require the discretisation of the boundary of the domain, making the setup of the simulation straightforward and lowering the number of degrees of freedom. However, while many parallel efficient libraries are available for the Finite Element Method (FEM), the implementation of scalable BEM solvers still poses many challenges. We present the open source framework π-BEM (where π stands for parallel): a novel boundary element method solver, combining distributed and shared memory paradigms to achieve high scalability. π-BEM exploits high performance libraries and graph partitioning tools to deliver a parallel solver employing automatic domain decomposition, high order elements, local refinement capabilities, and exact geometry-adaptivity (using CAD files). A preliminary fast multipole accelerator is included in the implementation. Every aspect of the library is modular and easily extendible by the community. We discuss the internal structure of the code, and present some examples to demonstrate the reliability and scalability of our implementation.

Motion response analysis of large-scale structures with small-scale cylinders under wave action

The influence of small-scale cylinders is always ignored in classical models for hydrodynamic calculations of truss spar platforms and other large-scale structures in published research. This study is focused on this influence on the motion response under the wave action so that the response can be forecast more accurately. A numerical model is proposed to calculate the wave-body interactions under the second-order frame by utilizing the time-domain higher-order boundary element method. The wave loads on small-scale cylinders are calculated by the Morison formula, in which the wave field disturbed by a large-scale structure is taken into account. Three-dimensional elastic rod theory is chosen to analyze the cable force to a platform, and the motion equation for the mooring system and hull are solved simultaneously using the Newmark-β method. A moored truss spar platform is modelled, and the numerical results demonstrate that the existence of small-scale cylinders may lead to a decrease of the motion response of the platform, while the natural frequency will move to a lower frequency. This phenomenon explains why small-scale cylinders, which can be modelled as truss spar play an important role in damping and change the added mass of the platform. Therefore, in order to get more accurate results for motion response, the wave loads of small-scale components cannot be ignored.

Fully nonlinear simulation of wave interaction with a cylindrical wave energy converter in a numerical wave tank

Convergence and accuracy of fully nonlinear time-domain simulation of wave interaction with a submerged cylindrical wave energy converter oscillating about an off-center axis which was conducted using a two-dimensional numerical wave tank are examined in this study. A high-order boundary element method is used to compute the fluid flow in the Eulerian frame. Fully nonlinear boundary conditions are applied to achieve the unique solution of the boundary value problem. Hence, kinematics of the fluid particles on the exact free surface boundary is calculated using the material node approach in the Lagrangian frame. Runge-Kutta time integration is employed to update the boundary values and the computational control volume. To implement the body condition, the time derivative of the velocity potential is computed implicitly using the acceleration potential. An artificial wave maker and two artificial sponge layers are provided to retain the open water condition. The power absorption efficiency of the submerged Bristol cylinder is compared with experimental data and numerical solutions for different incident waves to verify the present numerical model and the accuracy of the solution is also investigated. The convergence of the present fully nonlinear model in the simulation of the freely rotating cylinder about the off-centered axis as a wave energy converter is evaluated for different submergence depths, off-center distances, and a robust nonlinear incident wave.

Comparative study on steady wave-making problem using viscous and potential-flow methods

Accurate predictions of free surface and pressure are devoted to the design and optimization of ship hull. The present study focuses on a comparison of numerical results from different methods on steady wave-making problem of a standard Wigley hull, especially for pressure distribution on the hull. Two numerical approaches are employed: a higher-order boundary element method, which uses Rankine panel method based on potential flow theory, and a computational fluid dynamics model based on the software FINE/Marine. The computed results of the wave profile along the hull, wave pattern around the hull, pressure distribution on the hull and wave-making resistance are validated by comparing them with experimental measurements and show favorable agreements. Through above comparisons, it is confirmed that the inclusion of nonlinear and viscosity effects improves the accuracy of numerical results for predicting free surface and pressure distribution in spite of heavy computation.

Nonlinear analysis for ship-generated waves interaction with mooring line/riser systems

A time-domain hybrid finite element-boundary element (FE-BE) method is developed to investigate nonlinear interaction between ship waves and slender structures. The ship waves are modeled in the context of the three-dimensional potential theory including ship steady state motion, and the velocity potential are computed using a higher-order boundary element method (HOBEM). A special global coordinate-based FEM rod model, when using catenary solution as an initial static profile, is employed in the accurate analysis of slender structures. The internal flow is incorporated in the numerical model by adding a new term to the effective tension formulation while both the seabed support and friction effect are performed by using a nonlinear quadratic spring and a Coulomb friction model. The numerical model is verified with the published experimental and numerical results for the ship waves generation and dynamic responses of mooring line/riser. Numerical simulations are executed to determine the dynamic behavior of two example slender structures such as multi-component mooring line and deepwater steel catenary riser under ship waves. Then the coupled analysis program is applied for a truss Spar with its mooring lines subjected to ship waves, wind-generated waves and their combined action, respectively. The dependence of the top node tensions of the slender structures on ship path, ship speed, ship distance, internal flow and seabed friction, is also examined.

Numerical investigation of water entry of a wedge into waves with current effects using a fully nonlinear HOBEM

This paper concerns the hydrodynamic performance of a two-dimensional asymmetric wedge entering waves obliquely with gravity effect in the presence of a uniform current. A time-domain higher-order boundary element method (HOBEM) based on potential theory with fully nonlinear boundary conditions is developed to solve this problem. A stretched coordinate system is adopted in the spatial domain to avoid a large number of elements discretized in fixed computational domain at the initial stage. During mesh regriding and interpolation, a rotation scheme of the stretched coordinate system is employed to meet the requirement of continuity of flux at the intersection points. An auxiliary function is introduced to calculate pressure distribution on body surface. The present model is validated against the published numerical results for vertical wave entry in the absence of currents. Numerical calculations are conducted to analyze the dependence of the free surface elevation and the pressure distribution on the physical parameters, such as current velocity, incident wave amplitude, entry location, heel angle and oblique motion. It is found that the variation of current velocity has obvious effect on both the horizontal velocity of fluid and the wave steepness, which play a major role on free surface and pressure.

A quasi-cloaking phenomenon to reduce the wave drift force on an array of adjacent floating bodies

The mooring of offshore floating structures, such as offshore platforms, in large waves against drift forces and rotational moments is a challenging problem in offshore engineering. To accurately investigate such problems, called positioning problems, the time-averaged steady forces of the second order known as the wave drift forces must be taken into account. Fortunately, a cloaking phenomenon occurs under certain conditions and dramatically reduces the wave drift force acting on such a floating body, as previously reported by several researchers. In the diffraction problem of water waves, cloaking refers to the condition where there is no scattering in the form of radial outgoing waves. The reduction of wave drift force on a truncated cylinder with the occurrence of cloaking phenomenon has been numerically and experimentally confirmed. In this paper, the arrangement of several small circular cylinders at regular intervals in a circle concentric with a fixed floating body is considered as an effective means of reducing the wave drift force. Using a combination of a higher-order boundary element method (HOBEM) and wave interaction theory, the influences of the geometric parameters of the outer surrounding cylinders on the wave drift force and the total scattered-wave energy are systematically investigated and discussed. A quasi-cloaking phenomenon is first found and reported in the present study, which is beneficial and flexible for application in practical engineering. More than one quasi-cloaking trigger (where a trigger is an occurrence condition) can be found simply by varying the distance between the inner and outer floating bodies.

Nonlinear and viscous effects on the hydrodynamic performance of a fixed OWC wave energy converter

The hydrodynamic performance of a fixed Oscillating Water Column (OWC) device is experimentally and numerically investigated. Based on the time-domain higher-order boundary element method (HOBEM), by introducing an artificial viscosity term in the dynamic free surface boundary condition, a fully nonlinear numerical wave model is used to simulate the hydrodynamic performance of an OWC device. A set of comprehensive experiments for regular waves is carried out to validate the numerical results as well as to investigate the nonlinear effects on the hydrodynamic performance of OWC. The mechanism of the nonlinear phenomenon is investigated based on the analysis of the experimental and numerical results. The influence of the wave nonlinearity and the viscosity on the hydrodynamic efficiency is quantified by comparing the linear and nonlinear numerical results. It was found that the hydrodynamic efficiency increases with the nonlinearity and viscosity when the incident wave amplitude is small. When the incident wave amplitude is large, the hydrodynamic efficiency is reduced by the weakened transmission of the second-order harmonic wave component due to the strong wave nonlinearity. However, when the wave amplitude is between these two regimes, the wave is weakly nonlinear, the efficiency decreases with the wave amplitude due to the combined effect of the nonlinearity and viscosity.

Numerical Simulation of a Dual-Chamber Oscillating Water Column Wave Energy Converter

The performance of a dual-chamber Oscillating Water Column (OWC) Wave Energy Converter (WEC) is considered in the present study. The device has two sub-chambers with a shared orifice. A two-dimensional (2D) fully nonlinear numerical wave flume based on the potential-flow theory and the time-domain higher-order boundary element method (HOBEM) is applied for the simulation. The incident waves are generated by using the immerged sources and the air-fluid coupling influence is considered with a simplified pneumatic model. In the present study, the variation of the surface elevation and the water column volume in the two sub-chambers are investigated. The effects of the chamber geometry (i.e., the draft and breadth of two chambers) on the surface elevation and the air pressure in the chamber are investigated, respectively. It is demonstrated that the surface elevations in the two sub-chambers are strongly dependent on the wave conditions. The larger the wavelength, the more synchronous motion of the two water columns in the two sub-chambers, thus, the lager the variation of the water column volume.

Numerical and experimental investigation of hydrodynamic performance of a cylindrical dual pontoon-net floating breakwater

The use of a simple, inexpensive, and effective type of floating breakwater is increasingly becoming a necessity in shoreline and marine structure protection. This study concerns the hydrodynamic behavior of a dual pontoon floating breakwater (DPFB) when attached to one or more rows of plane net by using physical and numerical models. A two-dimensional (2D) fully nonlinear numerical wave tank (NWT) based on a time higher-order boundary element method (HOBEM) and mixed Eulerian-Lagrangian (MEL) approach is applied to obtain numerical solutions. In the model, Darcy's law is used to represent the porous media of the fluid-net interaction, and some auxiliary functions are introduced, instead of an iterative process using the acceleration potential method. Mesh regridding and interpolation combined with a double collocation node technique are implemented to tackle the mismatch between the meshes on the free surface and the wet body surface. In addition, the numerical model is verified with a series of corresponding experimental tests. Numerical solutions and measurement tests are executed to systematically examine the dependence of the reflection coefficient, transmission coefficient and motion responses on the design parameters, such as net number, net porosity, net height, wavelength and wave amplitude. It is found that the new floating breakwater exhibits a better performance with the optimal design parameters as compared with traditional DPFB, especially for long period and large amplitude waves.

Extreme wave run-up and pressure on a vertical seawall

The performance of coastal vertical seawalls in extreme weather events is studied numerically, aiming to provide guidance in designing and reassessing coastal structures with vertical wall. The extreme wave run-up and the pressure on the vertical seawall are investigated extensively. A time-domain higher-order boundary element method (HOBEM) is coupled with a mixed Eulerian-Lagrangian technique as a time marching technique. Focused wave groups are generated by a piston wave-maker in the numerical wave tank using a wave focusing technique for accurately reproducing extreme sea states. An acceleration-potential scheme is used to calculate the transient wave loads. Comparisons with experimental data show that the extended numerical model is able to accurately predict extreme wave run-ups and pressures on a vertical seawall. The effects of the wave spectrum bandwidth, the wall position and the wave nonlinearity on the wave run-up and the maximum wave load on the vertical seawall are investigated by doing parametric studies.