Higher-order Boundary Element Method Research Articles

Numerical Investigation on a High-Speed Transom Stern Ship Advancing in Shallow Water

A high-speed advancing ship will cause significant squats in shallow water, which could increase the risk of grounding. To this end, a program based on the Rankine higher-order boundary element method (HOBEM) is developed to investigate a high-speed displacement ship with a transom stern moving in shallow water. The nonlinear free surface condition is satisfied by adopting an iterative algorithm on the real free surface. The transom condition is considered by implementing a modified transom condition. Computations of wave-making resistance, sinkage and trim in deep water are first performed, and satisfactory agreement is achieved by comparing with the experimental results; the simulations are then extended to the shallow water case. It indicates that the present method can provide a suitable balance of practicability and robustness, which can be considered as an efficient tool for the guidance in ship design stage.

A boundary element analysis of quasi-potential inviscid incompressible flow in multiply connected airfoil wing

AbstractThe quasi-potential flow in a doubly linked airfoil wing with a sharp trailing edge can be solved using a high-order boundary-element method (BEM). A bicubic Hermite spline interpolation is used to express the unknown, with node derivatives given by finite difference method. The revolutionary traits apply to bleeding-edge settings; specifically, challenges relating to the three-dimensional problem of multiply connected domains are investigated. Numerical validation examples were utilized to assess the implemented BEM approach.

A Comparative Study on the Structural Response of Multi-Linked Floating Offshore Structure between Digital Model and Physical Model Test for Digital Twin Implementation

A digital twin is a virtual model of a real-world structure (such as a device or equipment) which supports various problems or operations that occur throughout the life cycle of the structure through linkage with the actual structure. Digital twins have limitations as a general simulation method because the characteristic changes (motion, stress, vibration, etc.) that occur in the actual structure must be acquired through installed sensors. Additionally, it takes a huge computing cost to output changes in the structure’s characteristics in real time. In particular, in the case of ships and offshore structures, simulation requires a lot of time and resources due to the size of the analysis model and environmental conditions where the wave load acts irregularly, so the application of a different simulation methodology from existing ones is required. The order reduction method, which accurately represents the system’s characteristics and expresses them in a smaller model, can significantly reduce analysis time and is an effective option. In this study, to analyze the applicability of the order reduction method to the development of digital twins for offshore structures, the structural responses of a multi-connected floating offshore structure were estimated by applying the order reduction method based on distortion base mode. The order reduction method based on the distortion base mode predicts the responses by constructing an order-reduced conversion matrix consisting of the selected distortion base mode, based on the mode vector’s orthogonality and autocorrelation coefficients. The predicted structural responses with the reduced order model (ROM) were compared with numerical analysis results derived using the higher order boundary element method and finite element method with in-house code owned by the Korea Research Institute of Ship & Ocean Engineering and measured responses with a model test. When compared with the numerical analysis results, the structural responses were predicted with high accuracy in the wave direction and wave frequency band of the selected distortion base mode, but there are differences due to changed characteristics of the structure when compared with the results of the model test. In addition, differences were also seen in reduced order model evaluation with different sensor locations, and it was confirmed that the more similar the extracted distortion base modes of input sensor location set is to the distortion base modes of predicted location set, the higher accuracy is in predicting the structural responses. As a result, the performance of the reduced order model is determined by the distortion base mode selection method, the locations of the sensor, and the prediction for the structural response.

Wave motion in a narrow gap coupling with the roll motion of a vessel next to an offshore terminal

Concerning the scenario of a vessel operating alongside an offshore terminal, the vessel being the floating body and the offshore terminal bottom-mounted. The closeness of the two bodies leads to the formation of a long narrow gap, within which the large amplification of the water-surface elevation could occur. The coupling effect between the roll motion of the vessel and the water-surface motion within the gap is primarily concerned. Under the assumption that the quay wall of the terminal is infinitely long and fully reflective, the imaging principle and a modified Green's function have been developed to investigate such an effect. Numerical models are developed based on the two proposed approaches, respectively, in conjunction with a higher-order boundary element method. For a small gap width, the p = 1 resonant mode of water-surface motion in the gap can apparently enhance the roll motion of the vessel. The radiation wave owing to the enhanced roll motion can in turn disturb the water-surface motion, increasing the frequency of the dominant peak while decreasing its amplitude. As the gap width increases, the water-surface elevation around the resonant frequency of roll motion gets more and more noticeable, gradually becoming the dominant peak. The effect of the mooring stiffness is also discussed, revealing the existence of a ꞌnon-resonance regionꞌ where no resonance occurs in this specific frequency region. It is also noted that the resonant water-surface motion of p = 1 mode can cause a remarkable amplification of the negative horizontal mean wave drift force, pushing the vessel away from the terminal.

Wave energy capturing of an offshore stationary platform equipped with three oscillating wave column devices

This paper deals with the hydrodynamic properties of an offshore stationary multi-functional platform consisting of triple oscillating water column (OWC) wave energy converters (WECs). The three OWCs are arranged on the vertices of an equilateral triangle, and one of them is forced to face the incident wave, and is labeled as the seaward OWC. Based on the potential flow theory, a second-order time-domain higher order boundary element method (HOBEM) model was established to investigate the wave energy capturing of this triple-OWCs-platform. It is found that there exists a quarter-cycle phase lag between the surface elevation and the air pressure measured in the chamber of the OWCs. In addition, the wave energy capturing capability of the seaward OWC is found to be improved significantly when the incident wave approaches the platform with an angle of 0 and π/3. Its maximum efficiency can reach up to 97.4% with the optimized device spacing. The averaged efficiency of the three OWCs at the resonant frequency is found to be enhanced when the device spacing is four times larger than the diameter of the OWCs.

Experimental and numerical investigation on hydrodynamic performance of a 3D land-fixed OWC wave energy converter

The hydrodynamic performance of a land-fixed oscillating water column (OWC) wave energy device was thoroughly investigated under various wave conditions and geometric parameters through both experimental and numerical studies in a three-dimensional (3D) wave tank. The study involved the measurement and comparison of water surface elevation, air pressure in the OWC chamber, and hydrodynamic efficiency with the results obtained from the 3D Higher-Order Boundary Element Method (HOBEM). The influences of 3D effects, incident wave amplitude, front wall draught, and incident wave angle on the hydrodynamic performance of the OWC device were analyzed. The results demonstrate that, under actual sea conditions, the 3D effects significantly affect the hydrodynamic efficiency of the OWC, making it exceed unity near the resonance frequency. On the other hand, the incident wave angle was found to have minimal effect on the resonant frequency and varied trend of efficiency. However, the hydrodynamic efficiency decreases with increasing incident wave angle. These results highlight the importance of the consideration of the 3D effects and incident wave angles in optimizing the design and performance of fixed OWC wave energy devices.

Numerical investigation on the hydrodynamic performance of a land-based OWC system with multi-chamber modules

Modular design of wave energy converters (WECs) can offer a promising scheme to improve the economic competitiveness of wave energy by reducing the levelized cost of energy. In this study, the hydrodynamic performance of a land-based oscillating water column (OWC) system with multi-chamber modules is evaluated to optimize the layout and design of the system, the OWC structure is extended to five-chamber modules. A two-dimensional wave flume is developed using the high-order boundary element method to simulate the interaction of the incident waves with the OWC device based on the fully non-linear potential flow theory. The proposed numerical model is validated by published experimental data. It is concluded that the two-chamber installation provides superior hydrodynamic performance compared with the single-chamber OWC. The five-chamber OWC system is more suitable for energy capture in low-frequency wave conditions, where the first and last chambers are dominant in energy extraction. From the perspective of energy conversion efficiency and economy, the two- and three-chamber systems are most suitable for targeting low and high frequency waves in the tested conditions, respectively. Additionally, the resonant frequency of the single-chamber module plays a decisive role in the energy capture of the OWC system, and a chamber with a low resonant frequency and a small orifice opening ratio should be selected as the module unit. This paper provides valuable guidance for the modular application of OWCs and the arrangement of multiple chambers.

Analysis of FPSO Motion Response under Different Wave Spectra

A variety of floating structures at sea play a vital role in the exploitation and utilization of marine resources. The study about interactions between waves and structures is necessary for the impact of the harsh marine environment on the motion and service life of structures. Currently, most studies about the seakeeping of structures are based on simplified regular waves. Because the regular waves do not truly restore the actual wave conditions at sea, the simulation of irregular waves has great practical importance to the study of interactions between waves and structures. Based on the potential flow theory and high-order boundary element method (HOBEM), a Fortran code is developed in this paper and named as SWBI (Solver for Wave–Body Interactions). This program consists of the following two parts: a time–domain numerical model about interactions between waves and 3D structures is based on weakly nonlinear method, and a numerical model about simulation of the nonlinear regular waves, the long-crested irregular waves, and the short-crested irregular waves. This Fortran code is used to simulate the motion of Floating Production Storage and Offloading (FPSO) under three different ocean wave spectra (including ITTC two-parameters spectrum, JONSWAP spectrum and the most likely spectral form of Ochi-Hubble) and found that: To a certain extent, the difference in the motion of FPSO under different wave spectra have a connection with different type of wave, sea conditions and incident angle. The difference in roll of FPSO is quite significant in short-crested irregular waves. The range of FPOS’s roll under the JONSWAP spectrum is the largest when the incident angle is 30°, and range of FPOS’s roll under the most likely spectral form of Ochi-Hubble is the largest when the incident angle is 90°.

Fully coupled aero-hydrodynamic modelling of floating offshore wind turbines in nonlinear waves using a direct time-domain approach

In standard operating conditions, a floating offshore wind turbine (FOWT) suffers from the combined action of nonlinear waves and wind. It is important to explore the effect of higher-order hydrodynamic loads induced by the nonlinear waves on its motion response. In this work, a fully coupled aero-hydrodynamic analysis method for FOWT is developed. The hydrodynamic part is based on the nonlinear potential flow theory and the perturbation approach, using a higher-order boundary element method in the time domain. Blade element momentum theory is applied to evaluate the aerodynamic load on the turbine rotor. The mooring system is modelled by the catenary theory. The developed method is verified against the published data of the NREL's 5-MW baseline wind turbine supported by the DeepCwind semi-submersible platform. The verifications are conducted stepwise in the following ways, including hydrodynamic wave excitation load, free-decay response, the load-displacement relationship of the mooring line, the steady-state response of the wind turbine, and fully coupled aero-hydrodynamic interaction. It is found that the nonlinear effect mainly influences the surge motion. The coupled configuration has little effect on the linear wave force and platform displacement but notably impacts the nonlinear wave force and displacement. The FOWT motion is found to be critical in changing the wave field, the maximum wave elevation position around the platform, and the nonlinear wave force acting on the floating platform. The results illustrate that the traditional indirect time-domain method using the Quadratic Transfer Function (QTF) based on assumed response cannot appropriately evaluate the nonlinear wave force of a FOWT in coupled motions. The results also reveal that nonlinear wave hydrodynamics is necessary to understand the motion response of a fully coupled FOWT.

The nonlinear wave interaction with a two-dimensional large-scale floating elastic plate

A two-dimensional fully nonlinear numerical wave tank is developed based on the higher order boundary element method and modal expansion to simulate the nonlinear interaction between waves and a finite horizontal floating elastic plate with draft. The large second harmonic displacement of a large-scale elastic plate near a second-order particular frequency is successfully calculated by this model. By the comparison with the results in a frequency model that involve perturbation expansion method, it is found that the large values of the second-order waves in a finite elastic plate near the particular frequency are not introduced by perturbation expansion. It is different from the case in infinite plate. The effect of the third-order particular triple frequency on the hydroelastic problem is small compared to the effect of the second-order particular frequency.

Numerical and Experimental Investigations on Non-Linear Wave Action on Offshore Wind Turbine Monopile Foundation

Monopiles are commonly utilized in offshore wind farms but are prone to non-linear wave loads and run-ups, significantly affecting their engineering design. Therefore, it is crucial to pursue a complete understanding of the non-linear wave action on monopile foundations. Both numerical and experimental investigations on the non-linear wave loads and run-ups on an offshore wind turbine monopile foundation are performed in this paper. The experiment is carried out at a scale of 1/30 in a wave flume at the State Key Laboratory of Coastal and Offshore Engineering, Dalian University of Technology, in which the wave loads and run-ups along the monopile are measured. Based on the second-order potential flow model and time-domain higher-order boundary element method (HOBEM), the related numerical tests are conducted to study the non-linear effects further. It is found that the present non-linear potential theory is sufficient for the simulation of wave force and run-ups on the monopile in the range of wave slope kA < 0.15 before wave breaking. “W” type distribution of wave run-up along the monopile is found, in which the peak value occurs at the frontward side (i.e., θ =180°) and is the maximum due to full reflection; the two symmetrical minimum amplitudes lie in the zones of (45° ≤ θ ≤ 90°) and (270° ≤ θ ≤ 315°), whose positions shift downward with the increase of wave non-linearity. Energy transfer among the fundamental wave component and higher-order components is also found, which is most apparent on the backward side. Besides, the transverse resonance occurs in the wave flume due to the wavelength being near the flume width, which induces the wave run-up at the backward position larger than that at the frontward position.

Time-domain simulation of sum-frequency hydrodynamic performance of a TLP-type floating wind turbine in multidirectional seas

Previous studies (Ren et al., 2022) by the authors have confirmed that the second-order diffracted wave loads are very sensitive to the wave directional spreading of the so-called storm-swell mixed seas where the storm-driven local seas combined with long-period swells propagating from different angles. In this paper, particular attention is paid to estimate the sum-frequency hydrodynamic forces and motion responses of a TLP-type floating wind turbine (FWT) in such complex sea states. A time-domain potential flow model is developed for the second-order interaction between the multidirectional irregular waves and moored structures. The boundary value problems for the scattered potential are solved by a higher-order boundary element method (HOBEM). The fourth-order Adams–Bashforth–Moulton method is used to update the wave surface and the Newmark method combined with the Newton–Raphson iteration scheme is adopted to solve the body motion equations.For validation, a freely floating hemisphere in unidirectional bichromatic waves is investigated and good agreement is found by comparing with Kim and Yue's frequency-domain results. Numerical tests are conducted on the TLP-type FWT with different combinations of incident angles and spreading factors of the storm-swell mixed seas. The effects of the wave headings and directional spreading on the sum-frequency hydrodynamic response are discussed. It is noted that when the swells approach the storm seas from large relative angles, the sum-frequency vertical response of the FWT can be up to seven times greater than that in the collinear mixed seas. In addition, the present model can extend the rotor-floater-tether coupled dynamic simulator of FWT to more realistic seas such as storm-swell mixed seas.

Numerical and experimental modeling of wave-induced forces on submarine pipeline buried in the soil of different engineering properties

In the present paper, the wave-induced forces on the submarine pipeline buried in porous soil having different properties are studied. The mathematical problem is solved using the higher-order boundary element method, and the numerical results are validated with experimental results obtained through rigorous model tests. Four different types of Kuwaiti marine soil: Al-Khiran, Al-Koot, Sabiya, and Shuaiba are considered in the present study. The effect of burial depth of the submarine pipeline in the soil seabed, incident wave period and wave height, and hydraulic conductivity of the soils on the wave forces acting on the submarine pipeline is analyzed in a detailed manner. The study reveals that for moderate burial depth of the pipeline in the soil bed, the wave force acting on the submarine pipeline can be reduced up to 50% for soil having less hydraulic conductivity. The measured wave forces and numerical predictions are matching well for a wide range of wave, soil and structural parameters.

Numerical investigations on nonlinear motions of ships sailing in large-amplitude waves based on hybrid HOBEM

This paper presents a nonlinear numerical approach based on hybrid higher-order boundary element method (HOBEM) to investigate the wave-induced motion responses of ships with forward speed. The radiation and diffraction forces in time domain are determined by convolution of the impulsive response functions computed from frequency-domain hydrodynamic data of the hybrid method using a combination of Rankine source and translating-pulsating Green's function. In its numerical implementation, steady flow effects are included in the inner domain encompassing the ship, 9-node bi-quadratic curvilinear elements are employed to discretize the boundaries and analytical quadrature formulas are derived to calculate the influence coefficients related to the Green's function over the control surface so as to improve the accuracy, efficiency and stability. Meanwhile, the nonlinear incident wave force and hydrostatic restoring force are calculated over the instantaneous wetted surface of the ship hull during the motion process to capture the nonlinear properties. Based on the method, a numerical program is developed to predict and analyze the motion response of a modified Wigley hull and a non-wall-sided S175 containership advancing in waves with different amplitudes and heading angles. By comparing the present computed results with experimental data and other numerical solutions, it is found that the nonlinear calculations can produce more accurate motion responses in large-amplitude waves with strong nonlinearity. And the present nonlinear method based on hybrid HOBEM is of higher accuracy and better stability than traditional nonlinear methods based on three-dimensional pulsating source (3DP) and translating-pulsating source (3DTP), especially for S175 with large flare near the waterline.

Numerical Analysis for Motion Response of Modular Floating Island in Waves

In recent years, modular-type floating islands have been considered as a promising option for future ocean space utilization. A modular floating island consists of a number of standardized pontoon-type modules and connectors between them. In this study, the motion responses of a modular floating island in waves was investigated based on frequency-domain numerical analysis. The numerical method is based on the potential flow theory and adopts a higher-order boundary element method with Green’s function. First, motion RAOs were directly compared with the model test data by reference to validate the present numerical method. Then, numerical investigations were conducted to analyze the motion characteristics of the floating island by considering various modules shapes and arrangements. It was found that motion responses were reduced in a single central module compared to when divided central modules were used. Finally, the effect of modular arrangement on the motion responses in irregular waves was discussed. It was confirmed that multiple-layer outer modules are more effective in calming the central module than using single-layer outer modules, except under very long period conditions.

Featured Cover

The cover image is based on the Research Article A novel high-order collocation indirect boundary element method based on the Leis formulation for three-dimensional high frequency exterior acoustic problems by Chao Zhang et al., https://doi.org/10.1002/nme.7138.

Numerical and experimental studies on a self-protected energy concentrator for water waves

Wave concentrators have significant value for ocean energy absorbers to reduce power generation costs. Besides, the protection problem of the concentrator itself should also be carefully considered. In this study, a self-protected energy concentrator (SPEC) for water waves consisting of several truncated cylinders arranged in a concentric circle is proposed by adopting the cloaking technology based on the idea of manipulating scattered waves. To design the SPEC, an optimization program built by a combination of the higher-order boundary element method, the wave interaction theory, and the real-coded genetic algorithm is utilized. To address the manipulation mechanism of the scattered waves as well as to investigate the performance of the SPEC against various wave steepness and a spectrum of waves, the scattered waves and the total waves both inside and outside the concentrator are numerical and experimentally analyzed. High energy density inside the concentrator and the self-protection of the concentrator are validated. It is found that the large wave steepness can reduce the efficiency of concentration and self-protection due to the wave breaking. The SPEC has a high performance at a moderate bandwidth. Besides, the wave frequencies perform a more significant effect on self-protection than the concentration characteristic.

Power performance and motion response of a floating wind platform and multiple heaving wave energy converters hybrid system

Integrating wave energy converters (WECs) on an offshore wind platform provides an economic and robust solution for co-located wind and wave power exploitation. The power performance of the hybrid system depends on the hydrodynamic interactions between the platform and WECs, but most of these effects remain unclear. In this study, through a hybrid system consisting of a semi-submersible platform and heaving point absorber WECs, this matter is addressed based on the higher-order boundary element method and multi-body constrained dynamics. Results show that a new prominent power peak appears while the platform and WECs are in a so-called synchronized mode regardless of the layout of the WECs, wherein the power of a single WEC can be increased by up to 41.4% and the total power by up to 26.7% and the heave motion of the platform increases. Out of the vicinity of the synchronized mode frequency, the WECs have a small impact on the surge motion and pitch motion of the platform, whereas reducing its resonant heave motion at the natural frequency. This work reveals useful insights into the dynamic and power performance of a wind-wave hybrid system, providing detailed configurations and potential guidance for the practical design and application.

A numerical approach for hydrodynamic performance evaluation of multi-degree-of-freedom floating oscillating water column (OWC) devices

A numerical approach has been developed in this study for hydrodynamic analysis of complex-shaped oscillating water column (OWC) converters. Physically, in the circumstance when the device is free to move or moored in waves, the airflow movement inside the chamber is coupled tightly with the changing of the internal fluid surface and the hydrodynamic response of the device. By taking these coupling effects into account, appropriate boundary integral equations are formulated with supplemental theoretical relations. The boundary value problem is then solved using a higher-order boundary element method (HOBEM). After obtaining the wave force and dynamic air force, the coupled motions and hydrodynamic efficiency are evaluated by integration. In particular, unlike conventional methods limited to fixed OWCs, the proposed method is proved applicable to those floating with coupled rigid-body motions. An optimal turbine parameter is hence mathematically derived to maximise the wave power absorption. Besides the linear quantities, nonlinear wave drift loads are evaluated via a newly derived formulation that accounts for the far-field contribution and the oscillating air pressure over the internal fluid surface. Based on the above methodology and the resulting tool, numerical studies are carried out for three cylindrical OWC scenarios that are floating in waves, attached by a reflector and connected to a submerged caisson, respectively. Different modes of body motion and typical free-surface oscillation modes in the chamber are analysed and discussed. It is found that, for an OWC with an attached arc-shaped reflector or connected to a submerged caisson, its wave power absorption can be characterised by a series of apparent peaks which are associated with the resonant internal fluid and in close proximity to the resonance frequencies of body motions involving surge, heave or pitch. Numerical results also indicate the benefits of floating OWCs from the coupled rigid-body motion modes and OWC’s resonance, expanding the frequency range of efficient conversion and improving the device’s adaptability to variable oceanic environments.

Hydrodynamic analysis of resonant waves within the gap between a vessel and a vertical quay wall

Concerning the side-by-side offloading operations at gravity-based terminals, the wave interaction with a vessel alongside a long vertical wall in both normal and oblique seas was investigated. The imaging principle was applied to convert the original to an equivalent wave-diffraction problem by two symmetrical vessels in open seas exposed to bi-directional incident waves. The boundary integral equation method, in conjunction with a higher-order boundary element method (HOBEM), was used to solve the equivalent problem. Besides the linear free-surface elevation and wave loads, the second-order mean wave drift loads on the vessel was calculated based on either the direct pressure integration or using a semi-cylindrical control surface. Moreover, the solution was also developed for the oblique wave interaction with a two-dimensional barge-wall system using the eigenfunction expansion matching method. Systemic numerical studies were then conducted for the cases of a rectangular barge and a Wigley hull in close proximity to a long wall, respectively. Under oblique incidence, both the odd and even modes of fluid motion within the gap can be induced. In the case of a rectangular barge, the linear wave force can be obviously amplified at the location of the (1, 0) resonant mode, while the linear wave moment at the (2, 0) mode. In addition, the negative mean force along the transverse direction, which tends to push the barge away from the wall, can be apparently enhanced at both the odd and even resonant modes. Besides, in oblique seas, the (1, 0) resonant mode can give rise to the significant amplification of the positive mean yaw moment around the centre of the cross-sectional area of the barge, which tends to pull the stern of the barge (that faces incident waves) closer to the wall, while pushing the bow away from the wall. Such amplification gets less noticeable for higher modes. Numerical results also indicate that the hull geometry imposes a significant impact on the resonant waves in the gap. In the case of a Wigley hull, the amplification of the free-surface response is much less apparent when compared with a rectangular barge.