Order Boundary Element Method Research Articles

Numerical investigation of triadic interactions during wave propagation over a submerged bar using a fully nonlinear numerical wave tank

This study models wave propagation over a submerged trapezoidal bar problem with a Higher Order Boundary Element Method (HOBEM) based on a two-dimensional fully nonlinear numerical wave tank (NWT). A Mixed Eulerian-Lagrangian technique is implemented, and the NWT is validated using the experimental results available from the existing literature. A numerical investigation is conducted on the individual wave harmonics as it propagates across the numerical wave tank over the submerged bar. The amplitude spectrum is obtained from the time histories of wave elevation across the tank at various locations over the trapezoidal bar and compared with the experimental results. The redistribution of the energy spectrum across the domain is recorded as the regular long wave propagates and undergoes deformations because of shoaling followed by de-shoaling. This phenomenon is investigated using a higher-order statistical analysis method viz. Bispectrum. The mechanics of the redistribution of energy among the harmonics is quantified through bispectrum analysis. Further, regular and irregular wave propagation over the submerged bar in the presence of the following and opposing currents is simulated. The effect of current is studied using the amplitude spectrum and the harmonic interaction for each case is compared using bispectrum analysis.

Two-dimensional diffraction and radiation problems of a floating body in varying bathymetry

In this study, a coupling method of the higher order boundary element method and the eigenfunction expansion method is applied to investigate the diffraction and radiation problems of a floating body over an uneven seabed under the assumption of linear small-amplitude wave theory. The wave exciting forces and hydrodynamic coefficients are calculated for a floating rectangular box. It is found that the curves of the wave exciting forces and hydrodynamic coefficients versus frequency become fluctuating as the horizontal length L and vertical height D of the sloping seabed gradually increase. The period of the fluctuations decreases with the increase of L, and the magnitude of the fluctuations increases with the increase of D. The mechanism of the fluctuation phenomenon is then examined, and found that it is associated with the wave reflection from the slope seabed. With increasing horizontal scale L and vertical scale D of the sloping seabed, the amplitude and phase of the reflection wave change, which leads to the combined action of the incident, scattering and reflection waves fluctuating with the wave frequency.

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 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.

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.

On numerical study of constrained coupled shape optimization problems based on a new shape derivative method

AbstractIn this article, we deal with a numerical method for the approximation of a class of coupled shape optimization problems, which consist in minimizing an appropriate general volume cost functional subjected to coupled boundary value problems by means of a Neumann boundary transmission condition. We show the existence of the shape derivative of the cost functional and express it by means of support functions, using a new formula of shape derivative on a family of convex domains. This allows us to avoid the disadvantages related to the classical shape derivative method using vectors field. Then the numerical discretization is performed using the dual reciprocity boundary element method in order to avert the remeshing task required for the finite element method. Finally, we give some numerical results, based on the gradient method, showing the efficiency of the proposed approach.

CVEM-BEM Coupling with Decoupled Orders for 2D Exterior Poisson Problems

For the solution of 2D exterior Dirichlet Poisson problems, we propose the coupling of a Curved Virtual Element Method (CVEM) with a Boundary Element Method (BEM), by using decoupled approximation orders. We provide optimal convergence error estimates, in the energy and in the weaker textit{L}^text {2}-norm, in which the CVEM and BEM contributions to the error are separated. This allows for taking advantage of the high order flexibility of the CVEM to retrieve an accurate discrete solution by using a low order BEM. The numerical results confirm the a priori estimates and show the effectiveness of the proposed approach.

Fully nonlinear dynamics of floating solar platform with twin hull by tubular floaters in ocean waves

Sustainability in harvesting solar power offshore depends on the survival of their platforms under environmental loads such as ocean waves. Therefore, a framework is established in this study to estimate the dynamics of floating solar platforms under ocean waves experimentally and numerically. Hence the platform is taken twin hull with double circular cylinders which exposed to a broad range of waves in the calibrated physical wave tank. On the other hand, the high order boundary element method and mixed Eulerian-Lagrangian approach are used in the numerical model to solve the fully nonlinear fluid field and the acceleration potential method is employed to approximate the response of the platform implicitly. To validate the framework, numerical and experimental results are compared for a single cylinder under the incident waves. Then, the comparison is extended for a twin hull platform. The effect of nonlinear free surface flow amongst the twin cylinders on the dynamics of the platform is studied parametrically. Afterwards, the performance of the present framework is assessed for a near breaking incident wave and an irregular wave. The stream function theory and flux of superposed wave components are used to emulate the high steep wave and the irregular wave respectively.

Wave-current interaction with three-dimensional bodies in a channel

A frequency-domain numerical model is developed to investigate the hydrodynamic properties of three-dimensional bodies of arbitrary geometry subjected to the action of waves and weak current in a channel based on higher order boundary element method. The problem is divided into a steady problem and an unsteady problem to be solved separately, where the unsteady potentials are given by two sets of integral equations based on linearizing the channel Green function and the velocity potential with the normalized current speed. The first-order wave force is obtained by the pressure integration over the body surface and the second-order drift force is evaluated by the far-field analysis. After examining the validation of the numerical model with some generalized relations derived in the present study, numerical studies for a Lewis-form ship and a four-column structure are performed to investigate the influence of channel walls on the hydrodynamic properties of bodies. Numerical results show that compared with the open-sea results, the side-wall effects become obvious when the oscillating frequencies are close to the transverse resonant frequencies of the channel, and compared with the cases without current, the current effects may enhance the interaction between the body and the reflected waves from the channel walls.

The singularity in the second-order solution of flexural-gravity waves and the influence of plate length on the second-order response of a finite elastic plate

Based on potential flow theory, the second-order interaction between nonlinear waves with infinite and finite elastic plates is examined by using the eigenfunction expansion method and Higher Order Boundary Element Method (HOBEM) coupling with the plate's Green function in the frequency domain. For infinite elastic plates, the singular phenomenon of second-order flexural-gravity waves is examined, and for finite elastic plates, the effects of plate length on the maximum second-order displacement and occurring frequency are studied. The maximum second-order displacement of the finite elastic plate increases with increasing plate length, and the frequency of the maximum displacement decreases and approaches the singular frequency of the corresponding infinite elastic plate with increasing plate length. In addition, the effects of the plate bending stiffness and water depth on the maximum second-order displacement are also investigated.

Inviscid flow passing a lifting body with a higher order boundary element method

A higher order boundary element method (HOBEM) is presented for inviscid flow passing a lifting body. In addition to the boundary integral equation used for the velocity potential, similar integral equation is derived and used for the tangential velocity on the body surface. Higher order elements are used to discretize the body surface, which ensures the continuity of slope at the element nodes. The velocity potential is also expanded using higher order shape function, in which the unknown coefficients involve the tangential velocity. The expansion then ensures the continuity of the velocity at element nodes and it also allows the Kutta condition to be imposed directly through the velocity. A particular shape function is also derived and used near the trailing edge to account for the continuity of the velocity and its sharp variations there. The unknown potential and tangential velocity are then found through solving their integral equations simultaneously. Through extensive comparison of the results for a Karman-Trefftz (KT) foil, it is shown that the present HOBEM is much more accurate than the conventional BEM, in particular for the velocity and local results near the trailing edge.

Comparison second order versus zero order boundary element method for tomography imaging

In this article a new version of the algorithm for Electrical Impedance Tomography is presented. By describing the problem with differential equations brought to integral equations the inverse problem for tomography has been defined. This algorithm can be used for many types of nonclassical tomography. This approach is particularly useful where it is not possible to formulate accurately the boundary conditions at the outer boundary of the region. The influence of the order of BEM effect on precision of imaging was shown.

Integration of multiple response signals into the probability of detection modelling in eddy current NDE of flaws

A new method of modelling the probability of detection (PoD) of a flaw parameter (flaw length) for the case of multiple correlated flaw response signals is proposed in this work. The PoD modelling is assisted by boundary element method in order to compute the required flaw features, the maximum change in the impedance of a coil, at two different frequencies. Experiments were performed using a LCR meter to verify the predictions of the boundary element model. The modelling was performed on a finite thickness aluminum-1050 plate with a narrow opening surface flaw. The features were computed for various flaw lengths and fit with a bi-variate Gaussian distribution, thus evading the problem of heteroscedasticity. On knowing the signal thresholds, the PoD of a flaw length can be computed from the Gaussian density. The obtained discrete PoD is then regressed by a generalized logistic function (GLF) in order to attain a continuous function for the PoD of the flaw length. The parameters of the GLF are estimated via non-linear least squares minimization by Levenberg-Marquardt algorithm.

The second-order interaction of monochromatic waves with three-dimensional bodies in a channel

A numerical model is developed to investigate the effect of the channel walls on the second-order hydrodynamic characteristics of three-dimensional bodies with arbitrary geometries in a wave channel based on the higher order boundary element method in the frequency domain. The second-order double-frequency radiation and diffraction potentials are obtained explicitly. In contrast to the second-order wave-body interaction problem in the open sea, the main difficulty stems from the boundary integral equation with the supplementary contribution of the up-wave and down-wave vertical surfaces of the channel in the second-order diffraction analysis. By exploiting the free-surface channel Green function and the known first-order scattering potential in eigenfunction expansion, the rapidly convergent integrals over the up-wave and down-wave vertical surfaces in the near fields can be evaluated analytically. Numerical results for the wave interaction with a bottom-mounted vertical circular cylinder, a fixed truncated vertical circular cylinder and a freely floating four-column structure are presented, and show that in comparison with the results in the open-sea condition, the second-order hydrodynamic properties of a body in a channel are affected more seriously than the linear ones due to the side-wall reflections.

Heaving wave energy converter-type attachments to a pontoon-type very large floating structure

This study proposes heaving wave energy converter (WEC) - type attachments to a pontoon-type very large floating structure (VLFS) for extracting wave energy and reducing the hydroelastic response of the VLFS. Each heaving WEC-type attachment consists of a linear Power Take-Off (PTO) system connected to VLFS at one end and to the seabed at the other end. The performance of heaving WEC-type attachments is investigated with respect to PTO damping coefficient and stiffness, locations and the number of PTO systems, VLFS flexural rigidity, VLFS module connection design, wave period and incident wave angle. For the fluid–structure interaction analysis, the Mindlin plate theory is adopted for modelling VLFS, while the linear wave theory is adopted for modelling fluid motions. The hybrid finite element – higher order boundary element method (FE-HOBEM) is employed to solve the fluid–structure interaction problem. It is found that, for long waves and large oblique waves, the present WEC-VLFS system outperforms the previously proposed WEC-VLFS systems in terms of the power produced.

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.

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.

Calculation of acoustic radiation modes by using spherical waves and generalized singular value decomposition.

Acoustic radiation modes (ARMs) have been widely used in noise control engineering owing to their convenient sound-power characteristics. However, the evaluation of ARMs for non-regular structures can be computationally intensive: it usually involves solving the boundary integral equation (BIE) by using the boundary element method in order to obtain the sound power radiation resistance matrix, followed by a singular value decomposition to obtain the radiation modes. The proposed procedure involves projecting spherical harmonics onto an enclosing surface, followed by the application of generalized singular value decomposition, with the result that the need to solve the BIE is eliminated, which potentially reduces the computational effort significantly.

Condition Monitoring a Coupled Differential Equation Using Order of Accuracy of Boundary Values in a Lid-Driven Cavity

The unsteady vorticity-transport equation and coupled stream function equation are discretised using Crank–Nicholson scheme in a finite difference mesh. The boundary conditions following Thom’s formula, Jenison’s formula of second and third order and computational boundary method are applied to solve the coupled equations. The geometry for the problem is a lid-driven cavity in which the coupled differential equations are solved. The order of accuracy of the boundary conditions affects largely the value of vorticity values at the centre of the primary vortex. At lower Reynolds number these values don’t have any impact; however at large Reynolds numbers those values are affected by large amount. Sometimes the computed values overshot the theoretical value of vorticity i.e. − 1.88596 with increase of Reynolds number. For this computation the grid meshes 129 × 129 and 257 × 257 are used. It is also observed that the computed vorticity value remain within the theoretical limit for the Jenison’s second order, third order and computational boundary element method.

Self-aligning behaviour of a passively yawing floating offshore wind turbine

ABSTRACTFloating offshore wind turbines are a promising concept for expanding offshore wind energy. In comparison with fix-founded offshore wind turbines, the overall costs are less dependent on water depths, which leads to a variety of potential locations and markets worldwide. Furthermore, floating platforms allow for new structural designs with the potential to save material and installation costs. In this paper, a self-aligning platform equipped with a 6 MW turbine is presented. The platform is moored on a single point and uses a turret buoy to be able to rotate freely around its anchor point. A downwind rotor and an airfoil-shaped tower induce self-aligning turning moments to passively follow changes of the wind direction. The first order boundary element method panMARE is used to simulate the motion behaviour considering aerodynamic, hydrodynamic and mooring loads. The self-aligning capability is demonstrated under partial turbine load for steady and dynamic conditions with waves and current.