Nonlinear Boundary Element Method Research Articles

Investigating the entry of symmetric/asymmetric elastic wedge into water

Understanding the impact of fluid–structure interaction during the entry of a marine vessel into water is crucial for its design. While numerous models have been proposed to investigate the penetration of symmetric elastic wedges into water, in this study, we propose a numerical model that combines the fully nonlinear boundary element method with a mode superposition method to analyze the penetration of symmetric and asymmetric elastic wedges into water. We derive the boundary conditions of the acceleration potential within the flow field for the problem of fluid–structure interaction and use auxiliary functions to decompose the acceleration potential and extract the instantaneous added mass. Following this, the differential equations of the modes of rigid motion and elastic deformation of the body are established and solved. This approach enables the simultaneous solution of the motion and vibrations of the wedge as well as the hydrodynamic pressure. The proposed model also accounts for flow separation while the wedge is submerged, thereby prolonging the simulation. We confirmed the validity of the model through comprehensive examinations based on semi-analytical, computational, and experimental data on the problem of the entry of a symmetric body into water. We then extended our study to encompass the free-falling entry of an asymmetric elastic wedge into water. The results revealed notable discrepancies in the evolution of deformation and jet flow between the right and left boundaries during the entry of the wedge into water.

Investigation of a weathervaning FPSO based on a fully nonlinear boundary element method

Investigation of a weathervaning FPSO based on a fully nonlinear boundary element method

Validation of a panel method with full-scale FOWT measurements and verification with engineering models

The present paper gives insights into the verification of different models of the Floatgen demonstrator by BW Ideol. This floating offshore wind turbine (FOWT) is based on a barge type-floater with moonpool. The moonpool shall dampen the platform’s motion and is challenging to model due to the increased complexity of the hydrodynamic behaviour. For the verification purposes, the results of two engineering methods (OrcaFlex, OpenFAST) equipped with frequency-dependent hydrodynamic solver and the nonlinear time-domain boundary element method panMARE are compared. The OrcaFlex model designed by BW Ideol is regarded as a reference because of a preceding calibration with measurement data. All models show fairly good agreement in most cases. Only in heave there are significant differences between engineering and the nonlinear models which is caused by different modelling of the moonpool hydrodynamics. In addition, a hybrid simulation approach with prescribed motion and constrained wind field based on motion sensors and LIDAR measurements on the Floatgen demonstrator was applied for an aerodynamic and load validation of panMARE. The validation shows good agreement based on time-domain comparison regarding the tower top loads and turbine operation.

A Nonlinear Fractional BEM Model for Magneto-Thermo-Visco-Elastic Ultrasound Waves in Temperature-Dependent FGA Rotating Granular Plates

The primary goal of this study is to create a nonlinear fractional boundary element method (BEM) model for magneto-thermo-visco-elastic ultrasound wave problems in temperature-dependent functionally graded anisotropic (FGA) rotating granular plates in a constant primary magnetic field. Classical analytical methods are frequently insufficient to solve the governing equation system of such problems due to nonlinearity, fractional order heat conduction, and strong anisotropy of mechanical properties. To address this challenge, a BEM-based coupling scheme that is both reliable and efficient was proposed, with the Cartesian transformation method (CTM) used to compute domain integrals and the generalized modified shift-splitting (GMSS) method was used to solve the BEM-derived linear systems. The calculation results are graphed to show the effects of temperature dependence, anisotropy, graded parameter, and fractional parameter on nonlinear thermal stress in the investigated plates. The numerical results validate the consistency and effectiveness of the developed modeling methodology.

Parametric Hull Form Optimization of Containerships for Minimum Resistance in Calm Water and in Waves

This paper described the process of generating the optimal parametric hull shape with a fully parametric modeling method for three containerships of different sizes. The newly created parametric ship hull was applied to another ship with a similar shape, which greatly saved time cost. A process of selecting design variables was developed, and during this process, the influence of these variables on calm water resistance was analyzed. After we obtained the optimal hulls, the wave added resistance and motions of original hulls and optimal hulls in regular head waves were analyzed and compared with experimental results. Computations of the flow around the hulls were obtained from a validated nonlinear potential flow boundary element method. Using the multi-objective optimization algorithm, surrogate-based global optimization (SBGO) reduced the computational effort. Compared with the original hull, wave resistance of the optimal hulls was significantly reduced for the two larger ships at Froude numbers corresponding to their design speeds. Optimizing the hull of the containerships slightly reduced their wave added resistance and total resistance in regular head waves, while optimization of their hulls hardly affected wave-induced motions.

Hydrodynamic performance of a cone falling into waves in 3DOFs free fall motion

This paper concentrates on the 3DOFs water entry of a cone into an incident wave. Undisturbed free surface is modeled with nonlinear 5th Stokes wave. Based on velocity potential theory, fully nonlinear boundary element method is used to simulate fluid field. Fluid-solid interaction between body's free fall motion and fluid loads is decoupled by auxiliary function method. Stretched coordinate system is constructed to make grids match the dramatically changing wetted surface. Distorted free surface is traced with improved Eulerian method. Water entries with different incident wave heights, initial vertical velocities and cone's densities are simulated respectively. 3DOFs motions including vertical, horizontal and rotational motion of cone with different initial conditions are compared and investigated. Pressure on the wetted surface is given, and discussions of pressure distribution along the cone's generatrices are provided.

Nonlinear IGABEM formulations for the mechanical modelling of 3D reinforced structures

The adequate coupling of dissimilar materials and structural element types provides effective engineering designs. Particularly, it enables high stiffness and low weight mechanical components, which are largely desired in numerous applications. Besides, Isogeometric Analysis (IGA) has recently stood out as a robust and accurate approach for the mechanical modelling of complex 3D engineering designs directly from Computer-Aided Design (CAD) tools. In this context, this study presents nonlinear Isogeometric Boundary Element Method (IGABEM) formulations for the effective mechanical modelling of nonhomogeneous 3D structural systems reinforced by embedded fibres. The IGABEM based on NURBS surfaces represents the solid whereas the 1D approach of BEM models the reinforcements, which characterises a 1DBEM/IGABEM coupling technique. The sub-region technique enables the nonhomogeneous system modelling whereas connection 1DBEM elements allow for crossing reinforcements along boundary interfaces without the need for surfaces remeshing. Thus, the CAD model information can be straightforwardly used in the mechanical model. Nonlinear time independent formulations are presented accounting for both elastoplastic and bond-slip behaviours. Besides, linear and nonlinear time dependent formulations are presented herein applying the Kelvin-Voigt and Boltzmann viscoelastic rheological models. Five applications illustrate the accuracy and robustness of the proposed formulations. The IGABEM formulations lead to accurate and stable results, in addition to efficient modelling. Besides, it demonstrates excellent performance considering the computational time consuming and the total number of degrees of freedom in each application.

Water entry of a floating body into waves with air cavity effect

Based on a fully nonlinear time-domain higher-order boundary element method (HOBEM), the present paper investigates a semi-circular floating body entering obliquely into Stokes wave with air entrapment. In this model, a fifth-order analytical solution is used to simulate the Stokes wave as incident boundary condition. A long and thin jet layer is simulated and assumed to separate from wetted body surface along its tangential direction, which avoids fluid particular leaving or invading body surface. When inner free surfaces on both sides of body collide with each other, a closured air cavity is formed based on the assumption of adiabatic. Dual local stretched coordinate systems are applied to resolve the impact with air cavity effect and downward motion of inner jet due to the local impacts between two water surface which is mainly dynamic effects. Some auxiliary functions are employed to obtain the pressure distribution induced by wave impact. The developed model is verified against the published numerical results for water entry with air cavity effect in the absence of waves. Then, numerical simulations are undertaken to investigate the mechanism of wave entry with air cavity formation through designed parameters, i.e. wave amplitude, wave length, initial wave phase, initial air pressure, prescribed floating body velocity. Numerical results indicate that the presence of wave delays the formation of air pocket and leads to more asymmetric shape of air pocket with the increase of wave nonlinearity when the body impacts the wave peak. After the occurrence of air pocket, the fluid pressure sharply increases.

Added resistance, heave and pitch for the KVLCC2 tanker using a fully nonlinear unsteady potential flow boundary element method

In this paper, a fully nonlinear unsteady potential flow method is used to predict added resistance, heave and pitch for the KVLCC2 hull in regular head waves at design speed. The method presents a nonlinear decomposition of the velocity potential and the wave field and an adaptive grid refinement. A formulation for the acceleration potential is used to obtain the pressure. To improve computational efficiency, a Barnes-Hut algorithm is introduced. A grid dependency study and a study on the impact of different time steps on the solution are performed. Numerical results have been compared with experimental data for the design speed. A general good agreement is found for added resistance, especially for longer waves. Heave and pitch are properly computed for all wave lengths in the range λ/Lpp=0.4 to 1.4.

A potential flow method combining immersed boundaries and overlapping grids: Formulation, validation and verification

A fully nonlinear two-dimensional numerical method based on potential-flow theory for water waves and their interaction with surface-piercing rigid bodies is presented. The harmonic polynomial cell (HPC) method is used to solve the Laplace equation for the velocity potential and its time derivative. The HPC method, which is a high-order method using analytical expressions (harmonic polynomials) to represent the solution inside overlapping cells, has previously been shown accurate and efficient. Supplementary research has shown that, in order to maximally benefit from the method’s accuracy, it is a requirement that square or close-to-square cells are used. Here, we use an immersed boundary method to model non-stationary boundaries such as the free surface or the surface of a rigid body, and overlapping, body-fixed grids that are locally Cartesian to refine the solution near moving bodies. Combining these two modelling concepts with the HPC method represents the main novel contribution in the present work. With this combined method, denoted as an immersed-boundary overlapping grid method (IBOGM), the challenge of generating boundary-fitted grids for complex boundaries is avoided. Moreover, square cells can be used throughout the domain and the solution can be refined locally without increasing the number of unknowns unnecessarily. The method is systematically validated and verified against analytical, experimental and numerical reference results. The cases studied include propagation of steep waves, forced heave motions of a semi-submerged circular cylinder and a fixed and freely floating ship section in beam-sea waves. For the freely floating ship section, the present method is compared in detail with results from a dedicated study performed with a fully nonlinear boundary element method for cases with roll motions up to 30°. The present results are generally in good agreement with reference results, even for the most challenging wave–body interaction cases considered. Based on this, we later intend to use the method to examine in depth the importance of nonlinear effects in the interaction between waves and rigid bodies.

Investigation on responses and wave climbing of a ship in large waves using a fully nonlinear boundary element method

In this study, a fully nonlinear method, in the time domain, is used to simulate ship motion caused by large waves, and the problem of wave climbing along the freeboard in large waves is studied. The method is based on a three-dimensional potential flow theory. Further, the nonlinear free surface condition is adopted, the incident potential and disturbance potential are separated from each other, and the problem of determining the disturbance potential is established. In order to consider the nonlinearity of the body surface, the wetted surface area of the hull is continuously updated with time and extended along the tangential direction of the uppermost grid when the free liquid level exceeds the height of the hull deck. In addition, an artificial damping layer is provided to eliminate the reflection of waves at the outflow boundary. Under the fully nonlinear boundary conditions, the perturbation potential is solved by the Rankine source boundary element method. Introducing auxiliary functions solves the coupling problem between motion and load in the process of solving rigid body motion equations. After the solution is completed, it is updated to the next time step using the Runge-Kutta 4th order method. In this study, the convergence of the computational grid of the numerical model is firstly verified. Then, the motion and load responses of a zero-speed ship in the regular wave of unit wave amplitude using this method are examined and compared with the linear method. The latter verifies the accuracy of this method in the linear field. Afterwards, in order to investigate the superiority of the fully nonlinear method when the nonlinear phenomenon is strong, the motion and load of the ship in the large amplitute wave are simulated and the influence of the nonlinear phenomenon is analyzed. Finally, the phenomenon of wave climbing generated in high waves is analyzed, and the relationship between wave and water heights is evaluated.

Three dimensional nonlinear BEM formulations for the mechanical analysis of nonhomogeneous reinforced structural systems

This study presents a coupling formulation for the mechanical modelling of nonhomogeneous reinforced 3D structural systems. The coupling of dissimilar reinforced materials and components provides efficient designs. Particularly, it enables high stiffness and low weight mechanical components, which are largely desired in several engineering applications. In the present study, the Boundary Element Method (BEM) mechanically represents 3D solids through elastic and viscoelastic approaches. The 1D truss BEM elements represent the reinforcements, which allow for elastoplastic behaviour. The sub-region BEM technique models the nonhomogeneous systems whereas connection 1DBEM elements allow for crossing reinforcements along materials interfaces. Bond-slip behaviour has been accounted in the formulation by a nonlinear scheme without special link elements. Different bond-slip laws can be accounted, which enables the mechanical modelling of different adherence behaviours. The proposed formulations are applied in the mechanical analysis of four complex 3D applications. The results achieved by the proposed formulations are compared against the responses of equivalent numerical methods and experimental results available in the literature. The proposed formulations lead to accurate and stable results. Besides, the applications complexity demonstrates it.

On the Comparative Seakeeping Analysis of the Full Scale KCS by Several Hydrodynamic Approaches

The main transport channel of the global economy is represented by shipping. Engineers and hull designers are more preoccupied in ensuring fleet safety, the proper operation of the ships, and, more recently, compliance with International Maritime Organization (IMO) regulatory incentives. Considerable efforts have been devoted to in-depth understanding of the hydrodynamics mechanism and prediction of ship behavior in waves. Prediction of seakeeping performances with a certain degree of accuracy is a demanding task for naval architects and researchers. In this paper, a fully numerical approach of the seakeeping performance of a KRISO (Korea Research Institute of Ships and Ocean Engineering, Daejeon, South Korea) container ship (KCS) container vessel is presented. Several hydrodynamic methods have been employed in order to obtain accurate results of ship hydrodynamic response in regular waves. First, an in-house code DYN (Dynamic Ship Analysis, “Dunarea de Jos” University of Galati, Romania), based on linear strip theory (ST) was used. Then, a 3D fully nonlinear time-domain Boundary Element Method (BEM) was implemented, using the commercial code SHIPFLOW (FLOWTECH International AB, Gothenburg, Sweden). Finally, the commercial software NUMECA (NUMECA International, Brussels, Belgium) was used in order to solve the incompressible unsteady Reynolds-averaged Navier–Stokes equation (RANSE) flow at ship motions in head waves. The results obtained using these methods are represented and discussed, in order to establish a methodology for estimating the ship response in regular waves with accurate results and the sensitivity of hydrodynamical models.

Cohesive crack propagation analysis using a dipole BEM formulation with tangent operator

This work deals with the use of Boundary Element Method (BEM) formulations in fracture problems in a 2D approach. BEM is a suitable method for solving this type of problem, since the absence of a domain mesh translates into a more efficient modeling of regions with high stress concentration. Besides, reducing the dimensionality of the mesh results in a convenient remeshing process. Regarding the nonlinear BEM formulation, an alternative to the classic dual formulation is used, with the introduction of an initial stress field to represent the cohesive zone, based on the concept of dipoles. This formulation is particularly interesting because it is able to represent mathematically the presence of the fracture process zone (FPZ) with only three algebraic equations (related to stress correction) per source point located in the crack path. In contrast, dual BEM formulation requires four algebraic equations (displacements and forces) per source point. As for the effects relevant to the nonlinear system, two distinct iterative solving algorithms are used, Constant Operator (CO) and Tangent Operator (TO). Only the elastic stiffness of the structure in the search of equilibrium point is considered in the first one, while the latter takes into account the mechanical degradation properties. Some examples are presented to validate the use of the Dipole BEM/TO formulation in crack propagation analysis and also in multiple crack analysis. Finally, this formulation proves to be a powerful alternative to other classic methods, considering the results presented.

Fully nonlinear investigations on performance of an OWSC (oscillating wave surge converter) in 3D (three-dimensional) open water

The hydrodynamic behavior of a hinged Oscillating Wave Surge Converter (OWSC) device in open water, including motion response, extracted power and capture efficiency, is numerically investigated through a developed three-dimensional (3D) fully nonlinear higher-order boundary element method (HOBEM) model. In this time-domain mode, an image Green function is adopted in the whole circular domain with radiation condition at far field to avoid re-reflection from the lateral walls and resonance of the transverse standing waves which both occur in 3D numerical wave tank. The calculated surface elevation and the structural response are compared well with the existing measured and numerical results. It can be proved from present results that wave nonlinearity would change the flap’s natural frequency and break the limit of highest possible efficiency 50%. An OWSC with thinner thickness and wider width becomes more efficient for extending the applicable extent of wave frequency for performance improvement. In addition, when the OWSC device is installed at the shallow water depth, the energy extraction can be magnified due to shoaling effect. The time series further reveal that the larger wave amplitude leads to the motion response dominated simultaneously by difference frequency, fundamental frequency and higher-order frequencies.

Boundary element and integral methods in potential flow theory: a review with a focus on wave energy applications

This paper presents a comprehensive review of boundary element methods for hydrodynamic modelling of wave energy systems. To design and optimise a wave energy converter (WEC), it is estimated that several million hours of WEC operation must be simulated. Linear boundary element methods are sufficiently fast to provide this volume of simulation and high speed of execution is one of the reasons why linear boundary element methods continue to underpin many, if not most, applied wave energy development efforts; however, the fidelity of the physics included is inadequate for some of the required design calculations. Judicious use of non-linear boundary element methods provides a route to increase the fidelity of the modeling while maintaining speed and other advantages over more computationally demanding alternatives such as Reynolds averaged Navier–Stokes (RANS) or smooth particle hydrodynamics (SPH). The paper presents some background to each aspect of the boundary methods reviewed, building up a relatively complete theoretical framework. Both linear and nonlinear methods are covered, and consideration is given to the computational complexity of the methods reviewed. The paper aims to provide a review that is useful in selection of the most appropriate techniques for the next generation of WEC design tools.

Fully nonlinear investigation on water entry of a rigid paraboloid

Oblique water entry of a rigid paraboloid is investigated by fully nonlinear boundary element method. The deadrise angle of the paraboloid increases gradually from zero at the bottom tip and a slenderness coefficient is introduced to symbolize the shape of the paraboloid. Convergence study with respect to time step and element size is carried on to assure the numerical procedure. The free surface elevation and pressure distribution are depicted in time domain. The present solution is compared with the linearized Wagner method and the composite solution with spray jets to investigate the influence of exact fully nonlinear boundary conditions. As there is no sharp edge or tip around the paraboloid, the water flows over the body both from lateral side and down side in oblique entry, the stagnation point relative to the body moves towards the flow-coming direction, and the negative pressure can be observed on both lateral side and down side of the paraboloid. The horizontal speed has great influence on horizontal total force and little influence on vertical total force, while the effect of the slenderness coefficient just has the opposite effect.

Numerical investigation on the effect of baffles on liquid sloshing in 3D rectangular tanks based on nonlinear boundary element method

The numerical simulation of liquid sloshing in the three-dimensional tanks under horizontal excitation and roll excitation was carried out, and the inhibition effect of different baffles on the sloshing phenomenon was investigated. The numerical calculations were carried out by the nonlinear Boundary Element Method (BEM) with Green's theorem based on the potential flow, which was conducted with the governing equation corresponding to the boundaries of each region. The validity of the method was verified by comparing with experimental values and published literatures. The horizontal baffle, the vertical baffle and the T-shaped baffle in the sloshing tanks were investigated respectively, and the baffles' position, dimension and the liquid depth were provided and discussed in detail. It is drawn that the baffle shape plays a non-negligible role in the tank sloshing. The vertical baffle is a more effective way to reduce the sloshing amplitude when the tank is under a horizontal harmonic excitation while the horizontal baffle is a more effective way when the tank is under a roll excitation. The amplitude of free surface elevation at right tank wall decreases with the increasing of the horizontal baffle length and the vertical baffle height. Although the T-shaped baffle has the best suppression effect on tank sloshing under horizontal excitation, it has limited suppression effect under roll excitation and will complicate the sloshing phenomenon when changing baffle height.

Numerical Investigation on Hydrodynamic Performance of an OWC Wave Energy Device in the Stepped Bottom

The efficiency response of an oscillating water column (OWC) wave energy converter is analysed by adopting a numerical approach and using various stepped bottom configurations for the seabed. A two-dimensional, fully non-linear higher-order boundary element method (HOBEM) model is developed to simulate the hydrodynamic characteristics and fluid structure interaction for a fixed OWC located in a numerical wave flume. A number of model tests are conducted for various regular incident wave conditions by modifying the wave amplitudes and wavelengths. The measured free surface elevation at the chamber centre and the oscillatory air pressure generated by the fluctuating free surface within the chamber are recorded. The hydrodynamic efficiency of the OWC is determined using these parameters. The simulation results are validated by comparing against previously published experimental and numerical data. Good agreement is observed in both cases. The geometric dimensions of the step are modified by altering the step height and step length. Furthermore, by altering the step length, the location of the front face of the step relative to the front wall of the OWC is adjusted. Therefore, the positional significance of the step can also be analysed in terms of its relative location to the OWC chamber and to the flow field development. It is shown that the geometry of the step and the position of the vertical face of the step relative to the OWC influences the hydrodynamic efficiency. The research demonstrates that by optimising the step geometry and position for a given wave condition a higher operational efficiency can be achieved.

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.