Higher-order Boundary Element Research Articles

Numerical and experimental investigation of nonlinear focused waves-current interaction with a submerged plate

This paper concerns the interaction of focused waves with uniform current and a submerged horizontal plate in the context of the two-dimensional nonlinear potential theory. A fully nonlinear NWT (numerical wave tank) based on a time domain HOBEM (higher-order boundary element method) is applied to the numerical solutions. In the model, the Goda's method is modified to discriminate incident and reflected waves, whereas the acceleration potential approach is introduced to determine the hydrodynamic forces on the structure. The corresponding experiments are also conducted in a two-dimensional glass-walled wave tank at Jiangsu University of Science and technology. The developed model is verified with published numerical results and validated with experimental test. Through a systematic parametric study, it is found that the maximum reflection or minimum transmission for different currents occur at the same plate width because of the compensating Doppler effects of current. The spectral analysis reveals that the energy dissipation occurs primarily in the fundamental waves when focused waves propagating over the plate. The current has a stronger influence on the wave forces than on wave surface elevation. Furthermore, the 'secondary loading cycle' phenomenon is also observed in the time series of the wave forces for a following current.

On the fast-multipole implementation of the simplified hybrid boundary element method

The present paper is part of a research line to implement, test and apply a novel numerical tool that can simulate on a personal computer and in just a few minutes a problem of potential or elasticity with up to tens of millions of degrees of freedom. We have already developed our own version of the fast-multipole method (FMM), which relies on a consistent construction of the collocation boundary element method (BEM), so that ultimately only polynomial terms are required to be integrated – and in fact can be given as a table of pre-integrated values – for generally curved segments related to a given field expansion pole and no matter how complicated the problem topology and the underlying fundamental solution. The simplified hybrid BEM has a variational basis and in principle leads to a computationally less intensive analysis of large-scale 2D and 3D problems of potential and elasticity – particularly if implemented in an expedite version. One of the matrix-vector products of this formulation deals with an equilibrium transformation matrix that comes out to be the transpose of the double-layer potential matrix of the conventional BEM. This in principle requires a reverse strategy as compared to our first developed (and reverse) FMM. The effective application of these strategies to any high-order boundary element and any curved geometry needs to be adequately assessed for both numerical accuracy and computational effort. This is the subject of the present investigations, which are far from a closure. A few numerical examples are shown and some initial conclusions can already be drawn. Keywords: Boundary elements, Hybrid boundary elements, Fast multipole method, Variational methods

Application of ALE to nonlinear wave radiation by a non-wall-sided structure

Water waves generated by the forced oscillation of a 3D non-wall-sided structure are analyzed based on the fully nonlinear potential-flow theory. In order to track an exact position of the free surface, Arbitrary Lagrangian-Eulerian (ALE) scheme is used in the computation. The feature of ALE scheme is that by introducing a prescribed path (curve) for each fluid marker on the free surface, the movement of fluid marker representing the deformation of free surface is confined along a path. With properly-designed configuration of those paths, this scheme brings about several advantages compared with the Mixed Eulerian-Lagrangian scheme and the Semi-Lagrangian scheme which are extensively used. In the computation, a higher-order boundary element method (HOBEM) and the 4th-order Runge-Kutta method are adopted as the solver of an initial boundary value problem (BVP). In the validation, the radiated waves generated by a truncated circular cylinder with flare oscillating in heave or surge are studied. Regarding the calculation of hydrodynamic forces, the temporal derivative of the velocity potential (ϕt) is evaluated in an exact but simple manner, i.e. ϕt is obtained by solving a reconstructed BVP without evaluating second-order derivatives of the velocity potential. Besides hydrodynamic forces, wave profiles at specific points and wave run-ups are also computed. Comparisons of the present results against published results of corresponding computations are made. To evaluate capability of the proposed ALE scheme, bodies with curved large flare are used in the computation.

Numerical investigation of solitary wave action on two rectangular boxes with a narrow gap

Based on the time-domain higher-order boundary element method (HOBEM), a two-dimensional numerical wave flume is developed to investigate solitary wave interaction with two rectangular boxes with a narrow gap. In the numerical model, the fully nonlinear boundary conditions are satisfied on the free surface, the mixed Eulerian-Lagrangian method is adopted to track the transient water surface and the fourth-order Runga-Kutta method is used to predict the velocity potential and wave elevation on the free surface. The acceleration potential technique is used to compute the transient wave forces along the wetted object surface. A piston-type wavemaker is used to generate solitary waves. The proposed model is validated by comparing the simulated wave run-up and the wave loads with the published experimental and numerical results of the reflection of a solitary wave from a vertical wall. Then, numerical experiments are performed to study the effects of the narrow gap and the size of each box on the wave run-ups at the two sides of the two-box system and in the narrow gap between two boxes, and the wave loads on the two boxes. The interaction between double solitary waves with a time interval between them with a two-box system is also investigated.

High order boundary and finite elements for 3D fracture propagation in brittle materials

The quasi-static propagation of fracture in brittle materials was studied in several recent publications. A variational formulation (Salvadori, 2008; Salvadori and Carini, 2011; Salvadori and Fantoni, 2013) led to three-dimensional crack tracking strategies (Salvadori and Fantoni, 2014 [2,3]; Salvadori and Fantoni, 2016). One of the complexities of this new class of algorithms is the evaluation of high-order terms of the expansion of the crack opening and sliding. In this paper, new types of finite and boundary elements are formulated that capture the near crack-front asymptotical displacement behavior up to the order 3/2. The use of these elements with the quasi-static crack propagation algorithms of the above references is demonstrated for a simple crack configuration.

Higher Harmonics Induced by Dual-Submerged Structures

ABSTRACT Ning, D.-z.; Li, Q.-x.; Chen, L.-f.; Zhao, M., and Teng, B., 2017. Higher harmonics induced by dual-submerged structures. Various types of submerged multibody systems are being inserted into and being devised in the real sea. Nonlinear diffraction of regular waves attributable to dual-submerged cylinders and boxes is investigated both experimentally and numerically in this study. The ultimate objective of the study is to provide best practice guidelines for the engineers in the field leading to the better design of the submerged structures with better hydrodynamic performance and higher survival characteristics. A two-dimensional fully nonlinear numerical wave flume based on a time-domain higher order boundary element method is used, and the comparisons between the numerical and experimental results are carried out for cross-checking both the numerical model and physical experiments. The influence of the submergence, gap distance, size, and geometric shape of the two submerged structures on the h...

Dynamic pressure distributions of semi-submersible

In the hydrodynamic study of Semi-submersibles, the wave pressure distribution on the wetted surface is critical input for structure analysis. Based on wave radiation and diffraction theory, a higher-order boundary element method (HOBEM) was applied to calculate the wave pressure distribution for a semi-submersible under specified wave direction and frequency. An integral system containing the FEM modeling of the semi-submersible, numerical calculation of hydrodynamic motion coefficients and data extraction and export for BEM analysis was established in the current study. A newly developed multiple and double nodes relocation method coupled with FEM-BEM model transformation is applied to remove the singularities along the sharp edges and corners of the semi-submersible. The numerical results confirm that the above algorithm significantly improves the numerical accuracy of wave pressure analysis for the semi-submersible.

2D Acoustic Design Sensitivity Analysis Based on Adjoint Variable Method Using Different Types of Boundary Elements

Continuous linear and quadratic boundary elements are often applied to numerical solution. Discontinuous higher-order boundary elements are developed for 2D acoustic problems to achieve higher accuracy in this paper. The Burton–Miller formulation is used to overcome the fictitious frequency problem when using a single Helmholtz boundary integral equation for exterior boundary-value problem. The strong singular integrals in Burton–Miller formulation using different types of element discretization are evaluated explicitly and directly, respectively. An example of scattering by an infinite rigid cylinder is presented to compare the performance of different types of elements. The effect of the position of nodes on the performance of discontinuous elements is studied, and an empirical value for optimal nodal position is concluded in this paper. Adjoint variable method is applied to evaluate the sensitivity value of the objective function, and the method of moving asymptotes is used for structural optimization analysis of noise barrier.

Interactions of fully nonlinear solitary wave with a freely floating vertical cylinder

Fully nonlinear numerical interaction of a transient wave with a three dimensional structure has been analysed based on a higher-order boundary element method (BEM). The BEM mesh on the free surface is generated through a combination of the structured and unstructured meshes. Through some auxiliary functions, the mutual dependence of fluid/structure motions is decoupled, which allows the body acceleration to be obtained without the knowledge of the pressure distribution. The solitary wave is used as the case study for the transient wave. It is obtained by the third order theory and the fully nonlinear theory. The accuracy of the present numerical model is verified through the steady propagation of a solitary wave and comparison with the published results for solitary wave interaction with a vertical wall. Simulations are then made to study solitary wave interaction with a truncated cylinder. Numerical results are provided for motions, forces and run-ups on the cylinder and comparison between results for the fixed cylinder and the freely floating cylinder is also made.

Computations of linear and nonlinear ship waves by higher-order boundary element method

A practical method for computing ship waves accurately and efficiently is favorable for hull form optimization in early stage of ship design. In the present study, Rankine source method incorporated with high-order boundary element (HOBEM) discretization is applied to solve linear ship waves at first. Numerical implementation is described in detail. An incomplete LU factorization preconditioned Generalized Minimal Residual method (GMRES) is employed to solve the resulting boundary integral equation in order to improve efficiency. A corresponding Fortran program is developed for the validation study. It is applied to solve ship waves of different hulls, including slender Wigley hull, KRISO Container Ship (KCS) with relatively full form and a fishery patrol boat. Ship wave drag, sinkage, trim and wave pattern over a wide range of Froude numbers are all well predicted.In order to further investigate nonlinear effects on ship waves, a fully nonlinear potential flow method based on stationary iteration is proposed. The same numerical approach of HOBEM is employed. The combined nonlinear free surface condition is solved in each iteration to evaluate the free surface. Numerical investigation for Wigley and KCS hull shows the present nonlinear method are accurate. Comparison study with linear and partial nonlinear solution are also carried out and nonlinear effects on ship waves are detailedly discussed.

Boundary element method based vibration analysis of elastic bottom plates of fluid storage tanks resting on Pasternak foundation

A higher-order boundary element procedure is presented for the free vibration analysis of flexible base plates of rigid fluid storage tanks resting on elastic foundation. The main principles of the procedure are replacing the biharmonic operator of the thin plate vibration problem by two successive harmonic operators, and representing the forcing terms (plate inertia, fluid loading and foundation influence) by applying the dual reciprocity boundary element formulation. The fluid effect on the plate dynamics is incorporated into the analysis by invoking another boundary element solution, which expresses the fluid pressure over the plate surface in terms of plate deflection. The performance of the method is thoroughly investigated and the nature of dynamic plate–foundation–fluid interaction is studied from several perspectives. The method provides excellent predictions, within the limits of Kirchhoff plate, potential flow, and Pasternak foundation models.

Experimental and Numerical Simulation of the Microcrack Coalescence Mechanism in Rock-Like Materials

Rocks and rock-like materials frequently fail under compression due to the initiation, propagation and coalescence of the pre-existing microcracks. The mechanism of microcrack coalescence process in rock-like materials is experimentally and numerically investigated. The experimental study involves some uniaxial compression tests on rock-like specimens specially prepared from portland pozzolana cement, mica sheets and water. The microcrack coalescence is studied by scanning electron microscopy on some of the prepared thin specimens. It is assumed that the mica sheets play the role of microcracks within the specimens. Some analytical and numerical studies are also carried out to simulate the experimentally observed microcrack coalescence phenomena within the specimens. A higher-order indirect boundary element method known as higher-order displacement discontinuity method implementing special crack tip elements to treat the singularities of stress and displacement fields near the crack ends is used to estimate the Mode I and Mode II stress intensity factors at the microcrack tips. The maximum tangential stress fracture criterion is implemented in a sophisticated computer code using the linear elastic fracture mechanics theory and the propagation and coalescence of random microcracks within the rock-like specimens are numerically simulated based on an iterative algorithm. The proposed analyses are validated by comparing the corresponding experimental and numerical results of microcrack coalescence phenomena of rock-like materials.

Realistic Design of a Floating Breakwater Design with Moonpools

In an attempt to obtain a 2D floating breakwater model with high performance in wave reflection, genetic algorithm (GA) was combined with boundary element method (BEM) in the previous study. The performance of the obtained model was verified with numerical relations as well as an experiment in towing tank. Moreover, its performance and characteristics in 3D case were also evaluated in the subsequent study. However, because the 3D model is formed by simply extruding the 2D shape in longitudinal direction, it only produces a model with uniform transversal shape which is considered to be less effective and efficient in terms of technical and economical points of view. Consequently, it is needed to modify the model to obtain a more realistic and efficient design without reducing significantly the high performance obtained previously. In the present study, several modifications of the original 3D model are performed which include placing moonpools inside the body. The performance and characteristics of the modified models in terms of wave elevations on the free surface are evaluated at various wavelengths by using higher order boundary element method (HOBEM). The accuracy of the computed results is confirmed with Haskind-Newman and energy conservation relations. From the modifications and evaluations of the models, it could be realized that the moonpools inside the body could be used to obtain a more realistic model without reducing the optimum performance of the original model shape.

Numerical study of resonance induced by wave action on multiple rectangular boxes with narrow gaps

By introducing a source term into the Laplace equation, a two-dimensional fully nonlinear time-domain numerical wave flume (NWF) is developed to investigate the resonance induced by the interaction between waves and multiple objects with narrow gaps. In the numerical model, the fully nonlinear kinematic and dynamic boundary conditions are satisfied on the instantaneous free surface and the constant artificial damping is employed in the gaps to approximate the viscous dissipation due to vortex motion and flow separation. The computational domain is discretized using a higher-order boundary element method (HOBEM). The proposed model is firstly validated against the published experimental data and numerical results of the wave height in the narrow gap between two boxes, the wave heights in the two gaps of three boxes, and wave loads on the boxes. Then, the extensive numerical experiments are performed to study the influences of the number of the boxes and the gap spacing on the resonant frequency, reflected and transmitted wave heights and wave loads on the boxes.

Irregular Nonlinear Wave Simulation and Associated Loads on Offshore Wind Turbines

The accuracy of predicted loads on offshore wind turbines depends on the mathematical models employed to describe the combined action of the wind and waves. Using a global simulation framework that employs a domain-decomposition strategy for computational efficiency, this study investigates the effects of nonlinear waves on computed loads on the support structure (monopile) and the rotor–nacelle assembly of a bottom-supported offshore wind turbine. The fully nonlinear (FNL) numerical wave solver is invoked only on subdomains where nonlinearities are detected; thus, only locally in space and time, a linear solution (and associated Morison hydrodynamics) is replaced by the FNL one. An efficient carefully tuned linear–nonlinear transition scheme makes it possible to run long simulations such that effects from weakly nonlinear up to FNL events, such as imminent breaking waves, can be accounted for. The unsteady nonlinear free-surface problem governing the propagation of gravity waves is formulated using potential theory and a higher-order boundary element method (HOBEM) is used to discretize Laplace’s equation. The FNL solver is employed and associated hydrodynamic loads are simulated in conjunction with aerodynamic loads on the rotor of a 5-MW wind turbine using the NREL open-source software, fast. We assess load statistics associated with a single severe sea state. Such load statistics are needed in evaluating relevant load cases specified in offshore wind turbine design guidelines; in this context, the influence of nonlinear wave modeling and its selection over alternative linear or linearized wave modeling is compared. Ultimately, a study such as this one will seek to evaluate long-term loads using the FNL solver in computations directed toward reliability-based design of offshore wind turbines where a range of sea states will need to be evaluated.

An efficient FEM–BEM coupling method in wave radiation problem analysis of oil platforms with complicated geometry

Real body model meshing and data preparation on body surface are two critical steps for the sea load calculation using boundary element method. In this study, an efficient procedure to solve these two issues is developed. The FEM type meshing model is used to construct real 3D platforms. Basic parameters such as mass and volume of platform are directly calculated from FEM model. A data extracting algorithm is developed to obtain the necessary data block on body surface of FEM model for the use of BEM method. A Double and Multiple Nodes Relocation Method (D&MNRM) is employed along sharp edges of FEM model to remove geometrical singularity. Based on the newly rearranged boundary information, shallow water Green function and higher-order boundary element method are used to solve the integral equations. A simple example for floating cylinder and a complex example for ETLP are used to validate the added mass and damping. The results show that the proposed method is efficient and can be extended to wave load analysis of any type of platforms with arbitrary shape.

Investigation of hydrodynamic performance of an OWC (oscillating water column) wave energy device using a fully nonlinear HOBEM (higher-order boundary element method)

Based on a time-domain HOBEM (higher-order boundary element method), a two-dimensional (2D) fully nonlinear NWF (numerical wave flume) is developed to investigate the hydrodynamic performance of a fixed OWC (oscillating water column) wave energy device. In the model, the incident wave is generated by the inner-domain sources to avoid the re-reflection at the inlet boundary. A self-adaptive Gauss integral method is introduced to tackle the mismatch between meshes on free surface and body surface. A simplified pneumatic model is used to determine the air pressure imposed on the free surface inside the chamber. The present model is validated against the published experimental and numerical results for OWCs over flat and sloping bottoms. Numerical model results indicate that the maximum air -pressure in the chamber does not occur at the same frequency as the maximum surface -elevation. For a fixed submerged depth of the OWC back wall, the peak efficiency increase with bottom slope initially then remains almost the same once the bottom slope reaches a certain value. The hydrodynamic efficiency attains a maximum value at a critical wave slope (wave slope kAi approximate 0.10 in the present study) and decrease from this value when the wave nonlinearity becomes either stronger or weaker.

Wave resonances in a narrow gap between two barges using fully nonlinear numerical simulation

The traditional potential flow theory to describe fully nonlinear waves is reformulated by separating the contributions from incident and scattered waves, in order to improve the computational efficiency. The nonlinear incoming wave is specified explicitly and the modified nonlinear free surface boundary conditions for the scattered wave are expressed in the full Lagrangian description. At each time step only the scattered wave is solved using a mixed Eulerian–Lagrangian scheme by a higher-order boundary element method. The accuracy of the newly developed model is illustrated by comparisons with existing experimental and numerical data in the case of wave diffraction around an array of circular cylinders. Wave resonances in the gap between two side-by-side barges in beam seas, as in Molin et al. [1], are simulated with the barges subjected to regular waves. To clearly understand the gap resonant responses, long time simulations are performed to achieve final steady states, and the resonant mode shapes of the gap surface are presented. The gap free surface RAOs (Response Amplitude Operators) in the case of mild waves are found to agree well with linear calculations. The nonlinear effects on the resonant response due to the free surface conditions are then investigated. The first resonant frequency is found to shift but the peak value is not changed much with increasing incoming wave steepness, which is known as stiff/soft spring behavior of a nonlinear system. Through the investigation of barges with different drafts, the stiff and soft spring behaviors are identified.

Numerical Investigation of Nonlinear Wave Scattering by a Horizontal Submerged Plate

Submerged plates are widely used as breakwater device and wave energy convertor in costal and ocean engineering. It is of practical importance to account for the effects of plate on wave transformation and nonlinear wave loads on the plate. In this paper, a 2D fully nonlinear numerical wave flume is developed to investigate the nonlinear interactions between a regular wave and a submerged horizontal plate by use of a higher-order boundary element method (HOBEM) based on potential theory. An acceleration potential method is adopted to calculate the transient wave loads on the plate. The proposed model is verified against the published experimental and numerical data for nonlinear wave scattering by a submerged plate. Then numerical tests are performed to investigate the high free harmonics induced by the submerged plate and nonlinear wave loads on it. The second-order analysis shows a resonant behavior which is related to the ratio of the plate length to the second beat length. The wave-load analysis shows that the pressure on the upper surface oscillates more frequently than that on the lower surface.

A numerical study of the second-order wave excitation of ship springing by a higher-order boundary element method

ABSTRACT This paper presents some of the efforts by the authors towards numerical prediction of springing of ships. A time-domain Higher Order Boundary Element Method (HOBEM) based on cubic shape function is first presented to solve a complete second-order problem in terms of wave steepness and ship motions in a consistent manner. In order to avoid high order derivatives on the body surfaces, e.g. mj-terms, a new formulation of the Boundary Value Problem in a body-fixed coordinate system has been proposed instead of traditional formulation in inertial coordinate system. The local steady flow effects on the unsteady waves are taken into account. Double-body flow is used as the basis flow which is an appropriate approximation for ships with moderate forward speed. This numerical model was used to estimate the complete second order wave excitation of springing of a displacement ship at constant forward speeds.