Multi-domain Boundary Element Method Research Articles

Numerical investigation of interaction of two internal solitary waves with different initial wave amplitudes

The present study investigates the collision of two internal solitary waves with different initial amplitudes using a multi-domain boundary element method. It calculates and discusses wave profiles, residence time, and phase shifting. Observations reveal that tail waves are induced during the collision process, with the amplitude of the tail wave increasing as the initial amplitude of the internal solitary wave increases. Residence time, defined in terms of the interaction between waves of different initial amplitudes, decreases as wave amplitudes increase, eventually stabilizing at a constant value. Additionally, phase shifting increases with wave amplitude, with the rate of increase slowing down as the wave profile becomes relatively large.

Oblique wave trapping by sinusoidal rippled barrier of finite thickness placed on closely spaced semi-circular seabed

The oblique wave trapping by a sinusoidal rippled barrier (SRB) of finite thickness placed far away from a partially reflecting seawall is examined under the assumption of linear potential flow theory. The sinusoidal ripples are identical, and the barrier is installed in the presence of a semi-circular bottom (SCB). As a special case, the wave trapping by a thick non-rippled barrier (NRB) is also studied to show the significance of the ripples on the wave trapping. The multi-domain boundary element method is employed to tackle the wave-trapping performance of the SRB satisfying Darcy's law. Various types of structural configurations, such as (a) SRB alone, (b) SRB placed on SCB with trapping chamber, (c) without trapping chamber, (d) NRB alone, and (e) NRB placed on SCB with trapping chamber, are proposed and evaluated under oblique wave incidence. The correctness of the study results is verified with the available results for specific structural configurations. The effect of wall reflection, chamber length, plate porosity, and plate submergence depth on wave-trapping coefficients (wave reflection and wave run-up) and force coefficients (horizontal force on the wall and vertical force on the barrier) are presented against relative wavelength. The enhanced wave trapping is obtained after installing a SRB when compared with the NRB of finite thickness. The chamber spacing minimizes the trapping coefficients when compared with the non-chamber rippled barrier. The study recommends the SRB with SCB of thickness 0.10≤D/h≤0.15, porosity 20%, width B/h=2, and chamber spacing 1≤L/h≤2 to replace the gravity-type breakwaters to be installed against intermediate and deep-water waves.

Surface gravity wave interaction with a submerged tunnel in the presence of a submerged wavy porous plate

The present study deals with the surface gravity wave interaction of a submerged composite wavy porous plate in the presence of a tunnel placed at a finite distance under the assumption of small amplitude theory and a two-dimensional framework. To solve the boundary value problem, a numerical method using the multi-domain boundary element method (MDBEM) is implemented. Hydrodynamic characteristics of interest such as reflection coefficient and transmission coefficient are computed and studied for a wide range of dimensionless wavenumber and relative plate lengths. Various parametric studies are conducted to emphasize the effect of structural parameters like the number of relative ripple wavelengths, relative ripple amplitude, and relative submergence depth. Dimensionless wave force coefficients acting on the plate and the tunnel are calculated and investigated. The study reveals that the presence of composite wavy porous plate has a significant effect in the mitigation of wave-exerted force on the tunnel. Optimized results are shown for the efficient design of coupled breakwater and tunnel models, which can be used beneficially to protect against the severity of wave action in similar applications of ocean engineering in marine environments.

Surface gravity wave interaction with a floating dock in the presence of a submerged composite wavy porous plate

A numerical model based on the multi-domain boundary element method is adopted to study surface gravity wave interaction with a finite floating dock in the presence of a submerged composite wavy porous plate in the two-dimensional Cartesian coordinate system. Physical parameters of interest such as reflection coefficient, transmission coefficient, dissipation coefficient, horizontal and vertical wave force coefficients of the wavy plate, vertical force and moment coefficients about the centre of the dock are calculated. The influence of structural parameters such as the number of ripple wavelengths, relative ripple amplitude, structural porosity and relative submergence depth of wavy plate, relative space between the plate and dock are computed and studied. An insight into the comparison of various types of plates such as flat plate, wavy plate and composite wavy plate is also presented. The study reveals that the mitigation of wave force on the dock is significant due to a wavy porous plate. Besides, the study exhibits that placing the wavy porous plate at a finite distance from the floating dock has a significant role in dissipating incident wave energy compared to that placing the submerged plate below the dock. Bragg reflection occurs due to the presence of the submerged wavy plate, which diminishes with an increase in structural porosity and depth of submergence of the wavy plate.

Hydrodynamic Performance of Fixed Floating Structures Coupled with Submerged Breakwaters Using the Multidomain Boundary Element Method

Hydrodynamic Performance of Fixed Floating Structures Coupled with Submerged Breakwaters Using the Multidomain Boundary Element Method

Wave Action Analysis of Multiple Bottom Fixed Semi-Circular Breakwaters in the Presence of a Floating Dock

Abstract The comprehensive usage of an array of natural or artificial semi-circular breakwaters (SCBs) as supporting structures to secure the floating/fixed structures are received increasingly more consideration in recent years. The performance characteristics of bottom-fixed SCB in the presence of a floating dock are investigated under the framework of linear wave theory. The edge conditions such as continuity of velocity and pressure along the SCB and zero-flow condition near rigid surfaces are adopted. The multi-domain boundary element method (MBEM) is used to examine the wave attenuation performance of the whole breakwater system in two dimensions. The correctness of the present study's numerical results is confirmed by performing the comparative study with the readily available experimental and analytical results reported by various researchers. The reflection, transmission, energy loss coefficients, and fluid force experienced by floating dock are examined as a function of incident wave properties and breakwater physical properties. The Bragg resonant reflection and performance comparison with other shapes are also studied. The study results strongly suggest that the damage of floating structures and leeward locales due to incident wave stroke is minimized with the introduction of porosity for SCB located on the seabed by dissipating the wave energy. A pair of bottom-fixed SCBs having 10–20% breakwater porosity placed far away from the dock is identified as the optimal choice to reduce the fluid force experienced by the dock and wave transmission coefficient.

Role of dual breakwaters and trenches on efficiency of an oscillating water column

In this paper, the effects of double-submerged breakwaters and trenches on the hydrodynamic performance of an oscillating water column (OWC) are investigated. The multi-domain boundary element method is used to tackle the physical problem of wave scattering and radiation from the device. The role of the height of the breakwaters, depth of the trenches, width of the breakwaters and trenches, spacing between the structures, length of the OWC chamber, and other wave and structural parameters is investigated on the efficiency of OWC. The study reveals that there is an oscillating pattern of the efficiency curve in the presence of single or double breakwater/trenches; this pattern is absent when the bottom is flat. Moreover, compared to single or no breakwaters/trenches, the occurrence of full OWC efficiency is higher in the presence of double breakwaters/trenches. Furthermore, the amplitude of the oscillating pattern in the efficiency curve increases with an increase in the height and depth of the breakwaters and trenches, respectively. For some particular wave and structural parameters, zero OWC efficiency occurs nearly k0h=3.4 within 0<k0h<5 (k0 wave number and h water depth). This zero efficiency moves toward small wave numbers as the spacing between OWC and rigid breakwater/trench increases. The radiation conductance of OWC decreases with an increase in the barrier height. The findings outline the structural criteria that can be employed to build and deploy an effective OWC device.

Finite element - Dual boundary element method for hydroelastic analysis of very large floating structures protected by perforated barrier

This paper presents a Finite Element – Dual Boundary Element Method (FE-DBEM) for hydroelastic analysis of very large floating structures protected by vertical perforated barrier. The structure and barrier are modelled based on the shell theory, while the linear potential flow theory is employed for modelling wave structure interactions. The algebraic equations of the fluid-structure interaction are obtained by using the Finite Element Method (FEM) and Dual Boundary Element Method (DBEM). In DBEM, an approximation method is proposed for calculating the hypersingular integrals associated with the perforated barrier. The proposed method is validated by comparing with published experimental and mathematical results. As compared to the previously proposed Finite Element – Multi-Domain Boundary Element Method, the method presented in this paper has an advantage that the fluid domain does not need to be decomposed into sub-domains in the formulation. In addition, the same mesh of elements can be adopted in the DBEM and FEM; and thus, these methods can be coupled easily for fluid-structure-interaction analysis.

Hydrodynamic performance of submerged breakwater in tandem with thin-walled as submerged reef structure

The interaction of gravity waves with submerged tandem breakwater of different structural configurations is analysed in finite water depth using the Multi-Domain Boundary Element Method (MDBEM). The wave transformation characteristics, wave forces and wave energy dissipation are analysed considering the presence of impermeable type thin-walled as reef structure in front of the primary submerged breakwater. The comparative study is performed for the submerged structures of various shapes (trapezoidal, triangular, rectangular and thin-walled) and types (rubble mound, permeable, impermeable) that are designed to function together as a tandem breakwater. The effect of varying angle of incidence, relative submergence depth, and relative gap between the reef structure and primary breakwater on wave reflection and transmission are derived for the suggested tandem breakwater models. Among all the impermeable-type models, the thin-walled as reef structure designed at a distance in front of thin-walled as a primary submerged breakwater as a tandem is observed to perform efficiently in terms of energy dissipation and also offers an optimum wave transmission for both short and long wave conditions. Further, the permeable and rubble mound type trapezoidal tandem breakwater offers higher energy dissipation in comparison with all other breakwaters. In view of the design considerations and structural stability of submerged breakwaters, the addition of a reef structure acts as a defence system for the primary breakwater and also creates an energy dissipation zone that allows the shore dynamics to be preserved, making tandem models more effective in the harbour region.

Three-dimensional elastic analysis of a bi-material system with a single domain boundary element method

The three dimensional fundamental solution of bi-materials is introduced to the boundary element method (BEM) for elastic analysis of a bi-material system instead of solving a multi-domain problem. Using the explicit form of the Green’s function for bi-materials, the boundary integral equations (BIE) are set up for a finite bi-layered material system with a plane interface. The Green’s function for a bi-material system combines the Kelvin’s solution and disturbed fields by image source. Because it exactly satisfies the jointed continuity conditions at the interface (x3=0), no internal mesh is required along the interface and a single domain BEM can be established. The potential numerical errors of the multi-domain BEM using the Kelvin’s solution due to the mesh quality at the interface can be avoided. This single domain BEM for bi-material saves the effort of domain discretization, predicts the singularity on the interface analytically, and enables an efficient, straightforward analysis similar to the homogeneous solids. It is particularly suitable to simulate thin film/substrate systems, overlay/base structures, and other similar layered materials. The numerical simulations verify the effectiveness and convergence of the method, and the case studies of wind turbine blades and solar panels demonstrate the industrial applications of this method.

Hydrodynamic analysis of floating tunnel with submerged rubble mound breakwater

The wave interaction with a Submerged Floating Tunnel (SFT) of two different shapes (rectangular and circular) in the presence of a submerged rubble mound breakwater (SRMB) is analyzed using Multi-Domain Boundary Element Method (MDBEM). Furthermore, three typical SFT cross-sections (rectangular, trapezoidal, and circular) of equal area and structural height in the presence of SRMB under similar operating conditions are investigated as comparative study to analyse the influence of SFT shape on hydrodynamic performance. The performance of the tunnel configurations is analyzed as a (a) measurement in terms of hydrodynamic efficiency and (b) criterion for tunnel structure safety. In both shallow and intermediate water depth regions, the critical wave number and the critical angle of incidence followed by resonant wave reflection are identified, and suitable structural parameters of SRMB such as structural porosity in the armour layer, relative crest width, relative gap width between the SFT and the SRMB, structural width and position (relative draft of tunnel structure measured from the free water surface) of SFT are investigated. The present parametric investigation of SFT with SRMB reveals an improved wave transformation properties for a specific range of water depth. The coupling of SRMB has resulted not only in a reduction of wave-induced force acting on SFTs, but also in improved performance in wave transformation characteristics as a coastal protection structure, which is substantially determined by SRMB structural properties. Due to the presence of SRMB, the SFT's safety is improved, which may also add stability to the SFT. A comparative study of different distinct cross-sections of SFTs indicates that, due to its shape, the circular SFT has a reduced reflection capability and lower wave-induced force with nearly the same wave transmission as the rectangular and trapezoidal SFT. The study performed on the coupled SFT and rubble mound breakwater may be useful in determining the suitability of breakwaters not only for maintaining shore dynamics but also for protecting important floating structures for underwater transit.

An approach for assessing the effects of porous materials on controlling the tire cavity resonance noise

Pasting porous material on the inner surface of a tire is an effective means to mitigate the tire cavity resonance noise. However, its performance is highly related to the material properties and geometry. This paper proposes an eigenvalue analysis approach based on the multi-domain boundary element method (MBEM) to evaluate this performance and further help determine the optimal parameters of the porous material lining in a tire. In this approach, the porous material is equivalent to a fluid, and the MBEM is used to model the sound fields in the air and equivalent fluid domains inside the tire cavity, based on which the eigenvalue analysis is conducted and the associated nonlinear eigenproblem is solved by using a contour integral method. The accuracy and validity of the proposed method are first verified by using a two-dimensional tire model with analytical solution, and then testified by comparing with the experimental results of a real tire lined with porous materials with different flow resistivities, thicknesses and widths. Furthermore, the effects of porous materials with different parameters are investigated based on the proposed method and the optimal parameters are suggested for a typical three-dimensional tire model.

Numerical study on the surface signature of internal solitary waves

The propagation of internal solitary waves (ISWs) underwater and their interactions with surface waves can affect the sea surface state and generate distinct surface signatures of bright–dark striped patterns in remote sensing imagery. These patterns have been one of the most important methods for wide coverage detection and observation of ISWs. This study establishes a fully nonlinear numerical model for ISW–surface wave interaction simulation based on the multi-domain boundary element method. By numerical simulation, the formation mechanism, characteristics, and influencing factors of ISW surface signatures are revealed. The physical essence of the ISW surface signature is the spatially inhomogeneous wavelength and roughness changes in the surface waves due to the convergence and divergence effects of the ISW. Based on the surface wave wavelength changes, the surface area affected by an ISW is divided into three regions from front to back—the converging, diverging, and restoring regions—corresponding to the regions where the surface waves are compressed, stretched, and recover, respectively. By comparing different cases, it is found that the ISW amplitude, surface wave amplitude, and surface wave wavelength are the main factors influencing the ISW surface signature. ISWs with larger amplitudes have stronger convergence and divergence effects on surface waves, making the overall scale of the ISW surface signature larger. Larger surface wave amplitudes provide stronger resistance to the ISW divergence effect and lead to smaller diverging regions, and larger surface wave wavelengths make the converging region larger and the diverging and restoring regions smaller.

Attenuation of wave force on a floating dock by multiple porous breakwaters

The impact of an array of porous breakwaters for reducing wave forces on a floating dock is studied based on the small-amplitude water wave theory in finite water depth. The breakwater is approximated by a porous bottom-standing rectangular structure and an impermeable floating rectangular structure is placed at a finite distance on the lee-side representing a dock. The physical problem is solved analytically using Eigenfunction expansion method and numerically using a hybrid element multi domain boundary element method. Two scenarios are considered: (i) wave scattering by N porous structures and the floating dock and (ii) wave reflection by a dock attached to the impermeable wall in presence of N porous structures. The reflection, transmission and dissipation coefficients are evaluated and analyzed for various structural and wave parameters. The study reveals that the Bragg reflection occurs in the case of wave scattering by an array of porous structures in the absence of floating dock. It is also found that in the absence of dock N−2 sub-harmonic peaks occurs whereas, in the presence of dock the effect of Bragg resonance is clearly evident on hydrodynamic coefficients. Also, the study concludes that for four structures with larger height, 100% of wave energy can be dissipated and thus mitigate the wave force on the floating rigid dock. Furthermore, the height and width of the porous structure and spacing between the structures play a vital role to protect the floating platform.

Finite element - Multi-domain boundary element method for hydroelastic analysis of large floating pontoons with perforated plates

This paper presents a Finite Element-Multi-Domain Boundary Element (FE-MDBE) method for the hydroelastic analysis of large floating pontoon structures with perforated plates. The structure is modelled by the shell theory while the fluid motion is modelled by using the linear wave theory. When fluid flows through the perforated plates, it is assumed that the velocity is unchanged but the fluid pressure drops by an amount that is determined by an empirical relationship involving the fluid velocity and the porosity of the perforated plate. The equations governing the fluid-structure interaction are solved by using the Finite Element Method (FEM) and Multi-Domain Boundary Element Method (MDBEM). In the MDBEM, the fluid domain is imaginarily decomposed into interior and exterior domains so as to avoid the singularity that arises when using the conventional BEM for solving hydroelastic problems involving thin submerged structures. The proposed FE-MDBE method is verified by comparing its results with experimental results obtained for a flexible platform and a fixed cylinder with perforated sidewall. The verified method is used for the hydrodynamic and hydroelastic analyses of floating breakwaters and large floating platforms with vertical perforated plates at their fore and/or aft.

Water wave scattering by a submerged horizontal semi-circular barrier in a two-layer fluid

This paper studies water wave scattering by a submerged horizontal bottom-mounted semi-circular barrier in a two-layer fluid based on the linear potential flow theory. Two cases, i.e., the barrier completely located in the lower fluid and the barrier intersecting the internal interface of two fluids, are considered independently. In the former case, an analytical solution for the wave scattering problem is derived by using the method of multipole expansions, whereas in the latter case, a numerical solution is developed by employing the multi-domain boundary element method (BEM). The conservation relations of wave energy for incident waves in surface and internal modes are derived, and the reflection and transmission energy coefficients as well as the wave forces acting on the barrier are calculated. Comparisons of calculated results between the analytical solution and the BEM numerical solution are conducted. Calculation examples are presented to examine the effects of various physical parameters (angular frequency, interface position, density ratio of two fluids and incident angle) on wave forces acting on the barrier and the energy transformation between two wave modes. It is found that there is a noticeable difference in the energy transformation between the barrier intersecting the fluid interface or not.

Numerical Simulation for the Evolution of Internal Solitary Waves Propagating over Slope Topography

In this study, the propagation and evolution characteristics of internal solitary waves on slope topography in stratified fluids were investigated. A numerical model of internal solitary wave propagation based on the nonlinear potential flow theory using the multi-domain boundary element method was developed and validated. The numerical model was used to calculate the propagation process of internal solitary waves on the topography with different slope parameters, including height and angle, and the influence of slope parameters, initial amplitude, and densities jump of two-layer fluid on the evolution of internal solitary waves is discussed. It was found that the wave amplitude first increased while climbing the slope and then decreased after passing over the slope shoulder based on the calculation results, and the wave amplitude reached a maximum at the shoulder of the slope. A larger height and angle of the slope can induce larger maximum wave amplitude and more obvious tail wave characteristics. The wave amplitude gradually decreased, and a periodic tail wave was generated when propagating on the plateau after passing the slope. Both frequency and height of the tail wave were affected by the geometric parameters of the slope bottom; however, the initial amplitude of the internal solitary wave only affects the tail wave height, but not the frequency of the tail wave.

Surface Gravity Wave Interaction With a Submerged Composite Wavy Porous Plate Attached to a Vertical Wall

Abstract The hydrodynamic performance of composite wavy submerged porous plate attached to a wall is investigated using numerical method multi-domain boundary element method in the linearized water wave theory in which wave past the porous barriers are modeled using Darcy’s law. Effect of the presence of wall on the hydrodynamic parameters like reflection coefficient, dimensionless wave height, wave force and moment acting on the composite wavy porous structure, and horizontal force on the vertical rigid wall is analyzed for various physical parameters like the number of ripple wavelength, structural porosity, submergence depth, and relative amplitude of composite wavy porous plate. The study demonstrates that the efficiency of hydrodynamic characteristics of the composite wavy porous plate attached to the wall is better compared to a horizontal porous plate attached to the wall of the same applicability conditions (around 27% reduction in wave reflection). Moreover, optimum performance of this kind of breakwater system is increased by considering the lower submergence, higher relative ripple amplitude, appropriate relative amplitude, and suitable moderate porosity of the structure in the range of wavenumber 1 < k0h < 5. Dimensionless wave height, horizontal load on the impermeable wall are reduced to zero, while substantial minimization of vertical load on the structure with suitable consideration of the other influencing parameters of porosity Gp = 0.3, relative amplitude Ds/h = 0.1, relative ripple wavelength m > 3, and submergence depth h1/h = 0.3. The present structural arrangement will be useful for attenuating wave effects on the sea wall.

Wave attenuation properties of rubble mound breakwater in tandem with a floating dock against oblique regular waves

This work examines the impact of oblique wave action on a combination of a rubble mound breakwater and a floating dock. The influence of the breakwater on safety of the floating dock is examined in various configurations. The breakwater is approximated by a submerged trapezoidal structure and the floating dock by a surface piercing rigid rectangular structure. A numerical solution is developed combining the eigenfunction expansion method and the multi-domain boundary element method while assuming linear water wave theory. Reflection and transmission coefficients along with wave forces on the dock are evaluated and analyzed over a wide parameter range. The study reveals that the sloping angle of the trapezoidal breakwater plays a vital role in dissipating the wave load on the floating dock and the case of a trapezoidal breakwater with a vertical lee side faces is also examined. The reflection and transmission of waves are found to be dominated by the floating dock parameters in the system, while the wave forces on the dock are reduced by increasing the length and breadth of the breakwater and frictional coefficient of the porous structure, respectively. The combination of two structures in different tandem configurations is proposed as an effective coastal protection system.

Iterative dual boundary element analysis of a wavy porous plate near an inclined seawall

In this work, wave trapping characteristics of a permeable composite wavy barrier located near an inclined seawall are examined under the framework of linear potential flow theory. For the waves past the thin barrier, a semi-empirical discharge equation having a quadratic pressure drop along with the velocity continuum is considered. The boundary value problem is solved using the dual boundary element method (DBEM). The accuracy of the numerical results is ascertained with a numerical model based on an independently developed multi-domain boundary element method (MBEM). Various numerical results such as wave reflection, wave run-up, the vertical force on the barrier and horizontal force on the wall are discussed considering various values of barrier porosity, number of ripples, ripple amplitude, relative spacing between the wall and barrier, relative submergence depth, seawall inclination, and seawall porosity. The numerical results suggest that the performance of the wavy barrier within sight of partial reflection leeward inclined wall is marginal for the design of the development of breakwaters as contrasted to the vertical and seaward inclined wall. The ripple amplitude minimizes the vertical force on the wavy barrier and the reduction is 63% for the non-dimensional ripple amplitude a/B=0.25 as compared to a flat plate where ‘a’ and ‘B’ refers to ripple amplitude and plate width respectively. The parametric study is expected to serve as an incentive for the design and development of composite wavy porous plates to secure the available seawalls.