Iterative Boundary Element Method Research Articles

A Fast Multi-Frequency Iterative Acoustic Boundary Element Method Solver Based on a Preconditioned Accelerated Recycling Strategy

Common multi-frequency solution strategies for acoustic systems are either associated with high computational costs or resort to model reduction strategies, which potentially induce considerable approximation errors. In that context, this work proposes a fast multi-frequency solver for acoustic Boundary Element Method (BEM) analyses that combines conventional preconditioners with an accelerated recycling strategy, achieving in that way a fast and accurate iterative solution. Specifically, by approximating both the system and the related preconditioner by individual affine expressions, an upfront Galerkin projection on a global deflation basis is facilitated, therefore enabling the deployment of the accelerated recycling scheme. The latter acts by accelerating the construction of the related deflation projectors, therefore leading to a significantly faster recycling procedure. Finally, the efficiency of the global deflation basis is guaranteed through an Automatic Krylov subspaces Recycling for Deflation (AKR-D) [D. Panagiotopoulos, W. Desmet and E. Deckers, An accelerated subspaces recycling strategy for the deflation of parametric linear systems based on model order reduction, Comput. Methods Appl. Mech. Eng. 403 (2023) 115765] algorithm and thus, the cumulative number of iterations required by an iterative solver within a frequency sweep is drastically decreased. The proposed multi-frequency solver is tested on two industrially relevant examples of an interior and an exterior scattering problem, benchmarked against the traditional solution strategies.

Numerical Investigation of Curved Tip Effect on the Performance of 3-D Cavitating Hydrofoils Moving under Free Surface

Curved tip effect on the performance of three-dimensional (3-D) cavitating hydrofoils moving steadily under a free water surface has been investigated numerically by an iterative boundary element method (IBEM) developed before. The IBEM has been modified and extended to solve this problem. The fluid is assumed to be inviscid, incompressible and the flow irrotational. All variables and equations are made non-dimensional to achieve a very quick and consistent numerical scheme. The IBEM is based on the Green’s theorem. The hydrofoil problem and the free surface problem are solved separately with the effects of each other via their potential values in an iterative manner. Both the 3-D hydrofoil surface and the free surface are modelled with constant strength source and constant strength doublet panels. The kinematic boundary condition is applied on the hydrofoil surface while the linearized kinematic and dynamic combined condition is applied on the free water surface. The method was validated extensively before for cavitating hydrofoils but not when combined with tip curvature. It is first applied to a tapered wing and the results are compared with those of experiments. Later, it is applied to a tapered hydrofoil with curved tip upwards or downwards. The effects of curved tip shape on cavitating hydrofoil performance have been investigated. It is found that the curved tip caused an increase in loading and cavitation volume in unbounded flow domain. It is also obtained that curved tip caused a decrease in loading on the hydrofoil for lower chord-based Froude numbers than 0.8.

Mathematical modelling of nonlinear pressure drops in arbitrarily shaped port utilizing dual boundary element method

A mathematical model is developed to analyze the wave-trapping characteristics of different types of porous and non-porous barriers in arbitrarily shaped domains utilizing the iterative Dual Boundary Element Method (DBEM) under the resonance conditions. In this analysis, Vertical, Wavy, Inverted V-shaped, and Inverted L-shaped barriers are utilized to investigate the effect of incident waves over the boundary of the harbor. The harbor is divided into an interconnected domain with variable bathymetry, and the Helmholtz equation is solved in each subdomain utilizing the prescribed boundary conditions. Further, the semi-empirical discharge equation with quadratic pressure drop is considered on the porous barrier to incorporate the effect of incident wave amplitude on the energy loss. The present numerical scheme is validated with existing studies and verified with convergence analysis. Further, the mathematical model is implemented on the Visakhapatnam port at four key locations with improved performance of different types of barriers. Moreover, the influence of the partial reflection coefficient is also investigated by considering the various combination of partially reflecting boundaries of the port. The wave trapping or absorbing characteristics of porous barriers proves an efficient tool to reduce the wave resonance inside the port.

PREDICTION OF HYDRODYNAMIC PERFORMANCE OF 3-D WIG BY IBEM

The hydrodynamic performance of three-dimensional WIG (Wing-In-Ground) vehicle moving with a constant speed above free water surface has been predicted by an Iterative Boundary Element Method (IBEM). IBEM originally developed for 3-D hydrofoils moving under free surface has been modified and extended to 3-D WIGs moving above free water surface. The integral equation based on Green's theorem can be divided into two parts: (1) the wing part, (2) free surface part. These two problems are solved separately, with the effects of one on the other being accounted for in an iterative manner. Both the wing part including the wake surface and the free surface part have been modelled with constant strength dipole and source panels. The effects of Froude number, the height of the hydrofoil from free surface, the sweep, dihedral and anhedral angles on the lift and drag coefficients are discussed for swept and V-type WIGs.

Numerical and experimental modelling of wave interaction with fixed and floating porous cylinders

We consider wave forces on fixed porous cylinders with and without a solid inner cylinder and wave-induced motions of floating cylinder with and without a porous outer cylinder. Comparisons between experimental measurements and numerical predictions from an iterative boundary element method (BEM) model are presented. The BEM model assumes that pressure drop across porous surface is proportional to the square of the velocity through the surface. It is shown that the BEM model is able to accurately predict the nonlinear variation of the forces with wave amplitude or motion amplitude. It is demonstrated that adding a porous outer cylinder to a solid vertical cylinder leads to increased excitation force on the combined structure. For floating cylinders adding a porous outer cylinder also leads to a corresponding increase in excitation force. However, the porous outer cylinder provides a larger increase in the damping, resulting in reduced motion response. Further numerical simulations indicate that placing the porous cylinder lower in the water column can lead to increased damping without the corresponding increase in excitation forces. It is shown that for low Keulegan Carpenter numbers, the damping coefficient for a porous cylinder is significantly higher than the viscous damping on a solid cylinder. The results suggest that porous materials could be beneficial for motion damping of floating structures.

Comparative study on ship wave making resistance via viscous and IBEM solvers

The present paper concerns the numerical modeling of the free surface flow around the Wigley hull using viscous and potential flow based solvers over a range of Froude Numbers from 0.22 to 0.48. Numerical results of the in-house iterative boundary element method (IBEM) code and unsteady Reynolds Averaged Navier-Stokes (URANS) CFD tool presented in comparison with the range and average of the experimental data from the literature. The study aims to identify the Froude number envelope in which potential and viscous solvers work well, and match the results with the physical behavior of the flow. The results reveal that both methods are capable of predicting the wave-making resistance in satisfactory agreement with the experiments at intermediate Froude Numbers. IBEM exhibits slightly better accuracy at very low operating velocities. When the ship is advancing with a forward speed at Fn > 0.4, boundary layer separation related viscous effects and enhancing turbulence in the wake region of the hull cause the IBEM to underestimate the resistance. For low Fns, the deviation from the mean of the experimental data is 5% and 17% for IBEM and URANS, respectively, while the relative errors are calculated as 7% for URANS and 23% for IBEM at the highest Fn. Similarly, when the Fn reaches up to 0.4, the URANS method predicts higher values of wave climbing near the bow region compared to those of the potential solver, while the computations of both methods are very close at lower Fn. Besides these, IBEM enables a greater saving of computational time while reaching through the lower values of Fn. The findings of the study reveal that the present IBEM approach provides quick and reliable numerical solutions till the effect of viscous stress becomes non-negligible (which is Fn < 0.4 for investigated Wigley case), and up to this range, the accuracy of the method is inversely related to the ship velocity.

Dual boundary element analysis for a pair of inverted T-type porous barriers having nonlinear pressure drop

In the present study, scattering of gravity waves by different wave barrier configurations is analyzed under the assumptions of small amplitude wave theory. The wave past barriers are assumed to follow a nonlinear pressure drop boundary condition . The boundary value problem is solved using iterative Dual Boundary Element Method (DBEM) which facilitates the problem-solving in a single domain. Based on the hydrodynamic characteristics, it is revealed that a pair of inverted T-type porous barrier is effective for wave damping application. Further, a parametric analysis is carried for the inverted T-type barrier to understand the effect of porosity, relative submergence of the barrier, relative spacing between the barriers and relative width of barriers on wave scattering, wave forces and moments. It is observed that increasing the porosity of the inverted T-type twin barrier increases the wave transmission but reduces the wave forces and moments. It is revealed that the relative depth of submergence h 1 / h significantly affects the wave scattering and wave loads on the twin inverted T barrier. Varying the relative spacing between the inverted T-type barrier, S / h from 0 to 1 does not change the wave transmission significantly but changes the wave forces and moments appreciably. It is recommended to increase the value of relative width of the wave barrier, B / h , only if wave transmission need to be reduced considerably for a selected porosity, the relative depth of submergence and relative spacing between the barrier, but at the expense of higher vertical wave force and wave-induced moments. The study results are expected to be useful for optimized design of inverted T-type wave barrier for a wide range of wave conditions.

Dual BEM for wave scattering by an H-type porous barrier with nonlinear pressure drop

In this paper, wave scattering by an H-type porous barrier having nonlinear pressure drop boundary condition is analysed within the framework of small amplitude water wave theory. The H-type barrier is constructed using multiple thin (near zero thickness) rigid porous plates which are termed degenerate boundaries. The boundary value problem is solved using an iterative dual boundary element method. Different barrier configurations are analysed and compared to demonstrate the improved hydrodynamic performance of the H-type barrier. Further, the effect of porosity, relative spacing, the relative depth of submergence of the horizontal plate, wave steepness, and the rotation of the horizontal plate is investigated parametrically. Several results such as scattering coefficients (reflection, transmission, and energy-loss) and force coefficients (horizontal, vertical, and moment) are presented to understand the feasibility of the H-type barrier in real field applications. It is revealed that the increase in the structure porosity consistently increases the wave transmission, but reduces wave reflection, and force coefficients. Further, an increase in the relative spacing between the vertical barriers reduces the wave transmission by 20% without increasing the horizontal wave force but at the expense of increasing the vertical force coefficient in the shallow water regime. The results of this study are expected to be useful for the appropriate selection of different structure parameters to optimize the design.

Surface gravity wave scattering by multiple slatted screens placed near a caisson porous breakwater in the presence of seabed undulations

In the present study, the effectiveness of using multiple slatted screens placed in front of a caisson porous breakwater to dissipate the incident wave energy is analyzed. To model the water flow through the slatted screens, a quadratic (nonlinear) pressure drop condition which involves both the inertial and drag effects is considered. Further, for thick porous structure, the well known Sollitt and Cross (1972) model is used. To handle the nonlinear boundary conditions, an iterative multi-domain boundary element method (BEM) is used. The numerical convergence of the multi-domain BEM based solutions is presented. Energy identities for flow past a single slatted screen, multiple slatted screens and for the present problem are derived. The study shows that the minimum reflection coefficient (<1%) and maximum wave energy dissipation (>98%) can be obtained by using four slatted screens. Further, for moderate values of perforation-effect Keulegan-Carpenter number (KC), the reflection coefficient attains maximum for b1λ≈n2,n=1,2,3,⋯ (b1 is the distance between the rigid caisson and the front slatted screen, λ is the incident wavelength) irrespective of the variations in the number of slatted screens. Further, the transmission coefficient, horizontal and vertical wave forces acting on the rigid caisson attain maximum for b1/λ≈0.55n, and minimum occurs for 2n+14<b1λ<3n+26,n=0,1,2,3,⋯. The Bragg resonance in the reflection coefficient occurs at Bragg value ≈1.6 irrespective of the variations of KC number. Moreover, the reflection coefficient increases around the Bragg value for higher values of KC number without altering the peak value in the reflection coefficient.

Hydrodynamic analysis of a seaside quarter-circular breakwater with an array of porous cages using DBEM

ABSTRACT The gravity wave interaction with an array of porous cages in the presence of a seaside submerged quarter-circular porous breakwater (QCPBW) is analyzed within the framework of linearized water wave theory. A quadratic pressure drop is adopted for the wave past the porous structures to capture the effect of wave height on the energy dissipation. The boundary value problem is solved using the iterative Dual Boundary Element Method (DBEM). The numerical code is validated with known results in the literature. One or upto four units of submerged and bottom fixed porous cages are considered. The effect of porosity on QCPBW and porous cages are investigated pragmatically to study the scattering and wave forces. It is found that introducing a single porous cage with 10% porosity at the rear side of a QCPBW with a porosity of 25% can reduce the wave transmission coefficient by about 50% for relative water depth . Increasing the porosity of the single cage from 10% to 40% has a significant reduction on the horizontal and vertical force coefficients on the cage, whereas the change is insignificant on QCPBW. The horizontal wave force is three to four times higher when compared to the vertical wave force for both QCPBW as well as on the porous cage. For a system comprising four units of cages, both the horizontal and vertical wave forces on cage 1, 2, 3, and 4 reduce progressively (cage 2 is at the lee side of cage 1 and so on). The results of the present study can be used for the optimal design of open sea porous systems.

WAVE FIELD GENERATED BY FINITE-SPAN HYDROFOILS OPERATING BENEATH A FREE SURFACE

The present paper focuses on the numerical investigation of the flow around the fully submerged 2D and 3D hydrofoils operating close to a free surface. Iterative boundary element method is implemented to predict the flow field. This study aims to investigate the aspect ratio effect on the free surface interactions and hydrodynamic performance of the hydrofoils under a free surface by using potential flow theory. Three different submergence depths and aspect ratios are studied in the wide range of Froude Numbers. In 3D cases, spanwise width of the numerical wave tank model is selected both equal and wider to the foil span, to observe the sidewall effects. Wave field seems to be two dimensional at low Froude numbers. On the other hand, signs of three dimensionalities are observed on the free surface structure for higher Fn, even the predicted wave elevations are very close to 2D calculations in the midsection. Increment in the Fn give a rise to the amplitude of the generated waves first, however a further increase in Fn has a lowering effect with the beginning of waves spill in the spanwise direction in the form of Kelvin waves. Free surface proximity and resultant wave field are also seeming to be linked with the lift force on the hydrofoil. As aspect ratio of the foil increase, 3D lift values are getting closer to those of 2D calculations. However, it is seen that, 3D BEM predictions of a hydrofoil under free surface effect cannot be considered two-dimensional even the aspect ratio is equal to 8.

Gravity wave interaction with multiple submerged artificial reefs

In the present study, gravity wave interaction with a series of submerged artificial permeable reefs is analysed within the framework of linearised water wave theory. For wave past porous walls of the artificial reefs, a quadratic pressure drop is assumed to account for the wave energy dissipation due to the changes in wave height. The physical problem is handled for a solution using a numerical model based on the iterative multi-domain boundary element method and the developed numerical model is validated with known results in the literature. The iterative model is compared with the numerical model based on linear pressure drop boundary condition (i.e., Darcy law). The study reveals that the wave transmission reduces with the increase in the number of reef units. It is demonstrated that the transmission coefficient can be reduced to less than 0.5 when the number of reef units is greater than or equal to three for a relative height greater than 0.7, reef porosities less than 20% and for 0.4&lt;k0h&lt;3.0. Bragg reflection is observed when the porosity is in the range of 0 to 10% and above which the recurrent nature of wave reflection gradually fades away due to dissipation effects. The number of peaks occurring in the shallow and intermediate water depths is equal to the number of reef units wherein the maxima occur at the first frequency. A relative increase in the base width improves the energy losses but the rate of increase in energy loss decreases. The scattering coefficient pattern is oscillatory when the relative spacing between the barriers is varied and their hydrodynamic performance is invariant for higher relative base width.

Iterative dual BEM solution for water wave scattering by breakwaters having perforated thin plates

This study develops an iterative dual BEM (Boundary Element Method) model for water wave scattering by breakwaters having perforated thin plates, which avoids setting artificial sub-domains near the thin plates and thus is very convenient to write general computer codes. In order to directly consider the effect of wave height on the wave energy dissipation by perforated plates, a quadratic (nonlinear) pressure drop condition is imposed on the plates. Then, iterative calculations are conducted to solve the nonlinear boundary condition in the dual BEM model. Also, an analytical method to calculate the integrals of kernel function, which is generated from the hypersingular boundary integral equation in the dual BEM model, is introduced to enhance the calculation efficiency and accuracy. Numerical results show that the iterative calculations in the newly developed dual BEM solution converge rapidly. As examples, wave scattering by several typical breakwaters involving perforated thin plates are solved using the iterative dual BEM model. The corresponding numerical results of the reflection and transmission coefficients are in excellent agreement with the predictions by the analytical solutions. The numerical results also agree reasonably well with experimental data in literature. The present iterative dual BEM model is valuable for analyzing the hydrodynamic performance of breakwaters with perforated thin plates in preliminary engineering design.

Gravity wave interaction with a wave attenuating system

In the present study, gravity wave interaction with a wave attenuating system (WAS) comprised of a series of surface piercing thin vertical porous plates fixed on a fully submerged rigid pontoon is analyzed. For the wave past porous barriers, a quadratic pressure drop is considered to account for the energy dissipation due to changes in wave steepness. A numerical model based on iterative multi-domain boundary element method is developed under the assumption of a small amplitude water wave theory. The scattering coefficients (reflection and transmission) and wave force coefficients (horizontal, vertical and moments) are computed for different wave and structural parameters such as the number of plates, relative plate width, the relative depth of submergence, plate porosities, and relative wave steepness. The results demonstrate that with the creation of trap zones through multiple porous plates on a submerged rigid pontoon, it is possible to reduce wave transmission by 60–70%. For optimum performance across all water depths, it is suggested for the WAS to have 1 ≤ B/h ≤ 1.5 and h1/h ≤ 0.3. Also, interestingly there is a 25–30% reduction in the vertical wave forces which reduces the load acting on the ancillary equipment. The WAS is found to perform well for porosity μ=20%, above which the wave transmission increases gradually. The study results can be useful in the cost-effective design of this type of special wave barrier for a range of coastal engineering field applications.

Comparison of analytical and numerical solutions for wave interaction with a vertical porous barrier

Analytical solutions for wave interaction with a vertical porous barrier are presented. The analytical solutions are derived using two different methods for taking the depth-average of the pressure drop across the porous barrier. Both solutions assume that the evanescent modes in the wave field can be neglected. The results from the analytical models are compared to results from an iterative boundary element method (BEM) model. The BEM model shows that neglecting evanescent modes is a reasonable assumption for long waves, but that for short waves the velocity through the porous wall from the evanescent modes can be up to 25% of the velocity from the progressive modes at the free surface. However, the effect of neglecting the evanescent modes has only a small effect on the depth-averaged velocity through porous wall and the analytical models derived using depth-averaged assumptions are shown to give good agreement with the BEM model for the reflection coefficient, horizontal force and overturning moment on the porous barrier.The analytical models are used to investigate the effects of the drag and inertia coefficients of the porous barrier on the behaviour of the solution. It is shown that for fixed values of the drag coefficient, wave frequency and amplitude, the solutions for the reflection coefficient lie on approximately semicircular arcs on the complex plane, with the position on the arc determined by the inertial coefficient. This places bounds on the size of the phase change in the reflected and transmitted wave that are possible. The analytical models are also used to derive the asymptotic behaviour of the solution in long and short waves. The implications of the results for more general cases of wave interaction with porous structures are discussed.

Wave interaction with multiple slotted barriers inside harbour: Physical and numerical modelling

Effective design of ports, harbours, and marinas is a major challenge due to the utilization of available space, material involvement, and capital investment. One of the major aspects of the design is the construction of cost-effective breakwater for the creation of a tranquillity zone and an increase in the usable area within the harbour. In the present study, the interaction of regular waves with such barriers with the impermeable back wall is investigated experimentally and a numerical model based on the direct iterative boundary element method is validated. In numerical modelling, a quadratic pressure drop is considered for the slotted walls for wave past the porous barriers to account for the effect of wave height on energy dissipation. Effects of wave height, wave period, porosity and the number of walls on the reflection coefficient are investigated. The study reveals that a number of optima are one less than the number of slotted barriers which occurs alternately and the number of minima is always one more than the number of maxima. It is found that the reflection from the slotted wall barrier reduces with an increase in the number of slotted walls, especially for longwave conditions and for porosity greater than 30%. Whilst for porosity less than 30%, adding more number of slotted walls do not contribute to a reduction in wave reflections. An array of slotted vertical barrier arranged in a zigzag manner is found to be an effective mechanism for enhancing the wave energy dissipation, more space creation and a significant reduction in material requirement and easy installation procedure.

Prediction of Hydrodynamic Performance of 3-D WIG by IBEM

The hydrodynamic performance of three-dimensional WIG (Wing-In-Ground) vehicle moving with a constant speed above free water surface has been predicted by an Iterative Boundary Element Method (IBEM). IBEM originally developed for 3-D hydrofoils moving under free surface has been modified and extended to 3-D WIGs moving above free water surface. The integral equation based on Green's theorem can be divided into two parts: (1) the wing part, (2) free surface part. These two problems are solved separately, with the effects of one on the other being accounted for in an iterative manner. Both the wing part including the wake surface and the free surface part have been modelled with constant strength dipole and source panels. The effects of Froude number, the height of the hydrofoil from free surface, the sweep, dihedral and anhedral angles on the lift and drag coefficients are discussed for swept and V-type WIGs.

Numerical Simulation of Cushioning Problem for Blunt Bodies Using Boundary Element Method

Abstract Induced air pressure and resulting free surface profile due to air cushioning layer is studied. The study is mainly focused on 2D blunt circular bodies with constant downward speed. The problem is first solved for the air flow between the body and the free surface of the water. Then the results are employed to solve the problem for the water problem, numerically. Both air and water problem are assumed to be governed by Laplace potential equation. Depending on the induced pressure and velocity of the escaping air flow from cushioning layer, compressibility of the air is also included in the modeling. Gravitational acceleration is also included in the model. An iterative boundary element method is used for numerical solution of both air and water problems. Instantaneous pressure distribution and free surface profile are evaluated for different bodies. The results of calculation for large blunt bodies show that inviscid potential method can fairly approximate the problem for large blunt bodies. Additionally, the behavior of the air pressure for the very blunt body is impulsive and the magnitude of the peak pressure is in order of impact pressure of water entry. The obtained results are compared with analytical method. The comparison shows that as the bluntness of a body increases, the better agreement is concluded.

Iterative boundary element method for crack analysis of two-dimensional piezoelectric semiconductor

Based on the well-developed boundary element methods for piezoelectric media and conductors, we present an iterative boundary element method to solve the boundary value problems in two-dimensional piezoelectric semiconductors (PSCs). The proposed method is verified by analyzing a piezoelectric semiconductor plate under multi-field load. Two typically important boundary value problems, a hole and a crack, are studied in PSC plates by using the proposed method. The stress concentration near the edge of an elliptical hole in a finite piezoelectric semiconductor plate is studied by using the single-domain boundary element method. Also, by using the sub-domain boundary element method, we analyzed how the mechanical load, electrical load, electric current density, and initial electron density affected the stress, electric displacement and electric current intensity factors near the crack tip.

Iterative multi-domain BEM solution for water wave reflection by perforated caisson breakwaters

This study develops a full solution for water wave reflection by a partially perforated caisson breakwater with a rubble mound foundation using multi-domain BEM (boundary element method). Regular and irregular waves are both considered. A quadratic pressure drop condition on caisson perforated wall is adopted, and direct iterative calculations are performed. Due to the use of quadratic pressure drop condition, the effect of wave height on the energy dissipation by the perforated wall is well considered. This study also develops an iterative analytical solution for wave reflection by a partially perforated caisson breakwater on flat bottom using matched eigenfunction expansion method. The reflection coefficients calculated by the multi-domain BEM solution and the analytical solution are in excellent agreement. The present calculated results also agree reasonably well with experimental data from different literatures. Suitable values of discharge coefficient and blockage coefficient in the quadratic pressure drop condition are recommended for perforated caissons. The effects of the wave steepness, the blockage coefficient of perforated wall and the relative wave chamber width on the reflection coefficient are clarified. The present BEM solution is simple and reliable. It may be used for predicting the reflection coefficients of perforated caisson breakwaters in preliminary engineering design.