Boundary Element Method Solution Research Articles

New analytical solutions of oblique wave scattering by submerged horizontal perforated plates using quadratic pressure drop condition

This study develops new analytical solutions for oblique wave scattering by horizontal perforated plates submerged below the water free surface using a quadratic pressure drop boundary condition. Both kinds of submerged plates are considered: a) A horizontal perforated plate as an offshore breakwater; b) A horizontal perforated plate attached to a vertical wall as a wave absorber. The velocity potential decomposition method combined with an efficient iterative algorithm is adopted to obtain the analytical solutions, and the hydrodynamic parameters of engineering interest for the reflection, transmission and energy-loss coefficients as well as the vertical wave force are considered. A new simple formula for predicting the discharge coefficient in the quadratic pressure drop condition, which is critical to make the analytical solutions to be practically applicable, is developed by fitting the analytical results and an amount of experimental data in literature. The present analytical solutions are in good agreement with previous analytical solution for special case and the numerical results of an independently developed multi-domain boundary element method solution. The reflection and transmission coefficients calculated by the present analytical solutions are also in good agreement with experimental data when adopting a well-predicted discharge coefficient. The present analytical solutions are valuable for fast assessing the hydrodynamic performance of horizontal perforated plates in the initial stage of engineering design. The present analytical model can also be readily extended to the multi-layer horizontal perforated plates and similar structures.

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.

A blind test on floats in extreme waves using a transient potential flow model

Numerical modelling of wave–structure interactions is a cost-effective alternative to physical models in understanding of marine energy device system behaviours and performance, though it is important to establish the reliability of the numerical model first. In this paper, experimental studies in which two surface-piercing floating buoys were subject to a series of focused waves have been modelled using open-source wave energy converter time-domain dynamics modelling software, WEC-Sim. The results were obtained as a blind test, with no knowledge of the experimental data and have not been tuned to match the experiment. Hydrodynamic forces and buoy motions are modelled in the time domain based on the experimental wave elevations, using linear frequency-domain hydrodynamic coefficients estimated by boundary element method solver, Nemoh. Drag coefficients were calculated through computational fluid dynamics, but found to be insignificant. When compared with the experimental results, the WEC-Sim model shows excellent agreement in its prediction of some motions, but less favourable in others. It is thought that non-linear fluid–structure interactions apparent in the experimental data are missed in the WEC-Sim predictions, creating differences in the results. This research study demonstrates that a significant benefit of linear models is their low computational expense when compared to other methods.

Exploiting the limit of BEM solvers in moonpool type floaters

Solvers based on Boundary Element Method (BEM) are fast and they have proven to provide accurate results in a wide range of applications. On the other hand, computational fluid dynamics (CFD) solvers are high-fidelity tools able to account for viscous effects. However, they are computationally demanding. In the present work, the limitations of BEM solvers are exploited considering the case study of a moonpool type floater in which the viscous effects near the sharp edges of the body (vortex shedding) are not negligible. The BEM results are compared with the results from an unsteady Reynolds averaged Navier-Stokes (URANS) CFD solver and experimental data, while viscous corrections of the BEM method are assessed.

A fast boundary element method using the Z‐transform and high‐frequency approximations for large‐scale three‐dimensional transient wave problems

SummaryThree‐dimensional (3D) rapid transient acoustic problems are difficult to solve numerically when dealing with large geometries, because numerical methods based on geometry discretization (mesh), such as the boundary element method (BEM) or the finite element method (FEM), often require to solve a linear system (from the spacial discretization) for each time step. We propose a numerical method to efficiently deal with 3D rapid transient acoustic problems set in large exterior domains. Using the ‐transform and the convolution quadrature method, we first present a straightforward way to reframe the problem to the solving of a large amount (the number of time steps, M) of frequency‐domain BEMs. Then, taking advantage of a well‐designed high‐frequency approximation, we drastically reduce the number of frequency‐domain BEMs to be solved, with little loss of accuracy. The complexity of the resulting numerical procedure turns out to be O(1) in regard to the time discretization and for the spacial discretization, the latter being prescribed by the complexity of the used fast BEM solver. Examples of applications are proposed to illustrate the efficiency of the procedure in the case of fluid‐structure interaction: the radiation of an acoustic wave into a fluid by a deformable structure with prescribed velocity and the scattering of an abrupt wave by simple and realistic geometries.

The Dual-Reciprocity Boundary Element Method Solution for Gas Recovery from Unconventional Reservoirs with Discrete Fracture Networks

SummaryIn this paper, we present a novel application of the dual-reciprocity boundary-element formulation (DRBEM) to model compressible (gas) fluid flow in tight and shale-gas reservoirs containing arbitrary distributed finite- or infinite-conductivity discrete fractures. Compared with the standard boundary-element method (BEM), the DRBEM transforms the nonlinear domain integrals at the righthand side (RHS) of BEM formulations for nonlinear partial differential equations into equivalent boundary integrals. This transformation allows retention of the domain-integral-free, boundary-integral-only character of standard BEM approaches. The proposed approach is based on coupling DRBEM with the finite-volume method (FVM) in which a multidimensional system is solved by integrating over a line with random fractures. The resulting system of equations is solved simultaneously for fracture and matrix boundary conditions by combining DRBEM and FVM without invoking any approximation for pressure-dependent nonlinear terms such as gas viscosity and compressibility. Numerical examples and field cases are presented to test the validity and showcase the capabilities of the proposed approach. The proposed model provides a general framework that can be applied to a variety of well and fracture geometries and operating schedules, and it is used to analyze production behavior for these complex systems. To the best of the authors’ knowledge, this is the first successful application of the dual-reciprocity principle to the BEM analysis of massively fractured horizontal wells (MFHWs) performance in natural-gas formations in which nonlinear, pressure-dependent gas properties are captured without approximation.

Oblique wave scattering by porous breakwaters/seawalls: Novel analytical solutions based on contour integral without finding complex roots

This article develops novel analytical solutions to oblique wave scattering by porous rubble-mound structures based on linear potential theory. The novelty of the solutions is that the usual complex root finding algorithm is avoided by applying a contour integral technique. In the solving procedure, the whole fluid domain is divided into multiple regions, and the series solution of the velocity potential in each region is obtained using the separation of variables. A linear equation system for the expansion coefficients is achieved by matching the interface conditions between adjacent regions. The matrix elements, which initially include the complex roots of complex dispersion relations, are evaluated by applying the contour integral technique. The evaluation process is free of complex roots, and thus the difficulties arising in solving the complex dispersion relations in traditional solutions are avoided. Case studies show that the series solution converge rapidly, and the calculation results of the new solutions agree well with those of previous analytical solutions with complex roots and multi-domain boundary element method (BEM) solutions. Besides providing a simple and elegant approach for analyzing the hydrodynamic performance of porous structures, the new solutions can be extended to water wave problems involving other dissipative coastal structures.

Analytical and experimental studies on Bragg scattering of water waves by multiple submerged perforated semi-circular breakwaters

This study develops analytical solutions for Bragg scattering of water waves propagating over a series of submerged perforated semi-circular breakwaters (bars) based on potential theory. The series solutions for velocity potentials in the external common fluid domain of the bars are expressed through multipole expansions, whilst those in the internal fluid domains are given using the separation of variables. The continuous condition of fluid velocity and a quadratic pressure drop condition on the perforated semi-circular arcs are applied to determine the unknowns in the series solutions. In the analytical solutions, obliquely and normally incident waves are independently considered. The calculation methods of the reflection, transmission and energy loss coefficients are given. Experimental tests are performed to measure the reflection and transmission coefficients of the breakwaters at different wave periods and water depths. The analytical solutions are in excellent agreement with the independently developed multi-domain BEM (boundary element method) solutions, and are also compared with the wide-spacing approximations and the experimental results. Calculation examples are presented to check the effects of various parameters (e.g., bar number, bar radius, bar spacing and incident wave angle) on the hydrodynamic quantities of the breakwaters. Main valuable results for engineering application are draw.

Iterative multipole solution for wave interaction with submerged partially perforated semi-circular breakwater

This study develops an iterative multipole solution for water wave interaction with a submerged partially perforated semi-circular breakwater based on the linear potential theory. Multipole expansion method and separation variable method are used for getting the velocity potential expressions of the flow fields, and the expansion coefficients in velocity potentials are determined by iteratively solving the non-linear pressure loss boundary condition on the partially perforated wall. The calculation results of the present multipole solution are in good agreement with the numerical results of multi-domain BEM (boundary element method) solution. The effects of the radius, porosity and perforated position on the reflection coefficient, transmission coefficient and wave force of breakwater are thoroughly discussed. Some meaningful suggestions are presented for engineering design.

Closed-Form Evaluation of Potential Integrals in the Boundary Element Method

A method is presented for the closed-form evaluation of the singular and near-singular integrals arising in the Boundary Element Method solution of the Helmholtz equation. An error analysis is presented for the numerical evaluation of such integrals on a plane element, and used to develop a criterion for the selection of quadrature rules. The analytical approach is based on an optimized expansion of the Green’s function for the problem, selected to limit the error to some required tolerance. Results are presented showing accuracy to tolerances comparable to machine precision.

Fast BEM Solution for 2-D Scattering Problems Using Quantized Tensor-Train Format

It is common to accelerate the boundary element method (BEM) by compression techniques [fast multipole method (FMM), H-matrix/adaptive cross approximation (ACA)] that enable a more accurate solution or a solution in higher frequency. In this article, we present a compression method based on a transformation of the linear system into the tensor-train format by the quantization technique. The method is applied to a scattering problem on a canonical object with a regular mesh and improves the performance obtained from existing methods.

Parametric model for the reconstruction and representation of hydrofoils and airfoils

In this paper, we present a geometric parametric model that has been mainly designed to accommodate the need for an appropriate analysis model in the context of IsoGeometric Analysis (IGA) [1], but can be also employed as a universal tool for reconstructing and representing a wide range of different airfoils' and/or hydrofoils' families. This model is an extension of the parametric model employed in optimization of hydrofoils' efficiency with the aid of an isogeometric boundary element method solver (IGA-BEM), presented in [2]. Apart from its apparent use in design evaluation and optimization, the parametric model can be further employed in a wider range of relevant applications. Specifically, in this work we begin by demonstrating its use, as an alternative approach, in efficiently reconstructing airfoil and hydrofoil shapes from large sets of points appearing, mainly, in the context of reverse engineering applications. This is further investigated and the parametric model's performance in accurately representing existing airfoils is extensively benchmarked against results from pertinent methods and approaches followed in the aeronautics and aerospace research community. Finally, we demonstrate the use of the parametric model in a foil shape optimization problem. We conclude this work with a discussion on future enhancements and possible research directions.

BEM solutions to a class of elliptic BVPs for anisotropic trigonometrically graded media

Boundary value problems (BVPs) governed by a class of elliptic equations for anisotropic trigonometrically graded media are solved using Boundary Element Method (BEM). The variable coefficient governing equation is transformed to a constant coefficient equation which is then transformed to a boundary integral equation. The results show the convergence, consistency, and accuracy of the BEM solutions.

Diffusion convection-reaction equation in exponentially graded media of incompressible flow: Boundary element method solutions

We setup boundary value problems (BVPs) for exponentially graded media. The BVPs are governed by a diffusion convection-reaction (DCR) equation of spatially varying coefficients (with an anisotropic diffusion coefficient). A boundary element method (BEM) is utilized to find numerical solutions to BVPs. The numerical solutions obtained verify the validity of the analysis used to derive the BEM with accurate and consistent solutions. The computation shows that the BEM procedure elapses very efficient time in producing the solutions. In addition, the results show the effect of anisotropy and inhomogeneity of the media on the solutions. To illustrate the application, an example of a layered material is presented.

Boundary value problems governed by Helmholtz equation for anisotropic trigonometrically graded materials: A boundary element method solution

Trigonometrically graded media of anisotropic diffusion coefficient are under consideration. Boundary value problems (BVPs) of such kind of media, governed by a Helmholtz type equation, are solved numerically using a boundary element method (BEM). A technique of transforming the variable coefficient governing equation to a constant coefficient equation is utilized for deriving a boundary integral equation. Some particular problems are considered to illustrate the application of the BEM. The results show convergence, accuracy and consistency between the scattering and flow solutions. The results also show efficiency of the BEM procedure for producing the solutions in a short elapsed computation time length. Moreover the results indicate the effect of large wave number on the accuracy and the impact of the inhomogeneity and anisotropy of the material on the solutions.

Numerical simulation of modified Helmholtz boundary value problems for anisotropic exponentially graded materials

In this paper we consider the modified Helmholtz type equation governing interior two-dimensional boundary value problems (BVPs) for anisotropic functionally graded materials (FGMs) with Dirichlet and Neumann boundary conditions. Persistently spatially changing diffusivity and leakage factor coefficients are involved in the governing equation. Both the anisotropic diffusivity and leakage factor coefficients vary according to an exponential gradation function. We use a technique of transforming the variable coefficient governing equation to a constant coefficient equation for deriving a boundary integral equation. And from the boundary integral equation obtained a standard boundary element method (BEM) is constructed to find numerical solutions to the BVPs. In order to illustrate the application of the BEM, some particular examples of BVPs are solved. The results show the convergence, accuracy, consistency between the scattering and flow solutions and efficiency (less computation time) of the BEM solutions. The results also show the impact of the inhomogeneity and anisotropy of the material on the solutions.

Numerical solutions to BVPs governed by the anisotropic modified Helmholtz equation for trigonometrically graded media

A Boundary Element Method (BEM) is used for obtaining solutions to anisotropic trigonometrically graded media (FGM) boundary value problems (BVPs) governed by the modified Helmholtz type equation. A technique of transforming the variable coefficient governing equation to a constant coefficient equation is utilized for deriving a boundary integral equation. Some particular problems are considered to illustrate the application of the BEM. The results show the convergence, consistency, and accuracy of the BEM solutions.

Numerical solutions to anisotropic BVPs for quadratically graded media governed by a Helmholtz equation

A Boundary Element Method (BEM) is used for obtaining solutions to anisotropic quadratically graded media boundary value problems (BVPs) governed by a Helmholtz type equation. A technique of transforming the variable coefficient governing equation to a constant coefficient equation is utilized for deriving a boundary integral equation. Some particular problems are considered to illustrate the application of the BEM. The results show the convergence, consistency, and accuracy of the BEM solutions.

A boundary element method solution to spatially variable coefficients diffusion convection equation of anisotropic media

The diffusion-convection equation with variable coefficients and for anisotropic media which were previously considered by other authors are discussed again in this paper to find their numerical solutions by using the boundary element method (BEM). The numerical results obtained show the consistency and accuracy of the BEM solutions. Also, the solutions exhibit the impact of the anisotropy and inhomogeneity (spatial variability) of the media.

Numerical solutions to anisotropic FGM BVPs governed by the modified Helmholtz type equation

The Boundary Element Method (BEM) is used for obtaining solutions to anisotropic functionally graded media (FGM) boundary value problems (BVPs) governed by the modified Helmholtz type equation. A technique of transforming the variable coefficient governing equation to a constant coefficient equation is utilized for deriving a boundary integral equation. Some particular problems are considered to illustrate the application of the BEM. The results show the convergence, consistency, and accuracy of the BEM solutions.