Boundary Integral Equation Technique Research Articles

Interfacial hydroelastic hydraulic falls and trapped waves over bottom obstacles

Steady nonlinear flexural-gravity hydraulic falls on the interface of a two-layer density stratified flow past a submerged obstruction on the bottom of a channel are considered. The fluid is assumed to be ideal, and the flow is irrotational. We extend the previous works (Dias and Vanden-Broeck, 2002; Wang et al., 2022) by including the effect of hydroelasticity. The interface is modeled as a thin elastic shell with the Cosserat theory. The boundary integral equation techniques are employed to find steady solutions by numerically solving the full Euler equations. New solutions characterized by subcritical flow upstream with different depth ratios (thick upper layer, thick bottom layer, or critical depth) are found, and the effects of the aspect ratio of obstruction are investigated. By introducing a second obstruction downstream, solutions characterized by a train of trapped waves are sought with wavelength coherent with the prediction of the linear dispersion relation. In addition, solutions with a soliton-like form and oscillatory decaying tails are found when the sheet rigidity is small and the second obstruction is placed upstream.

Analysis of Two-Dimensional Heat Transfer Problem Using the Boundary Integral Equation

In this paper, we examine the problem of two-dimensional heat equations with certain initial and boundary conditions being considered. In a two-dimensional heat transport problem, the boundary integral equation technique was applied. The problem is expressed by an integral equation using the fundamental solution in Green’s identity. In this study, we transform the boundary value problem for the steady-state heat transfer problem into a boundary integral equation and drive the solution of the two-dimensional heat transfer problem using the boundary integral equation for the mixed boundary value problem by using Green’s identity and fundamental solution.

Numerical solution of a cavity problem under surface tension effect

Free‐surface flow past two inclined plate is considered. The “Riabouchinsky model” has been chosen to close the cavitiy. The fluid is assumed to be inviscid and incompressible and the flow to be two dimensional, steady, and irrotational. Surface tension is included in the dynamic boundary conditions, but the effects of gravity are neglected. The problem is solved numerically using boundary integral equation techniques. More specifically, the numerical method used is based on an integro‐differential equation reformulation. Numerical solutions are found for different values of the angle of inclination γ and for various values of the inverse Weber number δ.

Free surface flows over a successive obstacles with surface tension and gravity effects

The problem of steady two-dimensional flow of a fluid of finite depth over a successive obstacles is considered. Both gravity and surface tension are taken into account in the dynamic boundary conditions. The fluid is assumed to be inviscid, incompressible and the flow to be irrotational. The flow is characterized by the two parameters, the Froude number $Fr$ and the inverse Weber number $\delta$. The fully non-linear problem is solved numerically by using the boundary integral equation technique. The numerical solutions for sub-critical ($Fr 1)$ are presented for various values of $Fr$ and $\delta$. The effects of surface tension and gravity on the shape of the free surface are discussed, and solution diagrams for all flow regimes are presented.

Analyzing bowtie structures with sharp tips by a vertical mode expansion method.

Bowtie structures of metallic nanoparticles are very effective in producing strong local fields needed in many applications. Existing numerical studies on bowtie structures are limited to those with rounded tips. Due to the field singularities at sharp edges and corners, accurate numerical solutions for bowtie structures with mathematically sharp tips are difficult to obtain. Based on an improved vertical mode expansion method (VMEM) that incorporates boundary integral equation techniques for domains with corners, we analyze bowtie structures with truly sharp tips. Numerical results are presented to reveal the effects of a few key factors including the distance between the tips, the apex angle and the substrate.

Solitary flexural-gravity waves in three dimensions.

The focus of this work is on three-dimensional nonlinear flexural–gravity waves, propagating at the interface between a fluid and an ice sheet. The ice sheet is modelled using the special Cosserat theory of hyperelastic shells satisfying Kirchhoff's hypothesis, presented in (Plotnikov & Toland. 2011 Phil. Trans. R. Soc. A 369, 2942–2956 (doi:10.1098/rsta.2011.0104)). The fluid is assumed inviscid and incompressible, and the flow irrotational. A numerical method based on boundary integral equation techniques is used to compute solitary waves and forced waves to Euler's equations.This article is part of the theme issue ‘Modelling of sea-ice phenomena’.

Overresolving in the Laplace Domain for Convolution Quadrature Methods

Convolution quadrature (CQ) methods have enjoyed tremendous interest in recent years as an efficient tool for solving time-domain wave problems in unbounded domains via boundary integral equation techniques. In this paper we consider CQ type formulations for the parallel space-time evaluation of multistep or stiffly accurate Runge--Kutta rules for the wave equation. In particular, we decouple the number of Laplace domain solves from the number of time steps. This allows us to overresolve in the Laplace domain by computing more Laplace domain solutions than there are time steps. We use techniques from complex approximation theory to analyze the error of the CQ approximation of the underlying time-stepping rule when overresolving in the Laplace domain and show that the performance is intimately linked to the location of the poles of the solution operator. Several examples using boundary integral equation formulations in the Laplace domain are presented to illustrate the main results.

Energy transfer in a liquid filled elemental passage of a porous medium for permeability enhancement due to pulsations of a vapor bubble

In this paper, a novel method which has been proposed during the last decade for increasing of the permeability of porous media of petroleum reservoirs by transferring of energy via ultrasound waves is investigated numerically. Increasing of permeability of porous media of petroleum reservoirs results in enhancing of oil recovery. This technique is based on the idea of transferring of energy to the liquid filled porous media via the ultrasound waves and consequently producing of pulsating vapor bubbles. The generated vapor bubbles transfer the energy of ultrasound waves in the liquid filled passages of a porous medium through velocity and pressure fields in the liquid domain and in turn apply impact forces on obstacles which have been made from accumulation of clays inside passages of the porous medium around the wellbore of an oil well after a long operating time. Removing of these obstacles, results in increasing of permeability of the porous medium and enhancing of oil recovery. The boundary integral equation technique is employed for numerical simulation of the growth and collapse of the cavitation bubbles and transferring of energy in the liquid domain by the bubbles pulsations. Three different geometries have been proposed as elemental passages of a porous medium. Numerical results show that the controlled growth and collapse of vapor bubbles induced by emission of the ultrasound waves in a liquid filled porous medium could lead to a powerful technology for enhancing oil recovery and consequently for industrial and economical development of the nations and countries having petroleum and other liquid reservoirs.DOI: http://dx.doi.org/10.5755/j01.mech.22.1.14238

The dual reciprocity boundary integral equation technique to solve a class of the linear and nonlinear fractional partial differential equations

In this paper, we apply the boundary integral equation technique and the dual reciprocity boundary elements method (DRBEM) for the numerical solution of linear and nonlinear time‐fractional partial differential equations (TFPDEs). The main aim of the present paper is to examine the applicability and efficiency of DRBEM for solving TFPDEs. We employ the time‐stepping scheme to approximate the time derivative, and the method of linear radial basis functions is also used in the DRBEM technique. This method is improved by using a predictor–corrector scheme to overcome the nonlinearity that appears in the nonlinear problems under consideration. To confirm the accuracy of the new approach, several examples are presented. The convergence of the DRBEM is studied numerically by comparing the exact solutions of the problems under investigation. Copyright © 2015 John Wiley & Sons, Ltd.

Bridged defects in continuously and discretely reinforced solids

The paper examines problems that relate to defects in elastic solids that are reinforced by aligned fibres. The category of problems deals with flaw bridging that can occur as a result of continuity of fibres across the defect in the matrix region. The defects can be a flaw of finite dimensions or cracks in the conventional sense. The presence of fibre continuity across a flaw exerts a displacement-dependent boundary condition at the faces of the crack that can alter the stress state at the boundary of the defect and contribute to fracture generation. The analysis of both a spheroidal flaw with fibre bridging and an idealized penny-shaped crack with fibre continuity across the faces of the crack leads to the conclusion that the stress amplification usually associated with extension of the flaw is suppressed by the fibre continuity. The second type of problem deals with the mechanics of flaws that can emanate from the extremities of an isolated cylindrical fibre in an elastic matrix of infinite extent. The problem is examined using a computational approach based on the boundary integral equation technique. The modelling is used to examine the role of the fibre–matrix elasticity mismatch on the stress intensity factors at the tip of penny-shaped cracks emanating from the ends of the fibre.

Time dependent hydraulic falls and trapped waves over submerged obstructions

We consider the classical problem of a free surface flowing past one or more disturbances in a channel. The fluid is assumed to be inviscid and incompressible, and the flow, irrotational. Both the effects of gravity and surface tension are considered. The stability of critical flow steady solutions, which have subcritical flow upstream of the disturbance and supercritical flow downstream, is investigated. We compute the initial steady solution using boundary integral equation techniques based on Cauchy integral formula and advance the solution forward in time using a mixed Euler-Lagrange method along with Adams-Bashforth-Moulton scheme. Both gravity and gravity-capillary critical flow solutions are found to be stable. The stability of solutions with a train of waves trapped between two disturbances is also investigated in the pure gravity and gravity-capillary cases.

Modeling of sound propagation in the vicinity of rigid porous ground by boundary element method

A standard boundary integral equation (BIE) technique takes advantage of the well-known Green’s function for the sound fields above an impedance ground. It leads to a simplified solution that enables the development of efficient two-dimensional boundary element numerical codes for computing sound fields above a locally reacting ground. The method is particularly useful if there are multiple reflecting surfaces and the presence of a mixed impedance ground in the proximity of sources and receivers. However, the BIE formulation cannot be applied readily for calculating the sound fields above a non-locally reacting ground because the corresponding Green’s function is generally not available. The so-called two-domain approach was frequently used to model the sound propagation inside the porous material by assuming it as a dissipative fluid medium. By matching the particle velocities and pressures at the interface, a two-domain BIE formulation can be developed for modeling the sound propagation above a rigid porous ground. Use of the two-domain approach has a significant impact on the required computational resources. The current paper presents an alternative method aiming to reduce the computational time by exploring the use of an accurate Green’s function for the development of a BIE formulation above a non-locally reacting ground.

Hydraulic falls under a floating ice plate due to submerged obstructions

AbstractSteady two-dimensional nonlinear flexural–gravity hydraulic falls past a submerged obstruction on the bottom of a channel are considered. The fluid is assumed to be ideal and is covered above by a thin ice plate. Cosserat theory is used to model the sheet of ice as a thin elastic shell, and boundary integral equation techniques are then employed to find critical flow solutions. By utilising a second obstruction, solutions with a train of waves trapped between two obstructions are investigated.

Decomposition in regular and singular parts (in domains with corners) and stability under perturbation of the geometry

We analyze the Dirichlet problem for the Laplacian in a polygonal domain where boundary and angles depend on a parameter. We use the boundary integral equation, localization and Mellin transformation techniques to show that the solution has a decomposition in regular and singular parts which blow up at certain exceptional angles. We derive a modified decomposition which depends continuously on the angle.

The influence of surface tension upon trapped waves and hydraulic falls

We consider steady two-dimensional free-surface flows past submerged obstructions on the bottom of a channel. The flow is assumed to be irrotational, and the fluid inviscid and incompressible. Both the effects of gravity and surface tension are considered. Critical flow solutions with subcritical flow upstream and supercritical flow downstream are sought using fully nonlinear boundary integral equation techniques based on the Cauchy integral formula. When a second submerged obstruction is included further upstream in the flow configuration in the absence of surface tension, solutions which have a train of waves trapped between the two obstacles before the critical flow have already been found (Dias and Vanden-Broeck 2004 [2]). We extend this work by including the effects of surface tension. Trapped wave solutions are found upstream for small values of the Bond number, for some values of the Froude number. Other types of trapped waves are found for stronger tension when the second obstruction is placed downstream of the hydraulic fall generated by the first obstacle.

Gravity-capillary water waves generated by multiple pressure distributions

Steady two-dimensional free-surface flows subjected to multiple localised pressure distributions are considered. The fluid is bounded below by a rigid bottom, and above by a free-surface, and is assumed to be inviscid and incompressible. The flow is assumed irrotational, and the effects of both gravity and surface tension are taken into account. Forced solitary wave solutions are found numerically, using boundary integral equation techniques, based on Cauchy integral formula. The integrodifferential equations are solved iteratively by Newton's method. The behaviour of the forced waves is determined by the Froude number, the Bond number, and the coefficients of the pressure forcings. Multiple families of solutions are found to exist for particular values of the Froude number; perturbations from a uniform stream, and perturbations from pure solitary waves. Elevation waves are only obtained in the case of a negatively forced pressure distribution.

Free-surface flow past arbitrary topography and an inverse approach for wave-free solutions

[Received on 8 December 2012; revised on 19 March 2013; accepted on 20 March 2013] An efficient numerical method to compute nonlinear solutions for 2D steady free-surface flow over an arbitrary channel bottom topography is presented. The approach is based on a boundary integral equation technique which is similar to that of Vanden-Broeck’s (1996). The typical approach for this problem is to prescribe the shape of the channel bottom topography, with the free surface being provided as part of the solution. Here we take an inverse approach and prescribe the shape of the free surface ap rioriand solve for the corresponding bottom topography. We show how this inverse approach is particularly useful when studying topographies that give rise to wave-free solutions, allowing us to easily classify 11 basic flow types. Finally, the inverse approach is also adapted to calculate a distribution of pressure on the free surface, given the free surface shape itself.

Numerical Simulations and Analysis of Wide-Band Range-Resolved HF Backscatter From Evolving Ocean-Like Surfaces

Low-grazing wide-band HF radar echoes from evolving ocean-like surfaces in 2-D space are simulated employing a first-principles boundary integral equation technique. The synthesized signals achieve range resolution of about 15 m and correspond to observations with a hypothetical narrow-beam system having a 10-MHz effective bandwidth. Range-time plots of backscatter magnitude appear to contain a pattern consistent with the long surface waves. The double Fourier transform of the signal magnitude or power in range and time reveals strong harmonics located along the dispersion curve of the propagating gravity waves. The effect is discernable due to the high range resolution employed and is not predicted by the first-order Small Perturbation Method (SPM) approximation. It is explained by considering the next, second order of the SPM under certain simplifying assumptions such as perfect surface conductivity (with the Norton attenuation factor set to 1). The derived analytical expression is then used to investigate the feasibility of the surface profile retrieval from such time-varying range-resolved HF backscatter (gathered in the course of the “numerical experiment” through the exact scattering simulations). For the case of perfectly conducting sea-like surface developed under 10-m/s wind conditions, the results are very encouraging. The approach is sensitive to the surface harmonics with wavelengths far exceeding the Bragg resonance scales of the HF band. Extending this type of analysis to cases with realistic finite conductivity and/or rougher surface (assuming that the second-order SPM is still valid) will require proper accounting for ground-wave propagation effects in perturbation expressions. The study offers a glimpse into ocean observation possibilities that the wide-band HF systems can provide if and when they become a reality.

An Efficient 1-D Periodic Boundary Integral Equation Technique to Analyze Radiation onto Straight and Meandering Microstrip Lines

A modeling technique to analyze the radiation onto arbitrary 1-D periodic metallizations residing on a microstrip substrate is presented. In particular, straight and meandering lines are being studied. The method is based on a boundary integral equation, more specifically on a mixed potential integral equation (MPIE), that is solved by means of the method of moments. A plane wave excites the microstrip structure, and according to the Floquet–Bloch theorem, the analysis can be restricted to one single unit cell. Thereto, the MPIE must be constructed using the pertinent 1-D periodic layered medium Green’s functions. Here, these Green’s functions are obtained in closed form by invoking the perfectly matched layer paradigm. The proposed method is applied to assess the radiation onto 1) a semi-infinite plate, 2) a straight microstrip line, and 3) a serpentine delay line. These three types of examples clearly illustrate and validate the method. Also, its efficiency, compared to a previously developed fast microstrip analysis technique, is demonstrated.

Semi-analytic treatment of the three-dimensional Poisson equation via a Galerkin BIE method

A systematic treatment of the three-dimensional Poisson equation via singular and hypersingular boundary integral equation techniques is investigated in the context of a Galerkin approximation. Developed to conveniently deal with domain integrals without a volume-fitted mesh, the proposed method initially converts domain integrals featuring the Newton potential and its gradient into equivalent surface integrals. Then, the resulting boundary integrals are evaluated by means of well-established cubature methods. In this transformation, weakly-singular domain integrals, defined over simply- or multiply-connected domains with Lipschitz boundaries, are rigorously converted into weakly-singular surface integrals. Combined with the semi-analytic integration approach developed for potential problems to accurately calculate singular and hypersingular Galerkin surface integrals, this technique can be employed to effectively deal with mixed boundary-value problems without the need to partition the underlying domain into volume cells. Sample problems are included to validate the proposed approach.