Boundary Integral Equation Approach Research Articles

On a time-discrete convolution—space collocation BEM for the numerical solution of two-dimensional wave propagation problems in unbounded domains

Abstract For the numerical solution of a classical two-dimensional Dirichlet exterior problem for the wave equation in the time domain, we consider a space-time boundary integral equation approach, based on its discretization by means of the coupling of a time discrete convolution quadrature of order 2, with the classical continuous piecewise linear boundary element method. The latter is either the Galerkin method or the collocation one, this being computationally much cheaper. For the efficient evaluation of most of the integrals generated by these two boundary element methods, but also for the subsequent analysis, we first define a new Gaussian quadrature. After recalling that stability and convergence have been proved for the Galerkin method, while for the collocation one we only have numerical evidences of these properties, to (partially) fill this gap, we first note that collocation can be obtained directly from the Galerkin system, after discretizing its inner product integral by using the composite trapezoidal rule, which in this case turns out to be the Gaussian rule mentioned above with $1$ node. Then, we show that for all (positive) time step-sizes $\varDelta _t$, when we let $h\rightarrow 0$, independently from $\varDelta _t$, the matrix of the linear system produced by the collocation method, properly normalized, converges, in the $\infty $-norm, to the corresponding Galerkin matrix. The behavior of the associated absolute and relative errors in the $\infty $-norm, as $h\rightarrow 0$, are $O\big (h^{\frac {5}{3}}+\varDelta _t^{-(1+\varepsilon )}h^{\frac {5+\varepsilon }{2}}[|\!\ln \varDelta _t|+|\!\ln h|]\big )$ and $O\big (h^{\frac {2}{3}}/|\!\ln \varDelta _t|+\varDelta _t^{-(1+\varepsilon )}h^{\frac {3+\varepsilon }{2}}[1+|\!\ln h|/|\!\ln \varDelta _t|]\big )$, respectively, with $\varepsilon>0$ as close as one likes to $0$ and the $O(\cdot )$ constant independent from $\varDelta _t, h$. Some possible extensions of the investigation we carried out are also mentioned.

New three-dimensional time-stepping transient fundamental solutions with applications

In this paper, alternative static-like fundamental solutions for transient dynamic analysis of three-dimensional elasticity are developed. The new formulation is based on modification of Navier's equations of elastodynamics by expressing the acceleration in terms of the displacements at different time steps using suitable finite difference scheme. The new fundamental solutions are derived via operator decoupling technique. The displacements from the previous steps are used to form the new generalized inertia terms. The new fundamental solutions are used to solve non-homogeneous media by dividing the continuous media into multi regions. The stiffness and load vector of each region are calculated using the boundary integral equation approach. Then, the whole multi-region system is solved as super finite elements. Numerical examples are solved to demonstrate the validation of the proposed solutions. The results of the proposed solutions are more stable and accurate than those of previous time domain and dual reciprocity formulations.

An integrated approach to the direct simulation of brasses in the performance environment

The present work deals with the synthesis of sounds produced by brass instruments through the direct physical modelling. The purpose is the development of an integrated methodology for the evaluation of the response of a wind instrument taking into account the properties of the surrounding environment. The identification of the frequency response of the resonator and the performing environment is obtained by means of a Boundary Integral Equation approach. The formulation produces the matrix transfer function between the inflow at the input section of the instrument bore and the signal evaluated at an arbitrary location, and can account for the response of any boundary and object present in the surroundings. The reflection function obtained from the above model is coupled to a simplified model of valve, used to represent the excitation mechanism behaviour. The exploited algorithm has demonstrated to be accurate and efficient in offline calculation, and the observed performance discloses the possibility to implement real–time applications.

A boundary integral equation approach to computing eigenvalues of the Stokes operator

The eigenvalues and eigenfunctions of the Stokes operator have been the subject of intense analytical investigation and have applications in the study and simulation of the Navier–Stokes equations. As the Stokes operator is second order and has the divergence-free constraint, computing these eigenvalues and the corresponding eigenfunctions is a challenging task, particularly in complex geometries and at high frequencies. The boundary integral equation (BIE) framework provides robust and scalable eigenvalue computations due to (a) the reduction in the dimension of the problem to be discretized and (b) the absence of high-frequency “pollution” when using Green’s function to represent propagating waves. In this paper, we detail the theoretical justification for a BIE approach to the Stokes eigenvalue problem on simply- and multiply-connected planar domains, which entails a treatment of the uniqueness theory for oscillatory Stokes equations on exterior domains. Then, using well-established techniques for discretizing BIEs, we present numerical results which confirm the analytical claims of the paper and demonstrate the efficiency of the overall approach.

Small deformation theory for two leaky dielectric drops in a uniform electric field

A small deformation theory for two non-identical spherical drops freely suspended in an ambient fluid and subjected to a uniform electric field is presented. The three phases are assumed to be leaky dielectric (slightly conducting) viscous incompressible fluids and the nonlinear effects of inertia and surface charge convection are neglected. The deformed shapes of the drops are linearized with respect to the electric capillary number that characterizes the balance between the electric stress and the surface tension. When the two drops are sufficiently far apart, their deformed shapes are predicted by Taylor’s small deformation theory—depending on Taylor’s discriminating function, the drops may become prolate, oblate or remain spherical. When the two drops get closer to each other, in addition to becoming prolate/oblate, they start translating and developing an egg shape. (Since there is no net charge, the centre of mass of the two drops remains stationary.) The extent of each of these ‘modes’ of deformation depends on the distance between the drops’ centres and on drop-to-ambient fluid ratios of electric conductivities, dielectric constants and viscosities. The predictions of the small deformation theory for two drops perfectly agree with the existing results of two-drop dynamics simulation based on a boundary-integral equation approach. Moreover, while previous works observed only three types of behaviour for two identical drops—the drops may either become prolate or oblate and move towards each other or become prolate and move away from each other—the small deformation theory predicts that non-identical drops may, in fact, become oblate and move away from each other when the drop-to-ambient fluid conductivity ratios are reciprocal and the drop-to-ambient fluid viscosity ratios are sufficiently large. The presented theory also readily yields an analytical insight into the interplay among different modes of drop deformation and can be used to guide the selection of the phases’ electromechanical properties for two-drop dynamics simulations.

Evaluation of the hypersingular boundary integral equation for acoustic wave-propagation simulations

We have developed a hypersingular boundary integral equation (BIE) approach to simulate wave scattering by inclusions such as karstic cavities and gypsum salts. The solution of conventional BIEs for exterior boundary value problems is of nonuniqueness and numerical instability at characteristic wavenumbers (corresponding to fictitious resonant frequencies). With the hypersingular boundary integral operator, we can handle this problem, but with the difficult-to-calculate hypersingular kernels. Our method to evaluate the singular part in the operator is more efficient than other singularity-elimination techniques. Numerical examples validate the method by comparing with the analytical solution.

Fast computation of electric field and capacitance matrix of transformer windings with boundary element method and adaptive cross approximation

Accurate and fast computation of electric fields is of great interest in transformer design. This article presents a fast and precise method of computation of electric field of transformer windings based on the boundary element method and integral equations approach. Additionally, computation of the capacitance matrix of a transformer geometry is presented. Proposed methodology depends on the knowledge of the transformer geometry. Improvement in computational costs and memory requirements of the developed electrostatic solver is achieved by implementation of the adaptive cross approximation algorithm. Results are benchmarked against commercial software based on the finite element method and applicability of the developed solver is proven.

Two boundary integral equation methods for linear elastodynamics problems on unbounded domains

We consider (transient) 3D elastic wave propagation problems in unbounded isotropic homogeneous media, which can be reduced to corresponding 2D ones. For their solution, we propose and compare two boundary integral equation approaches, both based on the coupling of a discrete time convolution quadrature with a classical space collocation discretization. In the first approach, the PDE problem is preliminarily replaced by the equivalent well known (vector) space–time boundary integral equation formulation, while in the second, the same PDE is replaced by a system of two (coupled) wave equations, each one of which is then represented by the associated boundary integral equation. The construction of these two approaches is described and discussed. Some numerical testing are also presented.

A Potential-Based Formalism for Modeling Local and Hydrodynamic Nonlocal Responses From Plasmonic Waveguides

In this paper, we propose a general boundary integral equation (BIE) approach for solving both exterior (e.g., scattering) and interior (e.g., guided wave propagation) problems involving general plasmonic waveguiding structures with arbitrary cross-sectional geometries and with a continuous translational symmetry in the direction of wave propagation. Contrary to the field-based approach which deals with the electric and magnetic fields, we employ a potential-based formalism instead, involving vector and scalar potentials, and match them at the media interfaces. The proposed approach can handle not only conventional plasmonic waveguides on the order of a few hundred nanometers but also those whose critical sizes are a few nanometers and in which the nonlocal hydrodynamic effects need to be accounted for. The BIEs describing the interaction of light with the plasmonic waveguide are solved by the method of moments (MoM) algorithm. Two illustrative examples, the first one of which deals with the scattering problem, while the second computes the dispersion diagram of a plasmonic cylinder, are considered for both local and nonlocal cases. An excellent agreement is observed between the numerical and theoretical results, with the latter being derived from the generalized Mie theory.

Method of singular integral equations in liquid vibration problems for coaxial shells

The paper deals with the problem of free vibrations of an ideal incompressible fluid in coaxial shells of revolution. It is assumed that the motion of the fluid is irrotational that allows us to introduce the velocity potential. In these suppositions the potential is satisfied to Laplace equation. The boundary conditions are formulated on the wetted surfaces of the shells and on the free liquid surface. The non-penetration conditions are applied to the wetted surfaces. On the free surface we consider dynamical and kinematical boundary conditions. The dynamical condition consists in equality of the liquid pressure on the free surface to the atmospheric one. The kinematic condition requires that total time derivative of the free surface elevation will be equal to zero at any instant. Regarding the potential of velocities, a boundary value problem is formulated that is further reduced to the eigenvalue problem. To solve the boundary value problem for the Laplace equation, the boundary element method is used in a direct formulation. The axial symmetric form of the shells allows us to reduce the obtained system of singular equations to one-dimensional equations. The kernels in singular operators of obtained integral equations are expressed on terms of elliptical integrals of the first and second kinds, and have the logarithmic singularities. The special numerical technique is elaborated to treat with such kind integral equations. The resulting one-dimensional singular equation is solved by the method of discrete singularities. The integration region contains the free surface of the fluid that in the case of coaxial shells is a ring. So, the possibility of using the boundary integral equation approach coupled with application of the discrete singularities method is established to solution of the singular integral equation with incoherent boundaries. A numerical study has been carried out that made it possible to determine the frequencies and modes of the liquid sloshing in the shells for different ratios of the inner and outer radii of cylindrical coaxial shells. The obtained modes of natural vibrations will be used for numerical simulation of forced liquid vibrations in the tanks and reservoirs.

Nonlocal impedance conditions in direct and inverse obstacle scattering

We discuss the use of nonlocal impedance conditions within the use of boundary integral equations for the solution of direct and inverse obstacle scattering problems for penetrable obstacles with constant index of refraction. In the first part, for the classical transmission problem we present an approach that leads to a two-by-two system of nonlinear integral equations in the spirit of the method initiated by Kress and Rundell (2005 Inverse Problems 21 1207–23) rather than the three-by-three system arising from the traditional boundary integral equation approach to the transmission conditions. In the second part we survey on recent work of Cakoni and Kress (2017 Appl. Anal. 96 23–38) on the use of boundary integral equations for the characterization and numerical computation of transmission eigenvalues. In particular, we modify and simplify the analysis by the use of a nonlocal rather than a local impedance condition as in the 2017 paper.

On the numerical solution of the exterior elastodynamic problem by a boundary integral equation method

A numerical method for the Dirichlet initial boundary value problem for the elastic equation in the exterior and unbounded region of a smooth closed simply connected 2-dimensional domain, is proposed and investigated. This method is based on a combination of a Laguerre transformation with respect to the time variable and a boundary integral equation approach in the spatial variables. Using the Laguerre transformation in time reduces the time-depended problem to a sequence of stationary boundary value problems, which are solved by a boundary layer approach resulting to a sequence of boundary integral equations of the first kind. The numerical discretization and solution are obtained by a trigonometrical quadrature method. Numerical results are included.

Rigorous Full-Wave Resistance and Inductance Computation of 3-D Interconnects

A full-wave technique to rigorously extract the resistance and the inductance of 3-D interconnects is proposed. A novel and full 3-D differential surface admittance operator and a boundary integral equation approach are combined in a circuit interpretation which provides a simple way to evaluate the characteristics of the interconnects. The method is accurate for various examples over a broad frequency range, thus conforming its appositeness for the modeling of 3-D interconnects.

Scattering problems of the SH Wave by Using the Null-Field Boundary Integral Equation Method

The scattering problem of seismic waves is an important issue for studying earthquake engineering. In this paper, the null-field boundary integral equation approach was used in conjunction with degenerate kernels and eigenfunction expansion to solve the SH-wave scattering problem of a circular or an elliptical-arc hill. The original problem is divided into subdomains by taking a free-body diagram. One region is an interior boundary value problem. The other is a canyon scattering problem. For the boundary value problem, not only a simply connected domain (elliptical-arc hill problem) but also a doubly connected domain (a circular-arc hill problem containing a circular tunnel or a circular inclusion) is considered. The canyon scattering problem may be addressed in an infinite domain with an artificial boundary of a full plane such that the degenerate kernel can be fully utilized. The null-field integral equation method is used to match boundary conditions. Numerical results are compared favorably with the available data.

Application of the dual reciprocity boundary integral equation approach to solve fourth-order time-fractional partial differential equations

ABSTRACTThis paper proposes a numerical approach to approximate the unknown solution of some high-order fractional partial differential equations. The main idea of this approach is to transform the original problem into an equivalent integral equation that depends only on the boundary values. The linear radial basis functions are used as the main tool for approximating the non-homogeneous terms and time derivative. Also the Caputo's sense is applied to approximate time derivatives. Numerical results demonstrate the order of time steps is and when and , respectively. Finally to overcome the nonlinear terms, predictor–corrector scheme is employed. The efficiency and usefulness of proposed method are demonstrated by some numerical examples.

Integral equations for free-molecule ow in MEMS: recent advancements

Abstract We address a Boundary Integral Equation (BIE) approach for the analysis of gas dissipation in near-vacuum for Micro Electro Mechanical Systems (MEMS). Inspired by an analogy with the radiosity equation in computer graphics, we discuss an efficient way to compute the visible domain of integration. Moreover, we tackle the issue of near singular integrals by developing a set of analytical formulas for planar polyhedral domains. Finally a validation with experimental results taken from the literature is presented.

Lightning impulse voltage distribution over voltage transformer windings — Simulation and measurement

This paper presents a fast and precise method for the calculation of internal voltage transients over the voltage instrument transformer windings. Lumped circuit parameters of the transformer winding are calculated using self-developed solvers based on the boundary element method and integral equations approach. A detailed equivalent circuit of the transformer winding is solved in time domain. Test model of the voltage instrument transformer is constructed with a number of measurement points along the windings. Results of the calculation are in a good agreement with the measured voltages.

An efficient and high‐order frequency‐domain approach for transient acoustic–structure interactions in three dimensions

This paper is concerned with numerical solution of the transient acoustic–structure interaction problems in three dimensions. An efficient and higher-order method is proposed with a combination of the exponential window technique and a fast and accurate boundary integral equation solver in the frequency-domain. The exponential window applied to the acoustic–structure system yields an artificial damping to the system, which eliminates the wrap-around errors brought by the discrete Fourier transform. The frequency-domain boundary integral equation approach relies on accurate evaluations of relevant singular integrals and fast computation of nonsingular integrals via the method of equivalent source representations and the fast Fourier transform. Numerical studies are presented to demonstrate the accuracy and efficiency of the method. Copyright © 2016 John Wiley & Sons, Ltd.

Numerical study on ring bubble dynamics in a narrow cylinder with a compliant coating

In this paper, the ring bubble contraction inside a narrow vertical rigid cylinder with a compliant coating filled with water is studied numerically. To simulate ring bubble dynamics numerically, in addition to computing the pressure and velocity fields of the surrounding fluid, an axisymmetric boundary integral equation approach is adopted alongside a finite difference method. The compliant boundary is modeled as a membrane with a spring foundation. During the ring bubble contraction and under the attraction of the cylinder wall due to the Bjerknes force, a horizontal ring jet is initiated and develops towards the cylinder wall. The numerical results represent the effects of the cylinder radius and two compliant coating characteristics, including its mass per unit area and the spring constant, on the ring bubble behavior. This investigation is motivated by the possibility of utilizing the ring jet in therapeutic cardiovascular applications.

Modeling of sound scattering by an obstacle located below a hardbacked rigid porous medium

The boundary integral equation (BIE) formulation takes advantage of the well-known Green’s function for the sound fields above a plane interface. It can then lead to a simplified numerical solution known as the boundary element method (BEM) that enables an accurate computation of sound fields above the plane interface with the presence of obstacles of complex shapes. The current study is motivated by the need to explore the acoustical characteristics of a layer of sound absorption materials embedded with equally spaced rigid inserts. In principle, this problem may be solved by a standard finite element program but it is found more efficient to use the BIE approach by discretizing only the boundary surfaces of the obstacles within the medium. The formulation is facilitated by using accurate Green’s functions for computing the sound fields above and within a layer of rigid porous medium. This paper reports a preliminary study to model the scattering of sound by an obstacle placed within the layered rigid porous medium. The two-dimensional Green’s functions will be derived and used for the development of a BEM model for computing the sound field above and within the rigid porous medium due to the presence of an arbitrarily shaped obstacle.