Boundary Integral Equation Approach Research Articles

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.

Simultaneous reconstruction of shape and generalized impedance functions in electrostatic imaging

Determining the geometry and the physical nature of an inclusion within a conducting medium from voltage and current measurements on the accessible boundary of the medium can be modeled as an inverse boundary value problem for the Laplace equation subject to appropriate boundary conditions on the inclusion. We continue the investigations on the particular inverse problem with a generalized impedance condition started in Cakoni and Kress (2013 Inverse Problems 29 015005) by presenting an inverse algorithm for the simultaneous reconstruction of both the shape of the inclusion and the two impedance functions via a boundary integral equation approach. In addition to describing the reconstruction algorithm and illustrating its feasibility by numerical examples we also provide some extensions to the uniqueness results in Cakoni and Kress (2013 Inverse Problems 29 015005).

Numerical investigation on the pulsating bubble dynamics in a narrow cylinder with a compliant coating

In this paper, the bubble dynamics in a narrow vertical rigid cylinder with a compliant coating filled with water is studied numerically. Considering the surface tension and the coincidence of bubble and cylinder centers, an adopted axisymmetric boundary integral equation approach besides a finite difference method is used for numerical simulation of bubble evolution accompanying with computing the pressure and velocity fields of fluid around it. The compliant boundary is modeled as a membrane with a spring foundation. It is found that the confinement of bubble by the cylinder increases greatly its lifetime and influences its behavior. The first stage of bubble contraction depicts the flattening of its top and bottom parts. A liquid counter-jet is then initiated and develops itself toward the bubble center; the bubble becomes biconcave at the end of the contraction phase. The bubble dynamics is affected by the ratio of cylinder radius to the maximum radius of the bubble. The results also represent the effects of two compliant coating characteristics including its mass per unit area and the spring constant on the bubble behavior. This investigation is motivated by the possibility of utilizing bubbles in the therapeutic cardiovascular applications.

Scattering of a plane harmonic SH wave by a rough multilayered inclusion of arbitrary shape

Steady-state scattering of a plane harmonic SH wave by a multilayered inclusion of arbitrary shape consisting of a general number of homogeneous isotropic layers, embedded within a half space, is investigated by using a direct boundary integral equation approach. The layer interfaces are assumed to be either smooth or periodically corrugated without any sharp corners being present. The peak corrugation amplitude is arbitrarily prescribed. The corresponding integral equations are derived in terms of the unknown boundary displacements and tractions. The integral equations are discretized using linear elements leading to a system of linear algebraic equations which are solved by using the standard Gauss elimination procedure.Testing of the numerical results is performed by comparing the surface response of a cavity with those available in the literature. The tests show excellent agreement between the two results for different frequencies and angles of incidence.The numerical results include surface response of multilayered inclusions and pipes. The inclusion layers are assumed to be either softer or stiffer from the half space. The pipe layers include both soft–stiff and stiff–soft combinations of the materials.The layered inclusion results demonstrate that for the soft materials the presence of layers may significantly change the surface response when compared to the corresponding homogeneous inclusion motion. For stiff layers this effect is greatly reduced. The same phenomenon has been observed for the inclusions with periodically corrugated interfaces as well.The surface response of a layered pipe is found to strongly depend upon the nature of the incident wave and the materials of the layers.

Two-Dimensional Modeling of Functionally Graded Viscoelastic Materials Using a Boundary Element Approach

In this paper, a 2D boundary element approach able to model viscoelastic functionally graded materials (FGM) is presented. A numerical implementation of the Somigliana identity for displacements is developed to solve 2D problems of exponentially graded elasticity. An FGM is an advanced material in which its composition changes gradually resulting in a corresponding change in properties of the material. The FGM concept can be applied to various materials for structural and functional uses. Our model needs only the Green’s function of nonhomogeneous elastostatic problems with material properties that vary continuously along a given dimension. We consider the material properties to be an exponential function of Cartesian coordinates x. As application, a numerical example is provided to validate the proposed boundary integral equation approach.

Some Treatments of Fictitious Volume Charges in Nonlinear Magnetostatic Analysis by BIE

The scalar potential formulation by the boundary integral equation approach is attractive for numerical analysis but has fatal drawbacks due to a multi-valued function in current excitation. We derive an all-purpose boundary integral equation with double layer charges as the state variable and apply it to nonlinear magnetostatic problems by regarding the nonlinear magnetization as fictitious volume charges. We investigate two approaches how to treat the fictitious charges. In discretization by the constant volume element, a surface loop current is introduced for the volume charge. By the linear volume element, the fictitious charges are evaluated on the condition that the divergence of the magnetic flux density is zero. We give a comparative study of these two approaches.

A New Boundary Integral Formulation for Stream Function and Vorticity in Axisymmetric Stokes Flow

A boundary integral equation approach is presented to solve the flow stream function and vorticity in the axisymmetric Stokes problem without swirl. The governing equations expressed in these variables require that the fundamental solutions of the involved differential operators be found. The fundamental solutions in a closed form, suitable for numerical treatment of integral equations, are obtained, and their equivalent forms in infinite integral are derived by using Fourier and Hankel transforms. A coupled system of boundary integral equations for the stream function and the vorticity is formulated by employing new fundamental solutions. Solution of these equations is realized by the collocation method. Numerical results are included for several bounded and unbounded domain problems to validate and demonstrate the effectiveness of the proposed approach.

Green's functions for Kirchhoff plates of irregular shape

A method is proposed for the construction of Green's functions for the Sophie Germain equation in regions of irregular shape with mixed boundary conditions imposed. The method is based on the boundary integral equation approach where a kernel vector function B satisfies the biharmonic equation inside the region. This leads to a regular boundary integral equation where the compensating loads and moments are applied to the boundary. Green's function is consequently expressed in terms of the kernel vector function B , the fundamental solution function of the biharmonic equation, and kernel functions of the inverse regular integral operators. To compute moments and forces, the kernel functions are differentiated under the integral sign. The proposed method appears highly effective in computing both displacements and stress components.

SH-wave scattering by a semi-elliptical hill using a null-field boundary integral equation method and a hybrid method

Following the success of seismic analysis of a semi-circular hill, the problem of SH-wave scattering by a semi-elliptical hill is revisited by using the null-field boundary integral equation method (BIEM). To fully use the analytical property in the null-field boundary integral equation approach in conjunction with degenerate kernels for solving the semi-elliptical hill scattering problem, the problem is decomposed to two regions to produce elliptical boundaries by using the technique of taking free body. One is the half-plane problem containing a semi-elliptical boundary. This semi-infinite problem is imbedded in an infinite plane with an artificial elliptical boundary such that degenerate kernel can be fully applied. The other is an interior problem bounded by an elliptical boundary. The degenerate kernel in the elliptic coordinates for two subdomains is used to expand the closed-form fundamental solution. The semi-analytical formulation in companion with matching boundary conditions yields six constraint equations. Instead of finding admissible wave-expansion bases, our null-field BIEM in conjunction with degenerate kernels has the five features over the conventional BIEM/BEM, (1) free of calculating principal values, (2) exponential convergence, (3) elimination of boundary-layer effect, (4) meshless and (5) well-posed system. All numerical results are compared well with those of using the hybrid method which is also described in this paper. It is interesting to find that a focusing phenomenon is also observed in this study.

Scattering of plane harmonic SH waves by multiple inclusions

SUMMARY Scattering of a plane harmonic SH wave by an arbitrary number of inclusions of general shape and placements within an elastic half-space is investigated by using a direct boundary integral equation approach. The integral equations have been reduced to a system of linear algebraic equations by using the linear elements. The structure of the resulting algebraic system is investigated and compared to the single-inclusion model. For convenience the multiple-inclusion models are assumed to consist of the reference and the auxiliary inclusions. The reference inclusion has the same shape, placement and the material properties as those of the corresponding single-inclusion model. The auxiliary inclusions then surround the reference inclusion assuming different patterns. Extensive testing of the proposed solution is performed. These tests include the transparency and the single-inclusion tests. In the former the material properties of the inclusions are assumed to be the same as those of the half-space and the surface response is found to be equal to the free field one. In the single-inclusion test the material properties of the auxiliary inclusions are assumed to be the same as those of the half-space while those of the reference inclusion are different. Then the resulting surface response is found to reduce to that of the single-inclusion model. For different number of multiple inclusions the surface motion atop the reference inclusion is compared with that of the corresponding single-inclusion model. In particular, a specific measure is introduced to assess accurately the importance of the multiple scattering upon the surface response. The effect of the auxiliary inclusion locations with respect to the reference inclusion upon the surface motion is investigated for two-, three- and nine-inclusion models. The results clearly show the importance of the interactions between the inclusions for a wide range of parameters present in the problem (e.g. nature of the incident wave, geometry of the model and the number of inclusions). For the problems considered here it appears that by ignoring these interactions, that is, using the single-inclusion model, will result in underestimation of the peak surface motion amplification.

Eigenmode-Based Capacitance Calculations With Applications in Passivation Layer Design

The design of high-speed metallic interconnects such as microstrips requires the correct characterization of both the conductors and the surrounding dielectric environment, in order to accurately predict their propagation characteristics. A fast boundary integral equation approach is obtained by modeling all materials as equivalent surface charge densities in free space. The capacitive behavior of a finite dielectric environment can then be determined by means of a transformation matrix, relating these charge densities to the boundary value of the electric potential. In this paper, a new calculation method is presented for the important case that the dielectric environment is composed of homogeneous rectangles. The method, based on a surface charge expansion in terms of the Robin eigenfunctions of the considered rectangles, is not only more efficient than traditional methods, but is also more accurate, as shown in some numerical experiments. As an application, the design and behavior of a microstrip passivation layer is treated in some detail.

A Primitive-Variable Boundary Integral Formulation Unifying Aeroacoustics and Aerodynamics, and a Natural Velocity Decomposition for Vortical Fields

The paper presents an integrated approach to the study of aeroacoustics and unsteady aerodynamics, which is based upon the use of the boundary integral equation approach. Accordingly, the boundary integral formulation in primitive variables is thoroughly developed, for Euler as well as Navier–Stokes equations. As a result, one obtains explicit time–domain expressions for velocity and pressure in terms of the Cauchy data of the problem (boundary integral), plus the contribution of the nonlinear terms (field integral). It is shown that the representation for the pressure is very closely related to the classical boundary integral formulations in aeroacoustics (such as the Ffowcs Williams and Hawkings equation and the Kirchhoff method). Similarly, the representation for the velocity is closely related to a formulation introduced by the author for unsteady viscous compressible aerodynamics. In the process, however, a new velocity decomposition of the type v = grad φ + w is uncovered. The novelty is that the vortical velocity w is not defined in terms of the vorticity ω; on the contrary, w is defined through its own governing equation. Such a decomposition (and the governing equation for w, which does not involve the pressure) arises in a very natural way from the mathematical approach used, and hence is here named the natural velocity decomposition. In addition, two innovative results are obtained. First, for all the cases addressed (incompressible and compressible, inviscid and viscous fields) there exist generalizations of the Bernoulli theorem (formally very similar to the classical Bernoulli theorems, with the potential φ of the irrotational portion of the velocity replacing the velocity potential φ). Second, it is shown that the velocity field may be obtained independently of the pressure, which may be evaluated a–posteriori in terms of the potential φ (akin to the evaluation of the pressure in a Venturi tube), through the appropriate generalized Bernoulli theorem. The paper is of theoretical nature – no numerical results are included; however, the schemes for computational implementations are presented and the advantages discussed.

A boundary integral equations approach for mixed impedance problems in elasticity

Direct scattering problems for partially coated obstacles in linear elasticity lead to interior and exterior mixed impedance boundary value problems for the equations of steady-state elastic oscillations. We employ the potential method and reduce the mixed impedance problems to equivalent boundary pseudodifferential equations for arbitrary values of the oscillation parameter. We give a detailed analysis of the corresponding pseudodifferential equations which live on a proper submanifold of the boundary of the elastic body and establish uniqueness and existence results for the original mixed impedance problems for arbitrary values of the oscillation parameter; this is crucial in the study of inverse elastic scattering problems for partially coated obstacles. We also investigate regularity properties of solutions near the curves where the boundary conditions change and establish almost best Holder smoothness results.

A Time Domain Inverse Method for Identification of Random Acoustic Sources at Launch Vehicle Lift-Off

In order to characterize the lift-off acoustic environment of launch vehicles, EADS has developed an inverse method via a time domain boundary integral equation approach using an optimal control method. An industrial application of the method is presented via the identification of acoustic sources of ARIANE 5 from lift-off in-flight measurements. As the lift-off acoustic field is random in nature, the optimization process is performed using the cross-correlation functions matrix measurement. After having characterized the sources from pressure measurements, the complete environment is rebuilt by an acoustic direct computation from the obtained acoustic sources. We finally estimate the loads created by the acoustic field by integrating the resulting pressures over all surfaces of the launchers.

SH-wave diffraction by a semi-circular hill revisited: A null-field boundary integral equation method using degenerate kernels

Following the success of seismic analysis of a canyon [1], the problem of SH-wave diffraction by a semi-circular hill is revisited using the null-field boundary integral equation method (BIEM). To fully utilize the analytical property in the null-field boundary integral equation approach in conjunction with degenerate kernels for solving the semi-circular hill scattering problem, the problem is decomposed into two regions to produce circular boundaries using the technique of taking free body. One is the half-plane problem containing a semi-circular boundary. This semi-infinite problem is imbedded in an infinite plane with an artificial full circular boundary such that degenerate kernel can be fully applied. The other is an interior problem bounded by a circular boundary. The degenerate kernel in the polar coordinates for two subdomains is utilized for the closed-form fundamental solution. The semi-analytical formulation along with matching boundary conditions yields six constraint equations. Instead of finding admissible wave expansion bases, our null-field BIEM approach in conjunction with degenerate kernels have five features over the conventional BIEM/BEM: (1) free from calculating principal values, (2) exponential convergence, (3) elimination of boundary-layer effect, (4) meshless and (5) well-posed system. All the numerical results are comparing well with the available results in the literature. It is interesting to find that a focusing phenomenon is also observed in this study.

Boundary integral equation approach for stokes slip flow in rotating mixers

In order to employ continuum models in the analysis of the flow behaviour of a viscous Newtonian fluid at micro scale devices, it is necessary to consider at the wall surfaces appropriate slip boundary conditions instead of the classical non-slip condition. To account for the slip condition at the nano-scale, we used the Navier's type boundary condition that relates the tangential fluid velocity at the boundaries to the tangential shear rate. In this work a boundary integral equation formulation for Stokes slip flow, based on the normal and tangential projection of the Green's integral representational formulae for the Stokes velocity field, which directly incorporates into the integral equations the local tangential shear rate at the wall surfaces, is presented. This formulation is used to numerically simulate concentric and eccentric rotating Couette mixers and a Single rotor mixer, including the effect of thermal creep in cases of rarefied gases. The numerical results obtained for the Couette mixers, concentric and eccentric, are validated again the corresponding analytical solutions, showing excellent agreements.

Shear Deformation Effect in Flexural-torsional Vibrations of Composite Beams by Boundary Element Method (BEM)

In this paper a boundary element method (BEM) is developed for the general flexural-torsional vibration problem of Timoshenko beams of arbitrarily shaped composite cross-section taking into account the effects of warping stiffness, warping and rotary inertia and shear deformation. The composite beam consists of materials in contact, each of which can surround a finite number of inclusions. The materials have different elasticity and shear moduli with same Poisson’s ratio and are firmly bonded together. The beam is subjected to arbitrarily transverse and/or torsional distributed or concentrated loading, while its edges are restrained by the most general linear boundary conditions. The resulting initial boundary value problem, described by three coupled partial differential equations, is solved employing a boundary integral equation approach. Besides the effectiveness and accuracy of the developed method, a significant advantage is that the displacements as well as the stress resultants are computed at any cross-section of the beam using the respective integral representations as mathematical formulae. All basic equations are formulated with respect to the principal shear axes coordinate system, which does not coincide with the principal bending one in a non-symmetric cross-section. To account for shear deformations, the concept of shear deformation coefficients is used. Six boundary value problems are formulated with respect to the transverse displacements, to the angle of twist, to the primary warping function and to two stress functions and solved using the Analog Equation Method, a BEM-based method. Both free and forced vibrations are examined. Several beams are analyzed to illustrate the method and demonstrate its efficiency and wherever possible its accuracy.

Shear deformation effect in flexural-torsional buckling analysis of beams of arbitrary cross section by BEM

In this paper a boundary element method is developed for the general flexural-torsional buckling analysis of Timoshenko beams of arbitrarily shaped cross section. The beam is subjected to a compressive centrally applied concentrated axial load together with arbitrarily axial, transverse and torsional distributed loading, while its edges are restrained by the most general linear boundary conditions. The resulting boundary value problem, described by three coupled ordinary differential equations, is solved employing a boundary integral equation approach. All basic equations are formulated with respect to the principal shear axes coordinate system, which does not coincide with the principal bending one in a nonsymmetric cross section. To account for shear deformations, the concept of shear deformation coefficients is used. Six coupled boundary value problems are formulated with respect to the transverse displacements, to the angle of twist, to the primary warping function and to two stress functions and solved using the Analog Equation Method, a BEM based method. Several beams are analysed to illustrate the method and demonstrate its efficiency and wherever possible its accuracy. The range of applicability of the thin-walled theory and the significant influence of the boundary conditions and the shear deformation effect on the buckling load are investigated through examples with great practical interest.

Interaction between a crack and a circular inhomogeneity with interface stiffness and tension

The interaction between a straight crack and a circular inhomogeneity with interface stiffness and energy is considered. The Gurtin and Murdoch model is adopted, wherein the interface between the inhomogeneity and the matrix is regarded as a material surface that possesses its own mechanical properties and surface tension. The elastostatics problem is decomposed into two complimentary problems for (1) a circular disk with unknown distributions of traction and displacements along its boundary and (2) a loaded isotropic plane containing a circular hole with unknown distributions of traction and displacements along its boundary. The unknown distributions are determined through the application of the constitutive relations at the material surface. For selected values of the dimensionless parameters that quantify the geometry, material properties and applied loading, the stress field, stress intensity factors and energy release rates are calculated using a complex boundary integral equation approach. Particular attention is paid to crack-tip shielding and anti-shielding that develops as a result of the stresses introduced by the material surface. An illustrative example involving a perforated plate loaded in tension suggests that the material surface produces a modest level of expected effective toughening.

Boundary integral equation methods in the theory of elasticity of hemitropic materials: A brief review

The theory of elasticity of hemitropic materials has recently been the object of rigorous mathematical analysis. In particular, the potential method and the theory of pseudodifferential equations have been used in studying the solvability in various function spaces of the main boundary value and transmission problems, in smooth and in Lipschitz domains. The main features and results of this boundary integral equations approach are briefly reviewed here.