Boundary Integral Equation Research Articles

Potential-flow computations in infinite or zero gravity limits via a weakly singular boundary integral equation

Potential flow computations, based on a weakly singular integral equation, are considered in the infinite-gravity limit g=∞ and the zero-gravity limit g=0. The weakly singular integral equation, obtained via a straightforward and rigorous application of Green’s basic identity, involves a distribution of weak dipoles that is continuous across the body surface. The integral equation is solved via a low-order panel method, which is validated for a half sphere and applied to the S60 ship model in translation along the x, y or z-axis. A systematic numerical analysis of the convergence of numerical predictions shows that the number of panels that is required to obtain a 0.1% accuracy is significantly different for the infinite-gravity or zero-gravity flows, and for translation of the S60 hull along the x, y or z-axis. In particular, a very large number of panels is required for translation of the S60 ship model along the horizontal x or y-axis in the zero-gravity limit.

A Boundary Integral Equation Method for Computing Numerical Conformal Mappings onto the Disk with Rectilinear Slit and Spiral Slits Regions

This article proposes a boundary integral equation method for computing numerical conformal mappings of bounded multiply connected region Ω onto the disk with rectilinear slit and spiral slits regions, Ω1 and Ω2 Initially, the process involves calculating the boundary value of the canonical region. Cauchy’s integral formula can then be used to compute the mapping of the interior values. The effectiveness of the proposed method is demonstrated using several numerical examples.

Numerical calculation of regular and singular integrals in boundary integral equations using Clenshaw–Curtis quadrature rules

The efficient and accurate calculation of integrals, especially singular and nearly singular integrals, poses a great challenge in boundary integral equation (BIE) methods. In this paper, the Clenshaw–Curtis integration technique is proposed to calculate both regular and singular integrals in the BIE in a unified way. Taking the meshless discretization of two-dimensional Helmholtz BIE by linear integration cell as an example, four types of Clenshaw–Curtis quadrature rules are established for regular, weakly singular, hypersingular, and nearly hypersingular integrals. Explicit expressions of integration weights in all quadrature rules are derived, and the degree of precision of these quadrature rules is analyzed theoretically. Integration points in all quadrature rules are the same, which is very useful for meshless analysis of BIEs. The values of integration points and weights for regular, weakly singular and hypersingular integrals are tabulated in the interval [−1, 1], which can be directly used to facilitate the practical application of BIE methods. Numerical results are provided to verify the effectiveness of these Clenshaw–Curtis quadrature rules.

On optimal radius of sub-domains in meshless LBIE method

Local weak based meshless methods construct weak form of governing equations on local sub-domains. In two dimensional domains, for the simplicity of computations, these sub-domains are taken as circles. In these methods, the optimal radius of sub-domains has been an open problem yet. This paper aims at solving this problem for meshless local boundary integral equation (LBIE) method to enhance its performance. It is proved that a sub-domain for which the Lebesgue constant takes its minimum over its boundary is the optimal sub-domain. In other words, the optimal sub-domain is one for which the solution of PDE is approximated on its boundary as accurate as possible. A comprehensive numerical study confirms the theory.

Impedance Sensitivity Analysis Based on Discontinuous Isogeometric Boundary Element Method in Automotive Acoustics

Abstract Acoustic sensitivity analysis is an essential technique to determine the direction of structural-acoustic optimization by evaluating the gradient of the objective functions with respect to the design variables. However, acoustic sensitivity analysis with respect to acoustic impedance, which is an important parameter representing the interior absorbent material in automotive acoustics, is lacking in the study. Moreover, acoustic sensitivity analysis implemented with conventional numerical methods is time and effort-consuming in automotive acoustics, due to the large-scale mesh generation. In this work, the impedance sensitivity analysis for automotive acoustics based on the discontinuous isogeometric boundary element method is presented. The regularized boundary integral equation with impedance boundary conditions is established, then the sensitivity is derived by differentiating the boundary integral equation. The efficiency of the proposed method is improved by employing the parallel technique and generalized minimal residual solver. A long duct example with an analytical solution validates the accuracy of the proposed method, and an automotive passenger compartment subjecting to impedance boundary conditions illustrates that the computing time of the proposed method is one order of magnitude less than the conventional method. This work presents an easily implementable and efficient tool to investigate acoustic sensitivity with respect to impedance, showing great potential in the application of automotive acoustics.

Dynamic Green’s Functions for an Infinite Acoustic Field with Multiple Spheres Subjected to the Robin Boundary Conditions

A semi-analytical approach is presented to solve three-dimensional dynamic Green’s function for an infinitely extended acoustic field with multiple spheres subjected to the Robin boundary conditions. The multipole expansions of the acoustic field induced by a time-harmonic point source are expanded with spherical wave functions. As an alternative to the complex addition theorem, the multipole expansion is computed in a straightforward way. By taking the finite number of terms, an algebraic system is constructed and is used to obtain Green’s function. This result of one sphere agrees with the available analytical solution. For the case of more than one sphere, the proposed results are verified by the numerical method such as the boundary element method (BEM). It indicates that the present solution is more accurate than that of the BEM and shows a fast convergence. Finally, the parameter study is performed to explore the influences of the exciting frequency of the point source, the surface admittance, the number and the separation of spheres on the dynamic Green’s functions. The proposed results can be applied to solve the acoustic scattering problems and to increase the application of boundary integral equation method in the way of numerical Green’s functions.

Computation of Transmission Eigenvalues by the Regularized Schur Complement for the Boundary Integral Operators

This paper is devoted to the computation of transmission eigenvalues in the inverse acoustic scattering theory. This problem is first reformulated as a two by two boundary system of boundary integral equations. Next, utilizing the Schur complement technique, we develop a Schur complement operator with regularization to obtain a reduced system of boundary integral equations. The Nystrom discretization is then used to obtain an eigenvalue problem for a matrix. We employ the recursive integral method for the numerical computation of the matrix eigenvalue. Numerical results show that the proposed method is efficient and reduces computational costs.

Intensity of Dynamic Stresses of Longitudinal Shear in a Periodically Layered Composite with Penny-Shaped Cracks

We consider the problem of harmonic torsional loading of an infinite elastic composite formed by alternating plane layers made of two different materials in the presence of a penny-shaped crack in one of the components of periodic structure. By using the integral representations for displacements and stresses in the frequency domain and satisfying the conditions of periodicity and perfect contact on the interfaces, we deduce a system of uncoupled boundary integral equations for the functions of tangential dynamic crack-opening displacement in a two-layered representative element of the composite. We perform the numerical analysis of the mode-III dynamic stress intensity factors in the vicinity of the analyzed crack depending on the wave number, the thicknesses of constituent layers, and the properties of their materials.

Analytical expressions for singular integrals arising from the 3D Laplace and Stokes kernels when using constant or linear triangular and quadrilateral boundary elements

In this paper we present analytical expressions for the computation of singular integrals obtained in the discretisation of boundary integral equations for the Laplace and creeping flow (Stokes) problems with triangular or quadrilateral boundary elements with linear interpolation of the potential and constant interpolation of the flux. We compare the singular integrals computed with the presented analytical expressions with the same integrals computed with numerical quadrature and find that a considerably larger computational effort has to be made for numerical quadrature to achieve high accuracy than with the analytical expression. Furthermore, we show that the accuracy in solving a Laplace test case and a creeping flow test case using analytical expressions for singular integrals is better than the accuracy achieved with numerical quadrature. The analytical expressions are listed in the appendix of the paper and their implementation in computer code is available online.

A new periodic FM-BEM for solving the acoustic transmission problems in periodic media

This paper presents a new fast multipole boundary element method (FM-BEM) for solving the acoustic transmission problems in 2-D periodic media. We divide the periodic media into fundamental blocks, and then construct the boundary integral equations in the fundamental block. The fast multipole algorithm is proposed for the square periodic and hexagonal periodic structures, the convergence and complexity of the algorithm are analyzed. Our approach is applied to solve the acoustic transmission problems for liquid phononic crystals, the acoustic band gaps of the phononic crystals are derived.

Analysis of near-interface cracks in three-dimensional anisotropic multi-materials by efficient BIEM

This paper presents the full analysis of near-interface cracks in three-dimensional linear elastic multimaterial bodies by a weakly singular boundary integral equation method (BIEM). The proposed technique was implemented in a general framework that allowed the treatment of finite cracked bodies with general configurations, material anisotropy, and mode-mixity. A system of integral equations governing unknown data on the boundary, crack surface, and material interface was established using a pair of weakly singular weak-form displacement and traction integral equations together with continuity along the material interface. A symmetric Galerkin boundary element method along with the finite element technique were implemented to solve the governing integral equations. In addition, the special near-front interpolation was employed to enhance the approximation of the relative crack-face displacement in the neighborhood of the crack front. The solved crack-face data were used directly to post-process the stress intensity factors and T-stresses along the crack front. An extensive numerical study was carried out not only to demonstrate the accuracy, convergence, and capability of the proposed technique, but also to explore the influence of material stiffness and distance to the material interface on fracture data along the crack front.

GPU acceleration of Nystrom type method for solving singular integral equations in H-polarized EM waves scattering by strip gratings

Abstract In this paper, we propose the use of the graphics processing units (GPUs) for acceleration of solution of the H-polarized electromagnetic waves scattering by strip gratings. The approach combines the boundary integral equation method with the Nystrom-type discretization of singular integral equations and is realized with the use of C++ and OpenCL. The elements of the matrix of the resulting system of equations can be calculated independently and, thus, in parallel. The calculation time of the proposed parallelized algorithm on GPU is compared to standard serial algorithm executed on CPU. We considered gratings of graphene strips in vacuum and inside a dielectric slab. The acceleration is up to 35–55 times for the gratings in vacuum and up to 80–140 times for the gratings of 5–20 strips inside the dielectric slab.

ACA-accelerated hybrid boundary node method applied to multi-domain steady heat conduction problems

This paper presents a fast implementation of the hybrid boundary node method (HBNM) to solve the multi-domain steady heat conduction problems using the Adaptive Cross Approximation (ACA). It is well known that the matrices arising from the discretization of boundary integral equation are fully populated and require special compression techniques for the efficient treatment. The ACA using only few of the original entries for the approximation of the whole matrix is a purely algebraic algorithm which is kernel-independent and easy to be applied directly for various problems. Accelerated by the ACA, several multi-domain steady heat conduction problems have been solved by the HBNM. The results demonstrate the potentials of the ACA-HBNM for solving multi-domain steady heat conduction problems.

Numerical analysis for isolation of vibrations by pile rows in transversely isotropic saturated media due to moving loads

An analytical layer-element–boundary-element coupling approach is presented to explore the isolation effect of pile rows in stratified transverse isotropic saturated media for moving-load induced vibration. The analytical layer-element basic solution for the soils is derived. The soil flexibility matrices are obtained using boundary integral equations for pile-soil. Then, the pile is modeled as a Bernoulli–Euler beam utilizing the finite difference approach. The dynamic equations for the piles are achieved by considering the pile-soil displacement coordination condition. Finally, to obtain the amplitude reduction ratio, the displacement of an observation point is calculated with and without pile isolation, respectively. The results are in good consistency with the existing solutions. The impacts of the soils’ transverse isotropy, the pile’s size and physical properties on the isolation effect are evaluated using numerical examples. It is observed that when the pile length exceeds a certain value, the increment in the pile length shows no obvious improvement in the vibration isolation effect. A stiffer pile has a greater impact on vibration reduction. The material anisotropy is also an important parameter for the pile isolation problem.

Design sensitivity analysis of three-dimensional acoustic problems using a fast multipole accelerated indirect BEM

In this paper, the design sensitivity formulations for three-dimensional acoustic problems are presented based on the indirect boundary element method (IBEM). The non-unique solution difficulty of the IBEM in solving exterior acoustic problems is avoided by applying a hybrid combination of single and double layer potentials. In order to accelerate the computational procedure easily, a set of non-singular indirect boundary integral equations (IBIEs) for the acoustic design sensitivity analysis are derived and then the fast multipole method (FMM) is employed to accelerate the matrix–vector products in the IBEM. The method is further extended to the design sensitivity analysis of thin-body acoustic problems above an infinite plane. Numerical examples are given to verify the effectiveness and accuracy of the method, and the simulations of thin-body acoustic problems show its potential in engineering applications.

Three-Temperature Boundary Element Modeling of Ultrasound Wave Propagation in Anisotropic Viscoelastic Porous Media

The main goal of this work is to develop a novel boundary element method (BEM) model for analyzing ultrasonic wave propagation in three-temperature anisotropic viscoelastic porous media. Due to the problems of the strong nonlinearity of ultrasonic wave propagation in three-temperature porous media, the analytical or numerical solutions to the problems under consideration are always challenging, necessitating the development of new computational techniques. As a result, we use a new BEM model to solve such problems. A time-stepping procedure based on the linear multistep method is obtained after solving the discretized boundary integral equation with the quadrature rule. The calculation of a double integral is required to obtain fundamental solutions, but this increases the total BEM computation time. Our proposed BEM technique is used to solve the current problem and improve the formulation efficiency. The numerical results are graphed to demonstrate the effects of viscosity and anisotropy on the nonlinear ultrasonic stress waves in three-temperature porous media. The validity, accuracy, and efficiency of the proposed methodology are demonstrated by comparing the obtained results to a corresponding solution obtained from the finite difference method (FDM).

Study of stress wave propagation across a non-persistent joint based on a boundary integral equation method

Studies on stress wave propagation across persistent joints have been conducted extensively. Nevertheless, there exists a consensus that non-persistent joints are widely and densely distributed, which have a profound impact on wave propagation in jointed rock masses. A boundary integral equation method is suggested in this paper to investigate the characteristics of transmitted wave field for the case of stress wave propagation across a single non-persistent joint. The displacement continuity and discontinuity boundaries are combined in the method. The method presented in the current study is applicable to the analysis of wave propagation across non-persistent joints with arbitrary incident angles. Then, taking a single non-persistent joint arranged with only one joint segment as an example, the applicability of the method in dealing with the problems of wave propagation is verified by comparing the results with those from the discrete element method and analytical methods. Subsequently, parametric studies are carried out, including the effects of joint-segment length, rock-bridge length, wave frequency and incident angle on the transmitted wave. The result indicates that the existence of non-persistent joint makes the transmitted displacement field different from that of persistent joint, because the scattered wave is produced during the process of wave propagation. The displacement amplitude may be amplified evidently in some regions and the spatial distribution pattern of the transmission coefficient is closely related to the joint-segment length, rock-bridge length and incident wavelength.

Surface Pressure Calculation Method of Multi-Field Coupling Mechanism under the Action of Flow Field

At present, the majority of fluid mechanisms are multi-field coupling mechanisms, and their function is also achieved in the flow field. Therefore, calculating the aerodynamic characteristics of the multi-field coupling mechanism in a symmetric flow field is very important. However, at present, the strong coupling algorithm in the domain has the problems of low accuracy and computational efficiency, which make it more difficult to solve the coupling problem. This article obtains the vector potential of the law of conservation of momentum using the tensor analysis method in a Cartesian coordinate system. Meanwhile, the generalized operator of Navier–Stokes equations and the fundamental solution of the generalized operator are obtained on this basis. Then, this article proposes the boundary integral equation of the Navier–Stokes equations by combining the fundamental solution of the Laplace equation with generalized potential theory. Based on this boundary integral equation, this article has developed a new calculation method that can help achieve integral calculation without domains, greatly reducing the problem’s difficulty. Finally, by comparing the ellipsoid example solution with the experimental results, the algorithm’s reliability in solving the incompressible problem is verified.

Volume Integral Equations and Single-Trace Formulations for Acoustic Wave Scattering in an Inhomogeneous Medium

AbstractWe study frequency domain acoustic scattering at a bounded, penetrable, and inhomogeneous obstacleΩ−⊂Rd\Omega^{-}\subset\mathbb{R}^{d},d=2,3d=2,3. By defining constant reference coefficients, a representation formula for the pressure field is derived. It contains a volume integral operator, related to the one in the Lippmann–Schwinger equation. Besides, it features integral operators defined on∂Ω−\partial\Omega^{-}and closely related to boundary integral equations of single-trace formulations (STF) for transmission problems with piecewise constant coefficients. We show well-posedness of the continuous variational formulation and asymptotic convergence of Galerkin discretizations. Numerical experiments in 2D validate our expected convergence rates.

The skeleton equation method for acoustic transmission problems with varying coefficients

AbstractIn this paper we describe the skeleton equation method, to transform a three‐dimensional acoustic transmission problem with variable coefficients and mixed boundary conditions to a non‐local equation on a two‐dimensional skeleton, without relying on an explicit knowledge of Green's function. This is achieved introducing and analyzing abstract layer potentials as solutions of auxiliary coercive full space transmission problems; they satisfy jump conditions across domain interfaces. The novelty in this paper is that we reduce acoustic scattering problems with possible variable coefficients to boundary integral equation on the domain skeleton, and we prove frequency exlicit continuity and coercivity estimates. The resulting equations are stated in a form such that standard boundary element methods can be applied for the discretization.