Solution Of Boundary Integral Equations Research Articles

Trapped acoustic waves and raindrops: High-order accurate integral equation method for localized excitation of a periodic staircase

We present a high-order boundary integral equation (BIE) method for the frequency-domain acoustic scattering of a point source by a singly-periodic, infinite, corrugated boundary. We apply it to the accurate numerical study of acoustic radiation in the neighborhood of a sound-hard two-dimensional staircase modeled after the El Castillo pyramid. Such staircases support trapped waves which travel along the surface and decay exponentially away from it. We use the array scanning method (Floquet–Bloch transform) to recover the scattered field as an integral over the family of quasiperiodic solutions parameterized by on-surface wavenumber. Each such BIE solution requires the quasiperiodic Green's function, which we evaluate using an efficient integral representation of lattice sum coefficients. We avoid the singularities and branch cuts present in the array scanning integral by complex contour deformation. For each frequency, this enables a solution accurate to around 10 digits in a few seconds. We propose a residue method to extract the limiting powers carried by trapped modes far from the source. Finally, by computing the trapped mode dispersion relation, we use a simple ray model to explain an acoustic chirp-like time-domain response that is referred to in the literature as the “raindrop effect.”

A highly accurate indirect boundary integral equation solution for three dimensional elastic scattering problem

In this paper, we propose a numerical formula for three dimensional elastic wave scattering problem. Different from the classical boundary integral equation method, the three dimensional elastic scattering problem is transformed into a coupled boundary value problem including a scalar Helmholtz equation and a vector Helmholtz equation by the decomposition of the displacement. The coupled boundary value problem can be solved by the boundary integral equations based on the Helmholtz equation. A denseness result is given to support this algorithm. A comparative study with the classical method of fundamental solutions getting from the boundary integral equation method is given to check the accuracy with different types of boundary conditions. From the numerical results, the present method can give more accurate results than the classical method of fundamental solutions method.

ReLU Neural Network Galerkin BEM

We introduce Neural Network (NN for short) approximation architectures for the numerical solution of Boundary Integral Equations (BIEs for short). We exemplify the proposed NN approach for the boundary reduction of the potential problem in two spatial dimensions. We adopt a Galerkin formulation-based method, in polygonal domains with a finite number of straight sides. Trial spaces used in the Galerkin discretization of the BIEs are built by using NNs that, in turn, employ the so-called Rectified Linear Units (ReLU) as the underlying activation function. The ReLU-NNs used to approximate the solutions to the BIEs depend nonlinearly on the parameters characterizing the NNs themselves. Consequently, the computation of a numerical solution to a BIE by means of ReLU-NNs boils down to a fine tuning of these parameters, in network training. We argue that ReLU-NNs of fixed depth and with a variable width allow us to recover well-known approximation rate results for the standard Galerkin Boundary Element Method (BEM). This observation hinges on existing well-known properties concerning the regularity of the solution of the BIEs on Lipschitz, polygonal boundaries, i.e. accounting for the effect of corner singularities, and the expressive power of ReLU-NNs over different classes of functions. We prove that shallow ReLU-NNs, i.e. networks having a fixed, moderate depth but with increasing width, can achieve optimal order algebraic convergence rates. We propose novel loss functions for NN training which are obtained using computable, local residual a posteriori error estimators with ReLU-NNs for the numerical approximation of BIEs. We find that weighted residual estimators, which are reliable without further assumptions on the quasi-uniformity of the underlying mesh, can be employed for the construction of computationally efficient loss functions for ReLU-NN training. The proposed framework allows us to leverage on state-of-the-art computational deep learning technologies such as TENSORFLOW and TPUs for the numerical solution of BIEs using ReLU-NNs. Exploratory numerical experiments validate our theoretical findings and indicate the viability of the proposed ReLU-NN Galerkin BEM approach.

A Hybrid Loop-Tree FEBI Method for Low-Frequency Well Logging of 3-D Structures in Layered Media

Electromagnetic simulation plays an essential role in formation conductivity determination by electromagnetic well-logging tools, especially when the nonvertical borehole and invasion zones are present in a layered earth. A hybrid finite-element boundary integral (FEBI) method is suitable for such well-logging simulations. The usage of layered medium Green's functions allows the simulation background to be a planar stratified medium to characterize the earth formation. However, conventional FEBI using the Rao-Wilton-Glisson (RWG) basis function produces inaccurate results due to the low-frequency breakdown of the boundary integral equation solution. The new contribution of this work is to apply the loop-tree (LT) basis functions in the FEBI method to reduce the numerical error at low frequencies. Such an LT-FEBI method can handle all specific requirements of well-logging simulations at low frequencies. A direct solver is applied to make it more efficient to obtain logging curves with multiple source locations. The LT-FEBI simulation results are validated in several numerical examples, including a challenging well-logging model with a deviated borehole and invasion zones in the Oklahoma formation having 28 layers.

A fourth-order Cartesian grid method for multiple acoustic scattering on closely packed obstacles

In this paper, we present a fourth-order Cartesian grid-based boundary integral method (BIM) for a multiple acoustic scattering problem on closely packed obstacles. We reformulate the exterior Helmholtz boundary value problems (BVPs) as a Fredholm boundary integral equation (BIE) of the second kind for some unknown density function. Unlike the traditional boundary integral method, a distinctive feature of our scheme is that we do not require quadratures and direct evaluations of nearly singular, singular or hyper-singular boundary integrals in the solution of BIEs. Instead, we reinterpret boundary integrals as solutions to equivalent simple interface problems in an extended rectangle domain, which can be solved efficiently by a fourth-order finite difference method coupled with numerical corrections, FFT based solution and interpolations. Extensive numerical experiments show that our method is formally high-order accurate, fast convergent and in particular insensitive to complexity of scatterers.

Solving Fredholm second-kind integral equations with singular right-hand sides on non-smooth boundaries

A numerical scheme is presented for the solution of Fredholm second-kind boundary integral equations with right-hand sides that are singular at a finite set of boundary points. The boundaries themselves may be non-smooth. The scheme, which builds on recursively compressed inverse preconditioning (RCIP), is universal as it is independent of the nature of the singularities. Strong right-hand-side singularities, such as 1/|r|α with α close to 1, can be treated in full machine precision. Adaptive refinement is used only in the recursive construction of the preconditioner, leading to an optimal number of discretization points and superior stability in the solve phase. The performance of the scheme is illustrated via several numerical examples, including an application to an integral equation derived from the linearized BGKW kinetic equation for the steady Couette flow.

The Laguerre transform of a convolution product of vector-valued functions.

The Laguerre transform is applied to the convolution product of functions of a real argument (over the time axis) with values in Hilbert spaces. The main results have been obtained by establishing a relationship between the Laguerre and Laplace transforms over the time variable with respect to the elements of Lebesgue weight spaces. This relationship is built using a special generating function. The obtained dependence makes it possible to extend the known properties of the Laplace transform to the case of the Laguerre transform. In particular, this approach concerns the transform of a convolution of functions.
 The Laguerre transform is determined by a system of Laguerre functions, which forms an orthonormal basis in the weighted Lebesgue space. The inverse Laguerre transform is constructed as a Laguerre series. It is proven that the direct and the inverse Laguerre transforms are mutually inverse operators that implement an isomorphism of square-integrable functions and infinite squares-summable sequences.
 The concept of a q-convolution in spaces of sequences is introduced as a discrete analogue of the convolution products of functions. Sufficient conditions for the existence of convolutions in the weighted Lebesgue spaces and in the corresponding spaces of sequences are investigated. For this purpose, analogues of Young’s inequality for such spaces are proven. The obtained results can be used to construct solutions of evolutionary problems and time-dependent boundary integral equations.

Approximation by interpolation spectral subspaces of operators with discrete spectrum

The Laguerre transform is applied to the convolution product of functions of a real argument (over the time axis) with values in Hilbert spaces. The main results have been obtained by establishing a relationship between the Laguerre and Laplace transforms over the time variable with respect to the elements of Lebesgue weight spaces. This relationship is built using a special generating function. The obtained dependence makes it possible to extend the known properties of the Laplace transform to the case of the Laguerre transform. In particular, this approach concerns the transform of a convolution of functions.
 The Laguerre transform is determined by a system of Laguerre functions, which forms an orthonormal basis in the weighted Lebesgue space. The inverse Laguerre transform is constructed as a Laguerre series. It is proven that the direct and the inverse Laguerre transforms are mutually inverse operators that implement an isomorphism of square-integrable functions and infinite squares-summable sequences.
 The concept of a q-convolution in spaces of sequences is introduced as a discrete analogue of the convolution products of functions. Sufficient conditions for the existence of convolutions in the weighted Lebesgue spaces and in the corresponding spaces of sequences are investigated. For this purpose, analogues of Young’s inequality for such spaces are proven. The obtained results can be used to construct solutions of evolutionary problems and time-dependent boundary integral equations.

On attached masses of panels oscillating in incompressible medium

The paper discusses small oscillations of a panel in an incompressible medium. Air can be considered an incompressible medium during modal tests of solar array panels for spacecraft deployed on the ground in a lab environment. A panel is represented as a two-sided boundary surface. Conditions are determined for applicability of the potential motion of the medium. Calculation of the attached mass is reduced to the solution of the Neumann boundary value problem. To solve the boundary value problem, the method of boundary elements is used in the piecewise constant approximation variant, which provides a solution of the hypersingular boundary integral equation. Numerical solutions are obtained for the three fundamental modes of rectangular panels. The obtained numerical values are refined using non-linear Shanks transformation. Dependence of attached mass on panel elongation and the amount of the gap between its fragments is studied. For any in-plane oscillation mode of a panel fragment, the attached mass is determined using the principle of linear superposition. An estimate is given of the effect of the distance from the panel to the wall on the attached mass value. Key words: oscillations, incompressible medium, air, attached mass, rectangular panels, boundary elements method.

ON ATTACHED MASSES OF PANELS OSCILLATING IN INCOMPRESSIBLE MEDIUM

The paper discusses small oscillations of a panel in an incompressible medium. Air can be considered an incompressible medium during modal tests of solar array panels for spacecraft deployed on the ground in a lab environment. A panel is represented as a two-sided boundary surface. Conditions are determined for applicability of the potential motion of the medium. Calculation of the attached mass is reduced to the solution of the Neumann boundary value problem. To solve the boundary value problem, the method of boundary elements is used in the piecewise constant approximation variant, which provides a solution of the hypersingular boundary integral equation. Numerical solutions are obtained for the three fundamental modes of rectangular panels. The obtained numerical values are refined using non-linear Shanks transformation. Dependence of attached mass on panel elongation and the amount of the gap between its fragments is studied. For any in-plane oscillation mode of a panel fragment, the attached mass is determined using the principle of linear superposition. An estimate is given of the effect of the distance from the panel to the wall on the attached mass value. Key words: oscillations, incompressible medium, air, attached mass, rectangular panels, boundary elements method.

An Introduction to Operator Preconditioning for the Fast Iterative Integral Equation Solution of Time-Harmonic Scattering Problems

The aim of this paper is to provide an introduction to the improved iterative Krylov solution of boundary integral equations for time-harmonic scattering problems arising in acoustics, electromagnetism and elasticity. From the point of view of computational methods, considering large frequencies is a challenging issue in engineering since it leads to solving highly indefinite large scale complex linear systems which generally implies a convergence breakdown of iterative methods. More specifically, we explain the problematic and some partial solutions through analytical preconditioning for high-frequency acoustic scattering problems and the introduction of new combined field integral equations. We complete the paper with some recent extensions to the case of electromagnetic and elastic waves equations.

Parallel Skeletonization for Integral Equations in Evolving Multiply-Connected Domains

This paper presents a general method for applying hierarchical matrix skeletonization factorizations to the numerical solution of boundary integral equations with possibly rank-deficient integral o...

A fast direct solver for nonlocal operators in wavelet coordinates

In this article, we consider fast direct solvers for nonlocal operators. The pivotal idea is to combine a wavelet representation of the system matrix, yielding a quasi-sparse matrix, with the nested dissection ordering scheme. The latter drastically reduces the fill-in during the factorization of the system matrix by means of a Cholesky decomposition or an LU decomposition, respectively. This way, we end up with the exact inverse of the compressed system matrix with only a moderate increase of the number of nonzero entries in the matrix.To illustrate the efficacy of the approach, we conduct numerical experiments for different highly relevant applications of nonlocal operators: We consider (i) the direct solution of boundary integral equations in three spatial dimensions, issuing from the polarizable continuum model, (ii) a parabolic problem for the fractional Laplacian in integral form and (iii) the fast simulation of Gaussian random fields.

On a problem of the elastic wedge-shaped body oscillations generated in its rib zone

In the paper there studies the wave process in the wedge-shaped body exited by the antiplane shear oscillation generators in the rib zone surface. Mathematical model construction is fulfilled by the mixed dynamic boundary value problem reduced to the boundary integral equation (BIE) about contact stresses in the domain of given oscillation generators. Previous author’s papers permit to construct BIE solution and reconstruct the wave field in the whole wedge-shaped medium. To estimate the correctness of application of the solution mentioned above the wedgeshaped medium with generators is transformed to the half-space. Then the solution of the mixed problem for the half-space is constructed by two independed methods: by the method based on BIE and by use the degenerated elliptical coordinates. Both solutions are in rigorous coincidence.

Subtraction singularity technique applied to the regularization of singular and hypersingular integrals in high-order curved boundary elements in plane anisotropic elasticity

The numerical solutions of boundary integral equations by the Boundary Element Method (BEM) have been applied in several areas of computational engineering and science such as elasticity and fracture mechanics. The BEM formulations often require the evaluation of complex singular and hypersingular integrals. Therefore, BEM requires special integration schemes for singular elements. The Subtraction Singularity Technique (SST) is a general procedure for evaluating such integrals, which allows for the integral over high-order curved boundary elements. The SST regularises these kernels through the Taylor expansion of the integral kernels around the source point. This study presents the expressions from Taylor expansions, which are required for the regularization of singular and hypersingular boundary integral equations of plane linear anisotropic elasticity. These expressions have been implemented into an academic BEM code, which enables the integration over high-order straight and curved boundary elements. The proposed scheme leads to excellent performance. The results obtained by the proposed scheme are in excellent agreement with reference responses available in the literature.

A fourth-order kernel-free boundary integral method for implicitly defined surfaces in three space dimensions

The kernel-free boundary integral (KFBI) method is a finite difference version of the traditional boundary integral method for elliptic and parabolic partial differential equations on complex domains. It evaluates boundary or volume integrals involved in the solution of boundary integral equations (BIEs) by solving equivalent but simple interface problems on regular grids, so that the integral kernel or Green function is never needed or computed. This is the essential difference of the KFBI method from the traditional ones. It takes advantage of the well-conditioning property of discrete BIEs so that the number of Krylov subspace iterations is essentially independent of discretization parameter or system dimension. This paper presents a fourth-order kernel-free boundary integral method for second-order elliptic partial differential equations on complex domains in three space dimensions, whose boundaries are given by implicitly defined surfaces. It represents the domain boundary and discretizes data on it by intersection points of the surface with Cartesian grid lines. The approach has a variety of advantages. The current work solves simple interface problems with corrected 27-point compact finite difference schemes and calculates the discrete equations with a fast Fourier transform based elliptic solver. Numerical examples show that the proposed method is efficient as well as accurate.

Exact solutions of boundary integral equation arising in vortex methods for incompressible flow simulation around elliptical and Zhukovsky airfoils

The problem of 2D incompressible flow simulation around airfoils using vortex methods is considered. An exact solution for the boundary integral equation with respect to a free vortex sheet intensity at the airfoil surface line that arises in such problems is obtained. The exact solution is constructed for flows around elliptical and Zhukovsky airfoils using the theory of complex potentials and conformal mappings technique. It is possible to take into account the influence of singularities in the flow domain — point vortices which simulate vortex wake. The obtained exact solutions can be used to verify and estimate the accuracy of numerical schemes for the boundary integral equation solution: such procedure is also described in details.

Boundary integral equations for anisotropic elasticity of solids containing rigid thread-like inclusions

The paper presents a novel approach in derivation and solution of boundary integral equations of anisotropic elasticity of solids containing thin rigid wires (thread-like inclusions). It proposes to model rigid thread-like inclusions as spatial curves, which can rotate as a rigid one and possess certain rigid displacement. Somigliana identity is written for solids containing rigid thread-like inclusions, which are modeled by spatial curves. Based on this identity the hypersingular boundary integral equations are derived, and it is shown that they also include the non-singular terms. The comparison is made between the models and integral equations for 3D rigid thread-like inclusions and 2D rigid line inclusions. A problem for a single rectilinear rigid thread-like inclusion in an infinite elastic medium is considered. Its solution strategy is proposed based on the boundary element approach. Distribution of forces along the inclusion modeling line and the displacement field near its tip are shown.

A High-Order Kernel-Free Boundary Integral Method for the Biharmonic Equation on Irregular Domains

This work proposes second-order and fourth-order versions of a Cartesian grid based kernel-free boundary integral (KFBI) method for the biharmonic equation on both bounded irregular domains and singly periodic irregular domains. It is further development of the previous KFBI method for second-order elliptic PDEs. It reformulates boundary value problems of the fourth-order PDE as boundary integral equations of the first kind but the solution never needs to know the fundamental solution or Green’s function of the elliptic operator. Evaluation of boundary or volume integrals in the solution of boundary integral equations is made by solving equivalent interface problems on Cartesian grids with standard finite difference methods and fast Fourier transform based solvers. The work decomposes the biharmonic equation into two Poisson equations. It assumes the solution to one Poisson equation, which has no boundary conditions, as the sum of a volume integral with a double layer boundary integral, and applies Green’s third identity to derive a scalar boundary integral equation from the other Poisson equation that are subject to two boundary conditions. In the solution of the scalar boundary integral equation, each volume or boundary integral is evaluated with the KFBI method. Numerical examples are presented to demonstrate the solution accuracy and algorithm efficiency. A remarkable point of the work is that the nine-point compact difference scheme in dealing with each split second-order elliptic interface problem on irregular domains yields fourth-order accurate solution for the biharmonic equation.

On hybrid temporal basis functions for stable numerical solution of time domain boundary integral equations

Problems in unsteady aerodynamics and aeroacoustics can sometimes be formulated as integral equations, such as the boundary integral equations. Numerical discretization of integral equations in the time domain often leads to so-called March-On-in-Time (MOT) schemes. In the literature, the temporal basis functions used in MOT schemes have been largely limited to low-order shifted Lagrange basis functions. In order to evaluate the accuracy and effectiveness of the temporal basis functions, a Fourier analysis of the temporal interpolation schemes is carried out. Based on the Fourier analysis, the spectral resolutions of various temporal basis functions are quantified. It is argued that hybrid temporal basis functions be used for interpolation of the numerical solution and its derivatives with respect to time. Stability of the proposed hybrid schemes is studied by a matrix eigenvalue method. Substantial improvement in accuracy and efficiency by using the hybrid temporal basis functions for time domain integral equations is demonstrated by numerical examples. Compared with the traditional temporal basis functions, the use of hybrid basis functions keeps numerical errors low for a larger frequency range given the same time step size. Conversely, for a given range of frequency of interest, a larger time step can be used with the hybrid temporal basis functions, resulting in an increase in computational efficiency and, at the same time, a reduction in memory requirement.