Discretization Of Boundary Integral Equations Research Articles

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.

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).

A fast direct boundary element method for 3D acoustic problems based on hierarchical matrices

The boundary element method (BEM) for acoustic problems is a numerical method based on solving the discretized boundary integral equation (BIE) corresponding to the Helmholtz equation. A fast direct BEM for 3D acoustic problems is proposed in this paper, which is more suitable for broadband acoustic simulation of complex structures, such as in the design and analysis of acoustic metamaterials. The main idea of the fast direct solver is based on the hierarchical off-diagonal low-rank (HODLR) matrix, randomized interpolative decomposition and fast matrix inversion formula. Several numerical examples in solving both interior and exterior acoustic problems are presented in this paper, including radiation and scattering problems with distributed and complex structures. The numerical results show that the same level of accuracy and higher computational efficiency can be achieved by using this fast direct BEM compared with the conventional direct BEM.The second and third authors (Liu and Ye) would like to dedicate this paper to Professor Subrata Mukherjee who had enormous influences on their academic lives and research work on the BEM.

Boundary integral equation methods for the solution of scattering and transmission 2D elastodynamic problems

Abstract We introduce and analyse various regularized combined field integral equations (CFIER) formulations of time-harmonic Navier equations in media with piece-wise constant material properties. These formulations can be derived systematically starting from suitable coercive approximations of Dirichlet-to-Neumann operators (DtN), and we present a periodic pseudodifferential calculus framework within which the well posedness of CIER formulations can be established. We also use the DtN approximations to derive and analyse OS methods for the solution of elastodynamics transmission problems. The pseudodifferential calculus we develop in this paper relies on careful singularity splittings of the kernels of Navier boundary integral operators, which is also the basis of high-order Nyström quadratures for their discretizations. Based on these high-order discretizations we investigate the rate of convergence of iterative solvers applied to CFIER and OS formulations of scattering and transmission problems. We present a variety of numerical results that illustrate that the CFIER methodology leads to important computational savings over the classical CFIE one, whenever iterative solvers are used for the solution of the ensuing discretized boundary integral equations. Finally, we show that the OS methods are competitive in the high-frequency high-contrast regime.

Corrected trapezoidal rules for boundary integral equations in three dimensions

The manuscript describes a quadrature rule that is designed for the high order discretization of boundary integral equations (BIEs) using the Nyström method. The technique is designed for surfaces that can naturally be parameterized using a uniform grid on a rectangle, such as deformed tori, or channels with periodic boundary conditions. When a BIE on such a geometry is discretized using the Nyström method based on the Trapezoidal quadrature rule, the resulting scheme tends to converge only slowly, due to the singularity in the kernel function. The key finding of the manuscript is that the convergence order can be greatly improved by modifying only a very small number of elements in the coefficient matrix. Specifically, it is demonstrated that by correcting only the diagonal entries in the coefficient matrix, \(O(h^{3})\) convergence can be attained for the single and double layer potentials associated with both the Laplace and the Helmholtz kernels. A nine-point correction stencil leads to an \(O(h^5)\) scheme. The method proposed can be viewed as a generalization of the quadrature rule of Duan and Rokhlin, which was designed for the 2D Lippmann–Schwinger equation in the plane. The techniques proposed are supported by a rigorous error analysis that relies on Wigner-type limits involving the Epstein zeta function and its parametric derivatives.

Fast solver for quasi-periodic 2D-Helmholtz scattering in layered media

We present a fast spectral Galerkin scheme for the discretization of boundary integral equations arising from two-dimensional Helmholtz transmission problems in multi-layered periodic structures or gratings. Employing suitably parametrized Fourier basis and excluding cut-off frequencies (also known as Rayleigh-Wood frequencies), we rigorously establish the well-posedness of both continuous and discrete problems, and prove super-algebraic error convergence rates for the proposed scheme. Through several numerical examples, we confirm our findings and show performances competitive to those attainedviaNyström methods.

Zeta correction: a new approach to constructing corrected trapezoidal quadrature rules for singular integral operators

A high-order accurate quadrature rule for the discretization of boundary integral equations (BIEs) on closed smooth contours in the plane is introduced. This quadrature can be viewed as a hybrid of the spectral quadrature of Kress (Math. Comput. Model. 15(3-5), 229–243 1991) and the locally corrected trapezoidal quadrature of Kapur and Rokhlin (SIAM J. Numer. Anal. 34(4), 1331–1356, 1997). The new technique combines the strengths of both methods, and attains high-order convergence, numerical stability, ease of implementation, and compatibility with the “fast” algorithms (such as the Fast Multipole Method or Fast Direct Solvers). Important connections between the punctured trapezoidal rule and the Riemann zeta function are introduced, which enable a complete convergence analysis and lead to remarkably simple procedures for constructing the quadrature corrections. The paper reports a detailed comparison between the new method and the methods of Kress, of Kapur and Rokhlin, and of Alpert (SIAM J. Sci. Comput. 20(5), 1551–1584, 1999).

Quotient-space boundary element methods for scattering at complex screens

A complex screen is an arrangement of panels that may not be even locally orientable because of junction lines. A comprehensive trace space framework for first-kind variational boundary integral equations on complex screens has been established in Claeys and Hiptmair (Integr Equ Oper Theory 77:167–197, 2013. https://doi.org/10.1007/s00020-013-2085-x) for the Helmholtz equation, and in Claeys and Hiptmair (Integr Equ Oper Theory 84:33–68, 2016. https://doi.org/10.1007/s00020-015-2242-5) for Maxwell’s equations in frequency domain. The gist is a quotient space perspective that allows to make sense of jumps of traces as factor spaces of multi-trace spaces modulo single-trace spaces without relying on orientation. This paves the way for formulating first-kind boundary integral equations in weak form posed on energy trace spaces. In this article we extend that idea to the Galerkin boundary element (BE) discretization of first-kind boundary integral equations. Instead of trying to approximate jumps directly, the new quotient space boundary element method employs a Galerkin BE approach in multi-trace boundary element spaces. This spawns discrete boundary integral equations with large null spaces comprised of single-trace functions. Yet, since the right-hand-sides of the linear systems of equations are consistent, Krylov subspace iterative solvers like GMRES are not affected by the presence of a kernel and still converge to a solution. This is strikingly confirmed by numerical tests.

Axisymmetric BEM analysis of one-layered transversely isotropic halfspace with cavity subject to external loads

This paper presents an axisymmetric boundary element analysis of one-layered transversely isotropic material of halfspace extent with either ellipsoidal or spherical cavity. This analysis uses the fundamental solution of a transversely isotropic bi-material fullspace subject to the body force uniformly concentrated at a circular ring. The finite and infinite boundary elements are, respectively, used to discretize the finite and semi-infinite regions of the external boundary and the finite boundary elements are used to discretize the internal boundary. Different types of the integrals in the discretized boundary integral equations are efficiently calculated. An improved traction recovered method is applied to calculate the stresses on the boundary. Using one set of BEM mesh, a total of 12 case studies are calculated for the elastic fields of the one-layered transversely isotropic solid by uniform traction on the external horizontal boundary. The solid has either an ellipsoidal, a spherical, or none cavity in the upper layer. Four combinations of two transversely isotropic metals are adopted for the one-layered solid. One metal is the stiff zinc and the other is the soft magnesium. The displacements for the cases with magnesium as the upper layer are larger than those with zinc as the upper layer. Besides, the effects of the lower solid materials to the displacements are limited. The BEM results also reveal the variations of the displacements and stresses within the halfspace with or without cavities for different cases.

Boundary element formulation of axisymmetric problems in vertically non-homogeneous solids subject to normal traction

This paper presents an efficient and accurate boundary element method (BEM) for the elastic analysis of axisymmetric problems in vertically non-homogeneous solids without or with cavity. This BEM uses the fundamental solution of the elastic field in a multilayered elastic solid induced by the body force uniformly concentrated at a circular ring. This solution is also called Yue's solution. The effective integration methods are used for dealing with the integrals in the discretized boundary integral equations. The discretization of the boundary surface uses one-dimensional boundary elements. It also adopts the infinite boundary element to take into account the influence of a far-field region on the boundary surface. Numerical verifications of displacements and stresses for three benchmark problems are conducted, which gives the excellent agreement with previously published results. Case studies are presented to numerically illustrate the influences of both vertically non-homogeneous elastic material properties and spherical cavity on the elastic fields induced by uniform tractions on the boundary surface. These numerical results show that this new BEM is a fast and simple numerical algorithm for accurately computing the axisymmetric elastic fields in vertically non-homogeneous solids with or without cavity induced by normal tractions.

Boundary element analysis of transversely isotropic bi‐material halfspaces with inclined planes of isotropy and interfaces

SummaryIn this paper, a single‐region boundary element method (BEM) is presented for the analysis of transversely isotropic bi‐material halfspaces with arbitrarily inclined planes of isotropy and material interfaces. The proposed BEM uses the fundamental solution of a transversely isotropic bi‐material fullspace and five boundary element techniques. Infinite boundary elements are introduced to consider the far‐fields of a transversely isotropic bi‐material halfspace. The effective integration methods are proposed for dealing with various integrals in the discretized boundary integral equation. The stresses at internal points are obtained using the coordinate transformation of kernel functions, and the stresses on the boundary surface are calculated using an improved traction recovered method. Numerical verifications of displacements and stresses for a benchmark problem are conducted, and excellent agreement with previously published results is obtained. Numerical examples are presented to illustrate the influence of non‐horizontal or horizontal planes of isotropy in bi‐material halfspaces on the displacements and stresses induced by the tractions on the horizontal boundary surface. Results reveal that the elastic fields vary clearly with the dip angle of the isotropic plane and the stresses across the bi‐material interface are closely related to the ratios of the elastic parameters of the bi‐material.

Fast direct isogeometric boundary element method for 3D potential problems based on HODLR matrix

A novel fast direct solver based on isogeometric boundary element method (IGABEM) is presented for solving 3D potential problems, which uses the hierarchical off-diagonal low-rank (HODLR) matrix structure arising from the discretization of boundary integral equations. Since the HODLR matrix can be factored into the product form of some diagonal blocks, we can use the Sherman–Morrison–Woodbury formula to solve the inverse of a HODLR matrix efficiently. For large scale problems, an accelerated adaptive cross approximation algorithm is developed to decompose the off-diagonal submatrices. In numerical implementation, bivariate NURBS basis functions are used to describe the geometry. Meanwhile, the same NURBS basis functions are also used to approximate the unknown boundary quantities. The present method is applied to some numerical examples, including an infinite space containing twenty spherical cavities. The numerical results clearly show that the fast direct solver developed in the paper can obtain accurate results with less CPU time.

Modeling and Optimization of Anisotropic Viscoelastic Porous Structures Using CQBEM and Moving Asymptotes Algorithm

The aim of this paper is to develop a novel numerical modeling optimization technique and implement it with the boundary element method (BEM) based on convolution quadrature method for studying optimization of anisotropic viscoelastic porous permeable structures. After applying the quadrature rule to the discretized boundary integral equation, a time stepping procedure based on the use of the linear multistep method is obtained. To obtain anisotropic fundamental solutions, the calculation of a double integral should be performed, but it increases the total computation time in the BEM. In order to overcome this problem and improve the computational efficiency of the formulation, our new proposed BEM technique is implemented. The method of implicit differentiation with respect to design variables has been implemented to calculate the displacements and pore pressure design sensitivities, and the effects of viscosity on these sensitivities are discussed. The resulting topology optimization problem has been solved using the method of moving asymptotes algorithm with adjoint variable method to optimize material distribution and find the influence of viscosity on the optimal design. The validity, efficiency and accuracy of the proposed BEM technique were confirmed by comparing obtained results for elliptical sandwich structure and multi-well reservoir with the corresponding results of finite difference method, lattice Boltzmann method and finite element method, which are special cases of our general and complex problem.

A Fourth-Order Kernel-Free Boundary Integral Method for the Modified Helmholtz Equation

Based on the kernel-free boundary integral method proposed by Ying and Henriquez (J Comput Phys 227(2):1046---1074, 2007), which is a second-order accurate method for general elliptic partial differential equations, this work develops it to be a fourth-order accurate version for the modified Helmholtz equation. The updated method is in line with the original one. Unlike the traditional boundary integral method, it does not need to know any analytical expression of the fundamental solution or Green's function in evaluation of boundary or volume integrals. Boundary value problems under consideration are reformulated into Fredholm boundary integral equations of the second kind, whose corresponding discrete forms are solved with the simplest Krylov subspace iterative method, the Richardson iteration. During each iteration, a Cartesian grid based nine-point compact difference scheme is used to discretize the simple interface problem whose solution is the boundary or volume integral in the BIEs. The resulting linear system is solved by a fast Fourier transform based solver, whose computational work is roughly proportional to the number of grid nodes in the Cartesian grid used. As the discrete boundary integral equations are well-conditioned, the iteration converges within an essentially fixed number of steps, independent of the mesh parameter. Numerical results are presented to verify the solution accuracy and demonstrate the algorithm efficiency.

Boundary element analysis of elastic fields in non-horizontally layered halfspace whose horizontal boundary subject to tractions

This paper presents a boundary element analysis of linear elastic fields in a layered halfspace whose material interface planes are not parallel to its horizontal boundary surface. This boundary element analysis uses the generalized Kelvin solution in a multilayered elastic solid (or the so-called Yue's solution) for taking into account the non-horizontally layered structures. It also adopts the infinite boundary elements for evaluating the influence of a far-field region. It further adopts both the discontinuous finite and infinite boundary elements to discretize the boundary surface around the strike lines of the inclined material interfaces. It uses Kutt's numerical quadrature to evaluate the strongly singular integral in the discretized boundary integral equation. Numerical examples are presented to illustrate the effects of the non-horizontally layered materials to the displacements and stresses induced by the tractions on the horizontal boundary surface. Two non-horizontally layered halfspace models are used for numerical analysis and illustrations. Numerical results show that across the material interface, the elastic displacements are non-smoothly continuous to different degrees and some stress components can have very high values at and around the interface planes, which can be important to tensile or shear failure in non-homogeneous materials.

On preconditioning the treecode-accelerated boundary integral (TABI) Poisson–Boltzmann solver

We recently developed a treecode-accelerated boundary integral (TABI) solver for solving Poisson–Boltzmann (PB) equation [1]. The solver has combined advantages in accuracy, efficiency, memory, and parallelization as it applies a well-posed boundary integral formulation to circumvent many numerical difficulties associated with the PB equation and uses an O(Nlog⁡N) treecode to accelerate the GMRES iterative solver. However, as observed in our previous work [2], occasionally when the mesh generator produces low quality triangles, the number of GMRES iterations required to solve the discretized boundary integral equations Ax=b could be large. To address this issue, we design a preconditioning scheme using preconditioner matrix M such that M−1A has much improved condition while M−1z can be rapidly computed for any vector z. In this scheme, the matrix M carries the interactions between boundary elements on the same leaf only in the tree structure thus is block diagonal with many computational advantages. The sizes of the blocks in M are conveniently controlled by the treecode parameter N0, the maximum number of particles per leaf. The numerical results show that this new preconditioning scheme improves the TABI solver with significantly reduced iteration numbers and better accuracy, particularly for protein sets on which TABI solver previously converges slowly. In addition, this preconditioning scheme potentially can improve the condition number of various multipole method accelerated boundary elements solvers in scattering, fluids, elasticity, etc.

Fast Solution of Boundary Integral Equations for Elasticity Around a Crack Network: A Comparative Study

Because of the non-local nature of the integral kernels at play, the discretization of boundary integral equations leads to dense matrices, which would imply high computational complexity. Acceleration techniques, such as hierarchical matrix strategies combined with Adaptive Cross Approximation (ACA), are available in literature. Here we apply such a technique to the solution of an elastostatic problem, arising from industrial applications, posed at the surface of highly irregular cracks networks.

A fast direct solver for boundary value problems on locally perturbed geometries

Many applications including optimal design and adaptive discretization techniques involve solving several boundary value problems on geometries that are local perturbations of an original geometry. This manuscript presents a fast direct solver for boundary value problems that are recast as boundary integral equations. The idea is to write the discretized boundary integral equation on a new geometry as a low rank update to the discretized problem on the original geometry. Using the Sherman–Morrison formula, the inverse can be expressed in terms of the inverse of the original system applied to the low rank factors and the right hand side. Numerical results illustrate for problems where perturbation is localized the fast direct solver is three times faster than building a new solver from scratch.

Simulation of electrical machines: a FEM-BEM coupling scheme

PurposeElectrical machines commonly consist of moving and stationary parts. The field simulation of such devices can be demanding if the underlying numerical scheme is solely based on a domain discretization, such as in the case of the finite element method (FEM). This paper aims to present a coupling scheme based on FEM together with boundary element methods (BEMs) that neither hinges on re-meshing techniques nor deals with a special treatment of sliding interfaces. While the numerics are certainly more involved, the reward is obvious: the modeling costs decrease and the application engineer is provided with an easy-to-use, versatile and accurate simulation tool.Design/methodology/approachThe authors present the implementation of a FEM-BEM coupling scheme in which the unbounded air region is handled by the BEM, while only the solid parts are discretized by the FEM. The BEM is a convenient tool to tackle unbounded exterior domains, as it is based on the discretization of boundary integral equations (BIEs) that are defined only on the surface of the computational domain. Hence, no meshing is required for the air region. Further, the BIEs fulfill the decay and radiation conditions of the electromagnetic fields such that no additional modeling errors occur.FindingsThis work presents an implementation of a FEM-BEM coupling scheme for electromagnetic field simulations. The coupling eliminates problems that are inherent to a pure FEM approach. In detail, the benefits of the FEM-BEM scheme are: the decay conditions are fulfilled exactly, no meshing of parts of the exterior air region is necessary and, most importantly, the handling of moving parts is incorporated in an intriguingly simple manner. The FEM-BEM formulation in conjunction with a state-of-the-art preconditioner demonstrates its potency. The numerical tests not only reveal an accurate convergence behavior but also prove the algorithm to be suitable for industrial applications.Originality/valueThe presented FEM-BEM scheme is a mathematically sound and robust implementation of a theoretical work presented a decade ago. For the application within an industrial context, the original work has been extended by higher-order schemes, periodic boundary conditions and an efficient treatment of moving parts. While not intended to be used under all circumstances, it represents a powerful tool in case that high accuracies together with simple mesh-handling facilities are required.

Modeling of Wave Propagation in the Unsaturated Soils Using Boundary Element Method

Problems of wave propagation in poroelastic bodies and media are considered. The behavior of the poroelastic medium is described by Biot theory for partially saturated material. Mathematical model is written in term of five basic functions – elastic skeleton displacements, pore water pressure and pore air pressure. Boundary element method (BEM) is used with step method of numerical inversion of Laplace transform to obtain the solution. Research is based on direct boundary integral equation of three-dimensional isotropic linear theory of poroelasticity. Green’s matrices and, based on it, boundary integral equations are written for basic differential equations in partial derivatives. Discrete analogue are obtained by applying the collocation method to a regularized boundary integral equation. To approximate the boundary consider its decomposition to a set of quadrangular and triangular 8-node biquadratic elements, where triangular elements are treated as singular quadrangular. Every element is mapped to a reference one. Interpolation nodes for boundary unknowns are a subset of geometrical boundary-element grid nodes. Local approximation follows the Goldshteyn’s generalized displacement-stress matched model: generalized boundary displacements are approximated by bilinear elements whereas generalized tractions are approximated by constant. Integrals in discretized boundary integral equations are calculated using Gaussian quadrature in combination with singularity decreasing and eliminating algorithms.