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A generalized boundary integral equation method for boundary value problems of two-dimensional isotropic lattice Laplacian is proposed in this paper. The proposed method is an extension of the classical boundary integral equation method with notable advantage. By utilizing the asymptotic expression of the fundamental solution at infinity, this method effectively addresses the challenge of numerical integration involving singular integral kernels. The introduction of Green’s formulas, Dirichlet and Neumann traces, and other tools which are parallel to the traditional integral equation method, form a solid foundation for the development of the generalized boundary integral equation method. The solvability of boundary integral equations and the solvability of lattice interface problem are important guarantees for the feasibility of this method, and these are emphasized in this paper. Subsequently, the generalized boundary integral equation method is applied to boundary value problems equipped with either Dirichlet or Neumann boundary conditions. Simple numerical examples demonstrate the accuracy and effectiveness of the generalized boundary integral equation method. [ABSTRACT FROM AUTHOR]

The paper is concerned with using boundary element methods (BEM) for the accurate evaluation of fields in structures with clusters. For large-scale problems, the BEM system is solved iteratively by speeding up matrix-to-vector multiplications by applying a kernel-independent fast multipole method. Multiplication starts with source-to-multipole (S2M) translations, whose accuracy predefines the overall accuracy. We aim to increase the accuracy of these translations. The intensities of sources are assembled into clusters by an algorithm suggested. Each of them is characterized by its representative source, whose intensity equals the sum of the intensities of cluster sources. Thus, with growing distance, its field tends toward the field of the cluster. The accuracy of S2M translations is increased by subtracting from and adding to the far field of the cluster the far field of its representative source, and by using the proposed modified kernel to evaluate the difference of the fields, which decreases faster than the field of the cluster itself. Numerical results for typical kernels show a notable increase in the accuracy provided by the modified S2M translations. Keeping in them merely the added field is acceptable for many practical applications. This simplifies the modified S2M translations by avoiding calculation and storing matrices specific to each of the clusters. The improved translations may be also used for multipole-to-multipole translations, performed on next, after leaves, levels in upward running a hierarchical tree. [ABSTRACT FROM AUTHOR]

The Navier equation is the governing equation of elastic waves, and computing its solution accurately and rapidly has a wide range of applications in geophysical exploration, materials science, etc. In this paper, we focus on the efficient and high-precision numerical algorithm for the time harmonic elastic wave scattering problems from cornered domains via the boundary integral equations in two dimensions. The approach is based on the combination of Nyström discretization, analytical singular integrals and kernel-splitting method, which results in a high-order solver for smooth boundaries. It is then combined with the recursively compressed inverse preconditioning (RCIP) method to solve elastic scattering problems from cornered domains. Numerical experiments demonstrate that the proposed approach achieves high accuracy, with stabilized errors close to machine precision in various geometric configurations. The algorithm is further applied to investigate the asymptotic behavior of density functions associated with boundary integral operators near corners, and the numerical results are highly consistent with the theoretical formulas. [ABSTRACT FROM AUTHOR]

Accurate evaluation of nearly singular integrals plays an important role in many boundary integral equation based numerical methods. In this paper, we propose a variant of singularity swapping method to accurately evaluate the layer potentials for arbitrarily close targets. Our method is based on the global trapezoidal rule and trigonometric interpolation, resulting in an explicit quadrature formula. The method achieves spectral accuracy for nearly singular integrals on closed analytic curves. In order to extract the singularity from the complexified distance function, an efficient root finding method is proposed based on contour integration. Through the change of variables, we also extend the quadrature method to integrals on the piecewise analytic curves. Numerical examples for Laplace and Helmholtz equations show that high-order accuracy can be achieved for arbitrarily close field evaluation. [ABSTRACT FROM AUTHOR]

This paper is concerned with the cavity scattering problem in an infinite thin plate, where the out-of-plane displacement is governed by the two-dimensional biharmonic wave equation. Based on an operator splitting, the scattering problem is recast into a coupled boundary value problem for the Helmholtz and modified Helmholtz equations. A novel boundary integral formulation is proposed for the coupled problem. By introducing an appropriate regularizer, the well-posedness is established for the system of boundary integral equations. Moreover, the convergence analysis is carried out for the semi- and full-discrete schemes of the boundary integral system by using the collocation method. Numerical results show that the proposed method is highly accurate for both smooth and nonsmooth examples. [ABSTRACT FROM AUTHOR]

In this paper, we find an equivalent boundary integral equation to the classical ∂ ̄ -Neumann problem. The new equation contains an equivalent regularity to the global regularity of the ∂ ̄ -Neumann problem. We also use the integral equations to observe a new-found rigidity property of the ∂ ̄ -Neumann problem on the unit ball in ℂ 2 . [ABSTRACT FROM AUTHOR]

This paper describes a locally corrected trapezoidal quadrature method for the discretization of singular and hypersingular boundary integral operators (BIOs) that arise in solving boundary value problems for elliptic partial differential equations. The quadrature is based on a uniform grid in parameter space coupled with the standard punctured trapezoidal rule. A key observation is that the error incurred by the singularity in the kernel can be expressed exactly using generalized Euler--Maclaurin formulas that involve the Riemann zeta function in 2 dimensions (2D) and the Epstein zeta functions in 3 dimensions (3D). These expansions are exploited to correct the errors via local stencils at the singular point using a novel systematic moment-fitting approach. This new method provides a unified treatment of all common BIOs (Laplace, Helmholtz, Stokes, etc.). We present numerical examples that show convergence of up to 32nd-order in 2D and 9th-order in 3D with respect to the mesh size. [ABSTRACT FROM AUTHOR]

This paper presents a novel boundary integral equation (BIE) formulation for the two-dimensional time-harmonic water-waves problem. It utilizes a complex-scaled Laplace free-space Green's function, resulting in a BIE posed on the infinite boundaries of the domain. The perfectly matched layer (PML) coordinate stretching that is used to render propagating waves exponentially decaying allows for the effective truncation and discretization of the BIE unbounded domain. We show through a variety of numerical examples that, despite the logarithmic growth of the complex-scaled Laplace free-space Green's function, the truncation errors are exponentially small with respect to the truncation length. Our formulation uses only simple function evaluations (e.g., complex logarithms and square roots), hence avoiding the need to compute the involved water-wave Green's function. Finally, we show that the proposed approach can also be used to find complex resonances through a linear eigenvalue problem since the Green's function is frequency-independent. [ABSTRACT FROM AUTHOR]

Let Ω be the multiply connected domain in the extended complex plane ℂ ¯ obtained by removing m non-overlapping rectilinear segments from the infinite strip S = { z : Im z < π / 2 } . In this paper, we present an iterative method for numerical computation of a conformally equivalent bounded multiply connected domain G in the interior of the unit disk D and the exterior of m non-overlapping smooth Jordan curves. We demonstrate the utility of the proposed method through two applications. First, we estimate the capacity of condensers of the form (S,E) where E ⊂ S is a union of disjoint segments. Second, we determine the streamlines associated with uniform incompressible, inviscid and irrotational flow past disjoint segments in the strip S. [ABSTRACT FROM AUTHOR]

We correct a sign error in the paper [ABSTRACT FROM AUTHOR]

In this paper, we present a fast boundary integral equation method for the numerical conformal mapping and its inverse of bounded multiply connected regions onto a disk and annulus with circular slits regions. The method is based on two uniquely solvable boundary integral equations with Neumann-type and generalized Neumann kernels. The integral equations related to the mappings are solved numerically using combination of Nyström method, GMRES method, and fast multipole method. The complexity of this new algorithm is O ((M + 1) n) , where M + 1 stands for the multiplicity of the multiply connected region and n refers to the number of nodes on each boundary component. Previous algorithms require O ((M + 1) 3 n 3) operations. The numerical results of some test calculations demonstrate that our method is capable of handling regions with complex geometry and very high connectivity. An application of the method on medical human brain image processing is also presented. [ABSTRACT FROM AUTHOR]

In this paper, a parallel probabilistic method using the Feynman--Kac formula of killed Brownian motions is proposed to solve the mixed boundary value problems (BVPs) of 3D Laplace equations. To avoid using reflecting Brownian motions and the calculation of their local time L(t) in the Feynman--Kac representation of solutions for Neumann and Robin BVPs, the proposed method uses an iterative approach to approximate the solutions where each iteration will use the Feynman--Kac formula to solve a pure Dirichlet problem, thus only involving killed Brownian motions. First, the boundary of the domain is decomposed with overlapping local patches formed by the intersection of hemispheres superimposed on the domain boundary. The iteration starts with an arbitrary initial guess for the Dirichlet data on the Neumann and Robin boundaries; then, using the Feynman--Kac formula for a pure Dirichlet problem with the current available Dirichlet data on the whole boundary, the solution over the hemispheres can be obtained by the Feynman--Kac formula for the killed Brownian motions, sampled by a walk-on-spheres (WOS) algorithm. Second, by solving a local boundary integral equation (BIE) over each hemisphere and a local patch on the domain boundary, the Dirichlet data on the Neumann and Robin boundaries can be updated. By continuing this process, the proposed hybrid probabilistic and deterministic BIE-WOS method gives a highly parallel algorithm for the global solution of any mixed-type BVPs of the Laplace equations. Numerical results of various mixed interior and exterior BVPs demonstrate the parallel efficiency and accuracy of the proposed method. [ABSTRACT FROM AUTHOR]

The complex variable boundary element-free method (CVBEFM) is a meshless method that takes the advantages of both boundary integral equations (BIEs) in dimension reduction and the complex variable moving least squares (CVMLS) approximation in element elimination. The CVBEFM is developed in this paper for solving 3D problems. This paper is an attempt in applying complex variable meshless methods to 3D problems. Formulations of the CVMLS approximation on 3D surfaces and the CVBEFM for 3D potential and Helmholtz problems are given. In the current implementation, the CVMLS shape function of 3D problems is formed with 1D basis functions, and the boundary conditions in the CVBEFM can be applied directly and easily. Some numerical examples are presented to demonstrate the method. [ABSTRACT FROM AUTHOR]

Fast and high-order accurate algorithms for three-dimensional elastic scattering are of great importance when modeling physical phenomena in mechanics, seismic imaging, and many other fields of applied science. In this paper, we develop a novel boundary integral formulation for the three-dimensional elastic scattering based on the Helmholtz decomposition of elastic fields, which converts the Navier equation to a coupled system consisted of Helmholtz and Maxwell equations. An FFT-accelerated separation of variables solver is proposed to efficiently invert boundary integral formulations of the coupled system for elastic scattering from axisymmetric rigid bodies. In particular, by combining the regularization properties of the singular boundary integral operators and the FFT-based fast evaluation of modal Green’s functions, our numerical solver can rapidly solve the resulting integral equations with a high-order accuracy. Several numerical examples are provided to demonstrate the efficiency and accuracy of the proposed algorithm, including geometries with corners at different wavenumbers. [ABSTRACT FROM AUTHOR]

This paper concerns an inverse elastic scattering problem which is to determine a rigid obstacle from time domain scattered field data for a single incident plane wave. By using the Helmholtz decomposition, we reduce the initial-boundary value problem for the time domain Navier equation to a coupled initial-boundary value problem for wave equations, and prove the uniqueness of the solution for the coupled problem by employing the energy method. The retarded single layer potential is introduced to establish a set of coupled boundary integral equations, and uniqueness is discussed for the solution of these boundary integral equations. Based on the convolution quadrature method for time discretization, the coupled boundary integral equations are reformulated into a system of boundary integral equations in the s -domain, and then a convolution quadrature based nonlinear integral equation method is proposed for the inverse problem. Numerical experiments are presented to show the feasibility and effectiveness of the proposed method. [ABSTRACT FROM AUTHOR]

In this paper, we consider the reconstruction of the shape and the impedance function of an obstacle from measurements of the scattered field at a collection of receivers outside the object. The data is assumed to be generated by plane waves impinging on the unknown obstacle from multiple directions and at multiple frequencies. This inverse problem can be reformulated as an optimization problem: that of finding band-limited shape and impedance functions which minimize the L2 distance between the computed value of the scattered field at the receivers and the given measurement data. The optimization problem is highly non-linear, non-convex, and ill-posed. Moreover, the objective function is computationally expensive to evaluate (since a large number of Helmholtz boundary value problems need to be solved at every iteration in the optimization loop). The recursive linearization approach (RLA) proposed by Chen has been successful in addressing these issues in the context of recovering the sound speed of an inhomogeneous object or the shape of a sound-soft obstacle. We present an extension of the RLA for the recovery of both the shape and impedance functions of the object. The RLA is, in essence, a continuation method in frequency where a sequence of single frequency inverse problems is solved. At each higher frequency, one attempts to recover incrementally higher resolution features using a step assumed to be small enough to ensure that the initial guess obtained at the preceding frequency lies in the basin of attraction for Newton’s method at the new frequency. We demonstrate the effectiveness of this approach with several numerical examples. Surprisingly, we find that one can recover the shape with high accuracy even when the measurements are generated by sound-hard or sound-soft objects, eliminating the need to know the precise boundary conditions appropriate for modeling the object under consideration. While the method is effective in obtaining high-quality reconstructions for many complicated geometries and impedance functions, a number of interesting open questions remain regarding the convergence behavior of the approach. We present numerical experiments that suggest underlying mechanisms of success and failure, pointing out areas where improvements could help lead to robust and automatic tools for the solution of inverse obstacle scattering problems. [ABSTRACT FROM AUTHOR]

The hot spots conjecture is only known to be true for special geometries. This paper shows numerically that the hot spots conjecture can fail to be true for easy to construct bounded domains with one hole. The underlying eigenvalue problem for the Laplace equation with Neumann boundary condition is solved with boundary integral equations yielding a non-linear eigenvalue problem. Its discretization via the boundary element collocation method in combination with the algorithm by Beyn yields highly accurate results both for the first non-zero eigenvalue and its corresponding eigenfunction which is due to superconvergence. Additionally, it can be shown numerically that the ratio between the maximal/minimal value inside the domain and its maximal/minimal value on the boundary can be larger than 1 + 10− 3. Finally, numerical examples for easy to construct domains with up to five holes are provided which fail the hot spots conjecture as well. [ABSTRACT FROM AUTHOR]

The Galerkin boundary element method (BEM), also known as the method of moments, is a powerful method for solving the Laplace equation in three dimensions. There are advantages to Galerkin formulations for integral equations, as they treat problems associated with kernel singularity, and lead to symmetric and better conditioned matrices. However, the Galerkin method requires the computation of double surface integral over pairs of triangles. There are many semianalytical methods to treat these integrals, which all have some issues and are discussed in this paper. Novel methods inspired by the treatment of these kernels in the fast multipole method are presented for computing all the integrals that arise in the Galerkin formulation to any accuracy. Integrals involving completely geometrically separated triangles are nonsingular, and are computed using a technique based on spherical harmonics and multipole expansions and translations, which require the integration of polynomial functions over the triangles. Integrals involving cases where the triangles have common vertices, edges, or are coincident are treated via scaling and symmetry arguments, combined with automatic recursive geometric decomposition of the integrals. The methods are validated, and example results are presented. [ABSTRACT FROM PUBLISHER]

This paper presents a numerical method for computing the conformal mappings onto the parabolic slit domain, the elliptic slit domain and the hyperbolic slit domain. The method relies on a boundary integral equation with the generalized Neumann kernel. For a given multiply connected domain of connectivity m+1, the proposed method requires O((m+1)nlogn) operations where n is the number of nodes in the discretization of each boundary component of the given domain. Several numerical examples are presented to illustrate the performance of the proposed method. [ABSTRACT FROM AUTHOR]

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). [ABSTRACT FROM AUTHOR]

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 operators. Rank-deficient operators arise in boundary integral approaches to elliptic partial differential equations with multiple boundary components, such as in the case of multiple vesicles in a viscous fluid flow. Our generalized skeletonization factorization retains the locality property afforded by the "proxy point method,"" and allows for a parallelized implementation where different processors work on different parts of the boundary simultaneously. Further, when the boundary undergoes local geometric perturbations (such as movement of an interior hole), the factorization can be recomputed efficiently with respect to the number of modified discretization nodes. We present an application that leverages a parallel implementation of skeletonization with updates in a shape optimization regime. [ABSTRACT FROM AUTHOR]

A fast multipole method (FMM)/graphics processing unit-accelerated boundary element method (BEM) for computational magnetics and electrostatics via the Laplace equation is presented. The BEM is an integral method, but the FMM is typically designed around monopole and dipole sources. To apply the FMM to the integral expressions in the BEM, the internal data structures and logic of the FMM must be changed. However, this can be difficult. For example, computing the multipole expansions due to the boundary elements requires computing single and double surface integrals over them. Moreover, FMM codes for monopole and dipole sources are widely available and highly optimized. This paper describes a method for applying the FMM unchanged to the integral expressions in the BEM. This method, called the correction factor matrix method, works by approximating the integrals using a quadrature. The quadrature points are treated as monopole and dipole sources, which can be plugged directly into current FMM codes. The FMM is effectively treated as a black box. Inaccuracies from the quadrature are corrected during a correction factor step. The method is derived, and example problems are presented showing accuracy and performance. [ABSTRACT FROM PUBLISHER]

In this paper, an algorithm for solving boundary value problems of elasticity theory suitable for solving contact problems and those whose deformation domain contains thin layers is presented. The solution is represented as a linear combination of auxiliary and fundamental solutions to the Lame equations. The singular points of the fundamental solutions are located in an outer layer of the deformation domain near the boundary. The linear combination coefficients are determined by minimizing deviations of the linear combination from the boundary conditions. To minimize the deviations, a conjugate gradient method is used. Examples of calculations for mixed boundary conditions are presented. [ABSTRACT FROM AUTHOR]

Resonance frequencies are complex eigenvalues at which the homogeneous transmission problems have nontrivial solutions. These frequencies are of interest because they affect the behavior of the solutions even when the frequency is real. The resonance frequencies are related to problems for infinite domains which can be solved efficiently with the boundary integral equation method (BIEM). We thus consider a numerical method for determining resonance frequencies with fast BIEM and the Sakurai--Sugiura projection method. However, BIEM may have fictitious eigenvalues even when one uses Müller or PMCHWT formulations which are known to be resonance free when the frequency is real valued. In this paper, we propose new BIEs for transmission problems with which one can distinguish true and fictitious eigenvalues easily. Specifically, we consider waveguide problems for the Helmholtz equation in two dimensions and standard scattering problems for Maxwell's equations in three dimensions. We verify numerically that the proposed BIEs can separate the fictitious eigenvalues from the true ones in these problems. We show that the obtained true complex eigenvalues are related to the behavior of the solution significantly. We also show that the fictitious eigenvalues may affect the accuracy of BIE solutions in standard boundary value problems even when the frequency is real. [ABSTRACT FROM AUTHOR]

The use of time-domain boundary integral equations has proved very e ective and ecient for three-dimensional acoustic and electromagnetic wave equations. In even dimensions and when some dissipation is present, time-domain boundary equations contain an in nite memory tail. Due to this. computation for longer times becomes exceedingly expensive. In this paper we show how oblivious quadrature, initially designed for parabolic problems. can be used to signi cantly reduce both the cost and the memory requirements of computing this tail. We analyze Runge{Kutta-based quadrature and conclude the paper with numerical experiments. [ABSTRACT FROM AUTHOR]

This paper presents stable magnetic-field and electric-field integral equation formulations for exterior electromagnetic scattering by either penetrable or perfectly conducting closed bodies of general shape. The development relies on an extension of the conventional reciprocity theorem allowing the fields produced by some combination of sources in one environment to be connected with the fields generated by another combination of sources in a different environment. By the use of lossy metamaterials, the internal-resonance problem inherent to the original versions is eliminated, and thus the new formulations are amenable to a unique, highly accurate, and well-conditioned solution. [ABSTRACT FROM PUBLISHER]

Boundary integral equations (BIE) are widely used in applied electromagnetics, with the boundary element method (BEM) typically employed for numerical solution. The paper extends the recently proposed boundary difference method (BDM)—an alternative to BEM—to electromagnetic scattering problems with perfect conductors and with corners. BDM shares several key features and advantages of BEM but avoids the singular kernels inherent in BIE. Convergence rates of the method established via numerical experiments are in agreement with the theoretical predictions. [ABSTRACT FROM AUTHOR]

In the paper [Eppler and Harbrecht, Control & Cybernetics 34: 203–225, 2005], the authors investigated the identification of an obstacle or void of perfectly conducting material in a two-dimensional domain by measurements of voltage and currents at the boundary. In particular, the reformulation of the given nonlinear identification problem was considered as a shape optimization problem using the Kohn and Vogelius criterion. The compactness of the complete shape Hessian at the optimal inclusion was proven, verifying strictly the ill-posedness of the identification problem. The aim of the paper is to present a similar analysis for the related least square tracking formulations. It turns out that the two-norm-discrepancy is of the same principal nature as for the Kohn and Vogelius objective. As a byproduct, the necessary first order optimality condition are shown to be satisfied if and only if the data are perfectly matching. Finally, we comment on possible consequences of the two-norm-discrepancy for the regularization issue. [ABSTRACT FROM AUTHOR]

This paper presents a method for the numerical computation of 3-D sensitivity coefficients of a target object in magnetic induction tomography (MIT). The sensitivity coefficient at a point is defined as the dot product of electromagnetic fields produced by unit current flowing in the excitation and the detector coil. In this paper, the fields are governed by a set of boundary integral equations (BIEs). Numerical results demonstrate that the fields on the boundary and interior volume domain of the target can be accurately represented by radial basis functions (RBFs). The paper compares numerical solutions of the BIEs based on RBFs with analytical solutions and boundary element solutions. [ABSTRACT FROM AUTHOR]

The hybrid finite element-boundary integral-multilevel fast multipole algorithm (FE-BI-MLFMA) is applied to solve the challenge problem of scattering by a large body with deep cavities in this paper. The hierarchical higher order curvilinear vector finite element (HCVFE) is employed to reduce the numerical dispersion error in the FEM and to efficiently model the geometry of the cavity. The coupling approaches are investigated at the interface between the FE region and the BI region for handling the problem of the different order basis functions and meshes used in the FE and BI region. The problems encountered with the previous decomposition algorithm of FE-BI-MLFMA are pointed out and analyzed in this paper. A special algorithm of FE-BI-MLFMA is designed based on the distinct geometry characteristics of deep cavity. Numerical results are presented to demonstrate the accuracy, efficiency and flexibility of the higher order FE-BI-MLFMA for scattering by a large body with big and deep cavities. [ABSTRACT FROM PUBLISHER]

We consider the wave equation in a boundary integral formulation. The discretization in time is done by using convolution quadrature techniques and a Galerkin boundary element method for the spatial discretization. In a previous paper, we have introduced a sparse approximation of the system matrix by cut-off, in order to reduce the storage costs. In this paper, we extend this approach by introducing a panel clustering method to further reduce these costs. [ABSTRACT FROM PUBLISHER]

In this paper, we investigate numerically the long-time dynamics of a two-dimensional dendritic precipitate. We focus our study on the self-similar scaling behavior of the primary dendritic arm with profile x ∼ t α¹ and y ∼ t α2, and explore the dependence of parameters α1 and α2 on applied driving forces of the system (e.g. applied far-field flux or strain). We consider two dendrite forming mechanisms: the dendritic growth driven by (i) an anisotropic surface tension and (ii) an applied strain at the far-field of the elastic matrix. We perform simulations using a spectrally accurate boundary integral method, together with a rescaling scheme to speed up the intrinsically slow evolution of the precipitate. The method enables us to accurately compute the dynamics far longer times than could previously be accomplished. Comparing with the original work on the scaling behavior α1 = 0.6 and α2 = 0.4 [Phys. Rev. Lett. 71(21) (1993) 3461–3464], where a constant flux was used in a diffusion only problem, we found at long times this scaling still serves a good estimation of the dynamics though it deviates from the asymptotic predictions due to slow retreats of the dendrite tip at later times. In particular, we find numerically that the tip grows self-similarly with α1 = 1/3 and α2 = 1/3 if the driving flux J ∼ 1/R(t) where R(t) is the equivalent size of the evolving precipitate. In the diffusive growth of precipitates in an elastic media, we examine the tip of the precipitate under elastic stress, under both isotropic and anisotropic surface tension, and find that the tip also follows a scaling law. [ABSTRACT FROM AUTHOR]

This paper presents a boundary integral method with the adjoint generalized Neumann kernel for conformal mapping of a bounded multiply connected region onto a disk with spiral slits region $$\varOmega _1$$ . This extends the methods that have recently been given for mappings onto annulus with spiral slits region $$\varOmega _2$$ , spiral slits region $$\varOmega _3$$ , and straight slits region $$\varOmega _4$$ but with different right-hand sides. This paper also presents a fast implementation of the boundary integral equation method for computing numerical conformal mapping of bounded multiply connected region onto all four regions $$\varOmega _1$$ , $$\varOmega _2$$ , $$\varOmega _3$$ , and $$\varOmega _4$$ as well as their inverses. The integral equations are solved numerically using combination of Nyström method, GMRES method, and fast multipole method (FMM). The complexity of this new algorithm is $$O((m + 1)n)$$ , where $$m+1$$ is the multiplicity of the multiply connected region and n is the number of nodes on each boundary component. Previous algorithms require $$O((m+1)^3 n^3)$$ operations. The algorithm is tested on several test regions with complex geometries and high connectivities. The numerical results illustrate the efficiency of the proposed method. [ABSTRACT FROM AUTHOR]

• Novel boundary-based ML method for 2D solid mechanics. • Requires only boundary discretization, enabling faster learning. • Replaces unstable differential operators by stable integral operators. • Higher precision and efficiency than traditional ML methods. In this paper, we make the first attempt to adopt the boundary integrated neural networks (BINNs) for the numerical solution of two-dimensional (2D) elastostatic and piezoelectric problems. The proposed BINNs combine the artificial neural networks with the exact boundary integral equations (BIEs) to effectively solve the boundary value problems based on the corresponding partial differential equations (PDEs). The BIEs are utilized to localize all the unknown physical quantities on the boundary, which are approximated by using artificial neural networks and resolved via a training process. In contrast to many traditional neural network methods based on a domain discretization, the present BINNs offer several distinct advantages. Firstly, by embedding the analytical BIEs into the learning procedure, the present BINNs only need to discretize the boundary of the problem domain, which reduces the number of the unknowns and can lead to a faster and more stable learning process. Secondly, the differential operators in the original PDEs are substituted by integral operators, which can effectively eliminate the need for additional differentiations of the neural networks (high-order derivatives of the neural networks may lead to instabilities in the learning process). Thirdly, the loss function of the present BINNs only contains the residuals of the BIEs, as all the boundary conditions have been inherently incorporated within the formulation. Therefore, there is no necessity for employing any weighting functions, which are commonly used in most traditional methods to balance the gradients among the different objective functions. Extensive numerical experiments show that the present BINNs are much easier to train and can usually provide more accurate solutions as compared to many traditional neural network methods. [Display omitted] [ABSTRACT FROM AUTHOR]

DC Resistivity Tomography is a non-linear inversion problem. So far there are mainly two kinds of inversion methods, based on the finite-element method and alpha centers method. In this paper, the disadvantages of these two kinds of methods were analysed, and a new method of forward modeling and inversion (Tomography) based on boundary integral equations was proposed. This scheme successfuly overcomes the difficulties of the two formarly methods. It isn’t necessary to use the linearization approximation and calculate the Jacobi matrix. Numerical modeling results given in this paper showed that the computation speed of our method is fast, and there is not any special requirement for initial model, and satisfying results of tomography can be obtained in the case of great contrast of conductivity. So it has wide applications. [ABSTRACT FROM AUTHOR]

This paper considers the mathematical formulation of the elastodynamic contact problem for a cracked body. The contact interaction of the crack edges in 2D is studied for the case of normal incidence of a harmonic tension–compression P-wave and shear-dilatational HS-wave, respectively. The problem is solved by the method of boundary integral equations along with a special iterative algorithm. The dependence of the stress intensity factor on the wave number is studied [ABSTRACT FROM AUTHOR]

This paper presents a class of boundary integral equation methods for the numerical solution of acoustic and electromagnetic time-domain scattering problems in the presence of unbounded penetrable interfaces in two spatial dimensions. The proposed methodology relies on convolution quadrature (CQ) schemes and the recently introduced windowed Green function (WGF) method. As in standard timedomain scattering from bounded obstacles, a CQ method of the user's choice is used to transform the problem into a finite number of (complex) frequency-domain problems posed, in our case, on the domains containing unbounded penetrable interfaces. Each one of the frequency-domain transmission problems is then formulated as a second-kind integral equation that is effectively reduced to a bounded interface by means of the WGF method--which introduces errors that decrease super-algebraically fast as the window size increases. The resulting windowed integral equations can then be solved by means of any (accelerated or unaccelerated) off the- shelf Nyström or boundary element Helmholtz integral equation solvers capable of handling complex wavenumbers with large imaginary part. A highorder Nyström method based on Alpert's quadrature rules is used here. A variety of CQ schemes and numerical examples, including wave propagation in open waveguides as well as scattering from multiple layered media, demonstrate the capabilities of the proposed approach. [ABSTRACT FROM AUTHOR]

This paper concerns an inverse elastic scattering problem which is to determine the location and the shape of a rigid obstacle from the phased or phaseless far-field data for a single incident plane wave. By introducing the Helmholtz decomposition, the model problem is reduced to a coupled boundary value problem of the Helmholtz equations. The relation is established between the compressional or shear far-field pattern for the elastic wave equation and the corresponding far-field pattern for the coupled Helmholtz equations. An efficient and accurate Nyström-type discretization for the boundary integral equation is developed to solve the coupled system. The translation invariance of the phaseless compressional and shear far-field patterns are proved. A system of nonlinear integral equations is proposed and two iterative reconstruction methods are developed for the inverse problem. In particular, for the phaseless data, a reference ball technique is introduced to the scattering system in order to break the translation invariance. Numerical experiments are presented to demonstrate the effectiveness and robustness of the proposed method. [ABSTRACT FROM AUTHOR]

The electromagnetic compatibility (EMC) analysis of electromagnetic interference (EMI) filter circuits using 3-D numerical modeling by the partial element equivalent circuit (PEEC) method represents the central topic of this paper. The PEEC-based modeling method is introduced as a useful tool for the prediction of the high frequency performance of EMI input filters, which is affected by PCB component placement and self- and mutual-parasitic effects. Since the measuring of all these effects is rather difficult and time consuming, the modeling and simulation approach represents a valuable design aid before building the final hardware prototypes. The parasitic cancellation techniques proposed in the literature are modeled by the developed PEEC-boundary integral method (PEEC-BIM) and then verified by the transfer function and impedance measurements of the L-C and C-L-C filter circuits. Good agreement between the PEEC-BIM simulation and the measurements is achieved in a wide frequency range. The PEEC-BIM method is implemented in an EMC simulation tool GeckoEMC. The main task of the presented research is the exploration of building an EMC modeling environment for virtual prototyping of EMI input filters and power converter systems. [ABSTRACT FROM AUTHOR]

Problem of electromagnetic wave scattering by ideal conductors with thin dielectric coatings is considered. The problem is reduced to a system of volume and surface hypersingular integral equations. We solve the system by Galerkin's method, using piecewise linear RWG functions for the surface equation, whilst for the volume equation we propose a combined system of piecewise linear and piecewise constant functions with prismatic supports. The method is tested on model problems, which showed the applicability of the proposed algorithm to the solution of scattering problems. [ABSTRACT FROM AUTHOR]

Electromagnetic scattering by penetrable bodies often is modelled by the Poggio-Miller-Chan-Harrington-Wu-Tsai (PMCHWT) integral equation. Unfortunately the spectrum of the operator involved in this equation is bounded neither from above or below. This implies that the equation suffers from dense discretization breakdown; that is, the condition numbers of the matrix resulting upon discretizing the equation rise with the mesh density. The electric field integral equation, often used to model scattering by perfect electrically conducting bodies, is susceptible to a similar breakdown phenomenon. Recently, this breakdown was cured by leveraging the Calderón identities. In this paper, a Calderón preconditioned PMCHWT integral equation is introduced. By constructing a Calderón identity for the PMCHWT operator, it is shown that the new equation does not suffer from dense discretization breakdown. A consistent discretization scheme involving both Rao-Wilton-Glisson and Buffa-Christiansen functions is introduced. This scheme amounts to the application of a multiplicative matrix preconditioner to the classical PMCHWT system, and therefore is compatible with existing boundary element codes and acceleration schemes. The efficiency and accuracy of the algorithm are corroborated by numerical examples. [ABSTRACT FROM AUTHOR]

The present paper is dedicated to the analytical computation of shape derivatives in the polarizable continuum model. We derive expressions for the interaction energy's sensitivity with respect to variations of the cavity's shape by means of the Hadamard representation of the shape gradient. In particular, by using the adjoint approach, the shape gradient depends only on two solutions of the underlying electrostatic problem. We further formulate boundary integral equations to compute the involved quantities. [ABSTRACT FROM AUTHOR]

Traditional boundary integral methods suffer from the singularity of Green's kernels. The paper develops, for a model problem of 2D scattering as an illustrative example, singularity-free boundary difference equations. Instead of converting Maxwell's system into an integral boundary form first and discretizing second, here the differential equations are first discretized on a regular grid and then converted to boundary difference equations. The procedure involves nonsingular Green's functions on a lattice rather than their singular continuous counterparts. Numerical examples demonstrate the effectiveness, accuracy and convergence of the method. It can be generalized to 3D problems and to other classes of linear problems, including acoustics and elasticity. [ABSTRACT FROM AUTHOR]