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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]

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]

• 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]

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]

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.