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The boundary element method is one of the most popular methods of numerical modeling for various problems in mechanics and physics. A big attraction for scientists is the possibility to consider not the region for which a problem needs to be solved but to consider the region's border instead. The boundary element method allows for significant simplification of the decision process, increasing the accuracy and reliability of the results.

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.

This paper presents a novel approach called the boundary integrated neural networks (BINNs) for analyzing acoustic radiation and scattering. The method introduces fundamental solutions of the time‐harmonic wave equation to encode the boundary integral equations (BIEs) within the neural networks, replacing the conventional use of the governing equation in physics‐informed neural networks (PINNs). This approach offers several advantages. First, the input data for the neural networks in the BINNs only require the coordinates of “boundary” collocation points, making it highly suitable for analyzing acoustic fields in unbounded domains. Second, the loss function of the BINNs is not a composite form and has a fast convergence. Third, the BINNs achieve comparable precision to the PINNs using fewer collocation points and hidden layers/neurons. Finally, the semianalytic characteristic of the BIEs contributes to the higher precision of the BINNs. Numerical examples are presented to demonstrate the performance of the proposed method, and a MATLAB code implementation is provided as supplementary material.

\"Widely accepted around the world, the BCU method is the only direct method used in the power industry. Direct Methods for Stability Analysis of Electric Power Systems presents a comprehensive theoretical foundation of the method and its numerical implementation. This book provides graduate students, researchers, and practitioners with theoretical foundations of direct methods, energy functions, and the BCU method as well as the group-based BCU method and its applications. Numerical studies on industrial models and data are also included\"-- \"This book describes the BCU method (Boundary of Stability Region Based Controlling Unstable Equilibrium Point method)\"--

The new crack front elements with Williams' eigenexpansion properties for 3D crack problems are proposed in this paper. In the presented method, the dual boundary integral equations are collocated on the uncracked boundary and one of the crack surfaces. The unknowns of the integral equations may be displacements, tractions and crack open displacements on the crack faces. To characterize the Williams' eigenexpansion properties of displacements around the crack front, the crack front elements with Williams' eigenexpansion properties are developed. The construction method of these elements is based on the first order of the Williams' series eigenexpansion, and the two-point formula is used to construct the elements by using the crack surface opening displacement and the relationship between crack opening displacement and stress intensity factor. Several numerical examples are given to validate our method. The numerical results of the proposed method agree very well with those of other methods and exact solutions.

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.

Transport of viscous fluid through porous media is a direct consequence of the pore structure. Here we investigate transport through a specific class of two-dimensional porous geometries, namely those formed by fluid-mechanical erosion. We investigate the tortuosity and dispersion by analyzing the first two statistical moments of tracer trajectories. For most initial configurations, tortuosity decreases in time as a result of erosion increasing the porosity. However, we find that tortuosity can also increase transiently in certain cases. The porosity-tortuosity relationships that result from our simulations are compared with models available in the literature. Asymptotic dispersion rates are also strongly affected by the erosion process, as well as by the number and distribution of the eroding bodies. Finally, we analyze the pore size distribution of an eroding geometry. The simulations are performed by combining a boundary integral equation solver for the fluid equations, a second-order stable time-stepping method to simulate erosion, and high-order numerical methods to stably and accurately resolve nearly touching eroded bodies and particle trajectories near the eroding bodies.

In this paper, we investigate a sink-driven three-layer flow in a radial Hele-Shaw cell. The three fluids are of different viscosities, with one fluid occupying an annulus-like domain, forming two interfaces with the other two fluids. Using a boundary integral method and a semi-implicit time stepping scheme, we alleviate the numerical stiffness in updating the interfaces and achieve spectral accuracy in space. The interaction between the two interfaces introduces novel dynamics leading to rich pattern formation phenomena, manifested by two typical events: either one of the two interfaces reaches the sink faster than the other (forming cusp-like morphology), or they come very close to each other (suggesting a possibility of interface merging). In particular, the inner interface can be wrapped by the other to have both scenarios. We find that multiple parameters contribute to the dynamics, including the width of the annular region, the location of the sink, and the mobilities of the fluids.

We consider the retarded potential boundary integral equation, arising from the 3D elastic (vector) wave equation problem, endowed with a Dirichlet condition on the boundary and null initial conditions. For its numerical solution, we employ a weak formulation related to the energy of the system and we discretize it by a Galerkin-type boundary element method (BEM). This approach, called energetic BEM, has been already applied in the context of time-domain acoustic (scalar) wave propagation and it has revealed accurate and stable even on large time intervals of analysis. In particular, when standard (constant) shape functions for time discretization are employed, the double integration in time can be performed analytically. Then, one is left with the task of evaluating double space integrals, whose integration domains are generally delimited by the wave fronts of the primary and the secondary waves. Since the accurate computation of the integrals involved in the numerical scheme is a key issue for the stability of the method, we propose an efficient evaluation strategy, based on the exact detection of the integration domain. The presented numerical tests show the effectiveness of the proposed approach.