Boundary Integral Equation Method Research Articles

Dynamic Ruptures on Bending Fault: Insights from Numerical Simulations of Transient Stress Field

ABSTRACT We delve into the spontaneous rupture propagation on bending faults by numerical simulations based on the boundary integral equation method with unstructured meshes. To study the effect of fault geometry on dynamic rupture propagation, special attention is paid to the role of the dynamic stress field. The numerical results demonstrate that the bending angle is a key geometrical factor influencing the rupture propagation because it affects both the initial stress distribution and the dynamic stress field on the bending branch. The rupture propagation on the bending branch can be separated into two distinct stages: first, the propagation from the main branch to the bending branch, which largely depends on the dynamic stress field near the bend; and second, a subsequent propagation stage primarily influenced by the initial stress state on the bending branch, with the influence of the dynamic stress field decreasing rapidly with distance from the bend. Geometrical smoothing of the bend can be regarded as a modification of the bending angle, which may significantly alter the behavior of rupture propagation near the bend. In theory, if the bending angle ranges between −120° and 60°, there is a potential for rupture to propagate onto the bending branch through the bend.

CALCULATION OF FLAT SHELLS BY THE BOUNDARY ELEMENT METHOD

This article describes the iterative process of solving the problem of the stress-strain state of a long flexible cylindrical panel. Stress-strain state and the study of the stability of flat shells taking into account the physical, mechanical properties of the material, the process of deformation change. It is based on sequential approximation methods and the boundary element method (MGE). The results of algorithmic weighting are presented in the form of a table showing what value each band represents. The equations are determined by solving twelve unknown values of functions at the ends of the segment. Linear problems of elasticity theory and plate theory fundamental solutions have a simple form, so the method is widely used here. For flat shells, the matrix of fundamental solutions is determined by complex volumetric expressions, and for flat shells - by special functions. Therefore, there is little research on solving problems in the theory of flat shells by the boundary element method. In this regard, an urgent topic of research is the development of methods of boundary integral equations for solving linear and nonlinear problems in the theory of flat shells based on the application of fundamental solutions defined by simple analytical expressions. The scientific novelty of the work consists in the development of a methodology for assessing the reliability of thin-walled spatial structures using the boundary element method.

Математическая модель в задаче идентификации формы летательного аппарата по диаграмме рассеяния звукового поля

This paper proposes a mathematical model to solve an inverse problem reconstructing the shape and size of possible models of an unmanned aerial vehicle based on its acoustic trail. To simplify the calculations, four two-dimensional simplest geometric shapes were chosen for numerical experiments - a circle, an ellipse, a square and a four-petal rose. The scattering diagrams of the sound field, as a solution to direct diffraction problem for each contour, were taken as the input data. A residual functional is constructed as a difference between the true data of the scattered sound field and the data obtained using the semi-analytical boundary integral equation method and the Fredholm integral equation of the second kind, in the direct diffraction problem. The minimization of this functional leads to the inverse diffraction problem. After discretizing the boundary curve, the problem is reduced to a linear algebraic equation system, from which a function is found that numerically describes the shape of the object.

On the boundary integral equation method of solving boundary value problems for the two dimensional Laplace equation

We consider approach based on the integral representation of solutionsin domain which consists of bounded and unbounded parts thatgives us opportunity to reduce different transmission type problems toconnected with them equivalent boundary equations of the first and the second kind.We suppose also that solutions of some of these boundary problems are unbounded at infinity.Interior and exterior Dirichlet and Neumann boundary valueproblems for Laplace equation are restrictions of the solutionsos more general this transmission problems.Interior Neumann and exterior Dirichlet boundary value problems we also can solve usingintegral equation of the second kind that have not unique solution.Corresponding modified equations are constructed in this case and solutions of obtained equations are unique.We also show correctness of all obtained boundary equations of the second typegiven on closed Lipschitz curve in some Hilbert spaceswithout compactness of corresponding integral operators.

Wave scattering in a cracked exponentially graded magnetoelectroelastic half‐plane

AbstractAn exponentially graded with respect to depth magnetoelectroelastic (MEE) half‐plane containing two line or curvilinear cracks under time‐harmonic SH wave is studied. The defined mechanical problem is described by boundary integral equations (BIEs) along the cracks boundaries. The computational tool based on the non‐hypersingular traction boundary integral equation method (BIEM) is developed, verified and inserted in numerical simulations. It is based on the analytically derived Green's function and free‐field wave motion solution for exponentially graded MEE half‐plane. The dependence of the generalized stress intensity factors (SIFs) on the material gradient parameters, on the dynamic load characteristics, on the cracks position and their shape, on the dynamic interaction between cracks and between them and half‐plane boundary is numerically analyzed.

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.”

Study on Ground Motion Amplification in Upper Arch Bridge Due to “W”-Type Deep Canyon Using Boundary-Integral and Peak Frequency Shift Methods

The study of the dynamic response characteristics of “W”-type deep canyon terrain to double-span concrete arch bridges under earthquake action holds great practical significance. In this research, a bridge in Sichuan Province is taken as the object of study. The boundary-integral equation method and peak frequency shift method are combined to apply an embedded linear time–history analysis algorithm to the finite element spatial dynamic calculation model of the entire bridge, resulting in an improved model. By comparing these two methods with model test results, the seismic response characteristics of the middle part of a “W” concrete arch bridge under different foundation depths and seismic intensities are examined. The boundary integral equation method was utilized to calculate ground motion response at any point on site, revealing a significant amplifying effect of increased seismic wave intensity on acceleration response at the top of the arch bridge. When input seismic wave intensity increased from 0.1 g to 0.3 g, maximum acceleration at buried depths of 3 m and 8 m in the middle of the arch bridge foundation increased by 102.63% and 79.16%, respectively, indicating that shallow buried depth structures are more sensitive to seismic wave intensity. Furthermore, using peak frequency shift rules for analyzing seismic wave propagation characteristics in “W”-type deep canyon topography confirms the sensitivity of shallow buried depth structures to seismic wave intensity and reveals the mechanism through which topography influences seismic wave propagation. This study provides a helpful method for understanding the propagation law and energy distribution characteristics of seismic waves in complex terrain. It was observed that the displacement at the top of the arch bridge increased significantly with an increase in seismic intensity. When subjected to 0.1 g, 0.2 g, and 0.3 g EI-Centro seismic waves, the maximum displacement at the top of the arch bridge model with a foundation buried depth of 3 m was 8 mm, 32 mm, and 142 mm, respectively. For arch bridge models with an 8-m foundation buried depth, these displacements were measured at 6.2 mm, 21 mm, and 68 mm, respectively. The results from model tests verified that increasing the depth of foundation burial effectively reduces the displacement at the top of the structure. Furthermore, by combining a boundary-integral equation method and peak-frequency shift method, this study accurately predicted significant influences on W-shaped double deep canyon topography from seismic response, and successfully captured stress concentration and seismic wave amplification/focusing effects on arch foot structures. The calculated results from both methods align well with model test data which confirm their effectiveness and complementarity when analyzing seismic responses under complex terrain conditions for bridge structures.

A new seismic wave input method for canyon sites based on the finite element method-indirect boundary integral equation method

The seismic input is the basis for the seismic safety analysis of dams, but current seismic input methods have not reasonably considered the influence of canyon scattering effects on the dynamic response of dams. In this paper, the total motion field of the canyon is deconstructed into the free field of a uniform half space and the scattering field of the canyon and they are obtained separately. The indirect boundary integral equation method (IBIEM) is used to determine the scattering field of the canyon, which is superimposed with the free field to obtain the seismic ground motion field (total motion field) of the canyon. The accuracy of the total motion field of the canyon is verified through reference solutions. In the finite element analysis of seismic ground motion field in the canyon, the total motion field at the truncation boundary of the canyon foundation is transformed into the equivalent seismic input loads and the interior of the canyon are simulated using the finite element method (FEM). Then, based on the wave input for the free field combined with the viscoelastic artificial boundary, a new wave input method for the total motion field combined with the viscoelastic artificial boundary is established. The seismic wave input method based on the finite element-indirect boundary integral equation method is applied to numerically determine the seismic response of a trapezoidal canyon. The differences in the seismic response of the canyon under free-field input and total motion field input are discussed. The results show that there is a certain deviation between the displacement and waveform obtained from the free-field input and the solutions obtained from the indirect boundary integral equation method. There is a large error for a large angle of incidence, and the errors increase with increasing seismic wave frequency. When the incident angle is 60°, the maximum error in the frequency domain reaches 92.3 %, and the maximum error in the time domain reaches 250 %. The displacement amplitude and waveform of the canyon obtained from the total motion field input showed agreement with the results of the indirect boundary integral equation method. The wave input method established in this paper has high calculation accuracy and reasonably considers the scattering effect of canyons, which provides a basis for more accurate predictions of the seismic response of dams on canyon foundations.

Об элементных фундаментах опор высоковольтных линий

Purpose: to substantiate the possibility of using the foundations of supports of high-voltage lines in the form of dispersed horizontal elements combined into a single structure for the perception of alternating loads. To determine by experimental and computational methods the effect of the mutual displacement of elements in the form of an increase in the bearing capacity of foundations and a decrease in their deformations under the influence of pressing and pulling loads. To show the possibility of replacing slab foundations of supports of high-voltage lines with structures made of prefabricated elements to reduce the volume and weight of products transported from the factory. Methods: application of the moire photogrammetric method to assess the development of soil compaction zones and angular deformations of the soil base in a tray with a glass wall under models of slab and elemental foundations. Using the method of boundary integral equations to obtain the dependence of precipitation on the step of elements for flexible and rigid foundations. Results: the effect of mutual influence of foundation elements during their convergence and dispersal in the form of a change in the stress-strain state in their base is illustrated. Differences in the depths of the distribution of compaction zones and areas of development of angular deformations are revealed. Quantitative data on the precipitation of slab and elemental foundations and the influence of the depth of the roof of the solid underlying layer on them have been obtained. Graphical dependences of the relationship of sediment elements with the distance (step) between them are presented. The effect of the division of the foundation on the rate of extinction of marginal tangential stresses in the base is shown. An illustration is made of an increase in the volume and surface area of the bulging body of the elemental foundation, which leads to an increase in its bearing capacity compared to the slab one when calculating for pulling out. Practical importance: the effectiveness of the use of an element foundation in comparison with the paid option in reducing the parameters of the stress-strain state of the base and facilitating logistical tasks during construction is shown. The experimental and computational methods used can be recommended for further refinement of the parameters of the elemental foundations.

Solving acoustic scattering problems by the isogeometric boundary element method

AbstractWe solve acoustic scattering problems by means of the isogeometric boundary integral equation method. In order to avoid spurious modes, we apply the combined field integral equations for either sound-hard scatterers or sound-soft scatterers. These integral equations are discretized by Galerkin’s method, which especially enables the mathematically correct regularization of the hypersingular integral operator. In order to circumvent densely populated system matrices, we employ the isogeometric embedded fast multipole method, which is based on interpolation of the kernel function under consideration on the reference domain, rather than in space. To overcome the prohibitive cost of the potential evaluation in case of many evaluation points, we also accelerate the potential evaluation by a fast multipole method which interpolates in space. The result is a frequency stable algorithm that scales essentially linear in the number of degrees of freedom and potential points. Numerical experiments are performed which show the feasibility and the performance of the approach.

Алгоритм визначення параметрів включення при розв’язуванні обернених задач геоелектророзвідки методом про

The paper aims to develop an algorithm for recognizing the physical and geometric parameters of inclusion, using indirect methods of boundary, near-boundary, and partially-boundary elements based on the data of the potential field. Methodology. The direct and inverse two-dimensional problems of the potential theory concerning geophysics were solved when modeling the earth's crust with a piecewise-homogeneous half-plane composed of a containing medium and inclusion that are an ideal contact. To construct the integral representation of the solution of the direct problem, a special fundamental solution for the half-plane (Green's function) of Laplace's equation, which automatically satisfies the zero-boundary condition of the second kind on the day surface, and a fundamental solution for inclusion were used. To find the intensities of unknown sources introduced in boundary, near-boundary, or partially-boundary elements, the collocation technique was used, i.e. the conditions of ideal contact are satisfied in the middle of each boundary element. After solving the obtained SLAE, the unknown potential in the medium and inclusion and the flow through their boundaries are found, considering that the medium and inclusion are considered as completely independent domains. Results. The computational experiment for the task of electric prospecting with a constant artificial field using the resistance method, in particular, electrical profiling, was carried out, while focusing on the physical and geometric interpretation of the data. Initial approximations for the electrical conductivity of the inclusion, its center of mass, orientation and dimensions are determined by the nature of the change in apparent resistivity. To solve the inverse problem two cascades of iterations are organized: the first one is to specify the location of the local heterogeneity and its approximate dimensions, the second one is to specify its shape and orientation in space. At the same time, the minimization of the functional considered on the section of the boundary, where an excess of boundary conditions is set, is carried out. Originality. The method of boundary integral equations is shown to be effective for constructing numerical solutions of direct and inverse problems of potential theory in a piecewise homogeneous half-plane, using indirect methods of boundary, near-boundary, and partial-boundary elements as variants. Practical significance. The proposed approach for solving the inverse problem of electrical exploration with direct current is effective because it allows fora step-by-step, "cascade" recognition of the shape, size, orientation, and electrical conductivity of the inclusion. We follow the principle of not using all the details of the model and not attempting to recognize parameters with little effect on the result, especially with imprecise initial approximations.

Boundary integral equation methods for Lipschitz domains in linear elasticity

Boundary integral equation methods for Lipschitz domains in linear elasticity

Generalized Boundary Integral Equation Method for Boundary Value Problems of Two-D Isotropic Lattice Laplacian

Generalized Boundary Integral Equation Method for Boundary Value Problems of Two-D Isotropic Lattice Laplacian

A boundary point interpolation method for acoustic problems with particular emphasis on the calculation of Cauchy principal value integrals

A boundary point interpolation method (BPIM) is presented in this paper for numerical solving acoustic problems. The BPIM is a boundary-type meshless method that combines the point interpolation technique for constructing approximate functions with boundary integral equations for reformulating boundary value problems. Since effective numerical integration techniques are crucial in the BPIM and boundary integral equation methods for calculating regular and singular integrals, two Clenshaw-Curtis integration techniques (CCITs) for regular and weakly singular integrals are derived in a unified manner with the same integration points on quadratic integration cells. In addition, a unilateral CCIT, a power series expansion technique, and an improved CCIT are proposed to effectively calculate strongly singular integrals or Cauchy principal value integrals in the BPIM. These integration techniques provide direct and efficient formulas for calculating boundary integrals on quadratic integration cells, and can be directly applied to the calculation of regular and singular integrals in the boundary element method and other meshless boundary integral equation methods. Explicit expressions of integration weights in all CCITs are derived, and the values of integration points and weights are listed. Numerical results are finally provided to validate the effectiveness of the present meshless method and integration techniques.

Potential method in the coupled theory of thermoelastic triple-porosity nanomaterials

In this paper, the coupled linear theory of thermoelasticity for nanomaterials with triple porosity is considered in which the combination of Darcy’s law and the volume fraction concept for three levels of pores (macro-, meso- and micropores) is provided. The 3D basic boundary value problems (BVPs) of steady vibrations of this theory are formulated and these BVPs are investigated using the potential method (boundary integral equation method) and the theory of singular integral equations. Namely, the formula of integral representation of regular vectors is obtained. The surface (single-layer and double-layer) and volume potentials are introduced and their basic properties are given. Some useful singular integral operators are defined for which Noether’s theorems are valid. The symbolic determinants and indexes of these operators are calculated. The BVPs of steady vibrations are reduced to the equivalent singular integral equations. Finally, with the help of the potential method, we prove the existence theorems for classical solutions of the aforementioned BVPs.

Dynamics of multi-mode nonlinear internal waves in fluids of great depth

This paper is mainly concerned with the dynamics of strongly nonlinear internal waves in two-dimensional fluids of great depth. A fully nonlinear model for a three-layer fluid of great depth containing topography, a Miyata–Choi–Camassa (MCC)-type system, is developed based on the generalization of the Ablowitz–Fokas–Musslimani global-relation formulation for free-surface water waves. Mode-1 internal solitary waves are numerically found in the MCC-type equation using the Petviashvili iteration technique and in the full Euler equations based on the boundary integral equation method. The comparison between the results validates the broad applicability of the MCC-type system. Mode-2 internal solitary waves are found in the MCC-type equations and are shown to be a type of gap solitary wave. Using the multi-mode internal solitary-wave solutions in the MCC-type equations, we apply the conformal mapping technique to calculate streamlines and particle trajectories. For unsteady simulations, we propose a pseudo-spectral algorithm to handle the time-coupled equations and investigate the generation mechanism of multi-mode internal solitary waves due to the resonance effect of local protrusion on the rigid wall, as well as their collisions when the wall is flat. For flows past protrusion on the rigid wall, the flow speed can be divided into subcritical, transcritical, and supercritical. The shedding of multi-mode internal solitary waves can occur only in the transcritical region of flow speed. We implement the carving of bifurcation curves with inflection to achieve the regional delimitation and, based on this, the interpretation of the generation mechanism from a physical point of view.

Fracture Analysis of Planar Cracks in 3D Thermal Piezoelectric Semiconductors

The study of piezoelectric semiconductor (PSC) materials and devices is experiencing rapid growth, giving rise to a burgeoning research domain known as piezotronics. Understanding the fracture behavior of PSC materials is essential for designing robust devices and ensuring their long-term functionality. A planar crack of arbitrary shape in the isotropic plane of a three-dimensional (3D) transversely isotropic thermal PSC body subjected to combined thermal-electrical-mechanical loadings is studied via the extended displacement discontinuity (EDD) boundary integral equation method. Under the framework of linearized basic equations and one-way thermal coupling, the hypersingular boundary integral equations with the volume integrals containing the temperature and carrier concentration are obtained and expressed by the EDDs which include displacement, electric potential, carrier concentration and temperature discontinuities across the crack surfaces and are the basic variables in analyzing crack problems. Based on the finite part of hypersingular integrals, the EDDs near the crack border are proved to have the classical behavior of O(r). Furthermore, the asymptotic solutions of extended stresses including the stress, electric displacement, current density and heat flux at the crack front are presented and display classical singular behavior of O(1/r). The extended stress intensity factors are defined and given in terms of EDDs. A 3D numerical model of a penny-shaped crack in the thermal PSC cylinder is established and used to verify the presented formulae.

A Boundary Integral Equation Method for the Complete Electrode Model in Electrical Impedance Tomography with Tests on Experimental Data

A Boundary Integral Equation Method for the Complete Electrode Model in Electrical Impedance Tomography with Tests on Experimental Data

Efficient convergent boundary integral methods for slender bodies

The interaction of fibers in a viscous (Stokes) fluid plays a crucial role in industrial and biological processes, such as sedimentation, rheology, transport, cell division, and locomotion. Numerical simulations generally rely on slender body theory (SBT), an asymptotic, nonconvergent approximation whose error blows up as fibers approach each other. Yet convergent boundary integral equation (BIE) methods which completely resolve the fiber surface have so far been impractical due to the prohibitive cost of layer-potential quadratures in such high aspect-ratio 3D geometries. We present a high-order Nyström quadrature scheme with aspect-ratio independent cost, making such BIEs practical. It combines centerline panels (each with a small number of poloidal Fourier modes), toroidal Green's functions, generalized Chebyshev quadratures, HPC parallel implementation, and FMM acceleration. We also present new BIE formulations for slender bodies that lead to well conditioned linear systems upon discretization. We test Laplace and Stokes Dirichlet problems, and Stokes mobility problems, for slender rigid closed fibers with (possibly varying) circular cross-section, at separations down to 1/20 of the slender radius, reporting convergence typically to at least 10 digits. We use this to quantify the breakdown of numerical SBT for close-to-touching rigid fibers. We also apply the methods to time-step the sedimentation of 512 loops with up to 1.65 million unknowns at around 7 digits of accuracy.

Bragg Resonance of Surface Gravity Waves by Surface-Piercing Vertical Multi-Plate Breakwaters

Abstract The interaction between surface gravity waves and surface-piercing vertical multi-plate breakwaters in finite water depth is investigated under the assumption of potential flow theory. The time-domain desingularized boundary integral equation method (DBIEM) is adopted to solve the boundary value problem. A decomposition method based on the superposition principle is used to obtain reflected waves. The effect of the vertical plates as a floating breakwater by making full use of Bragg resonance is analyzed. Initially, the waveforms calculated without breakwaters are compared with second-order Stokes regular wave to obtain the appropriate desingularization distance. Then, the interaction of surface gravity waves with the breakwaters with the single plate, double, and triple plates are analyzed. The transmission and reflection coefficients of the breakwater with the single plate are verified with the analytical solutions of previous literature. The transmission and reflection coefficients of the breakwaters with double and triple plates are verified with experimental results. Finally, the transmission and reflection coefficients of surface-piercing vertical multi-plate breakwaters with different plate numbers and drafts are also calculated. The occurring condition of Bragg resonance and its values affected by the plate number and the draft are analyzed. The present results are reference values for the optimal design of vertical plates as a floating breakwater for offshore and coastal engineering applications by fully using Bragg resonance.