Boundary Integral Equation Method Research Articles

GPU acceleration of Nystrom type method for solving singular integral equations in H-polarized EM waves scattering by strip gratings

Abstract In this paper, we propose the use of the graphics processing units (GPUs) for acceleration of solution of the H-polarized electromagnetic waves scattering by strip gratings. The approach combines the boundary integral equation method with the Nystrom-type discretization of singular integral equations and is realized with the use of C++ and OpenCL. The elements of the matrix of the resulting system of equations can be calculated independently and, thus, in parallel. The calculation time of the proposed parallelized algorithm on GPU is compared to standard serial algorithm executed on CPU. We considered gratings of graphene strips in vacuum and inside a dielectric slab. The acceleration is up to 35–55 times for the gratings in vacuum and up to 80–140 times for the gratings of 5–20 strips inside the dielectric slab.

Study of stress wave propagation across a non-persistent joint based on a boundary integral equation method

Studies on stress wave propagation across persistent joints have been conducted extensively. Nevertheless, there exists a consensus that non-persistent joints are widely and densely distributed, which have a profound impact on wave propagation in jointed rock masses. A boundary integral equation method is suggested in this paper to investigate the characteristics of transmitted wave field for the case of stress wave propagation across a single non-persistent joint. The displacement continuity and discontinuity boundaries are combined in the method. The method presented in the current study is applicable to the analysis of wave propagation across non-persistent joints with arbitrary incident angles. Then, taking a single non-persistent joint arranged with only one joint segment as an example, the applicability of the method in dealing with the problems of wave propagation is verified by comparing the results with those from the discrete element method and analytical methods. Subsequently, parametric studies are carried out, including the effects of joint-segment length, rock-bridge length, wave frequency and incident angle on the transmitted wave. The result indicates that the existence of non-persistent joint makes the transmitted displacement field different from that of persistent joint, because the scattered wave is produced during the process of wave propagation. The displacement amplitude may be amplified evidently in some regions and the spatial distribution pattern of the transmission coefficient is closely related to the joint-segment length, rock-bridge length and incident wavelength.

МАТЕМАТИЧЕСКОЕ МОДЕЛИРОВАНИЕ ДИНАМИЧЕСКИХ ЗАДАЧ МЕТОДОМ ГРАНИЧНЫХ УРАВНЕНИЙ.

The article describes the process of mathematical modeling of the dynamic problem of the theory of elasticity by the method of boundary integral equations. As a result of applying successive approximation, a system of algebraic equations is obtained, which is solved by a step method. The developed algorithm makes it possible to study the stress-strain state of a two-dimensional problem of the theory of elasticity from various dynamic influences.

Semi-analytical hybrid approach for modelling smart structures and guided wave-based SHM systems for a laminate with multiple delaminations and surface-mounted inhomogeneities

An advanced semi-analytical hybrid approach for simulating the dynamic behaviour of guided waves-based SHM system is proposed here. The proposed numerical method simulates wave excitation, scattering and sensing in a multi-layered elastic waveguide with a system of internal delaminations and piezoelectric/elastic inhomogeneities mounted on the surface the waveguide. The frequency domain spectral element method is applied to describe wave motion in surface-mounted obstacles (piezoelectric transducers, stringers, joints etc.), while the advanced boundary integral equation method is employed to calculate piezo-induced or scattered wave-fields propagating in the multi-layered waveguide with multiple delaminations. The crack opening displacements at the delaminations are interpolated via weighted Chebyshev polynomials of the second kind taking into account square-root behaviour at the tips of delaminations. The collocation and Galerkin methods are applied to satisfy boundary conditions in the contact area between surface-mounted inhomogeneities and the multi-layered waveguide. The convergence, the accuracy and the computational costs of the hybrid method are analysed, and the comparison with the finite element method is provided. The presented novel hybrid method grants opportunity to analyse resonance frequencies of a laminate with multiple delaminations and surface mounted inhomogeneties, calculate energy flow transferred from an actuator into the waveguide and provide fast parametric analysis of guided wave propagation in the considered structures.

External problems of steady vibrations in the theory of elastic materials with a triple porosity structure

AbstractIn this paper, the linear theory of elasticity for materials with a triple porosity (macro, meso and micro levels of porosities) structure based on the volume fraction concept is considered. The external boundary value problems (BVPs) of steady vibrations of this theory are investigated. Namely, the uniqueness theorem for classical solutions of these BVPs is established. Then, the basic properties of the surface (single‐layer and double‐layer) potentials are given. Finally, on the basis of the properties of potentials, the existence theorems for classical solutions of the above mentioned BVPs of steady vibrations are proved by means of the potential method (boundary integral equation method) and the theory of singular integral equations.

ЧИСЛЕННОЕ МОДЕЛИРОВАНИЕ ПЛОСКОГО НАПРЯЖЕННОГО СОСТОЯНИЯ ПРЯМОУГОЛЬНОЙ ПЛАСТИНЫ МЕТОДОМ ГРАНИЧНЫХ УРАВНЕНИЙ

The article presents the results of numerical simulation of a plane problem of elasticity theory by the method of boundary integral equations. The developed algorithm is implemented on the example of a square beam-wall from the action of a horizontal distributed load. The results of numerical simulation are compared with known solutions.

МATHEMATICAL MODEL OF ACOUSTIC EMISSION FROM MODE II CRACK

The dynamic problem of determining the wave field of displacements on the surface of an elastic half-space caused by the opening of an internal mode II crack has been solved. Using the method of boundary integral equations, a change in the amplitude-frequency characteristics of elastic oscillations on the surface of a solid body depending on the size of the defect, its depth, and the distance from the epicenter to the observation point was determined.

Smooth Solution to the Second Initial-Boundary Value Problem for a Parabolic System in a Domain with Nonsmooth Lateral Boundary

We consider the second initial-boundary value problem for a Petrovskii parabolic second order system with variable coefficients in a semibounded plane domain with nonsmooth lateral boundary. Using the boundary integral equation method, we show that the solution belongs to the class $${C}_{x,t}^{\mathrm{2,1}}$$ ( $$\overline{\Omega }$$ ) and construct the solution in the form of a special parabolic potential.

A boundary integral equation method for the fluid-solid interaction problem

<abstract><p>In this paper, a boundary integral equation method is proposed for the fluid-solid interaction scattering problem, and a high-precision numerical method is developed. More specifically, by introducing the Helmholtz decomposition, the corresponding problem is transformed into a coupled boundary value problem for the Helmholtz equation. Based on the integral equation method, the coupled value problem is reduced to a system of three coupled hypersingular integral equations. Semi-discrete and fully-discrete collocation methods are proposed for the singular integral equations. The presented method is based on trigonometric interpolation and discretized singular operators applied to differentiated interpolation. The convergence of the method is verified by a numerical experiment.</p></abstract>

Evaluation possibility of the turbine disks using of converted aircraft engines of the NK family

The article examines the possibility of using converted aircraft engines for gas pumping units using the example of the main parts of the GTE - the disks of the NK-8-2U aircraft engine and the NK-16ST and NK-16-18ST ground engines. For all disks of the second stage of low-pressure turbines of three engines, the stress-strain state calculation was performed using the boundary integral equations method (BEM) and using the FEM in the Ansys program. For three disk stress concentrators, the theoretical stress concentration coefficient was determined and the maximum equivalent stresses were calculated. During long-term operation, the main parameters of engine operation change due to degradation changes (wear, erosion and corrosion, development of seals, contamination and resizing of flow parts). As a result, the main parameters of engine operation change, in particular, the rotation speed of the rotors and the gas temperature in the turbine, which determine the stress-strain state, which must be taken into account when predicting durability. The compliance of the results of the calculated study of the stress-strain state of turbine disks with their real load was confirmed by the data of their metallurgical study after prolonged operation. Using the Manson formula for the disks of three engines, their cyclic durability was determined before the formation of a crack of low-cycle fatigue. Using the linear damage summation hypothesis, the total damage rates (under static loading and under fatigue loading) for the disks of three engines were calculated. It is established that after the conversion of aircraft engines that have spent their life in flight operation, it is advisable to continue the use of turbine disks on ground-based GTU.

RI-CDI-FDTD method and program implementation for electromagnetic characteristics simulation of lossy Debye dispersive medium

Dispersive media refer to a class of natural substances, including living organisms, composite materials, plasma and water, with diverse applications in areas such as biomedicine, microwave sensing, electromagnetic protection, and stealth technology. In the pursuit of investigating the electromagnetic properties of these media, time-domain numerical methods, including finite difference in time domain (FDTD), finite element method, and time domain boundary integral equation method, have been widely utilized. Time-domain numerical methods are preferred to their frequency-domain counterparts owing to their ability to handle nonlinear and wideband problems, as well as various material properties. The FDTD method, in particular, is a highly adaptable, robust, and easy-to-use numerical method that directly solves the Maxwell equations while also simulating the reflection, transmission, and scattering of electromagnetic waves in complex dispersion media. Nonetheless, the traditional FDTD method suffers low computational efficiency arising from the Courant-Friedrichs-Lewy (CFL) stability condition. To solve the problem of low computational efficiency, a new method, the complying divergence implicit finite-difference time-domain (CDI-FDTD) method with a one-step leapfrog scheme, is introduced for lossy Debye dispersive media. The Maxwell equations in the frequency domain form a starting point, and the Fourier transform is utilized to transform the electromagnetic field components from the frequency domain to the time domain. To approximate the integral terms arising from the frequency-to-time domain transformation, a recursive integration (RI) method is employed. Subsequently, the time-domain Maxwell equations and auxiliary variables are discretized with a one-step leapfrog implicit scheme. The iterative formula of the RI-CDI-FDTD algorithm for lossy Debye dispersive media is then derived. The RI-CDI-FDTD method does not change the formulas of the traditional CDI-FDTD method while only requiring to add auxiliary variables for updating field components to the dispersive medium region. The numerical implementation is straightforward, and the electromagnetic modeling is flexible. Moreover, the unconditional stability of the RI-CDI-FDTD algorithm is proven by using the von Neumann method. Finally, some numerical examples are presented to demonstrate the effectiveness and efficiency of the proposed method. In conclusion, our work contributes a crucial numerical simulation tool to accurately modeling complex dispersive media while providing a systemic stability analysis method for time-domain numerical methods.

Boundary Integral Equation Method for Electrostatic Field Prediction in Piecewise-Homogeneous Electrolytes

This article presents a boundary integral equation formulation for the prediction of electrostatic fields, potentials, and currents in regions comprising piecewise-homogeneous electrolytes. The integral equation is formulated in terms of the boundary electric potentials and normal electric current densities and is discretized using the locally corrected Nyström method. The method is validated by comparison to analytic solution data for both linear and nonlinear canonical problems. Solution convergence is investigated with respect to mesh discretization and basis order.

The Computation of Stiff and Elastic Features of Structures with Time Domain Analysis

Hydroelasticity of marine structures with and without forward speed is studied directly using time dependent Boundary Integral Equation Method with Neumann-Kelvin linearisation where the potential is considered as the impulsive velocity potential. The exciting and radiation hydrodynamic parameters are predicted in time with transient wave Green function whilst the structural analysis is solved with Euler-Bernoulli beam method at which modeshapes are defined analytically. The modal analysis is used to approximate the hydroelasticbehaviour of the floating systems through fully coupling of the structural and hydrodynamic analyses. As it is expected, it is found with numerical experience that the effects of the rigid body modes are greater than elastic modes in the case of stiff structures. The predicted numerical results of the present in-house computational tool ITU-WAVE are compared with experimental results for validation purposes and show the acceptable agreements.

Evaluation of seismic response of tunnels near slope surfaces and performance of anti-seismic measures

In valley areas, tunnels are commonly constructed near slope surfaces. The seismic behavior of the tunnel near a slope surface is less understood, especially under the combined effect of the geostatic stress and the seismic load. In this research, a boundary integral equation method is developed to analyze the seismic response of the tunnel near a slope surface. The effects of the canyon on the dynamic response of the tunnels with two cross-section shapes under the SV waves at different frequencies and different incident angles are analyzed. Numerical results indicate that the canyon has a manifest intensification effect on the acceleration and stress of the lining within a lower-frequency range. The effectiveness of two typical anti-seismic measures in improving the anti-seismic performance of the tunnel near a slope surface is also evaluated. The results indicate that the isolation layer can successfully improve the stress state of the circular tunnel, and it will intensify the tensile stress in the lining of the horseshoe tunnel. When the surrounding rock is strengthened, although the compressive stress in the lining of the circular or horseshoe tunnel decreases, the tensile stress increases.

Eliminating the irregular frequencies in wave-body interaction: An investigation for accuracy assessment with higher-order elements

The presence and elimination of irregular frequencies in wave-body interaction are reviewed by referring to the fundamental properties of the integral equations and a predictive measure that can be beneficial for wave-induced response assessment is proposed through numerical applications. By adopting higher-order elements and considering the hydroelastic response of the floating body, this study explores three interrelated questions: (i) how widely adopted irregular frequency suppression techniques perform, (ii) how, if possible, can these techniques be effectively applied, and (iii) how can the estimations be accepted as accurate for a given discretization, that is, finding a quantitative measure to relate the employed discretization to the frequency range for which the specified error threshold is not exceeded. The investigation involves three hull forms, a barge model along with the DTMB 5415 and KCS ships, and two techniques that extend the solution domain for suppression, namely the extended boundary integral equation method (EBIEM) and combined boundary integral equation method (CBIEM). The barge model is used for preliminary analyses to identify the existing patterns. DTMB 5415 and KCS allow the transfer of findings to practical applications and assess their validity. Three assessment measures in terms of frequency are employed by considering the hydrodynamic coefficients. In the present setting, it is observed that EBIEM is far more effective, and CBIEM cannot provide adequate suppression. The frequency envelope obtained by taking two quadratic elements per wave length is found to be a proper measure for estimating the frequency range that can be confidently studied.

Contact problems for a transversely isotropic layer

Two spatial, one axisymmetric and two plane contact problems are considered for a transversely isotropic elastic layer with one face subjected to sliding support. In the spatial and plane contact problems, the planes of isotropy may be either parallel or perpendicular to the layer faces. In the case of axial symmetry, the planes of isotropy are parallel to the layer faces. By using Fourier integral transforms, the contact problems are reduced to integral equations with respect to the contact pressure, the limiting cases of which are the well-known equations of the corresponding problems for an isotropic layer. For solving the spatial problems with unknown contact domains, the nonlinear boundary integral equations method is used, which make it possible to determine the contact pressure and the contact domain simultaneously. To extract the kernel principal part of the spatial problem integral equation when the isotropy planes are perpendicular to the layer faces, it is used the kernel of the integral equation of the corresponding contact problem for a transversely isotropic half-space obtained earlier without quadratures. The integral equation of the axially symmetric problem is reduced to a Fredholm integral equation of the second kind with the help of the method of pair equations, and the method of mechanical quadratures is used for numerical solutions. Plane problems are solved in a closed form based on special approximations of the kernel symbols. The approximations accuracy grows as anisotropy increases. Here, the anisotropy level can be characterized by the difference between ratio of a characteristic equation roots and unit because the unit value corresponds to the isotropic case. Mechanical characteristics as well as errors of the approximations are calculated for well-known transversely isotropic materials.

BIE formulation for axisymmetric transient heat conduction applications

Boundary integral equation method is proposed for axisymmetric transient heat transfer applications covering spatially varying heat sources. The radial integral approach using the normalized temperature, thus, eliminating temperature gradients involved, is utilized to switch two-dimensional integrals to the equivalent line-related integrals. The derivative of the temperature is approximated using the finite difference method. To evaluate time-depended equations, the time-marching scheme is adopted. The proposed boundary integral formulation is implemented into axisymmetric applications to assess its accuracy.

Antiplane spectral boundary integral equation method for an interface between a layer and a half-plane

A numerical scheme is proposed for simulating dynamic antiplane fracture at an interface between a layer and a half-plane. The scheme is a spectral boundary integral equation method which relates the shear stress and the displacement discontinuity at the interface in the spectral domain. In previous studies, numerical schemes have been developed for fracture problems in unbounded solids. This is the first work where a spectral boundary integral equation method has been developed for a system one of whose constituents is a finite solid, namely, a layer. The spectral scheme involves performing time convolution of the spectral amplitude of the shear stress at the interface of the layer and half-space. Numerically, conversion between the spectral domain and spatial domain is done by the Fast Fourier Transform. The numerical scheme is validated through comparison with known solutions of model problems. Use of the numerical method is illustrated by simulating 2D antiplane slip rupture propagation wherein the spectral boundary integral equation is coupled with a slip-weakening friction law. We illustrate the role of initial stress distribution on the modes of rupture propagation at the interface, which can be crack-like or pulse-like. The effect of the layer thickness on the critical nucleation size for dynamic rupture propagation is also studied numerically.

New solutions for interfacial capillary waves of permanent form

Progressive capillary waves on the interface between two homogeneous fluids confined in a channel with rigid walls parallel to the undisturbed interface are investigated. This problem is formulated as a system of integrodifferential equations that can be solved numerically via a boundary integral equation method coupled with series expansions of the unknown functions. With this highly accurate scheme and numerical continuation, we explore the global bifurcation of periodic travelling waves. It is found that there are two types of limiting profile, self-intersecting and boundary-touching, which appear either along a primary branch bifurcating from infinitesimal periodic waves or on an isolated branch existing above a certain finite amplitude. For particular sets of parameters, these two types of bifurcation curves can intersect, which can be viewed as a secondary bifurcation phenomenon occurring on the primary branch. Based on asymptotic and numerical analyses of the almost limiting waves, it is found that the boundary-touching solutions feature a circular geometry, i.e. the interface is pieced together by circular arcs of the same radius. A theoretical investigation yields the necessary conditions for the existence of these extreme waves, whereby we can predict the limiting configurations for most parameter sets. The comparisons between theoretical predictions and numerical results show good agreement.

Windowed Green function method for wave scattering by periodic arrays of 2D obstacles

AbstractThis paper introduces a novel boundary integral equation (BIE) method for the numerical solution of problems of planewave scattering by periodic line arrays of two‐dimensional penetrable obstacles. Our approach is built upon a direct BIE formulation that leverages the simplicity of the free‐space Green function but in turn entails evaluation of integrals over the unit‐cell boundaries. Such integrals are here treated via the window Green function method. The windowing approximation together with a finite‐rank operator correction—used to properly impose the Rayleigh radiation condition—yield a robust second‐kind BIE that produces superalgebraically convergent solutions throughout the spectrum, including at the challenging Rayleigh–Wood anomalies. The corrected windowed BIE can be discretized by means of off‐the‐shelf Nyström and boundary element methods, and it leads to linear systems suitable for iterative linear algebra solvers as well as standard fast matrix–vector product algorithms. A variety of numerical examples demonstrate the accuracy and robustness of the proposed methodology.