Boundary Integral Equation Method Research Articles

The factorization method for inverse scattering by a two-layered cavity with conductive boundary condition

Abstract In this paper we consider the inverse scattering problem of determining the shape of a two-layered cavity with conductive boundary condition from sources and measurements placed on a curve inside the cavity. First, we show the well-posedness of the direct scattering problem by using the boundary integral equation method. Then, we prove that the factorization method can be applied to reconstruct the interface of the two-layered cavity from near-field data. Some numerical experiments are also presented to demonstrate the feasibility and effectiveness of the factorization method.

Interference of internal waves due to two point vortices: linear analytical solution and nonlinear interaction.

In this work, we consider steady two-dimensional interfacial waves in a two-layer stratified fluid, which is induced by a vortex pair located in the lower layer of the fluids. A mathematical model based on the boundary integral equation method and the potential-flow theory is established. The linear analytical solution for the linearized model is given in the form of Cauchy integral and then asymptotic behaviour for large x is presented. The fully nonlinear model is solved by the Jacobian-free Newton–Krylov (JFNK) method numerically. Nonlinear characteristics of wave profiles are identified compared with the linear results under different vortex strengths and the distance between the vortex pair. The amplitude of steady downstream waves is found to vary periodically with respect to the distance of the vortex pair, which can be regarded as the interference between waves produced by each vortex. For equal-strength counter- and co-rotating pairs, the downstream wave heights of linear solutions can be eliminated for some special values of the distance between point vortices, namely, the destructive interference occurs. Meanwhile, the wave only exists between the vortex pair like trapped waves. So does the nonlinear counterpart for counter-rotating pairs, but it could not be diminished with any distance.

Stress analysis of composites with interfacial bridged cracks and damaged zones

Stress analysis of composite structures with bridged cracks and damaged zones is performed with the direct boundary integral equations method in axisymmetric formulation. It is assumed that: a) the crack faces interact in some zones behind of the cracks edges (bridged zones); b) these zones width can be comparable to the whole crack size; c) adjacent parts of the structure can be connected by mechanical ligaments (including interfaces ahead of cracks fronts) and the relative displacements of these parts can occur under the loading (these zones consider as damaged zones). The boundary element software to solve axisymmetric problems of elasticity with interfacial bridged cracks and damaged zones was developed. The results for different properties of composite materials and bonds are presented. Applications the bridged crack approach to modeling of cracks self-healing is remarked.

Метод пересчета результатов модельных испытаний гребного винта на натурные условия

Object and purpose of research. This paper discusses numerical method for simulation of propeller parameters in model and full-scale conditions. The purpose is to develop a similar methods for extrapolation of propeller model test data to the full-scale. Materials and methods. Propeller parameters are simulated as per the improved numerical method of E. Papmel. The flow around cylindrical section profiles of blades and hub is simulated as per the methods of boundary integral equations and integral calculation methods for the boundary layer with consideration of laminary-to-turbulent transition. Main results. Development of the method taking into account scale effect for propellers. The method has been applied to extrapolate model test data to the full scale. Conclusion. The method is of practical value for modern propeller design.

Helmholtz equation for a Neumann condition on a curved boundary: spurious resonances

The 2D canonical case of a semi-circular canyon is addressed via two types of boundary-integral equations (BIE) methods which are shown, via Galerkin procedures, to lead to matrix equations with kernels that vanish at two different sets of (what appear as resonant) frequencies. This physically impossible result is interpreted as being the sign that the two BIE methods are generally defective and that the resonances are ‘spurious’. This ‘disease’ is cured by combining two BIE's into one which gives rise to a closed-form solution identical to the exact separation-of-variables solution devoid of resonances.

Accelerated Calderón preconditioning for Maxwell transmission problems

We investigate a range of techniques for the acceleration of Calderón (operator) preconditioning in the context of boundary integral equation methods for electromagnetic transmission problems. Our objective is to mitigate as far as possible the high computational cost of the barycentrically-refined meshes necessary for the stable discretisation of operator products. Our focus is on the well-known PMCHWT formulation, but the techniques we introduce can be applied generically. By using barycentric meshes only for the preconditioner and not for the original boundary integral operator, we achieve significant reductions in computational cost by (i) using “reduced” Calderón preconditioners obtained by discarding constituent boundary integral operators that are not essential for regularisation, and (ii) adopting a “bi-parametric” approach [1,2] in which we use a lower quality (cheaper) H-matrix assembly routine for the preconditioner than for the original operator, including a novel approach of discarding far-field interactions in the preconditioner. Using the boundary element software Bempp (www.bempp.com), we compare the performance of different combinations of these techniques in the context of scattering by multiple dielectric particles. Applying our accelerated implementation to 3D electromagnetic scattering by an aggregate consisting of 8 monomer ice crystals of overall diameter 1 cm at 664 GHz leads to a 99% reduction in memory cost and at least a 75% reduction in total computation time compared to a non-accelerated implementation.

On the direct and inverse electromagnetic scattering by chiral media

Consider the direct and inverse electromagnetic scattering problems by a penetrable cylindrical chiral medium in a homogeneous achiral background medium. The scattering problem can be modeled by a pair of the two-dimensional Helmholtz equations with coupled transmission boundary conditions. The boundary integral equation method is applied to solve the direct scattering problem in a suitable Sobolev space. Then we investigate the linear sampling method for recovering the unknown scatterer from far-field pattern of the scattered field. The coupled boundary conditions bring new challenges during the procedure, and some novel techniques are needed in the theoretical analysis. Finally, we perform several numerical experiments to demonstrate the efficiency and effectiveness of this method.

Numerical investigation of the Fourier–Kochin theory for wave-induced response estimation of floating structures

The Fourier–Kochin theory—an indirect solution approach for the boundary integral representations of free-surface flows that rely on free-surface Green function—is investigated by considering effective computation and reliable prediction. The present study addresses wave–body interaction with zero forward speed by adopting a higher-order approximation. Three distinct hull forms, the DTMB 5415, a scaled model reproduced from it, and a barge are used for practical application. The scaled frigate and barge models are used for comparative analysis to assess the performance of different computation techniques, modeling elements, and solution methods and to find the best course; the purpose of studying DTMB 5415 is to provide a realistic application based on a full-scale hull form by using the findings of the earlier benchmark studies. The analyses that cover both the rigid-body and elastic responses indicate that an adaptive approach for the computation of Fourier components has the potential to eliminate the numerical drawbacks of the presented implementation the Fourier–Kochin theory, but also stringently needs parallelism, preferably using the Single Program Multiple Data model. Using discontinuous elements for free-surface discretization when applying the Extended Boundary Integral Equation Method, the featured technique for irregular frequency suppression, is also promoted.

Numerical investigation on evaluating nonlinear waves due to an air cushion vehicle in steady motion by a higher order desingularized boundary integral equation method

The problem of nonlinear ship waves generated by an air cushion vehicle is studied and modelled in the framework of potential flow theory. A higher order desingularized boundary integral equation method (HODBIEM) using Rankine source is developed. A stationary iterative method is adopted to solve the nonlinear free surface which is discretized into the inner area disturbed by cushion pressure and the outer undisturbed area. Techniques for numerical implementation are described in detail. To save computing time, grid density and domain size are refined.Both straight-ahead and yawed cases are considered for validations. Wave-making resistance versus various speeds in deep and shallow water are calculated for straight-ahead vehicles. The resistance curves obtained by HODBIEM agree well with results of nonlinear CFD simulations. The linear theory predicts higher peak value of resistance than nonlinear methods. The experimental data fit at low Froude number but deviate from numerical predictions when exceeding primary hump speed. Effects of speed and water depth are analyzed by wave patterns and wave profiles. Resistance, side force and yaw moment are calculated at Fn = 0.533 for drift angle from 0° to 90°. Satisfactory results are given by present method. Details about wave patterns and wave angles are discussed.

Plane Boundary Value Problems of Thermoelasticity and Their Generalized Solutions

Nonstationary boundary value problems of uncoupled thermoelasticity are considered. A method of boundary integral equations in the initial space-time has been developed for solving boundary value problems of thermoelasticity by plane deformation by use general functions method (GFM). The generalized solutions of boundary value problems are constructed in the space of generalized vector functions. Fundamental Green tensor, stress tensor and their antiderivatives over time are used for construction their regular integral representations. By use analogue of Gauss formula for fundamental stress the singular boundary integral equations are constructed to determine the unknown boundary functions. The resulting formulas have important engineering application. They make possible to determine the thermal stressed state of the medium by the boundary values of stresses, displacements, temperature and heat flux, without solving singular boundary integral equations. Because for real engineering problems these physical characteristics can be experimentally measured at the boundary. Moreover, the formulas allow to calculate the influence of each of these characteristics of the process on its stressstrain state. The last one is very important in designing structures made of thermoelastic materials.

A Boussinesq-Type Model for Nonlinear Wave-Heaving Cylinder Interaction

In the present work, a Boussinesq-type numerical model is developed for the simulation of nonlinear wave-heaving cylinder interaction. The wave model is able to describe the propagation of fully dispersive and weakly nonlinear waves over any finite water depth. The wave-cylinder interaction is taken into account by solving simultaneously an elliptic equation that determines the pressure exerted by the fluid on the floating body. The heave motion for the partially immersed floating cylinder under the action of waves is obtained by solving numerically the body’s equation of motion in the z direction based on Newton’s law. The developed model is applied for the case of a fixed and a free-floating circular cylinder under the action of regular waves, as well as for a free-floating cylinder undergoing a forced motion in heave. Results (heave and surge exciting forces, heave motions, and wave elevation) are compared with those obtained using a frequency domain numerical model, which is based on the boundary integral equation method.

Rigorous accounting diffraction on non-plane gratings irradiated by non-planar waves

The modified boundary integral equation method (MIM) is considered a rigorous theoretical application for the diffraction of cylindrical waves by arbitrary profiled plane gratings, as well as for the diffraction of plane/non-planar waves by concave/convex gratings. This study investigates 2D diffraction problems of the filiform source electromagnetic field scattered by a plane lamellar grating and of plane waves scattered by a similar cylindrical-shaped grating. Unlike the problem of plane wave diffraction by a plane grating, the field of a localised source does not satisfy the quasi-periodicity requirement. Fourier transform is used to reduce the solution of the problem of localised source diffraction by the grating in the whole region to the solution of the problem of diffraction inside one Floquet channel. By considering the periodicity of the geometry structure, the problem of Floquet terms for the image can be formulated so that it enables the application of the MIM developed for plane wave diffraction problems. Accounting of the local structure of an incident field enables both the prediction of the corresponding efficiencies and the specification of the bounds within which the approximation of the incident field with plane waves is correct. For 2D diffraction problems of the high-conductive plane grating irradiated by cylindrical waves and the cylindrical high-conductive grating irradiated by plane waves, decompositions in sets of plane waves/sections are investigated. The application of such decomposition, including the dependence on the number of plane waves/sections and radii of the grating and wave front shape, was demonstrated for lamellar, sinusoidal and saw-tooth grating examples in the 0th and −1st orders as well as in the transverse electric and transverse magnetic polarisations. The primary effects of plane wave/section partitions of non-planar wave fronts and curved grating shapes on the exact solutions for 2D and 3D (conical) diffraction problems are discussed.

Numerical Predictions of Elastic and Stiff Characteristics of Marine Structures

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 hydroelastic behaviour 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.

Three-Dimensional Frequency-Domain Green’s Functions of a Finite Fluid-Saturated Soil Layer Underlain by Rigid Bedrock to Interior Loadings

This paper presents the three-dimensional frequency-domain Green’s functions of a saturated poroelastic soil layer with incompressible constituents resting on rigid base due to interior time-harmonic point-, ring-, and disc-loadings with uniform distribution being composed of three effective stress source components and one pore fluid pressure source. The set of Green’s functions can provide complete fundamental solutions for relevant boundary-value problem studies by the method of boundary integral equations. In developing these solutions, the dynamic property of the porous medium is described by Boer’s poroelastic model. Four independent wave equations with definite physical meaning are obtained by introducing four scalar displacement potentials to uncouple the equations of motion of the layer and then resolved by the Fourier–Hankel integral transformations. By imposing the boundary and load interfacial conditions of the layer, the Green’s function solutions of all field variables corresponding to the point-, ring-, and disc-loadings are derived. The obtained solutions are then validated by comparing with the existing special solutions and the finite-element model (FEM) calculation results. Numerical examples with disc loading cases are also performed to examine the effects of the permeability and the thickness of the poroelastic layer on its dynamic characteristic.

Математичне моделювання напруженого стану в’язкопружної напiвплощини з включеннями

The application of the method of boundary integral equations is considered for studying the stress state of flat viscoelastic bodies with inclusions. The method is based on the use of complex potentials and the apparatus of generalized functions. An analytical solution of the problem is obtained for a half-plane with inclusions of arbitrary shape. For a numerical study of the change in the stress state depending on the time and geometry of the inclusions, a discrete analogue of the system of boundary-time integral equations has been developed.

Mixed Problems for Plane Parabolic Systems and Boundary Integral Equations

We consider the mixed problem for a one-dimensional (with respect to the spatial variable) second-order parabolic system with Dini-continuous coefficients in a domain with nonsmooth lateral boundaries. Using the method of boundary integral equations, we find a classical solution of this problem. We investigate the smoothness of the solution as well.

THE NONLOCAL CONJUGATION PROBLEM FOR A LINEAR SECOND ORDER PARABOLIC EQUATION OF KOLMOGOROV'S TYPE WITH DISCONTINUOUS COEFFICIENTS

In this paper, we construct the two-parameter Feller semigroup associated with a certain one-dimensional inhomogeneous Markov process. This process may be described as follows. At the interior points of the finite number of intervals $(-\infty,r_1(s)),\,(r_1(s),r_2(s)),\ldots,\,(r_{n}(s),\infty)$ separated by points $r_i(s)\,(i=1,\ldots,n)$, the positions of which depend on the time variable, this process coincides with the ordinary diffusions given there by their generating differential operators, and its behavior on the common boundaries of these intervals is determined by the Feller-Wentzell conjugation conditions of the integral type, each of which corresponds to the inward jump phenomenon from the boundary. The study of the problem is done using analytical methods. With such an approach, the problem of existence of the desired semigroup leads to the corresponding nonlocal conjugation problem for a second order linear parabolic equation of Kolmogorov’s type with discontinuous coefficients. The main part of the paper consists in the investigation of this parabolic conjugation problem, the peculiarity of which is that the domains on the plane, where the equations are given, are curvilinear and have non-smooth boundaries: the functions $r_i(s)\,(i=1,\ldots,n)$, which determine the boundaries of these domains satisfy only the Hölder condition with exponent greater than $\frac{1}{2}$. Its classical solvability in the space of continuous functions is established by the boundary integral equations method with the use of the fundamental solutions of the uniformly parabolic equations and the associated potentials. It is also proved that the solution of this problem has a semigroup property. The availability of the integral representation for the constructed semigroup allows us to prove relatively easily that this semigroup yields the Markov process.

Lichtenstein’s integral equation for the Stokes problem via conformal mapping

We investigate Lichtenstein’s boundary integral equation method for the Stokes problem on a planar simply connected domain by transforming it onto the unit disc via conformal mapping. We also present some simple numerical experiments utilizing the conformal mapping as a prerequisite.

On the Generalized Calderόn Formulas for Closed- and Open-Surface Elastic Scattering Problems

The Calderon formulas (i.e., the combination of single-layer and hyper-singular boundary integral operators) have been widely utilized in the process of constructing valid boundary integral equation systems which could possess highly favorable spectral properties. This work is devoted to studying the theoretical properties of elastodynamic Calderon formulas which provide us with a solid basis for the design of fast boundary integral equation methods solving elastic wave problems defined on a close-surface or an open-surface in two dimensions. For the closed-surface case, it is proved that the Calderon formula is a Fredholm operator of second-kind except for certain circumstances. Regarding to the open-surface case, we investigate weighted integral operators instead of the original integral operators which are resulted from dealing with edge singularities of potentials corresponding to the elastic scattering problems by open-surfaces, and show that the Calderon formula is a compact perturbation of a bounded and invertible operator. To complete the proof, we need to use the well-posedness result of the elastic scattering problem, the analysis of the zero-frequency integral operators defined on the straight arc, the singularity decompositions of the kernels of integral operators, and a new representation formula of the hyper-singular operator. Moreover, it can be demonstrated that the accumulation point of the spectrum of the invertible operator is the same as that of the eigenvalues of the Calderon formula in the closed-surface case.

RESEARCH OF ACOUSTIC EMISSIONS FROM THE SYSTEM OF COMPLANARY CRACKS

The dynamic problem of the displacement field in an elastic half-space caused by the time-steady displacement of the surfaces of the system of disc-shaped coplanar cracks is solved. The solutions are obtained by the method of boundary integral equations. The dependences of elastic displacements on the surface of the half-space on the wave number, the number of defects and the depths of their occurrence are constructed.