System Of Boundary Integral Equations Research Articles

Enhanced radiation of a dipole placed between a metallic surface and a nanoparticle

Enhanced dipole radiation in the presence of a flat metallic surface and a metal nanoparticle is considered on the basis of Maxwell's equations. For the case of axi-symmetrical illumination the initial problem is reduced to a system of boundary integral equations for the angular component of the magnetic field and its normal derivative. A boundary element method is used to solve the system of integral equations. The scattering of convergent cylindrical electromagnetic waves from a nanoparticle placed near a surface is calculated. The dipole placed between the nanoparticle and the surface is excited by the enhanced field in the gap and re-radiates electromagnetic waves of the same frequency into space. This dipole radiation in turn is enhanced by the nanoparticle/surface system. Two intensity enhancement factors are calculated: (1) the enhancement of the local electric field at the dipole position by the nanoparticle/surface system; and (2) the increase in dipole radiation due to the presence of a metallic nano-object. For very small gaps (1 nm) between the surface and nanoparticle, these factors reach very large values. At some frequencies the enhancement factors exhibit large resonance peaks which can be explained as plasmon resonances in the nanoparticle/surface system. For various shapes of metal nanoparticles and for different distances in the particle/dipole/surface configuration, the total intensity enhancement factor (the product of the two factors described above) is calculated using the developed model. The very large enhancement factors obtained in our calculations can be considered as a theoretical basis for single molecule Raman spectroscopy.

Boundary integral approximation of a heat-diffusion problem in time-harmonic regime

In this paper we propose and analyse numerical methods for the approximation of the solution of Helmholtz transmission problems in the half plane. The problems we deal with arise from the study of some models in photothermal science. The solutions to the problem are represented as single layer potentials and an equivalent system of boundary integral equations is derived. We then give abstract necessary and sufficient conditions for convergence of Petrov–Galerkin discretizations of the boundary integral system and show for three different cases that these conditions are satisfied. We extend the results to other situations not related to thermal science and to non-smooth interfaces. Finally, we propose a simple full discretization of a Petrov–Galerkin scheme with periodic spline spaces and show some numerical experiments.

On the boundary integral equation method for a mixed boundary value problem of the biharmonic equation§

This article is concerned with weak solution of a mixed boundary value problem for the biharmonic equation in the plane. Using Green's formula, the problem is converted into a system of Fredholm integral equations for the unknown data on different parts of the boundary. Existence and uniqueness of the solutions of the system of boundary integral equations are established in appropriate Sobolev spaces.

NUMERICAL PREDICTIONS OF SOUND PROPAGATION FROM A CUTTING OVER A ROAD-SIDE NOISE BARRIER

A boundary integral equation is described for the prediction of acoustic propagation from a monofrequency coherent line source in a cutting with impedance boundary conditions onto surrounding flat impedance ground. The problem is stated as a boundary value problem for the Helmholtz equation and is subsequently reformulated as a system of boundary integral equations via Green's theorem. The numerical solution of the coupled boundary integral equations by a simple boundary element method is then described. Predictions of A-weighted insertion losses for a traffic noise spectrum are made illustrating the effects of depth of the cutting and the profile of the associated noise barrier.

Determination of the Crack Configuration in an Anisotropic Elastic Medium

The problem of the identification of a single internal crack in an anisotropic elastic body is investigated. Using the dislocation theory approach, a system of boundary integral equations for the crack opening functions is constructed and studied by the boundary element method. A crack identification method is developed on the basis of the crack parametrization by a finite number of parameters, with their subsequent determination through the minimization of a certain nonquadratic residual functional. The problem of identifying a transverse tunnel crack in an orthotropic layer is solved for the cases of plane and antiplane deformations.

An integral equation method for the numerical analysis of gravity waves in a channel with free boundary

We consider the numerical solution of an evolution equation of second order on a boundary in R 2 that arises in the theory of gravity waves in fluids. First, by Laguerre transformation with respect to time we reduce the non-stationary problem to a sequence of operator equations. Then we handle the cross-section and longitudinal section cases of the channel in different ways. The potential theory for Laplace equation in a bounded domain with corners and the Green's function technique in a semi-infinite domain lead to systems of boundary integral equations of the second kind. For a full discretization we use a Nyström method. In the case of cross-section the trigonometrical quadrature rules in combination with special mesh grading transformation for the accounting of densities singularities in corners are applied. In the longitudinal section case the quadratures are based on sinc-approximation. The results of numerical experiments are presented.

Novel Boundary Integral Equations for Two-Dimensional Isotropic Elasticity: An Application to Evaluation of the In-Boundary Stress

A new nonsingular system of boundary integral equations (BIEs) of the second kind for two-dimensional isotropic elasticity is deduced following a recently introduced procedure by Wu (J. Appl. Mech., 67, pp. 618–621, 2000) originally applied for anisotropic elasticity. The physical interpretation of the new integral kernels appearing in these BIEs is studied. An advantageous application of one of these BIEs as a boundary integral representation (BIR) of tangential derivative of boundary displacements on smooth parts of the boundary, and subsequently as a BIR of the in-boundary stress, is presented and analyzed in numerical examples. An equivalent BIR obtained by an integration by parts of the integral including tangential derivative of displacements in the former BIR is presented and analyzed as well. The resulting integral is only apparently hypersingular, being in fact a regular integral on smooth parts of the boundary.

THE INDIRECT BOUNDARY INTEGRAL METHOD FOR CURVED CRACKS IN PLANE ELASTICITY

For curved crack problems in plane elasticity, subjected to the traction conditions on the crack faces, we present a system of boundary integral equations. The procedure is based on the indirect boundary integral method in terms of real variables. For efficient mathematical analysis, we decompose the singular kernel into the Cauchy singular part and the regular one. As a result, solvability of the presented system is proved and availability of the present approach is shown by the numerical example of a circular arc crack.

Analysis and optimization of energy flows in structures composed of beam elements – Part I: problem formulation and solution technique

The paper addresses the analysis of in-plane coupled flexural and longitudinal vibrations of a structure composed of elastic tubular beams. Emphasis is put on optimization of structural performance in terms of minimization of total power flow over a broad-banded frequency range, i.e. minimization of the emitted/trans- mitted energy of vibrations. The objective function is chosen as an energy outflow (a structural intensity) integrated within the given frequency range at a given remote point (cross-section) of the tubular structure. To gain insight into the physical mechanisms of energy transportation, the structure is decomposed into a set of elementary dynamical systems. These elementary dynamical systems (subsystems) are chosen as one-dimensional wave-guides carrying either longitudinal or flexural waves. Vibrations of each dynamical subsystem are described by a boundary equation method, a novel method in structural dynamics which ideally fits the substructuring concept since it deals with physical variables at boundaries between substructures, no matter what type of excitation is applied at individual substructures. A system of boundary integral equations is accomplished by continuity conditions at interfaces between subsystems and by boundary conditions. Since Green's functions used in the boundary integral formulation satisfy the radiation principle, the governing system of equations is equally applicable for analysis of vibrations of both finite-length structures and the structures having infinitely long elements jointed with elements of finite length.

Preconditioning of the second‐kind boundary integral equations for 3D eddy current problems

AbstractStarting from the time‐harmonic Maxwell equations at low‐frequency eddy current approximation the H–ϕ formulation is presented. An equivalent system of boundary integral equations of the second kind on the conductor surface (resp. the conductor/dielectric) is derived. Discretizing these equations with a boundary element method (BEM) yields a block linear equation system \[ \left[\begin{array}{cc} A_{1} & B_{1} \\ \tfrac{\mu}{\mu_{0}}\tilde{B}_{2} & A_{2}\end{array} \right] \left( \begin{array}{c} j \\ \sigma \end{array} \right)=\left(\begin{array}{c} b_{1} \\ b_{2}\end{array} \right) \] with a fully populated stiffness matrix. This system is non‐symmetric and, for large μ/μ0 (of interest in practice) ill‐conditioned. Iterative solvers like GMRES converge very slowly, in general. We propose a new preconditioner which depends on μ. We present numerical results with a serial and parallel version of this preconditioner also on large industrial eddy current problems with compex geometry. The performance of preconditioned GMRES is found to be practically independent of μ/μ0. Copyright © 2001 John Wiley & Sons, Ltd.

The diffraction of surface waves by an elastic platform floating on shallow water

The scattering of long gravitational waves by a floating elastic plate is investigated using linear shallow-water theory. For a plate of arbitrary shape, the solution of the problem is reduced to a system of boundary integral equations. Using the example of a rectangular plate, the solution obtained is compared with existing theoretical and experimental results. The behaviour of the buckling of a rectangular plate and of a strip of constant width is compared for oblique incidence of surface waves.

Numerical Solution of a System of Boundary Integral Equations with a Logarithmic Singularity for Problems in the Theory of Shells with Holes

A quadrature formula for integrals having a logarithmic singularity is investigated. The formula permits the method of mechanical quadratures to be used in solving a system of boundary integral equations with an analogous singularity in problems associated with the theory of perforated shells.

The Green Matrix for Strictly Hyperbolic Systems with Second-Order Derivatives

Mathematical modeling of dynamic problems of continuum mechanics in domains of complicated geometry for various types of boundary conditions usually leads to boundary value problems for hyperbolic systems. The method of boundary integral equations is a convenient tool for solving such problems; it allows one to reduce the original differential boundary value problem in a domain to a system of boundary integral equations on the boundary, thus reducing the dimension of the equations, improving the stability of numerical solution methods, and so on. Now this method is widely used for a large class of problems in mechanics and mathematical physics. The construction of fundamental solutions and kernels of boundary integral equations of the system is a key point of the method of boundary integral equations. The fundamental solutions of classical equations of elliptic and hyperbolic types have been studied quite comprehensively [1, 2]. Less is known about fundamental solutions of systems of partial differential equations. Related papers mainly deal with specific equations of continuum mechanics [3–7]. In the present paper, we consider strictly hyperbolic systems of M equations with second-order derivatives in an (N + 1)-dimensional space. For such systems, we construct the Green matrix and analyze its properties. We also study systems invariant under the orthogonal group.

Opening-Function Simulation of the Three-Dimensional Nonstationary Interaction of Cracks in an Elastic Body

Integral relations between three-dimensional dynamic displacements (stresses) in an infinite elastic body with arbitrarily located plane cracks and discontinuities in the displacements of the opposite crack faces are presented. The influence of opening cracks on each other is considered in the problem on crack faces loaded by pulse forces. This problem is reduced to a system of boundary integral equations of the wave-potential type in a time domain. The dynamic mode I stress intensity factors are determined for two coplanar elliptic cracks under forces in the form of the Heaviside function

Investigation of the interaction of flat surface cracks in a half-space by BIEM

The approach for determination of the stress–strain state of a half-space with flat surface cracks of various shapes is proposed. The surface cracks are assumed to be under the external load. The problem is reduced to the system of boundary integral equations, in which the displacement jumps across the crack faces are the unknowns. The algorithm for solution of such equations is based on ignoring the motionless singularity. The influence of interaction of closely located surface cracks of various sizes on the stress intensity factors is investigated numerically for various distances between the cracks. Two penny-shaped surface cracks, two surface cracks with front of the Pascal curve and two cracks with various fronts (penny-shaped and elliptical) are considered as the examples.

The first-kind and the second-kind boundary integral equation systems for solution of frictionless contact problems

An original approach to the numerical solution of displacement boundary integral equation (BIE) and traction hypersingular boundary integral equation (HBIE) by the boundary element method (BEM) for contact problems is given. The main point is to show, how the contact conditions are used to formulate the first-kind and the second-kind BIE systems in the case of frictionless two-body elastic contact. The solution of the first-kind BIE is performed by symmetric Galerkin BEM; the second-kind BIE is solved by an appropriate collocation BEM. The contact problem in itself is solved by the method of subsequent approximations of contact region. Both forms of BIE system are compared in several numerical examples. This comparison is made for different kinds of contact problem. The major emphasis is put on the evaluation of contact pressure. The obtained results are compared with referenced numerical and with the analytical ones.

Existence and uniqueness of weak solutions for a thin plate with elastic boundary conditions

Robin-type problems are studied for thin elastic plates with transverse shear deformation. These problems are reduced to analogous ones for the corresponding homogeneous equilibrium equation, whose solutions are then represented as single and double layer potentials. The unique solvability of the systems of boundary integral equations yielded by this procedure is discussed in Sobolev spaces.

Schwarz method for systems of boundary integral equations

We analyze the additive Schwarz method for preconditioning linear systems arising from the Galerkin method for systems of first kind integral equations. We deal with non-symmetric, indefinite systems and focus on high order methods for problems in three dimensions. Numerical results for a system governing time-harmonic Maxwell's equations are given.

Boundary integral equations for notch problems in an anisotropic half-space

A crack or a hole embedded in an anisotropic half-plane space subjected to a concentrated force at its surface is analyzed. Based on the Stroh formalism and the fundamental solutions to the half-plane solid due to point dislocations, the problem can be formulated by a system of boundary integral equations for the unknown dislocation densities defined on the crack or hole border. These integral equations are then reduced to algebraic equations by using the properties of the Chebyshev polynomials in conjunction with the appropriate transformations. Numerical results have been carried out for both crack problems and hole problems to elucidate the effect of geometric configurations on the stress intensity factors and the stress concentration.

Boundary-integral formulation of dynamic nonstationary problems for an elastic space with a mathematical cut along a nonclosed Lyapunov surface

By satisfying the boundary conditions on the discontinuity surfaces using specially constructed integral representations of the solutions, we obtain a system of boundary integral equations for the functions of the opening of the cut.