The method of boundary integral equations for designing high-frequency resonators

Acoustic wave scattering by scatterers in layered media is studied using a new method that combines the hybrid ray–mode and boundary integral equation methods. The latter is used to formulate the scattering process and the former is used to provide the Green’s function of the layered environment. Following standard procedures of the boundary integral equation method, integral equations are formulated first for field distributions along the boundaries of the obstacles. By expressing these unknown distributions in terms of appropriate basis functions, the integral equations are then reduced to algebraic equations that are solved numerically. In the integral equations, it is necessary to compute the Green’s function of the layered environment for various arrangements of locations of source and receiver. None of these conventional approaches (rays, modes, and spectral integration) to evaluate the Green’s function are satisfactory for all possible arrangements. Only the newly developed hybrid ray–mode method is best suited for this purpose because it combines rays and modes self-consistently within a single framework and optimizes the advantages of each. Numerical implementation illustrates these aspects.

The principles and development of the boundary integral equation (BIE) method for solving engineering problems are reviewed, with particular emphasis on applications in three-dimensional stress analysis. Its use in problems of this type is illustrated with the aid of three examples, one a classical stress concentration problem, the other two involving cracked components. Compared with a finite-element method, the BIE approach is shown to be capable of being both more accurate and more economical to use in terms of competing resources required and the cost of data preparation. These advantages may, however, be less clear-cut when predictions of stresses and displacements at a large number of points within a component are required, or when the component concerned has a relatively high surface-to-volume ratio.

A quantative comparison between the boundary integral equation (BIE) method and the finite difference (FD) method is presented in which each technique is applied to an elliptic boundary‐value problem (BVP) containing a boundary singularity. Two types of singularity have previously been analysed theoretically, namely those due to a discontinuous boundary potential, which we shall refer to as S1, and those due to a sudden change from the specification of boundary potential flux, an S2 singularity. In this paper the analysis is presented for a third type of boundary singularity, namely an S3 singularity: that arising from a discontinuous boundary flux. Such a condition is frequently encountered in the field of heat transfer where, for example, a system or pipe has a change of lagging material.In general, it is found that the BIE method is superior, with regards to the computational time required to achieve a certain degree of accuracy, over standard FD methods even when there is a boundary singularity. Further, the BIE method determines the solution near the singularity much more accurately than the FD method. The FD method does, however, have advantages for a very restrictive class of problems; for example, when the boundary conditions are of the Dirichlet type and the boundary geometry is rectangular. In this case an optimum relaxation parameter can easily be obtained. A soon as Neumann conditions are prescribed, the BIE is far more efficient than the FD, whatever the boundary geometry.It is concluded that, for fast, accurate solutions of general Laplacian boundary‐value problems, the BIE is appreciably superior to the FD and this is even more pronounced when there is a boundary singularity.

One popular approach to solve the sea surface scattering and propagation in a ducting environment is the parabolic wave equation (PWE) method. An alternative method is the boundary integral equations (BIE) method. The implementation of the BIE in inhomogeneous media (ducting environments) is not straightforward, however, since the Green's function for such a medium is not usually known. In this paper, a closed-form approximation of the Green's function for a two-dimensional (2-D) ducting environment made up of a duct having a linear-square refractive index profile below a medium of constant refractive index, recently published, is used. This paper demonstrates how the BIE method can model the combined effects of surface roughness and medium inhomogeneity. Furthermore, it illustrates its capability of accurately predicting scattering in all directions including backscattering. Then, the PWE combined with the Split-Step Fourier (SSF) method is compared with this method.

On solving external boundary value problems of the theory of elasticity using the method of boundary integral equations

We study a modification of the qualocation method for boundary integral equations on smooth curves, which allows the same high-order convergence as the original qualocation method but with reduced smoothness assumptions on the exact solution. This ‘tolerant qualocation’ method differs from the original one only in that the exact inner product is used on the righthand side of the qualocation equation, whereas in the original method a specially designed approximate inner product is used on both sides. The modified method achieves exactly the same error estimates as the Petrov-Galerkin method, at greatly reduced cost.

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.

Application of the boundary integral equation (BIE) method to transient response analysis of inclusions in a half space

In this paper the collocation method for first-kind boundary integral equations, by using piecewise constant trial functions with uniform mesh, is shown to be equivalent to a projection method for second-kind Fredholm equations.In a certain sense this projection is an interpolation projection.By introducing this technique of analysis, we particularly consider the case of polygonal boundaries.We give asymptotic error estimates in L2 norm on the boundaries, and some superconvergence results for the single layer potential.

Eddy current nondestructive evaluation (NDE) of airframe structures involves the detection of electromagnetic field irregularities due to non-conducting inhomogeneities in an electrically conducting material. Usually, the eddy current NDE problem can be formulated by the boundary integral equations (BIE) and discretized into matrix equations by the method of moments (MoM) or the boundary element method (BEM). The fast multipole method (FMM) is a well-established and effective method for accelerating numerical solutions of the matrix equations. Accelerated by the FMM, the BIE method can now solve large-scale electromagnetic wave propagation and diffusion problems. The traditional BIE method requires O(N) operations to compute the system of equations and another O(N) operations to solve the system using direct solvers, with N being the number of unknowns; in contrast, the BIE method accelerated by the two-level FMM can potentially reduce the operations and memory requirement to O(N). Moreover, several approaches have been proposed for the field calculation in the presence of flaws in three dimensional NDE; however, seldom work has been done in applying efficient methods to seek rapid solution in eddy current NDE simulation. As elaborated in the dissertation, we introduce a fast multipole BIE method for two-dimensional diffusive scalar problem and an efficient BIE method for three-dimensional eddy current NDE. Firstly, we work with the two-dimensional Helmholtz equation with a complex wave number for non-trivial boundary geometry. We describe the FMM acceleration procedure of the BIE method and its features briefly, explaining that the FMM is not only efficient in meshing complicated geometries, accurate for solving singular fields or

New boundary integral equation method which was previously proposed have been examined in detail and extended to the computer‐aided design (CAD) of asymmetrical dielectric slab waveguide bends. Many numerical results of symmetric waveguide discontinuities and asymmetric waveguide bends confirm the validity of the new method. They show that all numerical results satisfy the energy conservation law within an accuracy of 1% and indicate that the new integral equation method is very suitable for the basic theory or a solver of CAD software for various open dielectric waveguide circuits.

Fast and accurate simulation of responses of electromagnetic logging-while-drilling tools in 2D and 3D formations is important for further development of the proactive geosteering service, which is currently based on the 1D layered-model inversion. In search of an optimal method of forward simulation, we have developed boundary integral equations in a 2D formation model with planar boundaries. A peculiarity of the method is that the number of grid nodes required for reaching a certain accuracy, and hence the computation time increases as the transmitter approaches the boundary. To retain efficiency in the case of transmitter-boundary proximity, we use the following modification of the method. The main term of the calculated electromagnetic field, namely, the solution for the two-layered model corresponding to the closest boundary, is calculated by the well-known explicit formulas, whereas the anomalous field is found from the system of integral equations. The modified method is implemented and tested on complex realistic 2D formation models as well as basic models to be used in 2D inversion. Numerical simulations of real propagation and azimuthal resistivity tools show that software based on the modified method can be successfully used for synthetic log simulation in complex formations. However, the proposed method has two shortcomings: poor efficiency when the transmitter is close to the boundary endpoints and limitation to isotropic formations. Both are topics for further research. Currently, the singularity near the endpoints is treated by simply refining the grid, which deteriorates efficiency. Generalization to the anisotropy case would include modification of the basic method and derivation of the closed-form solution for the layered model without the assumption of transverse isotropy.

The direct problem of antiplane oscillations of a layer with delamination in the context of the gradient theory of elasticity is considered. The gradient model proposed by Aifantis is taken as a mathematical model. The main attention has been paid to the analysis of mechanical fields at the crack bank and at its tips - stress concentrators. The study is carried out using the method of boundary integral equations (BIE). The BIE for the gradient of displacement field on the crack is constructed. The analysis of the constructed BIE is carried out and the cubic singularity is explicitly revealed. The solution of singular BIE is constructed using approximating Chebyshev polynomials. A study for cracks of small relative length - asymptotic approach is carried out, simple semi-analytical expressions for constructing the crack swap function are obtained, the range of efficiency of the asymptotic approach is obtained. The stress fields in the area of the crack tips are constructed. Numerical results of computational experiments are presented.

We consider the Dirichlet problem for Laplace's equation on planar domains with corners.A Nyström method is used to solve the corresponding double layer boundary integral equation.The Nyström method is based on the trapezoidal rule with a graded mesh method as in [13], but we adopt a generalized mesh grading.An improved convergence is observed.

Based on the created generalized apparatus of vector-tensor analysis, integral representations of the main dynamic and kinematic characteristics of the problem of viscous gas flow around force systems of arbitrary spatial shape are constructed. The boundary value problem of the interaction of such systems with a viscous gas flow is reduced to a system of linear, conditioned by physical boundary conditions, boundary integral equations regarding the kinematic and dynamic characteristics of the problem. It is proven that all the obtained characteristics depend on the newly obtained irrotational vector potential of the momentum, which significantly simplifies the integral representations of solutions and their numerical implementation. On the basis of the created generalized apparatus of vector-tensor analysis, integral representations of the main dynamic and kinematic characteristics of the problem of the flow of a viscous gas flow around supporting systems of satisfactory spatial form have been constructed. The boundary value problem of the interaction of such systems with a viscous gas flow is reduced to a system of linear, conditioned by physical boundary conditions, boundary integral equations regarding the kinematic and dynamic characteristics of the problem. It is proven that all the obtained characteristics depend on the newly obtained, vortex-free vector potential of the momentum, which significantly simplifies the integral representations of the solutions and their numerical implementation.