New integral equation method for CAD of open waveguide bends

In this article, the conduction of transverse electric/transverse magnetic (TE/TM) propagation modes in nanomaterial graphene utilization of asymmetric dielectric slab waveguide was studied in terms of their applicability in optical fiber communication. Nano-graphene film was deposited on silica substrate and air cladding. Three layers were exposed to electromagnetic waves with a 1.55 µm wavelength and 1 µm thick nanographene film, silica, and air covering to study the performance and optical properties of graphene nanomaterials. The models were influenced by factors such as attenuation wavenumbers, cut-off frequency, mode number, propagation wavenumbers, skin depth, and angles of total internal reflection. The impact of these parameters was assessed through numerical analysis, revealing significant correlations. The modes generated ten numbers of TE and nine numbers of TM mode dependent upon the structure of nanographene, with low loss propagation wavenumber. As the TE/TM modes increase, the decay wavenumbers of the cladding and substrate parameters diminish. This indicates that the magnetic and electric fields undergo extremely rapid decay beyond the film region, which leads to the skin depth for higher TE/TM modes traveling longer distances in the cladding and substrate than do lower-order modes. Therefore, nanographene exhibits good optical properties due to its low absorption loss. The angles of internal reflection are greater in magnitude than the substrate and cladding key angles. Still, in the tenth TE mode, the internal reflection angles are extremely close to the critical angle as a result of the small substrate decay parameter and the near operating frequency. The tenth TE mode is weakly characterized. The characteristics of TE/TM modes in an asymmetric three-layer slab waveguide structure are realized for various mode orders and various parameters of nanographene material. The field profiles of the TE/TM modes are submitted to comprehend these characteristics.

Electromagnetic waves in a three-dimensional asymmetric dielectric slab waveguide

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.

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 Boundary Integral Equation Method (BIEM) is applied to transient water wave problems. Only two-dimensional linearized waves are considered. As is general practice, free-surface boundary conditions are applied at the equilibrium surface rather than the actual free surface; thus the problems become fixed-boundary problems rather the free-surface problems. For the cases in which fluid domain is unbounded in the horizontal direction, a radition condition is formulated such that waves pass through the computational boundaries without reflection. The stability limits and frequency distortion of the numerical method are examined and given. Numerical results are compared with analytical solutions or experimental data in three examples. Excellent agreement is observed.

We study the problem of the scattering of the first TM guided mode from an abruptly ended strongly asymmetrical slab waveguide by an improved iteration technique, which is based on the integral equation method with "accelerating" parameters. We demonstrate that the values of these parameters are related to the variational principle, and we save approximately 1-2 iterations compared with the case in which these parameters are not employed. The tangential electric-field distribution on the terminal plane, the reflection coefficient of the first TM guided mode, and the far-field radiation pattern are computed. Furthermore, a simple technique based on the Aitken extrapolation procedure is employed for faster computation of the higher-order solutions of the reflection coefficient. Numerical results are presented for several cases of abruptly ended waveguides, including systems with variational profile, while special attention is given to the far-field radiation pattern rotation and its explanation.

Computation of dynamic stress intensity factors by the time domain boundary integral equation method—II. Examples

The paper considers generalization of Mode Basis Method for calculation of pulse signal propagation in a circular open dielectric waveguide. The mode functions are calculated by Integral Equations Method. It is shown that the circular dielectric waveguide possesses only continuous spectrum. Then an integral-differential System of Evolutionary Equations is obtained that describes mode amplitudes evolution in time with propagation. Its solution by the method of moments is proposed. Propagation of some pulse signal in a circular dielectric waveguide is calculated.

Applying the concept of minimum potential energy and the variational method proposed in this paper, one can derive the governing equation and transversality conditions for the critical slip surface of a cohesive land slope described by simplified Janbu's model where both horizontal and vertical inter-slice forces are neglected. The governing equation, transversality conditions and boundary conditions were solved by the boundary integral equation method, which is a one dimensional BIEM, so that the critical slip surface and its associated minimal factor of safety can be determined effectively. By comparison of the results gotten from the boundary integral equation method and other numerical methods, it can be concluded, that, for some simplified cases, by using the boundary integral equation method on slope stability analysis of a cohesive land slope, a more reasonable result can be obtained.

A hybrid formulation of the 2.5D elastodynamic scattering problem combining the finite-element method and boundary integral equation method is presented and validated. The formulation of the 2.5D boundary integral equation method that is used was presented in detail by Papageorgiou and Pei (1998) and is an extension of the discrete wavenumber boundary integral equation method originally proposed by Kawase (1988) for 2D scattering problems. Modeling of the wave field in the domain of the scatterer is based on the variational principle of virtual displacements, and discretization of the domain is accomplished using the finite-element method.The formulation may be used to study the wave field in models of sedimentary deposits (e.g., valleys) or topography (e.g., canyons or ridges) with a 2D variation in structure but obliquely incident plane waves. The hybrid method exploits the versatility of the finite element method for modeling the scatterer and the effectiveness of the boundary integral equation method for taking care of the radiation condition in the half-space. The advantage of the 2.5D formulation is that it provides the means for calculations of 3D wave fields in scattering problems by requiring a storage comparable to that of the corresponding 2D calculations.

In this paper the new integral equation method proposed and developed in our previous papers on boundary-value problems is applied to eigenvalue problems. Two formulations by the integral equation expression are performed from the same manner that we adopted on the boundary value problems. As the result, whether the derived integral equation is single or not, the integral kernels do not include the unknown eigen-parameter. Through this notable feature in contrast with the boundary integral equation method, eigenvalue problems on continuous fields can be reduced to the standard eigenvalue problems of matrix equation. Finally, in order to show the applicability and validity of the proposed methodology and procedure, numerical calculations are performed for three illustrative examples, which are eigenvalue problems of Sturm-Liouville type equation, Helmholtz equation and biharmonic type equation. The approximate values of an eigenvalue parameter of each problem demonstrate a sufficient accuracy through the comparison of another values.

Boundary integral equation (boundary element) methods have the advantage over other commonly used numerical methods that they do not require values of the unknowns at points within the solution domain to be computed. Further benefits would be obtained if attention could be confined to information at one small part of the boundary, the particular region of interest in a given problem. A local boundary integral equation method based on a Taylor series expansion of the unknown function is developed to do this for two-dimensional potential problems governed by Laplace's equation. Very accurate local values of the function and its derivatives can be obtained. The method should find particular application in the efficient refinement of approximate solutions obtained by other numerical techniques.

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.

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