Complementary Discrete Geometric <inline-formula> <tex-math notation="LaTeX">$h$ </tex-math> </inline-formula>-Field Formulation for Wave Propagation Problems

To simulate or design a reasonably large system, fast and simple models are necessary. To verify the design versus the specifications, more detailed (and costly) calculations can be performed and final adjustments made. In wave propagation problems, circuit analogs provide a powerful, yet simple, means of computing the desired response of the system, such as reflection or transmission coefficients. The reason circuit analog models are good for wave propagation problems, is that they are exact for one-dimensional wave propagation, regardless of whether we consider acoustic or electromagnetic waves. Typically, wave propagation through homogeneous media is modeled as a transmission line with propagation constant β and characteristic impedance Z, whereas obstacles such as thin sheets are modeled as lumped elements. If the sheets are lossless, the circuit models contain only reactive elements such as capacitors and inductors. Modeling complex wave propagation problems with circuit analogs was to a large extent developed in conjunction with the development of radar technology during the Second World War. Many of the results from this very productive era are collected in the Radiation Laboratory Series and related literature, in particular (Collin, 1991; 1992; Marcuvitz, 1951; Schwinger & Saxon, 1968). Further development has been provided by research on frequency selective structures (Munk, 2000; 2003). In recent years, the circuit analogs have even been used in an inverse fashion: by observing that wave propagation through a material with negative refractive index could be modeled as a transmission line with distributed series capacitance and shunt inductance, i.e., the dual of the standard transmission line, the most successful realization of negative refractive index material is actually made by synthesizing this kind of transmission line using lumped elements (Caloz & Itoh, 2004; Eleftheriades et al., 2002). This chapter is organized as follows. In Section 2 we show that propagation of electromagnetic waves in any material, regardless how complicated, boils down to an eigenvalue problem which can be solved analytically for isotropic media, and numerically for arbitrary media. From this eigenvalue problem, the propagation constant and characteristic impedance can be derived, which generates a transmission line model. In Section 3, we show how sheets with or without periodic patterns can be modeled as lumped elements connected by transmission lines representing propagation in the surrounding medium. The lumped elements can be given a firm definition and physical interpretation in the low frequency limit, and in Section 4 we show how these low frequency properties

Mesh refinement for anisotropic diffusion in magnetized plasmas

Modeling concept and numerical simulation of ultrasonic wave propagation in a moving fluid-structure domain based on a monolithic approach

PurposeThe purpose of this paper is to show the applicability of a discrete Hodge operator in the context of the De Rham cohomology to bicomplex-valued electromagnetic wave propagation problems. It was applied in the finite element method (FEM) to get a higher accuracy through conformal discretization. Therewith, merely the primal mesh is needed to discretize the full system of Maxwell equations.Design/methodology/approachAt the beginning, the theoretical background is presented. The bicomplex number system is used as a geometrical algebra to describe three-dimensional electromagnetic problems. Because we treat rotational field problems, Whitney edge elements are chosen in the FEM to realize a conformal discretization. Next, numerical simulations regarding practical wave propagation problems are performed and compared with the common FEM approach using the Helmholtz equation.FindingsDifferent field problems of three-dimensional electromagnetic wave propagation are treated to present the merits and shortcomings of the method, which calculates the electric and magnetic field at the same spatial location on a primal mesh. A significant improvement in accuracy is achieved, whereas fewer essential boundary conditions are necessary. Furthermore, no numerical dispersion is observed.Originality/valueA novel Hodge operator, which acts on bicomplex-valued cotangential spaces, is constructed and discretized as an edge-based finite element matrix. The interpretation of the proposed geometrical algebra in the language of the De Rham cohomology leads to a more comprehensive viewpoint than the classical treatment in FEM. The presented paper may motivate researchers to interpret the form of number system as a degree of freedom when modeling physical effects. Several relationships between physical quantities might be inherently implemented in such an algebra.

A new h-version adaptive refinement scheme is developed for the analysis of Thin-walled structures in elastostatic range. The basic concepts of adaptivity for the finite element method are discussed and a brief review of the related theories and techniques is presented. Focus is devoted to three essential ingredients for the whole adaptive refinement scheme: automatic mesh generation, error estimation on the FE solution and adaptive refinement scheme design.

The effect of liquid on the propagation of waves in a micropolar elastic layer with voids immersed in an infinite liquid has been investigated. In case I the frequency equations for symmetric and antisymmetric vibrations have been obtained and some special cases have been deduced. Case II is concerned with wave propagation in a micropolar elastic solid with voids. Reflection of micropolar elastic waves at the free surface of a micropolar elastic body with voids is discussed. In case III, propagation of Rayleigh waves in a micropolar elastic body with voids and particle motion for Rayleigh waves are discussed. Numerical computations of the amplitude ratios for different reflected waves have been performed and plotted against the angle of emergence. The frequency equation of Rayleigh wave propagation is obtained and phase velocity as a function of Poisson’s ratio, frequency ratio, non-dimensional micropolar constant and normalized wave number have been plotted graphically for a particular model.

Dynamic spectrum access technology enables two types of users to operate on a channel–primary users and secondary users, which can use the channel when it is not in use by the primaries. We consider a scenario in which multiple primaries own bandwidth in a large region which is divided into smaller locations. A primary that has unused bandwidth in a time slot would like to lease it out to secondaries in return for a fee. This results in price competition among the primaries. In prior work, this price competition has only been studied under the approximation, made for analytical tractability, that the price of each primary takes values from a continuous set. However, in practice, the set of available prices is discrete. In this paper, we investigate the fundamental question of how the behavior of the players involved in the price competition changes when this continuity assumption is removed. Our analysis reveals several important differences between the games with continuous and discrete price sets. However, we show that as the number of available prices becomes large, the strategies of the primaries under every symmetric NE converge to the unique NE strategy of the game with continuous price sets.

The effect of random height variations associated with a conducting surface upon the characteristics of reflected wave energy is ascertained by the methods of physical optics. Average received powers its variance, angular and frequency power spectra, and the field correlation pattern are determined in terms of the statistical parameters of the surface. Volume type problems are treated by ascertaining the effect of refractive index fluctuations within a slab upon an emergent wavefront, and then generalizing to a continuous medium. The results are applied to various problems encountered in tropospheric and ionospheric wave propagation.

In some supply chains, retailers are relatively small and averse to taking risk. In such a situation, traditional methods of contracting, that typically assume risk neutrality on retailers, might not suffice to maximize the seller’s/distributor’s expected profit. We present tools for analyzing and solving such a problem from the viewpoint of a (risk-neutral) seller/distributor. We present two types of models that can be used to create contracts, one set in a discrete setting and the other in a continuous setting. In both settings, individual retailers are characterized with different degrees of risk aversion. We first explain in the discrete setting how the varying degrees of risk aversion present hurdle for the design of a uniform contract for all retailers. We then show how to mitigate the problem using ideas from the theory of mechanism design. We offer a simple solution to the contract design problem and show how it can be easily implemented. We next show that in the continuous setting in which the distribution of retailers is continuous, which could be viewed as a limiting case of the discrete setting, the contract design problem actually simplifies. In this continuous setting, we show that it becomes relatively easy to design contracts and establish their optimality from the seller’s/distributor’s viewpoint. We conclude the chapter with a summary of problems that are still open in this area.

In this paper we prove uniform oscillation estimates on L^p, with pin (1,infty ), for truncated singular integrals of the Radon type associated with the Calderón–Zygmund kernel, both in continuous and discrete settings. In the discrete case we use the Ionescu–Wainger multiplier theorem and the Rademacher–Menshov inequality to handle the number-theoretic nature of the discrete singular integral. The result we obtained in the continuous setting can be seen as a generalisation of the results of Campbell, Jones, Reinhold and Wierdl for the continuous singular integrals of the Calderón–Zygmund type.

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

Parallel discontinuous Galerkin unstructured mesh solvers for the calculation of three-dimensional wave propagation problems

[1] The problem of wave propagation in a two-dimensional waveguide with vertical absolutely thin barrier and ideally conducting walls is solved in the paper. This is a model approximating the radio wave propagation in the Earth–ionosphere waveguide. The main attention in the model is drawn to the influence of irregularities of the Earth surface including the influence of singular points of an angular type. The presence of the smooth irregularity of the upper wall, i.e., ionosphere, is also taken into account. Two methods of the solution are used: the method of semi-inversion developed by the authors and the method of quasi-static Green function. Using these methods, the diffraction of the field of a dipole at the vertical barrier in a waveguide with curved upper wall and in a waveguide with plain upper wall is calculated. On the basis of the obtained solutions a conclusion is drawn on the influence of ionospheric disturbances on the field.

Similarity analysis of hyperbolic type of partial differential equations arising in wave propagation problems in uniaxial noniinear-nonhomogeneous systems is complicated by the presence of a moving boundary at the wave front. In order that the original system be totally represented by a similarlty representation, location of similarity coordinate at the wave front is required. For this purpose, similarity coordinate is established at the moving front by making use of the property of its invariance and that of the characteristics of the original system under the same group of transformations. The similarity representation so obtained reduces the original hyperbolic system into a boundary value problem consisting of an ordinary differential equation with one boundary condition at the near end and the other at the wave front. This analysis is applied to wave propagation in axially noniinear-nonhomogeneous systems. Even though, the problems of wave propagation in a nonlinear axial system on the one hand and a nonhomogeneous axial system on the other have been dealt with by various authors, hardly any literature seems to be available on wave propagation in a system which is simultaneously nonlinear and nonhomogeneous. Similarity analysis augmented by s i m i l a r i t y c h a r a c t e r i s t i c r e l a t i o n s h i p , thus helps to so lve problems which are difficult to track by the known similarity methods.

The paper focuses on the problem of electromagnetic wave propagation in plane shielded dielectric waveguide. The waveguide is characterized by an isotropic inhomogeneous permittivity and constant permeability. We study symmetric guided waves that characterised by a pair of (coupled) propagation constants. Using an operator pencil approach, we prove a discreteness property for the searched-for propagation constants.