Unknown Expansion Coefficients Research Articles

Analysis and optimization of matching two-dimensionally periodic dielectric structures with cone-shaped cavities

The problem of scattering of plane electromagnetic waves by two-dimensionally periodic dielectric structures with cone-shaped cavities placed on a homogeneous dielectric substrate is considered. A modified hybrid projection method for solving this problem, which combines the coupled-wave method (incomplete Galerkin method), the projection field matching on domains’ boundaries, and the one-dimensional finite-element method, is proposed. The modified method enables one to avoid the use of the excessive number of unknown expansion coefficients, typical for the previous versions. The method is used for calculating the effective permittivity of a dielectric layer of finite thickness with cylindrical holes and for optimizing the parameters of conical cavities providing the matching of a dielectric half-space for various incidence angles.

Global and Regional Gravity Field Determination from GOCE Kinematic Orbit by Means of Spherical Radial Basis Functions

We present global and regional gravity field models to degree 130 based on the GOCE kinematic orbit from the period 01 November 2009 to 11 January 2010. The gravity field models are parameterized in terms of the Shannon and Kaula's spherical radial basis functions. The relation between the unknown expansion coefficients and the kinematic orbit of the satellite is established by the acceleration approach. We show that our global GOCE-only solutions free from prior information can compete with unconstrained spherical harmonic models in terms of accuracy. Furthermore, we utilize our low-degree global GOCE-based models to introduce prior information into the least-squares adjustment. This procedure substantially improves the zonal and near-zonal spherical harmonic coefficients, which are usually degraded due to the polar gap problem. As an unwanted side effect, low-pass filtering of the geopotential may occur, but this can be adjusted by the spectral content of the prior information. We show that the regional enhancement of the global solutions reduces noise in the final model between degrees 70 and 130 by ~10 % in terms of RMS error. In general, our Shannon-based solutions systematically outperform the Kaula-based ones. To validate our results, we use the EIGEN-6S model, which is superior to the solutions from kinematic orbits at least by one order of magnitude. Both the global and the regional models satisfy the GOCE-only strategy.

Ground motions around a semicircular canyon with a dipping edge under SH plane wave incidence

In order to explore the spatial distribution and temporal variation of ground motions near a semicircular canyon with a dipping edge, a simplified mathematical model is constructed. Based on the region-matching technique, a Fourier-Bessel series solution for the plane SH-wave excitation is derived and then applied to theoretically simulate the seismic response of the canyon. The use of the adequate wavefunctions and a newly derived Graf’s addition formula can solve the unknown expansion coefficients. Parametric analyses with respect to the frequency of input motion, angle of incidence, and canyon geometry are illustrated. Both frequency- and time-domain computations are presented. The canonical case, a completed semicircular canyon, which has the exact analytical solution, and the horizontally truncated case analyzed in previous works are considered as particular cases of the proposed general model. Comparisons with boundary-element solutions show good agreement. Steady-state results show that the phenomenon of wave focusing tends to be significant when the incident angle bends toward the horizontal ground surface. Propagation and attenuation of scattered waves that originated from the surficial anomaly are exhibited in transient-state simulations.

A subspace-based identification of Wiener–Hammerstein benchmark model

This paper develops a subspace-based method of identifying the Wiener–Hammerstein system, where a nonlinearity is sandwiched by two linear subsystems. First, a state space model of the best linear approximation of it is identified by using a subspace identification method and the poles of the best linear model are allocated between two linear subsystems by a state transformation. Unknown system parameters and coefficients of a basis function expansion of the nonlinearity are estimated by using the separable least-squares for all possible allocations of poles, so that there is a possibility that many iterative minimization problems should be solved. Finally, the best Wiener–Hammerstein system that yields the minimum mean square error is selected. Numerical results for a benchmark model show the applicability of the proposed method.

Scattering of a Plane Wave by a Homogeneous Anisotropic Elliptic Cylinder

A formal series solution to the problem of scattering of a normally incident plane wave from an infinite homogeneous anisotropic elliptic cylinder is presented. The formulation of the problem is accomplished by assuming that the permittivity and the permeability tensors of the anisotropic material referred to the elliptic cylindrical coordinate axes are mainly biaxial and diagonal, and expanding each field associated with the problem in terms of a series of suitable Mathieu functions and expansion coefficients. The incident field expansion coefficients are known, but those associated with the scattered and transmitted fields are unknown. The unknown expansion coefficients are obtained by imposing appropriate boundary conditions at the surface of the anisotropic cylinder. Numerical results are presented as normalized bistatic and backscattering widths for elliptic cylinders of different sizes and permeabilities to show the effects of the material anisotropy on scattering.

Modeling lightning strike behavior in the near field of elevated systems

The modeling of lightning strike behavior and estimation of the subsequent electric discharge is of great practical importance. In this study, a complete two-dimensional physics-based analytic formulation is presented for elevated grounded systems that can be envisioned to be contained within two non-concentric circular domains. The inner circle encompasses the body or system of interest, and the periphery of the outer circle addresses the cloud coverage. The potential field between the circular domains is modeled as the sum of two separate contributions. The first is formulated in terms of an eigen-function expansion involving simple radial functions and Legendre polynomials, while the second contribution is developed using two different approaches. The first approach utilizes an eigen-function expansion incorporating spherical Bessel functions and Legendre polynomials, while the second approach uses a Green’s function expansion also involving orthogonal polynomial functions. Each of the contributions to the total potential field leads to linear systems of equations that are solved for the unknown series expansion coefficients. The accuracy of the potential field solution is investigated with regard to convergence, stability, and error compared with an exact solution. The potential field solution is then used as the basis to evaluate leader formation, as a function of elevation, extent of cloud coverage, and propagation angle of the downward leader. Regions of high risk to lightning strikes on the grounded structure are developed in terms of joint probability functions. The electric discharge is estimated using electric current and is shown to be in the range of presently available information.

A generalized case of the electromagnetic scattering from an array of ferrite cylinders

A boundary value type solution of the generalized case of scattering from an array consisting of a finite number of axially magnetized ferromagnetic circular cylinders is presented in this paper. Proposed analysis is carried out by matching the tangential fields at the surface of each cylinder to find the unknown expansion coefficients of the scattered field. A planar array having size and a linear array having size consist of ferromagnetic microwires are considered to obtain the numerical results for and polarizations in terms of the variation in scattered field components of the near field and scattering cross section with respect to the angle of incidence, radius of microwires, spacing among the microwires, number of microwires in the array, and operating frequency. For validation, theoretical results of the method already available in the literature are simulated for a linear array and compared to match with the numerical results of proposed analysis for the same array. In order to prove the generalization of the solution in terms of the number of microwires, simulated results specialized to the case of single ferromagnetic microwire are shown to be matched with the results available in the literature for the special case.

New ultraspherical wavelets collocation method for solving 2nth-order initial and boundary value problems

Abstract In this paper, a new spectral algorithm based on employing ultraspherical wavelets along with the spectral collocation method is developed. The proposed algorithm is utilized to solve linear and nonlinear even-order initial and boundary value problems. This algorithm is supported by studying the convergence analysis of the used ultraspherical wavelets expansion. The principle idea for obtaining the proposed spectral numerical solutions for the above-mentioned problems is actually based on using wavelets collocation method to reduce the linear or nonlinear differential equations with their initial or boundary conditions into systems of linear or nonlinear algebraic equations in the unknown expansion coefficients. Some specific important problems such as Lane–Emden and Burger’s type equations can be solved efficiently with the suggested algorithm. Some numerical examples are given for the sake of testing the efficiency and the applicability of the proposed algorithm.

Coherent scattering by a collection of randomly located obstacles – An alternative integral equation formulation

Scattering of electromagnetic waves by discrete, randomly distributed objects is addressed. In general, the non-intersecting scattering objects can be of arbitrary form, material and shape. The main aim of this paper is to calculate the coherent reflection and transmission characteristics of a slab containing discrete, randomly distributed scatterers. The integral representation of the solution of the deterministic problem constitutes the underlying framework of the stochastic problem. Conditional averaging and the employment of the Quasi Crystalline Approximation lead to a system of integral equations in the unknown expansion coefficients. Of special interest is the slab geometry, which implies a system of integral equations in the depth variable. Explicit solutions for tenuous media and low frequency approximations can be obtained for spherical obstacles.

Many-body microhydrodynamics of colloidal particles with active boundary layers

Colloidal particles with active boundary layers—regions surrounding the particles where non-equilibrium processes produce large velocity gradients—are common in many physical, chemical and biological contexts. The velocity or stress at the edge of the boundary layer determines the exterior fluid flow and, hence, the many-body interparticle hydrodynamic interaction. Here, we present a method to compute the many-body hydrodynamic interaction between N spherical active particles induced by their exterior microhydrodynamic flow. First, we use a boundary integral representation of the Stokes equation to eliminate bulk fluid degrees of freedom. Then, we expand the boundary velocities and tractions of the integral representation in an infinite-dimensional basis of tensorial spherical harmonics and, on enforcing boundary conditions in a weak sense on the surface of each particle, obtain a system of linear algebraic equations for the unknown expansion coefficients. The truncation of the infinite series, fixed by the degree of accuracy required, yields a finite linear system that can be solved accurately and efficiently by iterative methods. The solution linearly relates the unknown rigid body motion to the known values of the expansion coefficients, motivating the introduction of propulsion matrices. These matrices completely characterize hydrodynamic interactions in active suspensions just as mobility matrices completely characterize hydrodynamic interactions in passive suspensions. The reduction in the dimensionality of the problem, from a three-dimensional partial differential equation to a two-dimensional integral equation, allows for dynamic simulations of hundreds of thousands of active particles on multi-core computational architectures. In our simulation of 104 active colloidal particle in a harmonic trap, we find that the necessary and sufficient ingredients to obtain steady-state convective currents, the so-called ‘self-assembled pump’, are (a) one-body self-propulsion and (b) two-body rotation from the vorticity of the Stokeslet induced in the trap.

Three-Body Protonium Formation in a Collision Between a Slow Antiproton ( $${\bar{\rm p}}$$ p ¯ ) and Muonic Hydrogen: $${{\rm H}_{\mu}}$$ H μ —Low Energy $${\bar{\rm p} + ({\rm p} \mu^-)_{1s} \rightarrow (\bar{\rm p} {\rm p})_{1s} + \mu^-}$$ p ¯ + ( p μ - ) 1 s → ( p ¯ p ) 1 s + μ - Reaction

A bound state of a proton, p, and its counterpart antiproton, $${\bar{\rm p}}$$ , is a protonium atom $${Pn = (\bar{\rm p} {\rm p})}$$ . The following three-charge-particle reaction: $${\bar{\rm p} +({\rm p} \mu^-)_{1s} \rightarrow (\bar{\rm p} \rm{p})_{1s} + \mu^-}$$ is considered in this work, where $${\mu^-}$$ is a muon. At low-energies muonic reaction $${Pn}$$ can be formed in the short range state with α = 1s or in the first excited state: α = 2s/2p, where $${\bar{\rm p}}$$ and p are placed close enough to each other and the effect of the $${\bar{\rm p}}$$ –p nuclear interaction becomes significantly stronger. The cross sections and rates of the Pn formation reaction are computed in the framework of a few-body approach based on the two-coupled Faddeev-Hahn-type (FH-type) equations. Unlike the original three-body Faddeev method the FH-type equation approach is formulated in terms of only two but relevant components: $${{\it \Psi}_1}$$ and $${\it \Psi_2}$$ , of the system’s three-body wave function $${\it \Psi}$$ , where $${{\it \Psi}={\it \Psi}_1+{\it \Psi}_2}$$ . In order to solve the FH-type equations $${\it \Psi_1}$$ is expanded in terms of the input channel target eigenfunctions, i.e. in this work in terms of the $${(\rm{p} \mu^-)}$$ eigenfunctions. At the same time $${\it \Psi_2}$$ is expanded in terms of the output channel two-body wave function, that is in terms of the protonium $${(\bar{\rm{p}} \rm{p})}$$ eigenfunctions. A total angular momentum projection procedure is performed, which leads to an infinite set of one-dimensional coupled integral–differential equations for unknown expansion coefficients.

A highly accurate Jacobi collocation algorithm for systems of high‐order linear differential–difference equations with mixed initial conditions

In this paper, a shifted Jacobi–Gauss collocation spectral algorithm is developed for solving numerically systems of high‐order linear retarded and advanced differential–difference equations with variable coefficients subject to mixed initial conditions. The spatial collocation approximation is based upon the use of shifted Jacobi–Gauss interpolation nodes as collocation nodes. The system of differential–difference equations is reduced to a system of algebraic equations in the unknown expansion coefficients of the sought‐for spectral approximations. The convergence is discussed graphically. The proposed method has an exponential convergence rate. The validity and effectiveness of the method are demonstrated by solving several numerical examples. Numerical examples are presented in the form of tables and graphs to make comparisons with the results obtained by other methods and with the exact solutions more easier. Copyright © 2015 John Wiley & Sons, Ltd.

An Efficient Legendre Spectral Tau Matrix Formulation for Solving Fractional Subdiffusion and Reaction Subdiffusion Equations

In this work, we discuss an operational matrix approach for introducing an approximate solution of the fractional subdiffusion equation (FSDE) with both Dirichlet boundary conditions (DBCs) and Neumann boundary conditions (NBCs). We propose a spectral method in both temporal and spatial discretizations for this equation. Our approach is based on the space-time shifted Legendre tau-spectral method combined with the operational matrix of fractional integrals, described in the Riemann–Liouville sense. The main characteristic behind this approach is to reduce such problems to those of solving systems of algebraic equations in the unknown expansion coefficients of the sought-for spectral approximations. In addition, this approach is also investigated for solving the FSDE with the variable coefficients and the fractional reaction subdiffusion equation (FRSDE). For conforming the validity and accuracy of the numerical scheme proposed, four numerical examples with their approximate solutions are presented. Also, comparisons between our numerical results and those obtained by compact finite difference method (CFDM), Box-type scheme (B-TS), and FDM with Fourier analysis (FA) are introduced.

Curvature Effects on the Vibration Characteristics of Doubly Curved Shallow Shells with General Elastic Edge Restraints

Effects of curvature upon the vibration characteristics of doubly curved shallow shells are assessed in this paper. Boundary conditions of the shell are generally specified in terms of distributed elastic restraints along the edges. The classical homogeneous boundary supports can be easily simulated by setting the stiffnesses of restraining springs to either zero or infinite. Vibration problems of the shell are solved by a modified Fourier series method that each of the displacements is invariably expressed as a simple trigonometric series which converges uniformly and acceleratedly over the solution domain. All the unknown expansion coefficients are treated equally as a set of independent generalized coordinates and solved using the Rayleigh-Ritz technique. The current method provides a unified solution to the vibration problems of curved shallow shells involving different geometric properties and boundary conditions with no need of modifying the formulations and solution procedures. Extensive tabular and graphical results are presented to show the curvature effects on the natural frequencies of the shell with various boundary conditions.

Study on underwater acoustic scattering of a Bessel beam by rigid objects with arbitrary shapes

Study on underwater acoustic scattering is very important for detection, location, and recognition of underwater targets. In the past decades, most investigations in this respect were focused on the case of plane wave incidence. But the Bessel beam is a kind of approximate non-diffracting beam with an excellent directing property, so more attention should be paid on it. So far, according to the literature, the studies about underwater acoustic scattering of a Bessel beam mainly focused on spheres and spherical shells using the partial wave series form. When the scatterers become complex objects, the partial wave series form fails to deal with these kinds of problems. To overcome this shortage, the T-matrix method has been introduced to calculate the underwater scattering of a Bessel beam by complex rigid objects. #br#In this paper, the underwater acoustic scattering of a Bessel beam by rigid objects with arbitrary shapes calculated by T-matrix method is studied. By means of the harmonic expansion of Bessel beam, the expression of the incident coefficient can be derived. Through the transmission matrix that relates the known coefficients of expansion of an incident wave to the unknown expansion coefficients of the scattered field, the acoustic scattering formula of a Bessel beam by a rigid scatterer with arbitrary shape is established. In this paper, the backscattering fields of rigid spheroids and finite cylinders with two spheroidal endcaps are discussed, and the backscattering form function modulus |F| is curved as a function of dimensionless frequency ka. Subsequently, the peak to peak intervals in backscattering form function caused by the interference of the specular wave and the Franz wave are also analyzed in geometry. The calculated results show that the frequency interval obtained from the curves agrees well with those obtained by geometric analysis for the rigid objects. Meanwhile, for both the rigid spheroid and finite cylinder, the highlight model is successfully applied to explain the phenomenon in which the amplitude of backscattering form function changes with the cone angle of the Bessel beam. From the above numerical results and analysis, the T-matrix method has been successfully introduced to calculate the acoustic scattering of the Bessel beam by complex objects, which extends the application of the T-matrix method and provides a useful tool to explore the characteristics of the Bessel beam.

The Mode Method as a Framework for Theoretical Studies of Ultrasonic Waves Diffraction in Non-homogeneous Layered Structures

Fundamentals of the mode method including orthogonality conditions between different modes and derivations of coupled equations systems for unknown expansion coefficients are given in the paper for the case of 2D isotropic structures. Using the mode method, several concrete problems of ultrasonic waves diffraction in non-homogeneous layered structures are presented in the paper such as: Scholte-Stoneley wave excitation and conversion at the edge of a liquid loaded plate, interaction of Lamb waves with delaminations in plates coated by highly absorbing materials and nonlinear modulation of Lamb modes by clapping delaminations.

Scattering of electromagnetic plane wave by an impedance strip embedded in homogeneous isotropic chiral medium

In this article, Kobayashi potential method has been used to investigate scattering from an impedance strip embedded in a homogeneous isotropic chiral medium. The discontinuous properties of the Weber-Schafheitlin's integral have been applied to incorporate the edge conditions. A set of matrix equations are obtained, using the projection properties of Jacobi's polynomials, and solved for unknown expansion coefficients of the fields. Monostatic and bistatic scattering widths have been determined for left and right handed Beltrami field excitations. Effects of variation in values of chirality, incident angle, impedance and width of strip on scattered fields have been investigated. The numerical results of the scattered fields for various parameters have been produced for comparison.

The Rayleigh hypothesis and the region of applicability of the extended boundary condition method in electrostatic problems for nonspherical particles

The electrostatic problem of excitation of a homogeneous axisymmetric particle in a constant electric field is considered. The approach is based on the surface integral equations arising in the extended boundary condition method (EBCM). The electrostatic fields are related to the scalar potentials, which are represented as expansions in the eigenfunctions of the Laplace operator in the spherical coordinate system. Unknown expansion coefficients are determined from infinite systems of linear algebraic equations (ISLAE). Analytic analysis of the solvability of the ISLAE is performed, and the convergence radii of expansions are obtained. It is shown that the EBCM can be applied in the far zone of a particle, i.e., the T matrix, can be constructed even if the Rayleigh hypothesis (the expansions converge up to the boundary of the particle) is not satisfied. However, a weaker restriction appears, which can be reduced to the requirement that there exists a spherical shell inside which the expansions of excited and internal fields simultaneously converge. The case of spheroids, as well as pseudospheroids that arise from spheroidal particles by inversion, is studied in detail. It is shown that the EBCM is applicable to spheroids for any ratio a/b of semiaxes and to pseudospheroids for a/b < 1 + \(\sqrt 2 \). The external Rayleigh hypothesis, i.e., the convergence of the expansion of a “scattered” field up to the surface of a particle is valid for a spheroid if a/b < \(\sqrt 2 \). The internal hypothesis, i.e., the convergence of an internal field, is always valid because the field is uniform inside the spheroid. For a pseudospheroid, the hypotheses are valid for a/b < \(\sqrt 2 \) and a/b < \(\sqrt 2 \), respectively. The relation and similarity between the results obtained for the wave and electrostatic problems are discussed.

Scattering of on-axis Gaussian beam by a uniaxial anisotropic object.

Based on the extended boundary condition method, a semi-analytical solution to the scattering of an on-axis Gaussian beam by an arbitrarily shaped uniaxial anisotropic object is constructed. For the on-axis incident Gaussian beam, scattered fields as well as internal fields are expanded in terms of appropriate spherical vector wave functions, and the unknown expansion coefficients of the scattered fields are determined by using Schelkunoff's equivalence theorem and continuous boundary conditions. Numerical results of the normalized differential scattering cross section are presented, and the scattering characteristics are discussed concisely.

Transmission of Gaussian beam through a coated chiral cylinder

Based on the generalized Lorenz–Mie theory (GLMT) framework, an exact analytical solution to the transmission of a Gaussian beam through a chiral cylinder with dielectric coating is formulated, by expanding the incident Gaussian beam, internal beam as well as scattered beam in terms of cylindrical vector wave functions (CVWFs). The unknown expansion coefficients are determined by a system of equations derived from the boundary conditions. For a localized beam model, numerical results of the normalized field intensity distributions are presented, and the propagation characteristics are discussed concisely.