Spectral Iterative Method Research Articles

An analytical approach to rotor blade modulation

Scattering of electromagnetic plane waves normally incident on a flat perfect electric conductor strip in arbitrary rotational motion in its own plane is investigated in the context of Hertzian Electrodynamics. This canonical problem is an analytical-asymptotic approach that reveals the parametric dependence of interference of electromagnetic waves with rotating blades, a subject which is known as rotor blade modulation. Scattering by a stationary perfect electric conductor strip under TE and TM incidences stand as two sub-problems that are incorporated in the main formulation. These sub-problems are formulated by Wiener–Hopf Technique and solved using a spectral iteration method that converges rapidly in the solution of corresponding coupled dual Fredholm type integral equations in case of electrically wide strips. Numerical implementations that simulate electromagnetic scattering by helicopters blades and wind turbines are provided and discussed.

Error estimates of generalized spectral iterative methods with accurate convergence rates for solving systems of fractional two - point boundary value problems

The main purpose of this study is to provide an efficient spectral iterative method based on fractional interpolants for solving a class of linear and nonlinear fractional two - point boundary value problems involving left and right fractional derivatives. In order to achieve this goal, by composition of iterative method and spectral method, in each step, the unknown solution of system is expanded by using two classes of generalized Jacobi functions to obtain numerically coefficients. For this system of fractional differential equations on the interval [−1,1], the singularities of the solution are considered of types (1+t)β and (1−x)α, where 0 < β, α < 1 are left and right singularities indexes. Then, these types of singularities can be well resolved with the mentioned functions. A rigours convergence analysis of the proposed method with accurate spectral rate of convergence is extensively discussed in L2−norm. Finally, some numerical results are given to demonstrate the effectiveness and applicability of the proposed method and accuracy of the presented convergence rates.

Spectral iterative method and convergence analysis for solving nonlinear fractional differential equation

In this study a new spectral iterative method (SIM) based on fractional interpolation is presented for solving nonlinear fractional differential equations (FDEs) involving Caputo derivative. This method is equipped with a pre-algorithm to find the singularity index of solution of the problem. This pre-algorithm gives us a real parameter as the index of the fractional interpolation basis, for which the SIM achieves the highest order of convergence. In comparison with some recent results about the error estimates for fractional approximations, a more accurate convergence rate has been attained. We have also proposed the order of convergence for fractional interpolation error under the L2-norm. Finally, general error analysis of SIM has been considered. The numerical results clearly demonstrate the capability of the proposed method.

A spectral iterative method for solving nonlinear singular Volterra integral equations of Abel type

In this paper, a spectral iterative method is employed to obtain approximate solutions of singular nonlinear Volterra integral equations, called Abel type of Volterra integral equations. The Abel's type nonlinear Volterra integral equations are reduced to nonlinear fractional differential equations. This approach is based on a combination of two different methods, i.e. the iterative method proposed in [7] and the spectral method. The method reduces the fractional differential equations to systems of linear algebraic equations and then the resulting systems are solved by a numerical method. Finally, we prove that the spectral iterative method (SIM) is convergent. Numerical results comparing this iterative approach with alternative approaches offered in [4,8,24] are presented. Error estimation also corroborate numerically.

A Spectral Iterative Method for the Computation of Effective Properties Of Elastically Inhomogeneous Polycrystals

AbstractWe report an efficient phase field formalism to compute the stress distribution in polycrystalline materials with arbitrary elastic inhomogeneity and anisotropy The dependence of elastic stiffness tensor on grain orientation is taken into account, and the elastic equilibrium equation is solved using a spectral iterative perturbation method. We discuss its applications to computing residual stress distribution in systems containing arbitrarily shaped cavities and cracks (with zero elastic modulus) and to determining the effective elastic properties of polycrystals and multilayered composites.

On a spectral method of solving the Stokes equation

A method of solving the Stokes equation for a spherical mantle model by expansion in spherical harmonics was developed by Hager and O’Connell [1979]. However, this method is applicable only if the viscosity depends solely on depth. In this case, the Stokes equation reduces to a system of independent equations for each harmonic. Given lateral variations in viscosity, the Stokes equation contains terms in the form of products of harmonics, which invalidates all advantages of harmonic expansion. Zhang and Christensen [1993] developed a perturbation method for the case when terms containing products of lateral viscosity variations are small. These terms are first calculated from the preceding iteration and are then expanded in a series of harmonic functions. As a result, equations for harmonics remain independent. An evident advantage of the spectral method is the simplicity of the technique of incorporating the self-gravitation and compressibility effects. Moreover, this method partially eliminates difficulties related to the singularities at poles. As yet, it has not been applied in practice, possibly because the equations presented in [Zhang and Christensen, 1993] contain misprints that have not been elucidated in the literature. In the present work, a system of equations is derived for the spectral-iterative method of solving the Stokes equation and the errata present in formulas of Zhang and Christensen [1993] and significantly affecting results of calculations are analyzed.

Dynamic response simulation of planetary gears by the iterative spectral method

A planetary gear is modeled in this work. The equations of motion and the eigenfrequencies are recovered. The computation of the dynamic response is made using the spectral iterative method. The procedure is based on a modal approach with developments in the frequency domain. This technique has been successfully used and its convergence was quickly reached. Good agreement was obtained compared with the standard Newmark method. The dynamic response of the planetary gear is given directly in the frequency domain. The inverse Fourier transform gives the time response of the system.

Microwave Properties of Cascaded Arbitrarily Oriented Metallic Gratings and of Gratings Cascaded with Lossy Dielectric Slabs

The theory of scattering matrices and microwave network analysis are employed to solve the problem of scattering from two cascaded infinite gratings with variable orientation at the second layer. The total reflection and transmission coefficients are calculated as functions of the angle of incidence, orientation of the second layer, distance of layer separation and grating size. Furthermore, the problem of cascading a grating with a lossy dielectric is formulated and solved. The Secant-Corrector Spectral Iteration method is used as a numerical technique for calculating the desired currents and fields. Finally, calculated values of the reflection and transmission coefficients are presented and compared with reported experimental and theoretical data.

Dual series solution to the scattering of plane waves from a binary conducting grating

The problem of scattering of electromagnetic waves from a perfectly conducting grating with a periodic groove structure is considered. A system of dual series equations is derived by enforcing the electromagnetic boundary conditions; this leads to a boundary-value problem that is solved. The mathematics leading to the solution of the dual series system is derived from the equivalent Riemann-Hilbert problem in complex-variable theory and its solution. The solution converges absolutely and makes it possible to obtain analytical results, even where other numerical methods, such as the mode-matching method and spectral iteration methods, are numerically unstable. Consideration is also given to the relative phase values for the diffracted fields. The phase differences between the scattered fields resulting from two orthogonally polarized incident plane waves can be explicitly determined for any incidence angles and for any groove dimensions. Comparisons with results from the mode-matching method and the spectral-iteration method are also presented. >

Application of the Spectral Method to Solve the Meridional Circulation Equation in Spherical-Sigma Coordinates

The equation governing the zonally averaged meridional circulation (W̄, v̄ in spherical-sigma coordinates is formulated including the semigeostrophic terms. This equation and the mass continuity equation are spectrally transformed in terms of Legendre polynomials. Finite differences are used to represent vertical derivatives. An iterative procedure is then used to solve the resulting spectral equations in which the quasi-geostrophic terms serve as the predictors and the semigeostrophic terms as the corrections. It is shown that this spectral iterative method is capable of solving the meridional circulation equation exactly when the forcing functions in this equation are prescribed in such a way that they are spectrally consistent with those in the zonally averaged momentum and thermodynamic energy equations. The computation time for the solution is comparable to that required for the conventional grid relaxation method. The discretization error incurred by the grid method is, however, noticeable when used with spectral model data. Given sufficiently coarse spectral truncation, such as to eliminate convectively unstable scales of atmospheric motion, it is shown that the meridional circulation equation can be solved when the moist static stability is used in place of the dry static stability and therefore the equation is hyperbolic over part of the domain.

On a class of finite step iterative methods (Conjugate directions) for the solution of an operator equation arising in electromagnetics

A class of finite step iterative methods for the solution of linear operator equations is presented. Specifically, the basic principles of the method of conjugate directions are developed. Gaussian elimination and the method of conjugate gradients are then presented as two special cases. With an arbitrary initial guess, the method of conjugate gradient always converges to the solution in at most <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">N</tex> iterations, where <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">N</tex> is the number of independent eigenvalues for the operator in the finite dimensional space in which the problem is being solved. The conjugate gradient method requires much less storage ( <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">\sim 5N</tex> ) than the conventional matrix methods ( <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">\sim N^{2}</tex> ) in the solution of problems of higher complexity. Also, after each iteration the quality of the solution is known in the conjugate gradient method. The conjugate gradient method is also superior to the spectral iterative method as the latter does not always converge and it doubles the complexity of a given problem, unnecessarily. Four versions of the conjugate gradient method are presented in detail, and numerical results for a thin wire scatterer are given to illustrate various properties of each version.

A spectral-iteration technique for analyzing a corrugated-surface twist polarizer for scanning reflector antennas

An analysis of the corrugated-surface twist polarizer which finds application in the design of scanning reflector antennas is presented. The spectraI-interation technique, a novel procedure which combines the use of the Fourier transform method with an iterative procedure is employed. The first step in the spectral-iteration method is the conversion of the original integral equation for the interface field into a form which is suitable for iteration using a method developed previously. An important feature of the technique is that it takes advantage of the discrete Fourier transform (DFT) type of kernel of the integral equation and evaluates the integral operators efficiently using the fast Fourier transform (FFT) algorithm. Thus, in contrast to the conventional techniques, e.g., the moment method, the spectral-iteration approach requires no matrix inversion and is capable of handling a large number of unknowns. Furthermore the method has a built-in check on the satisfaction of the boundary conditions at each iteration.