Standard Chebyshev Research Articles

Single-layer fully-planar extended-composite right-/left-handed transmission lines based on substrate integrated waveguides for dual-band and quad-band applications

The implementation and application of single-layer fully-planar extended-composite right-/left-handed transmission lines (E-CRLH TLs) in substrate-integrated waveguide (SIW) technology are presented. The multiband CRLH behavior of these artificial lines is explained by considering the lumped element equivalent circuit model. The potential of these lines for dual-band and quad-band applications is demonstrated by designing and fabricating a quad-band Y-junction power divider and two dual-band band-pass filters. The main relevant advantage of SIW-based E-CRLH TLs over other E-CRLH lines is fabrication simplicity, since only a single metal layer must be etched and lumped elements are avoided. The fabricated prototypes exhibit very reasonable performance. It is remarkable that for dual-band band-pass filters, standard Chebyshev responses can be obtained to a very good approximation.

Direct trajectory optimization based on a mapped Chebyshev pseudospectral method

In view of generating optimal trajectories of Bolza problems, standard Chebyshev pseudospectral (PS) method makes the points’ accumulation near the extremities and rarefaction of nodes close to the center of interval, which causes an ill-condition of differentiation matrix and an oscillation of the optimal solution. For improvement upon the difficulties, a mapped Chebyshev pseudospectral method is proposed. A conformal map is applied to Chebyshev points to move the points closer to equidistant nodes. Condition number and spectral radius of differentiation matrices from both methods are presented to show the improvement. Furthermore, the modification keeps the Chebyshev pseudospectral method’s advantage, the spectral convergence rate. Based on three numerical examples, a comparison of the execution time, convergence and accuracy is presented among the standard Chebyshev pseudospectral method, other collocation methods and the proposed one. In one example, the error of results from mapped Chebyshev pseudospectral method is reduced to 5% of that from standard Chebyshev pseudospectral method.

A modified Chebyshev pseudospectral DD algorithm for the GBH equation

In this paper, a Chebyshev spectral collocation domain decomposition (DD) semi-discretization by using a grid mapping, derived by Kosloff and Tal-Ezer in space is applied to the numerical solution of the generalized Burger’s–Huxley (GBH) equation. To reduce roundoff error in computing derivatives we use the above mentioned grid mapping. In this work, we compose the Chebyshev spectral collocation domain decomposition and Kosloff and Tal-Ezer grid mapping, elaborately. Firstly, the theory of application of the Chebyshev spectral collocation method with grid mapping and DD on the GBH equation is presented. This method yields a system of ordinary differential algebraic equations (DAEs). Secondly, we use a fourth order Runge–Kutta formula for the numerical integration of the system of DAEs. Application of this modified method to the GBH equation show that this method (M-DD) is faster and more accurate than the standard Chebyshev spectral collocation DD (S-DD) method.

Direct Synthesis of Parallel-Connected Symmetrical Two-Port Filters

This brief presents a general technique for the synthesis of parallel-connected two-port networks. This is based on the even- and odd-mode admittances of a standard Chebyshev network. The expressions for the even/odd-mode admittance are used to synthesize the low-pass network into branches of subnetworks connected in parallel between the source and the load. These together will have the same characteristics as a ladder network. It is also shown how a filter of a particular order will have the same parallel branch topology regardless of how many transmission zeros are used in the filter's transfer function. This design technique can be beneficial for various implementation techniques such as multilayer substrate and dual-mode resonators. Examples of fourth- and sixth-order degree networks with various numbers of transmission zeros are presented.

Multivariate modified Fourier series and application to boundary value problems

In this paper we analyse the approximation-theoretic properties of modified Fourier series in Cartesian product domains with coefficients from both full and hyperbolic cross index sets. We show that the number of expansion coefficients may be reduced significantly whilst retaining comparable error estimates. In doing so we extend the univariate results of Iserles, Norsett and S. Olver. We then demonstrate that these series can be used in the spectral-Galerkin approximation of second order Neumann boundary value problems, which offers some advantages over standard Chebyshev or Legendre polynomial discretizations.

Reprint of: Physics-Based Computational Methods for Aero-Acoustics

This paper provides a review of several physics-based methods developed for predicting sound propagation in flows. The basic principle of physics-based methods is to incorporate some known properties of the underlying physics into the numerical model. For instance, instead of using standard polynomials or Chebyshev polynomials to interpolate the solution, physics-based methods generally use canonical solutions such as Green's functions or plane waves to construct a local description of the solution. The methods described in this paper include the Green's function discretisation, the partition of unity finite element method and the wave-based discontinuous Galerkin method. The principles of these methods are described and their performance and shortcomings are discussed. A key issue that emerges in several of these methods is that the canonical solutions are only valid for uniform coefficients, while the methods are intended to be used with strongly inhomogeneous propagation media.

Physics-based computational methods for aero-acoustics

This paper provides a review of several physics-based methods developed for predicting sound propagation in flows. The basic principle of physics-based methods is to incorporate some known properties of the underlying physics into the numerical model. For instance, instead of using standard polynomials or Chebyshev polynomials to interpolate the solution, physics-based methods generally use canonical solutions such as Green’s functions or plane waves to construct a local description of the solution. The methods described in this paper include the Green’s function discretisation, the partition of unity finite element method and the wave-based discontinuous Galerkin method. The principles of these methods are described and their performance and shortcomings are discussed. A key issue that emerges in several of these methods is that the canonical solutions are only valid for uniform coefficients, while the methods are intended to be used with strongly inhomogeneous propagation media.

Adaptive rational spectral methods for the linear stability analysis of nonlinear fourth-order problems

This paper presents the application of adaptive rational spectral methods to the linear stability analysis of nonlinear fourth-order problems. Our model equation is a phase-field model of infiltration, but the proposed discretization can be directly extended to similar equations arising in thin film flows. The sharpness and structure of the wetting front preclude the use of the standard Chebyshev pseudo-spectral method, due to its slow convergence in problems where the solution has steep internal layers. We discuss the effectiveness and conditioning of the proposed discretization, and show that it allows the computation of accurate traveling waves and eigenvalues for small values of the initial water saturation/film precursor, several orders of magnitude smaller than the values considered previously in analogous stability analyses of thin film flows, using just a few hundred grid points.

Spectral radial basis functions for full sphere computations

The singularity of cylindrical or spherical coordinate systems at the origin imposes certain regularity conditions on the spectral expansion of any infinitely differentiable function. There are two efficient choices of a set of radial basis functions suitable for discretising the solution of a partial differential equation posed in either such geometry. One choice is methods based on standard Chebyshev polynomials; although these may be efficiently computed using fast transforms, differentiability to all orders of the obtained solution at the origin is not guaranteed. The second is the so-called one-sided Jacobi polynomials that explicitly satisfy the required behavioural conditions. In this paper, we compare these two approaches in their accuracy, differentiability and computational speed. We find that the most accurate and concise representation is in terms of one-sided Jacobi polynomials. However, due to the lack of a competitive fast transform, Chebyshev methods may be a better choice for some computationally intensive timestepping problems and indeed will yield sufficiently (although not infinitely) differentiable solutions provided they are adequately converged.

Fourier spectral and wavelet solvers for the incompressible Navier–Stokes equations with volume-penalization: Convergence of a dipole–wall collision

In this study, we use volume-penalization to mimic the presence of obstacles in a flow or a domain with no-slip boundaries. This allows in principle the use of fast Fourier spectral methods and coherent vortex simulation techniques (based on wavelet decomposition of the flow variables) to compute turbulent wall-bounded flow or flows around solid obstacles by simply adding one term in the equation. Convergence checks are reported using a recently revived, and unexpectedly difficult dipole–wall collision as a benchmark computation. Several quantities, like the vorticity isolines, truncation error, kinetic energy and enstrophy are inspected for a collision of a dipole with a no-slip wall and compared with available benchmark data obtained with a standard Chebyshev pseudospectral method. We quantify the possible deteriorating effects of the Gibbs phenomenon present in the Fourier based schemes due to continuity restrictions of the penalized Navier–Stokes equations on the wall. It is found that Gibbs oscillations have a negligible effect on the flow evolution allowing higher-order recovery of the accuracy on a Fourier basis by means of postprocessing. An advantage of coherent vortex simulations, on the other hand, is that the degrees of freedom of the flow computation can strongly be reduced. In this study, we quantify the possible reduction of degrees of freedom while keeping the accuracy. For an optimal convergence scenario the penalization parameter has to scale with the number of Fourier and wavelet modes. In addition, an implicit treatment of the Darcy drag term in the penalized Navier–Stokes equations is beneficial since this allows one to set the time step independent from the penalization parameter without additional computational or memory requirements.

Spectral method for matching exterior and interior elliptic problems

A spectral method is described for solving coupled elliptic problems on an interior and an exterior domain. The method is formulated and tested on the two-dimensional interior Poisson and exterior Laplace problems, whose solutions and their normal derivatives are required to be continuous across the interface. A complete basis of homogeneous solutions for the interior and exterior regions, corresponding to all possible Dirichlet boundary values at the interface, are calculated in a preprocessing step. This basis is used to construct the influence matrix which serves to transform the coupled boundary conditions into conditions on the interior problem. Chebyshev approximations are used to represent both the interior solutions and the boundary values. A standard Chebyshev spectral method is used to calculate the interior solutions. The exterior harmonic solutions are calculated as the convolution of the free-space Green’s function with a surface density; this surface density is itself the solution to an integral equation which has an analytic solution when the boundary values are given as a Chebyshev expansion. Properties of Chebyshev approximations insure that the basis of exterior harmonic functions represents the external near-boundary solutions uniformly. The method is tested by calculating the electrostatic potential resulting from charge distributions in a rectangle. The resulting influence matrix is well-conditioned and solutions converge exponentially as the resolution is increased. The generalization of this approach to three-dimensional problems is discussed, in particular the magnetohydrodynamic equations in a finite cylindrical domain surrounded by a vacuum.

Lattice BGK direct numerical simulation of fully developed turbulence in incompressible plane channel flow

Over the last decade, lattice Boltzmann methods have proven to be reliable and efficient tools for the numerical simulation of complex flows. The specifics of such methods as turbulence solvers, however, are not yet completely documented. This paper provides results of direct numerical simulations (DNS), by a lattice Boltzmann scheme, of fully developed, incompressible, pressure-driven turbulence between two parallel plates. These are validated against results from simulations using a standard Chebyshev pseudo-spectral method. Detailed comparisons, in terms of classical one-point turbulence statistics at moderate Reynolds number, with both numerical and experimental data show remarkable agreement. Consequently, the choice of numerical method has, in sufficiently resolved DNS computations, no dominant effect at least on simple statistical quantities such as mean flow and Reynolds stresses. Since only the method-independent statistics can be credible, the choice of numerical method for DNS should be determined mainly through considerations of computational efficiency. The expected practical advantages of the lattice Boltzmann method, for instance against pseudo-spectral methods, are found to be significant even for the simple geometry and the moderate Reynolds number considered here. This permits the conclusion that the lattice Boltzmann approach is a promising DNS tool for incompressible turbulence.

Novel microstrip bandpass filters based on complementary split-ring resonators

In this paper, a new methodology for the design of compact planar filters in microstrip technology is proposed. This is based on cascading filter stages consisting of the combination of complementary split-ring resonators (CSRRs), recently proposed by the authors, series capacitive gaps, and grounded stubs. By this means, we achieve the necessary flexibility to simultaneously obtain quite symmetric frequency responses, controllable bandwidths, and compact dimensions. Two prototype device bandpass filters are provided to illustrate the potentiality of the proposed approach. In the first prototype, the structure is periodic (i.e., composed of identical cells) and behaves as a left-handed transmission line with controllable bandwidth. In the second prototype device, periodicity is sacrificed with an eye toward the synthesis of a standard (Chebyshev) approximation. The measured frequency responses point out low insertion losses in the passband, as well as high-frequency selectivity with small dimensions. As compared to conventional parallel coupled line filters, reduction of device length by a factor of 2.4 is demonstrated. This is the first time that planar filters with controllable bandwidth based on CSRRs are achieved. These structures can be of interest in those applications where miniaturization and compatibility with planar circuit technology are key issues

A Rational Spectral Collocation Method with Adaptively Transformed Chebyshev Grid Points

A spectral collocation method based on rational interpolants and adaptive grid points is presented. The rational interpolants approximate analytic functions with exponential accuracy by using prescribed barycentric weights and transformed Chebyshev points. The locations of the grid points are adapted to singularities of the underlying solution, and the locations of these singularities are approximated by the locations of poles of Chebyshev-Pade´ approximants. Numerical experiments on two time-dependent problems, one with finite time blow-up and one with a moving front, indicate that the method far outperforms the standard Chebyshev spectral collocation method for problems whose solutions have singularities in the complex plane close to $[-1,1]$.

Superconducting spiral filters with quasi-elliptic characteristic for radio astronomy

To produce a filter small enough to fit a 2-in wafer at 408 MHz while maintaining high-quality performance, half-wavelength single spiral microstrip resonators are introduced. New coupling structures make both positive and negative coupling available by changing the directions of spiral winding. An eight-pole high-temperature superconducting bandpass spiral filter with 3.7% bandwidth at 408-MHz band is presented for radio astronomy applications at the Jodrell Bank Radio Observatory, Macclesfield, Cheshire, U.K. A quasi-elliptic characteristic with four transmission zeros is realized by adding three cross-couplings to the standard Chebyshev filter. The filter shown is designed and fabricated on a 32 mm /spl times/ 18 mm /spl times/ 0.508 mm MgO substrate. The untuned measured results of the filter at 30 K show a maximum passband insertion loss 0.35 dB (ripple 0.27 dB), a minimum return loss 13.2 dB, and minimum out-of-band rejection of 65 dB, which have good agreement with its electromagnetic full-wave simulation results.

The Legendre Galerkin-Chebyshev collocation method for Burgers-like equations

A Legendre Galerkin–Chebyshev collocation method for Burgers‐like equations is developed. This method is based on the Legendre–Galerkin variational form, but the nonlinear term and the right‐hand term are treated by Chebyshev–Gauss interpolation. Error estimates of the semi‐discrete scheme and the fully discrete scheme are given in the L2‐norm. Numerical results indicate that our method is as stable and accurate as the standard Legendre collocation method, and as efficient and easy to implement as the standard Chebyshev collocation method.

Spectral Schemes on Triangular Elements

The Poisson problem with homogeneous Dirichlet boundary conditions is considered on a triangle. The transformation from square to triangle is realized by mapping an edge of the square onto a corner of the triangle. Then standard Chebyshev collocation techniques can be implemented. Numerical experiments demonstrate the expected high spectral accuracy for smooth solutions. Furthermore, it is shown that finite difference preconditioning can be successfully employed to construct an efficient iterative solver. Then the convection–diffusion equation is considered. Here finite difference preconditioning with central differences leads to instability. However, using the first-order upstream scheme, we obtain a stable method. Finally, a domain decomposition technique is applied to the patching of rectangular and triangular elements.

Spectral Schemes on Triangular Elements

AbstractFirst we consider the Poisson Problem with homogeneous Dirichlet boundary conditions on a triangle. The mapping between square and triangle is realized by mapping an edge of the square onto a corner of the triangle. Then standard Chebyshev collocation techniques can be applied. Numerical experiments demonstrate the expected high spectral accuracy. Further, it is shown that finite difference preconditioning can be successfully applied in order to construct an efficient iterative solver. Finally, a domain decomposition technique is applied to the patching of rectangular and triangular elements.

Approximation properties of a mapped Chebyshev method

We analyze a mapped Chebyshev technique to approximate derivatives recently developed by Kosloff and Tal-Ezer. The technique is based on a one-parameter family of mappings. Earlier numerical experimentation suggests that, for suitable choices of the parameter, this mapped technique presents several advantages over the standard Chebyshev technique. Among them, better accuracy than the classical Chebyshev method. We obtain error bounds for the new technique that, when particularized to the choices of the parameter suggested in the literature, show a low order of approximation, so that, asymptotically, the new method is much less accurate than the standard Chebyshev method. These error bounds are corroborated by numerical experimentation. However, the low order behaviour of the mapped technique only appears for levels of accuracy much more stringent than those in usual computations.

An Efficient Method for the Solution of the Incompressible Navier–Stokes Equations in Cylindrical Geometries

The article presents a fast pseudo-spectral Navier–Stokes solver for cylindrical geometries, which is shown to possess exponential rate of decay of the error. The formulation overcomes the issues related to the axis singularity, by employing in the radial direction a special set of collocation points together with standard Chebyshev polynomials. A multi-domain technique with patching interfaces yields significant improvements in the conditioning of the algebraic problems arising from the discretization procedure and allows for an enhanced near wall resolution of wall bounded shear flows. The elliptic kernel enjoys the efficiency of an analytic expansion of the harmonic extension. The method is tested by computing the formation of Taylor vortices in a rotating Couette flow for both axisymmetric and non-axisymmetric configurations. A direct numerical simulation of a turbulent pipe flow at moderate Reynolds number demonstrates the effectiveness of the method in as much as the axis singularity is concerned. Results compare well with reference experimental and numerical data.