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The Solution of Waveguides Eigenvalues with Arbitrary Shapes by FD-FD Algorithm in Curvilinear Co-Ordinates
Abstract
A finite-difference frequency-domain algorithm has been developed in non-orthogonal curvilinear co-ordinates. The full vector Maxwell's equations are discretized on boundary-fitted meshes. Because the algorithm is formulated in a non-orthogonal co-ordinate system, it is not restricted to any a special orthogonal co-ordinate system, and can over the conventional FD-FD algorithm in the orthogonal co-ordinate system. To demonstrate the method, several of the lowest eigenvalues of waveguides with arbitrary cross-section shapes have been computed. The subspace iterative algorithm is used to solve the unsymmetric eigen-equation. Boundary-orthogonal meshes are introduced in order to eliminate the error generated by non-orthogonal meshes on which the boundary conditions for the electromagnetic fields are not satisfied.
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A discretization ansatz for Maxwell's equations is described that enables treatment of all possible homogeneous and inhomogeneous problems from nonlinear magnetostatics and electrostatics to time-dependent field problems that include charges moving freely at any speed. The same method can be used for magnet design, for cavity-mode investigations and wake-force computations in time and frequency domains, antenna problems, and waveguide structures. The method makes direct use of the electric and magnetic field as unknowns, thus yielding uniquely defined vectors in combination with a suitable grid definition. This 'natural' ansatz avoids problems arising from the use of artificial functions such as vector potentials or Hertz potentials. Using this ansatz, many computer codes have been developed and applied to various problems. Applications in the field of accelerator physics include magnet and solenoid design; design of accelerating rf-cavities and beam-tracking calculations for instability studies.
The characteristics of the waves guided along a plane [I] P. S. Epstein, “On the possibility of electromagnetic surface waves, ” Proc. Nat’l dcad. Sciences, vol. 40, pp. 1158-1165, Deinterface which separates a semi-infinite region of free cember 1954. space from that of a magnetoionic medium are investi- [2] T. Tamir and A. A. Oliner, “The spectrum of electromagnetic waves guided by a plasma layer, ” Proc. IEEE, vol. 51, pp. 317gated for the case in which the static magnetic field is 332, February 1963. oriented perpendicular to the plane interface. It is [3] &I. A. Gintsburg, “Surface waves on the boundary of a plasma in a magnetic field, ” Rasprost. Radwvoln i Ionosf., Trudy found that surface waves exist only when w,<wp and NIZMIRAN L’SSR, no. 17(27), pp. 208-215, 1960. that also only for angular frequencies which lie bet\\-een [4] S. R. Seshadri and A. Hessel, “Radiation from a source near a plane interface between an isotropic and a gyrotropic dielectric,” we and 1/42 times the upper hybrid resonant frequency. Canad. J. Phys., vol. 42, pp. 2153-2172, November 1964. The surface waves propagate with a phase velocity [5] G. H. Owpang and S. R. Seshadri, “Guided waves propagating along the magnetostatic field at a plane boundary of a semiwhich is always less than the velocity of electromagnetic infinite magnetoionic medium, ” IEEE Trans. on Miomave waves in free space. The attenuation rates normal to the Tbory and Techniques, vol. MTT-14, pp. 136144, March 1966. [6] S. R. Seshadri and T. T. \Vu, “Radiation condition for a maginterface of the surface wave fields in both the media are netoionic medium. ” to be Dublished. examined. Kumerical results of the surface wave characteristics are given for one typical case.
This article describes a time-domain finite-difference algorithm for solving Maxwell's equations in generalized nonorthogonal coordinates. We believe this approach would be most useful for applications where a uniform, uncurved, but oblique, meshing scheme could be applied in lieu of staircasing. This algorithm is similar to conventional leapfrog-differencing schemes, but now an additional leapfrogging between covariant and contravariant field representations becomes necessary.
The nonorthogonal finite-difference-time-domain (FDTD) technique
is used to compute the resonant frequencies of dielectric-filled
cylindrical cavities. Because the method is based on the nonorthogonal
coordinate system, it is not restricted to specific geometries, e.g.
rectangular or axially symmetric geometries, and is suitable for
analyzing cavities of arbitrary shape. The advantages of this technique
over the conventional FDTD algorithm with a staircase grid are shown in
a convergence study, where the two methods are used to compute the
dominant resonant frequency of a cylindrical cavity. The accuracy of the
technique for calculating the resonant frequencies of the first few
modes is demonstrated by comparing the results obtained with this
technique with those derived by using two versions of the finite element
method in the frequency domain
A generalization of the finite-difference time-domain (FDTD)
algorithm adapted to nonorthogonal computational grids is presented and
applied to the investigation of three-dimensional discontinuity
problems. The nonorthogonal FDTD uses a body-fitted grid for meshing up
the computation domain and, consequently, is able to model the problem
geometry with better accuracy than is possible with the staircasing
approach conventionally employed in the FDTD algorithm. The stability
conditions for the nonorthogonal FDTD algorithm are derived in two an
three dimensions. Numerical results, including an H-plane waveguide
junction, a circular waveguide with a circular iris, a circular
waveguide with a rectangular iris, and a microstrip bend discontinuity,
are presented to validate the approach
An application of a numerical method of finite differences in the
time domain (FDTD) coupled with the discrete Fourier transform is
presented to determine the resonant frequencies of the TE0
and TM0 modes of axially symmetric dielectric resonators
closed in a cavity. The technique is conceptually and computationally
simple, and it allows access at once to information on the entire modal
spectrum by means of the fast Fourier transform (FFT) applied to the
time series. The cylindrical cavity dielectrically loaded at the base
and the resonant frequency of the TE01δ mode are
analyzed in two systems: a cylindrical cavity with a cylindrical
dielectric resonator of variable radius, and the shielded dielectric
resonator on a microstrip substrate. The results obtained are compared
with the rigorous (exact) theoretical solutions and with experimental
results
The frequency-dependent characteristics of the microstrip
discontinuities have previously been analyzed using full-wave
approaches. The time-domain finite-difference (TD-FD) method presented
here is an independent approach and is relatively new in its application
for obtaining the frequency-domain results for microwave components. The
validity of the TD-FD method in modeling circuit components for MMIC CAD
applications is established
The finite-difference time-domain (FDTD) algorithm for the
solution of electromagnetic scattering problems is formulated in
numerically defined generalized coordinates in three dimensions and
implemented in a code with the lowest order Bayliss-Turkel radiation
boundary condition expressed in spherical coordinates. It is shown that
the algorithm is capable of accurately tracking the progress of a pulse
of electromagnetic radiation through the curvilinear mesh generated by a
body of revolution, the only problems occurring in the vicinity of the
rotation axis, which represents a coordinate singularity. A simple
method to deal with this singular line is presented and discussed, and
its is shown that, at least for the test problem, this approximation is
sufficient. The algorithm discussed is useful for the solution of the
exterior problem in the presence of conductors and dielectrics with
complicated shapes and electrical compositions, and for near-field
problems such as cavity penetration problems. The far fields are
obtained by replacing the scatterer with a virtual surface enclosing all
sources
The finite-difference-time-domain (FDTD) algorithm for the
solution of electromagnetic scattering problems is formulated in
generalized coordinates in two dimensions and implemented in a code with
the lowest-order Bayliss-Turkel radiation boundary condition expressed
in cylindrical coordinates. It is shown that, for a perfect conductor,
such a formulation leads to a stable, well-posed algorithm and that, in
regions where the curvature of the coordinate lines is not great, the
dispersion and anisotropy effects are negligible. Such effects become
more pronounced in regions of high curvature, leading to unphysical
phase shifts. The magnitude of such shifts and the amount of wavefront
distortion is studied via numerical experiments using a cylindrical
mesh. Near-field results are given for two canonical shapes for each
polarization: the circular cylinder and cylinders of square and
rectangular cross sections. These results are compared with those
obtained by exact eigenfunction expansion techniques, with
method-of-moments (MM) solutions, and with solutions obtained from an
alternate FDTD approach. In each case, agreement is excellent. The
propagation of a plane wave through a polar space in the absence of a
scatterer is also examined, and it is shown that the FDTD algorithm is
capable of tracking the incident wave closely
Applications of the finite-difference time-domain (FD-TD) method
for numerical modeling of electromagnetic wave interactions with
structures are reviewed, concentrating on scattering and radar cross
section (RCS). A number of two- and three-dimensional examples of FD-TD
modeling of scattering and penetration are provided. The objects modeled
range in nature from simple geometric shapes to extremely complex
aerospace and biological systems. Rigorous analytical or experimental
validations are provided for the canonical shapes, and it is shown that
FD-TD predictive data for near fields and RCS are in excellent agreement
with the benchmark data. It is concluded that, with continuing advances
in FD-TD modeling theory for target features relevant to the RCS
problem, and with continuing advances in vector and concurrent
supercomputer technology, it is likely that FD-TD numerical modeling
will occupy an important place in RCS technology in the 1990s and beyond
Three-dimensional simulations of frequency-phase measurement of arbitrary coupled-cavity rf circuits
- F Kanttowitz
- I Tammara
Microwave engineer's handbook
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