Stephen M. Wandzura

Electric Current Basis Functions for Curved Surfaces

ABSTRACT Differential geometry is used to construct basis functions for representing currents on curved surfaces. These basis functions are appropriate for method of moments solution of boundary integral equations derived from Maxwells equations. They maintain the essential properties of the basis functions of Rao, Wilton, and Glisson, while allowing higher order basis functions (more variables per patch). The use of these functions is expected to result in a large reduction in the computational resources required to solve a given problem to a fixed level of accuracy.

Spatial correlation of phase-expansion coefficients for propagation through atmospheric turbulence

Spatial correlation functions of phase-expansion coefficients are derived for phase fluctuations of a wave that has propagated through a layer of atmospheric turbulence. First, a general expression is derived giving the correlation of the coefficients of phase-expansion functions orthogonal over an arbitrary circularly symmetric weighting function for an isotropic turbulence spectrum. Second, the correlations are evaluated analytically for a Gaussian weight and numerically for a uniform circular weight. Finally, some of the correlations for a layer are integrated over the Hufnagel 1974 structure constant model to obtain results applicable to ground-to-space propagation.

Numerical solution of 2-D scattering problems using high-order methods

We demonstrate that a method of moments scattering code employing high-order methods can compute accurate values for the scattering cross section of a smooth body more efficiently than a scattering code employing standard low-order methods. Use of a high-order code also makes it practical to provide meaningful accuracy estimates for computed solutions.

Efficient high-order discretization schemes for integral equation methods

A high-order method is a method that provides extra digits of accuracy with only a modest increase in computational cost. A number of method of moment (MoM) techniques based on high-order basis and testing functions have been presented in the literature. Characteristically, these methods result in a substantial increase in precomputational cost principally due to the expensive numerical integration required for near interactions. This can be accelerated through the use of specialized quadrature schemes when available. Unfortunately, performing the double integration numerically over high-order functions can still be quite computationally intensive. A novel high-order technique based on a locally-corrected Nystrom scheme combined with advanced quadrature schemes is presented. It is shown that this method truly demonstrates high-order convergence for the solution of electromagnetic scattering problems with comparable computational cost to low-order schemes. The elegance of this technique is in its simplicity and ease of implementation. However, the power of the method is its ability to inexpensively provide true high-order convergence.

Scattering computation using the fast multipole method

The fast multipole method (FMM) computes scattering cross sections from large targets with several orders of magnitude reduction in the CPU time and memory storage compared to traditional method of moments (MoM) techniques. The authors compare the memory usage and CPU times for FMM with traditional MoM results using both direct and iterative solvers. It is shown that dramatic reductions in CPU and memory requirements can be realized by using the FMM to compute scattering and radiation from large objects. In the test cases considered, 2D objects with perimeters longer than about 12 wavelengths required less memory than using traditional MoM techniques. Using the FMM to compute scattering from objects with perimeters larger than about 30 wavelengths required less CPU time than using iterative solvers with a dense Z matrix.<<ETX>>

3D method of moments scattering computations using the fast multipole method

The fast multipole method (FMM) dramatically reduces the time and memory required to compute radar cross sections and antenna radiation patterns compared to dense matrix techniques. We have implemented the FMM in a method of moments (MoM) program to compute electromagnetic scattering from large bodies of arbitrary shape. We compare the memory and time required using the FMM to that for direct and iterative solutions using a dense impedance matrix.<<ETX>>

Electromagnetic scattering computations using high-order basis functions in the method of moments

Using high order basis functions in Galerkin method of moments calculations permits a significant reduction in the number of unknowns needed to achieve a given accuracy in modeling a scatterer or antenna [Hamilton et al., 1993]. The present authors have developed a set of vector valued basis functions of arbitrary order with a sparse overlap matrix. They demonstrate the reduction of computational effort for a given accuracy by computing the scattering from a sphere.

The importance of accurate surface models in RCS computations

The authors investigate the magnitude of the errors in the scattering cross section made by approximating 2-D targets by flat segments and by quadratic splines. These results were compared to computations using exact representations of the surface. A research program for scattering computation called Fast Scat was instrumental in computing the results. The importance of precise surface models for computing accurate scattering amplitudes and radiation patterns is shown. For the particular example of TM (transverse magnetic) scattering from an infinite PEC (perfectly electrically conducting) circular cylinder, the error made in approximating the surface by linear segments or quadratic splines was computed.<<ETX>>

Integral equations and discretizations for waveguide apertures

We present integral equations and their discretizations for calculating the fields radiated from arbitrarily shaped antennas fed by cylindrical waveguides of arbitrary cross sections. We give results for scalar fields in two dimensions with Dirichlet and Neumann boundary conditions and for (vector) electric and magnetic fields in three dimensions. The discretized forms of the equations are cast in identical format for all four cases. Feed modes can be TM, TE, or transverse electromagnetic (TEM). A method for numerically computing the modes of an arbitrarily shaped, cylindrical waveguide aperture is also given.

Variational improvements to scattering amplitude calculations

Abstract By applying a variational principle to radar cross section calculations, one can improve the convergence of iterative solutions and increase the accuracy of the results. We exhibit a variational formula for calculating scattering amplitudes that allows a decrease in the number of iterations required to obtain a given accuracy of the scattering amplitude by a factor of three or more. We also interpret the biconjugate gradient method as an iterative solution technique that makes this formula stationary within a Krylov subspace. We have demonstrated the improvement obtained using this variational formula by calculating the radar cross section of a 3λ cone-sphere and 2.5λ plate.