Vijaya Shankar

A Time-Domain, Finite-Volume Treatment for the Maxwell Equations

ABSTRACT For computation of electromagnetic scattering from layered objects, the differential form of the time-domain Maxwells equations are first cast in a conservation form and then solved using a finite-volume discretization procedure derived from proven Computational Fluid Dynamics (CFD) methods 1 . The formulation accounts for any variations in the material properties (time, space, and frequency dependent), and can handle thin resistive sheets and lossy coatings by positioning them at finite-volume cell boundaries. The time-domain approach handles both continuous wave (single frequency) and pulse (broadband frequency) incident excitation. Arbitrarily shaped objects are modeled by using a body-fitted coordinate transformation. For treatment of complex internal/external structures with many material layers, a multizone framework with ability to handle any type of zonal boundary conditions (perfectly conducting, flux through, zero flux, periodic, nonreflecting outer boundary, resistive card, and lossy ...

Computation of electromagnetic scattering and radiation using a time-domain finite-volume discretization procedure

Abstract Maxwells equations in the time domain are first cast into a conservation form. They are then solved using a finite-volume discretization procedure proven very successful in solving some hyperbolic partial differential equations in computational fluid dynamics such as Euler or Navier-Stokes equations. The Lax-Wendroff explicit scheme is used to solve the discrete Maxwells equations and as a result, second-order accuracy is achieved in both time and space. Multizoning is used to facilitate treatment of objects with internal and/or external sophistication. Body-fitted coordinates are used to map each zone independently from the rest of the object, allowing for nonmatching grid lines at the interface. Body-fitted coordinates facilitate accurate implementation of various boundary conditions on the object. The density of the computational mesh can vary and may be specified according to the local wavelength in the material. Also in the exterior region and away from the body, the mesh density is gradually decreased to reduce the total number of the grid points and the unknown field vectors. The formulation accounts for any variations in the material properties as a function of space, time, or frequency. In addition to perfect conductors, impedence surfaces and resistive sheets are handled. At the outer boundary of the computational domain a first-order, nonreflecting outer boundary condition is used. The Lax-Wendroff scheme, while providing second-order accuracy in the time and space, does not require, for its stability, an interlaced mesh for the electric and magnetic fields or the material properties. The method can be used for both time-harmonic and pulse excitations. At the end of the time-marching, the computed field in the time domain near the object and tangential to a conveniently chosen contour is stored. This information is converted into the frequency domain using fast Fourier transforms. If far field quantities such as radar cross-section or antenna radiation pattern are desired, they may be computed from the frequency-domain information using a standard free-space Greens function.

Aeroelastic computations of flexible configurations

Abstract An aeroelastic package consisting of (1) an aerodynamic solver, (2) a structural response mode, and (3) a grid generation/update program has been developed to study both static and dynamic response of flexible aerospace configurations. The aerodynamics at present is a full potential solver but can be an Euler/Navier-Stokes solver also. The structural response is based on a generalized modal representation. The grid generation package not only sets up the initial grid, but also updates the grid position in time to follow the instantaneous position of a flexible configuration using a ‘grid speed’ procedure. The aeroelastic package can handle (1) static rigid, (2) dynamic rigid, (3) static flexible, and (4) dynamic flexible cases. Also, the aerodynamic solver is based on an unsteady formulation and can handle subsonic, transonic, and supersonic flow conditions. Results are presented for rigid and flexible configurations at different Mach numbers ranging from subsonic to supersonic conditions. The dynamic response of a flexible wing below and above its flutter point is demonstrated.

Diffraction of a Shock Wave by a Compression Corner: Part II - Single Mach Reflection

The two-dimensional time-dependent Euler equations which govern the flow field resulting from the interaction of a planar shock with a compression corner are solved for initial conditions which result in single Mach reflection of the incident planar shock. The Euler equations are first transformed to include the self-similarity of the flow field. A second transformation is employed to normalize the distances between the ramp and the reflected shock and between the wall and the Mach stem. The resulting equations in strong conservation-law form are solved using a second-order discontinuity-fitting finite-difference approach. The results are compared with experimental interferograms and existing first-order shock-capturing numerical solutions.

Unstructured Grid-Based Discontinuous Galerkin Method for Broadband Electromagnetic Simulations

A parallel, unstructured, high-order discontinuous Galerkin method is developed for the time-dependent Maxwells equations, using simple monomial polynomials for spatial discretization and a fourth-order Runge–Kutta scheme for time marching. Scattering results for a number of validation cases are computed employing polynomials of up to third order. Accurate solutions are obtained on coarse meshes and grid convergence is achieved, demonstrating the capabilities of the scheme for time-domain electromagnetic wave scattering simulations.

Treatment of supersonic flows with embedded subsonic regions

A nonlinear method based on the full potential equation in conservation form, cast in an arbitrary coordinate system, has been developed to treat predominantly supersonic flows with embedded subsonic regions. This type of flow field occurs frequently near the fuselage/canopy junction area and wing leading-edge regions for a moderately swept fighter configuration. The method uses the theory of characteristics to accurately monitor the type-dependent flowfield. A conservative switching scheme is developed to handle the transition from the supersonic marching algorithm to a subsonic relaxation procedure, and vice versa. An implicit approximate factorization scheme is employed to solve the finite differenced equation. Results are shown for a few configurations, including a wing/body/wake realistic fighter model having embedded subsonic regions.

Numerical solution to Maxwell's equations in the time domain on nonuniform grids

In most integration schemes for the Maxwells equations, damping and distortion errors are strongly dependent on the size of the time step in relation to the size of the spatial discretization Δx. The disadvantage of strong dependence on this ratio becomes evident when one computes the solution on nonuniform meshes. A systematic way for arriving at a scheme that can operate accurately on nonuniform meshes is presented here. Performance of a higher-order scheme is compared with that of another recently developed scheme on a ramp grid.

Computational Transonic Inverse Procedure for Wing Design

A computational transonic inverse procedure for three-dimensional wings in which shapes are determined supporting prescribed pressures is presented. The method is based on modified small disturbance (MSD) theory and can handle wing design in the presence of a fuselage. A consistent analysis-inverse differencing is implemented at the wing slit grid points to ensure recovery of specified pressures in the analysis mode. Formation of an open or a fishtail trailing edge is avoided by a systematic alteration of the velocity potential in front of the leading edge of span stations under inverse mode, until closure is achieved. To lend support to the numerical procedure, an analogous incompressible two-dimensional problem is studied analytically. As an illustration of the usefulness and versatility of the method, the development of a laminar flow control (LFC) wing from a given base wing geometry is presented along with the analysis verification.

Algorithmic aspects and supercomputing trends in computational electromagnetics

Accurate and rapid evaluation of radar signature for alternative aircraft/ store configurations would be of substantial benefit in the evolution of integrated designs that meet RCS requirements across the threat spectrum. Finite-volume time domain methods offer the possibility of modeling the whole aircraft, including penetrable regions and stores, at longer wavelengths on today’s supercomputers and at typical airborne radar wavelengths on the teraflop computers of tomorrow. To realize this potential, practical means must be developed for the rapid generation of grids on and around the aircraft, and numerical algorithms that maintain high order accuracy on such grids must be constructed.

CFD spinoff - Computational electromagnetics for radar cross section (RCS) studies

A finite-volume discretization procedure derived from proven CFD methods is used to solve the conservation form of the time-domain Maxwells equations, in order to compute EM scattering from layered objects. This time-domain approach handles both single-frequency/continuous wave and broadband-frequency/pulse incident excitation. Arbitrarily shaped objects are modeled by means of a body-fitted coordinate transformation; complex internal/external structures with many material layers are treated through the implementation of a multizone framework capable of handling any type of zonal boundary condition. Results are presented for various two- and three-dimensional problems.