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Algorithmic Aspects And Supercomputing Trends In ComputationalElectromagnetics

Vijaya Shankar

William F. Hall

Alireza Mohammadian

Chris Rowell

Rockwell International Science Center

Thousand Oaks, California

Accurate and rapid evaluation of radar signatures for alternative aircraft/store configurations would be of substantial benefit in the evolution of integrated designs that meet radar cross-section (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. A structured grid and an unstructured grid-based finite-volume, time-domain Maxwell's equation solver have been developed incorporating modeling techniques for general radar absorbing materials. Using this work as a base, the goal of the computational electromagnetics (CEM) effort is to define, implement, and evaluate various issues suitable for rapid prototype signature prediction addressing many issues related to (1) the physics of electromagnetics, (2) efficient and higher-order accurate algorithms, (3) boundary condition procedures, (4) geometry and gridding (structured and unstructured), (5) computer architecture (SIMD and MIMD), and (6) validation.

INTRODUCTION

Computational Electromagnetics

The ability to predict radar return from complex structures with layered material media over a wide frequency range (100 MHz to 20 GHz) is a critical technology need for the development of stealth aerospace configurations. Traditionally, radar cross section (RCS) calculations have employed one of two



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Large-Scale Structures in Acoustics and Electromagnetics: Proceedings of a Symposium 5 Algorithmic Aspects And Supercomputing Trends In ComputationalElectromagnetics Vijaya Shankar William F. Hall Alireza Mohammadian Chris Rowell Rockwell International Science Center Thousand Oaks, California Accurate and rapid evaluation of radar signatures for alternative aircraft/store configurations would be of substantial benefit in the evolution of integrated designs that meet radar cross-section (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. A structured grid and an unstructured grid-based finite-volume, time-domain Maxwell's equation solver have been developed incorporating modeling techniques for general radar absorbing materials. Using this work as a base, the goal of the computational electromagnetics (CEM) effort is to define, implement, and evaluate various issues suitable for rapid prototype signature prediction addressing many issues related to (1) the physics of electromagnetics, (2) efficient and higher-order accurate algorithms, (3) boundary condition procedures, (4) geometry and gridding (structured and unstructured), (5) computer architecture (SIMD and MIMD), and (6) validation. INTRODUCTION Computational Electromagnetics The ability to predict radar return from complex structures with layered material media over a wide frequency range (100 MHz to 20 GHz) is a critical technology need for the development of stealth aerospace configurations. Traditionally, radar cross section (RCS) calculations have employed one of two

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Large-Scale Structures in Acoustics and Electromagnetics: Proceedings of a Symposium methods: high-frequency asymptotics, which treats scattering and diffraction as local phenomena; or solution of an integral equation (in the frequency domain) for radiating sources on (or inside) the scattering body, which couples all parts of the body through a multiple scattering process. A third approach is the direct integration of the differential or integral form of Maxwell's equations in the time domain. The time-domain Maxwell's equations represent a more general form than the frequency-domain vector Helmholtz equations, which are usually employed in solving scattering problems. A time-domain approach can, for instance, handle continuous wave (single frequency) as well as a single pulse (broadband frequency) transient response. Frequency-domain-based methods usually provide the RCS response for all angles of incidence at a single frequency, while time-domain-based methods provide solutions for many frequencies from a single transient calculation. Also, in a time-domain approach, one can consider time-varying material properties for treatment of active surfaces. By using Fourier transforms, the time-domain transient solutions can be processed to provide the frequency-domain response. Frequency-dependent (dispersive) and anisotropic material properties can also be included within the time-domain formulation. CEM is a critical technology in the advancement of future aerospace development through supercomputing. As we make the transition from the present gigaflops to the next-generation teraflops computing, CEM will become integral to aerospace design not only as a standalone technology but also as part of the multidisciplinary coupling that leads to well-optimized designs. Objectives Toward establishing a computational environment for performing multidisciplinary studies, the initial goal is to advance the state of the art in CEM with the following specific objectives: Apply algorithmic advances in Computational Fluid Dynamics (CFD) to solve Maxwell's equations in general form to study scattering (radar cross section), radiation (antenna), and a variety of electromagnetic environmental (electromagnetic compatibility, shielding, and interference) problems of interest to both the defense and commercial communities. (Mohammadian et al., 1991) Establish the viability of MIMD massively parallel architectures for tackling large-scale problems not amenable to present-day supercomputers. Develop the CEM technology to the point of being able to perform coupled CFD/CEM optimization design studies. CEM Issues Proper development of a CEM capability appropriate for all aspects of aerospace design must consider various issues associated with electromagnetics. Some of them are addressed in the following seven subsections. Maxwell's Equations In order to apply conservation principles (for example, in fluid dynamics, mass, momentum, and energy are conserved), many of the governing equations representing appropriate physical processes are written in conservation form. The general form of a differential conservation equation can be written as

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Large-Scale Structures in Acoustics and Electromagnetics: Proceedings of a Symposium where Q is the solution vector and E, F, and G are the fluxes in x, y, and z coordinate directions, respectively. The conservation form readily admits weak solutions such as shock waves. The integral form of the conservation laws that can easily be derived from the differential form by integrating (5.1) with respect to x,y, z over any conservation cell whose volume is V is This can be rewritten in vector notation as In the above, Applying the Gauss divergence theorem, we can convert the volume integral into a surface integral In the above equation, the cell average of the dependent variables are denoted by . The outward unit normal at any point of the boundary surface of a cell has been denoted by , The integral form of the conservation laws given by (5.5) defines a system of equations for the cell average values of the dependent variables. Maxwell's equations in their vector form are and

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Large-Scale Structures in Acoustics and Electromagnetics: Proceedings of a Symposium The divergence conditions and are derived directly from Maxwell's equations, where The vector quantities and are the electric and magnetic field intensities, D=(Dx,Dy,Dz) is the electric displacement, B=(Bx,By,Bz) is the magnetic induction, and is the current density and p is the charge density. The subscripts x, y, z in the vector representation of refer to components in respective directions. In order to apply conservation-law form finite-volume methods, (5.7) and (5.8) are rewritten in the form of (5.1), In what follows, the permittivity coefficient e and the permeability coefficient μ are taken to be isotropic, scalar material properties and satisfy the following relationship: , . Generalization to tensor ε and μ is rather cumbersome but straightforward. The current density J is usually represented by σ, where σ is the material electrical conductivity. For treatment of complex geometries, a body-fitted coordinate transformation is introduced to aid in the application of boundary conditions. Under the transformation of coordinates implied by equation (5.9) can be rewritten as where

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Large-Scale Structures in Acoustics and Electromagnetics: Proceedings of a Symposium where is the Jacobian of the transformation, and, e.g., . The quantities and in (5.11) represent tangential magnetic and electric fields at a constant surface. Thus, the fluxes and are nothing but the tangential fields. Maxwell's equations can also be cast in integral form as where the six components of in (5.5) are (). In general, the differential form, (5.11), will be employed for finite-volume schemes using a structured grid arrangement, and the integral form, (5.12), will be used for unstructured grid cell arrangements using finite-element-like finite-volume schemes. Finite-Volume Treatment Space/Time Discretization The major feature of the present discretization approach that distinguishes it from other finite-volume and finite-difference procedures is that the electric and magnetic field unknowns are co-located in both space and time, rather than being assigned to two interpenetrating spatial grids and separated a half-step in time. These field unknowns are the volume averages of E and H within each cell in the space-filling grid. Staggered-grid methods automatically achieve second-order accuracy in space, while co-located field algorithms require near-neighbor corrections. However, there is a fundamental equivalence between these methods in terms of the achievable accuracy and stability of the integration process. Both approaches typically use explicit time integration, which means that the upper limit on the allowable size of the time step Δt is determined by the physical size and shape of the smallest cells, corresponding roughly to the time that light takes to cross one of these cells. Implicit integration schemes can choose larger time steps, but they require the inversion of a banded matrix the size of the whole grid, and their ability to preserve phase information is not known. The unstructured algorithm developed here is applicable to any grid that fills the computational domain with polyhedral cells. In particular, necessary bookkeeping procedures are implemented to deal with hexahedra (such as cubes), tetrahedra, and prisms (translations of a triangle out of its plane of definition). Each polyhedron in the computational domain is specified by the location (x,y,z) of its vertices in physical space. From these locations, all the necessary geometrical quantities, including areas, surface normals, and centroidal locations are computed. As stated earlier, each field unknown attributed to a given polyhedron is considered to be an average of the field over the volume of the polyhedron. The six components of E and H at one time level are thus stored according to an index α that runs over all

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Large-Scale Structures in Acoustics and Electromagnetics: Proceedings of a Symposium polyhedra. Quantities related to the polyhedral faces, such as face normals, are stored according to another index that runs over all faces. The interior faces of a given polyhedron are kept distinct from those of the neighboring polyhedra that share faces with it in a purely geometrical sense. This allows, for instance, for any type of impedance boundary condition to be applied at the boundary between cells. Thus, each polyhedral face has a co-face with a distinct face index, and each such co-face belongs to its own polyhedron. Polynomial Representation and Least Squares To go beyond representation of the fields as simple volume averages, we have chosen initially to implement linear polynomial functions for both E and H. Higher-order polynomial representations will follow the same general procedures. The essential question is how the higher-order terms in these polynomials are to be determined from near-neighbor data, so as to achieve the desired level of accuracy within each cell. In our unstructured approach, this evaluation is closely tied to the time integration procedure through the Riemann fluxes at each interface. Ultimately, this preserves time accuracy as well as accuracy in space. In the first step of this method, first-order Riemann fluxes are constructed at each interface of a cell from the volume-averaged fields on either side of the interface. For Maxwell's equations, these fluxes are the tangential field components just inside the boundaries of the cell. To complete the specification at the cell surface, the normal components of E and H are taken to be the normal components of the volume-averaged fields. This maintains overall charge conservation within the cell. These boundary data are sufficient to determine all the terms in a linear polynomial fit to either E or H by a procedure such as least-squares minimization of the fitting error integrated over the boundary. If we denote the vector polynomial to be fitted as and its boundary values as , then the quantity to be minimized is Taking derivatives of e with respect to each polynomial coefficient in results in a nonsingular set of linear equations for these coefficients. For consistency, one constrains the constant term in to be equal to the known volume average of over the cell. A separate set of equations is obtained for each Cartesian component of . If one writes, e.g., where the angular brackets denote volume averaging and, e.g., , then these equations become

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Large-Scale Structures in Acoustics and Electromagnetics: Proceedings of a Symposium where we have denoted the cell boundary as . These equations can be solved by inverting the matrix M whose elements are the quadratic moments where and are unit vectors in the respective coordinate directions. For a linear polynomial fit, there is a simpler alternative procedure that we have implemented to evaluate these linear terms. From the divergence theorem, the average value of any derivative over the cell volume can be rewritten as a surface integral: where is the unit outward normal on the boundary and Vα is the cell volume. In particular, if ρ is a linear function of , then is constant and equal to this volume average, which can be calculated just from the values of ρ on the boundary. For every component of A, we can replace its boundary values by the corresponding component of to obtain the approximation which is equivalent to using as the weight in the method of weighted residuals applied to the difference . The quantity Kα is a vector dyadic. Since we have chosen , we can make use of the vector identity to rewrite the integral as

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Large-Scale Structures in Acoustics and Electromagnetics: Proceedings of a Symposium which will be more convenient to compute in terms of the tangential components of . This particular weighting can be shown to result from a variational principal that assumes each Cartesian component of is the boundary value of a solution of Laplace's equation inside the cell. The Unstructured Second-Order Algorithm An algorithm that maintains second-order accuracy in both space and time can be constructed from the linear polynomial representation as follows: Here we have written Maxwell's equations symbolically as and the solution of the Riemann problem just inside a cell interface is denoted . This solution depends only on the values of immediately on either side of the interface. These are the cell-average values for and the linear polynomial values for . Material Properties The primary design variables affecting RCS are the shape of the conducting structure and the electric and magnetic polarizabilities of the materials that cover various parts of this structure. Because the illuminating fields are weak, the response of the materials can be taken as linear, so that an effective dielectric permittivity ε and magnetic permeability μ can be defined, both of which will in general be complex at the illuminating frequency. The treatment of various limiting material conditions relevant to radar absorbing structures within the framework of time-domain electromagnetics is described below.

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Large-Scale Structures in Acoustics and Electromagnetics: Proceedings of a Symposium Resistance Cards A thin conducting sheet causes a jump in the tangential magnetic field proportional to the electric current in the sheet, which is given by , where σ is the electric conductivity of the sheet, d is its thickness, is the local normal to the surface of the sheet, and is the instantaneous local electric field. Perfectly Conducting Surface At typical radar frequencies, the electrical conductivity of metals and other aircraft structural composites is sufficiently high that they can be treated as perfect electrical conductors. In this limit, the electromagnetic fields do not penetrate the conducting surface, and the components of the electric field tangent to the surface vanish at every point. This relation, , thus appears as a boundary condition on the solution of Maxwell's equations. Lossy Materials and Dispersive Media The imaginary parts of e and p represent energy absorption. For a time harmonic radar wave at radian frequency co, their effects are exactly equivalent to adding instantaneous electric and magnetic current conductivities equal to ω Im(ε) and ω Im(μ), respectively, in the Maxwell curl equations: For a transient pulse, the frequency dependence of ε and μ (dispersive media) must be properly modeled over the bandwidth of the pulse. This leads to an integration over the past history of the fields, but the value of this integral at every time step can be accurately updated using only the immediately previous values of the field and the integral. Impedance Boundaries When the product me is large compared to , the electrical wavelength in the material is correspondingly reduced from its free-space value. To achieve the same accuracy inside a layer of such a material as is needed on the outside, the number of grid points per unit area on the inside surface must be increased by the factor () compared to the outside surface. In extreme cases, orders of magnitude employing more grid points would be used in the layer than in all the space surrounding the target. We avoid this problem by eliminating the points inside the layer through the use of an impedance boundary condition applied on its outer surface. The implementation of this condition in the time domain involves an integral over past history that is carded out by the same method that we use for transient-pulse integration with frequency-dependent ε or μ.

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Large-Scale Structures in Acoustics and Electromagnetics: Proceedings of a Symposium Anisotropic Media In general, the polarization induced in a material by an applied field need not be parallel to the direction of the applied field. The electrical permittivity ε, relating , and the magnetic permeability μ, relating , are therefore tensors rather than scalar quantities, and there will typically be different characteristic wave speeds along the three principal axes associated with three tensors. Aside from this formal complication, Maxwell's equations as given in (5.7) and (5.8) still apply. Chiral Media Materials that are predominantly composed of elements of a single parity, such as right-handed helices, exhibit electrical polarization in response to an applied magnetic field and magnetic polarization in response to an applied electric field. At a given frequency, one can define a chiral admittance tensor ξ, in terms of which and the propagation of electromagnetic fields through such a material will again be governed by (5.7) and (5.8). As in the case of dispersive media, an integration over the past history of the fields will be required to calculate the response to a transient pulse. Cracks, Gaps, and Wires Another limiting condition in which resolution requirements become intolerable is for a metal structure having one very small dimension. For instance, putting a grid cell inside a narrow crack can produce excessively large execution times. Fortunately, the local behavior of the fields near these singular geometries can be well approximated analytically. Formulas for the fluxes into reasonably sized grid cells neighboring the singularity can be derived from these asymptotic forms and used to update the local fields. Boundary Conditions Proper implementation of various boundary conditions associated with material properties such as perfectly conducting walls, resistive sheets, material interface, impedance boundaries, as well as computational boundary conditions such as nonreflecting outer boundaries are very crucial to accurately modeling problems in electromagnetics. In fact, higher-order accurate implementation of boundary conditions in any computational simulation in any discipline is the number one computational issue. The physical boundary conditions on the electric and magnetic fields at a material interface follow directly from the requirement that Maxwell's equations be satisfied on the interface. In the limiting case of a perfect electrical conductor, these fields vanish inside the conductor, and a sheet of electric current and charge at the interface provides the necessary field discontinuities between the outer and inner surfaces of the conductor.

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Large-Scale Structures in Acoustics and Electromagnetics: Proceedings of a Symposium At the outer limits of the computational domain, the true behavior of the scattered wave is that it propagates outward without reflection. An approximate outgoing-wave condition that can be applied locally at points on the outer boundary will cause some reflection, but errors from this source can be minimized as discussed below. Perfect Conductors In a finite-volume scheme, unknown field values are normally computed only at the centroids of the grid cells. However, to capture potentially rapid field variations along a conducting surface, one will have to solve Maxwell's equations fight on the conducting surface satisfying the boundary condition on the tangential electric field, n × E = 0, to the same accuracy as any field points (preferably to at least second-order accuracy). A rigorous boundary condition implementation procedure based on characteristic theory can be applied to solving Maxwell's equations right on the body points. This step is crucial to capturing the right surface currents accounting for traveling waves. The final RCS results that are obtained from the surface currents are only as accurate as the values one computes for n × H on the conducting surface. The boundary condition procedure for perfectly conducting walls can also be appropriately modified and applied for impedance walls, where the surface tangential electric field, instead of being 0, is proportional to the tangential magnetic field. Outer Boundaries Along a given direction in space, the local electromagnetic fields can be grouped into forward and backward propagating combinations. One can develop a hierarchy of nonreflecting boundary conditions using the characteristic theory of signal propagation. A simple first-order condition imposes the requirement that the incoming scattered field signal normal to the outer boundary be 0. Though this is sufficient for many scattering problems, research in numerical algorithms needs to address the development of higher-order nonreflecting boundary conditions. This will allow one to place the outer boundary very close to the scatter and minimize the number of grid cells in the computational domain. Material Interface Across a material interface where the material properties ε and μ can be different on either side, certain boundary conditions on the tangential fields are to be satisfied. For example, without the presence of any lossy medium at the interface, the tangential fields n × E and n × H are continuous even though the solution vectors D and B will be discontinuous. When a resistive medium is present at the interface, then appropriate jumps in n × E and n × H must be accounted for in the boundary condition implementation. In order to compute the right wave reflection and transmission at an interface, the boundary conditions will have to be satisfied to the same level of accuracy as the order of the scheme in the field points. For schemes with order of accuracy greater than two, developing corresponding higher-order material interface boundary condition procedures will be quite challenging.

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Large-Scale Structures in Acoustics and Electromagnetics: Proceedings of a Symposium Geometry/Gridding Problems in CEM involve arbitrarily shaped three-dimensional geometries that need to be represented properly in the computer simulation. In addition to the external shape, CEM also requires modeling the interior of the penetrable structure. Depending on the formulation (differential or integral), one may choose either a structured grid or an unstructured grid setup. Two gridding issues that need to be addressed in EM computations are number of grid points per wavelength to properly represent the fields in and around a scatterer, and how far the outer boundary should be placed from the scattering object to adequately simulate the nonreflecting boundary condition. In general, the number of points/wavelength is not determined by wavelength alone, and involves the body dimensions (characteristic body size with respect to wavelength) also. The outer boundary location, theoretically, can be right on the body surface itself; however, the computational implementation of nonreflecting boundary conditions requires the outer boundary at a few (2 to 5) wavelengths away from the surface. Again, if one can construct higher-order accurate implementations of nonreflecting boundary conditions, the outer boundary can be brought very close to the scattering surface. In general, the necessary grid resolution is provided only around and near the body surface. Between the body and the outer boundary, the mesh is allowed to stretch resulting in very crude (3 to 5 points per wavelength) meshes near the outer boundary regions. The free space wavelength is reduced to smaller values inside a material (as e and μ become large, the speed of propagation, , goes down, causing the wavelength to scale accordingly). Thus, the grid resolution must take into account material properties to adequately resolve the fields inside material zones. The number of grid points per wavelength required depends on the order of accuracy of the numerical scheme. A second-order accurate scheme usually requires at least 10 grid points per local wavelength. One may be able to use a higher-order scheme and minimize the number of grid points. However, as the order of accuracy goes up, the scheme will also require more computations per grid point, which may offset the execution savings with fewer grid points. The requirement that the fields be resolved accurately with proper grid resolution makes CEM problems computationally intensive, requiring large-scale supercomputing. For example, to compute the radar cross section of a typical aircraft at 1 GHz, even if one uses 10 grid cells per wavelength, will require tens of millions of grid points. Massively Parallel Computing With the emergence of massively parallel computing architectures with potential for teraflops performance, any code development activity must effectively utilize the computer architecture in achieving the proper load balance with minimum internodal data communication. Some of the massively parallel computing architecture issues addressed in the present study are Domain decomposition and load balancing Internodal message passing with minimum communication delays

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Large-Scale Structures in Acoustics and Electromagnetics: Proceedings of a Symposium Synchronization for time-accurate computation Storage of data for FFT processing of large RCS pulse cases Measure of MFLOP rating Scalability measure Pre- and post-processing A 512-node nCUBE and a 208-node Intel Paragon are currently used to study these issues in the development of the CEM code. Validation Once a CEM code is developed, the results must be validated against known exact solutions and carefully tailored experimental data. There are many computational issues such as grid resolution, location of the outer boundary, and accuracy of the boundary condition procedures that can only be addressed through a careful study of many validation cases. The Electromagnetic Code Consortium (EMCC) has a list of validation cases comprising many target shapes specifically designed for validating codes. PRESENT CEM CAPABILITY Both a structured grid version and an unstructured grid version of the CEM code are in development. The structured grid version is in a relatively advanced mature stage, while the unstructured grid version is the subject of state-of-the-art algorithm and code development. Figure 5.1 shows the progression of various CEM developments at Rockwell. Some of the salient features of the current CEM capabilities are Time-domain Maxwell's equations Proven algorithms from CFD Single pulse (multiple frequency, transient) or continuous incident wave (single frequency, time harmonic steady state) Numerical grid generation-structured multizone grid or unstructured grid Lossy or lossless material properties Frequency- and time-dependent properties Thin structures (resistive card, lossy paint) Vector/parallel code architecture—2 GFLOPS demonstrated on the CrayYMP with 8 processors, and 10 GFLOPS on the Cray-C90 with 16 processors. Scalable performance demonstrated on both the NCUBE and the Intel Paragon. Received the 1990 CRAY Gigaflop Performance Award Received the 1993 Computerworld Smithsonian Award

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Large-Scale Structures in Acoustics and Electromagnetics: Proceedings of a Symposium Figure 5.1 Structured grid and unstructured grid-based CEM development. Pre- and post-processor graphics/animation Application to scattering (RCS), radiation (antenna), EMP/EMI/EMC, and bioelectromagnetics problems Ideal for CFD/CEM optimization studies The CEM code has been extensively tested for the following geometries: Canonical objects such as spheres, cylinders, ogives, thin rods, cones, airfoils, and a circular disc Almond-shaped target Inlets of various shapes (square, circular, curved, . . .) including the presence of infinite ground plane Flat plates of various planeforms Double sphere Complete wing geometries with layers Finned projectile and cone-cylinder combinations Scattering from ship-like targets Complete fighter targets

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Large-Scale Structures in Acoustics and Electromagnetics: Proceedings of a Symposium Three-Dimensional Sphere of IncreasingKa. 500 time steps # of Grid Points 6 Nodes 24 Nodes 96 Nodes 384 Nodes 6354 Ka=2.5 389 sec (1) 11 × 11 × 9 per node 135 sec (4) 6 × 6 × 9 per node      23818 Ka=5 1300 sec (3) 21 × 21 × 9 per node 389 sec (6) 11 × 11 × 9 per node 135 sec (14) 6 × 6 × 9 per node   90774 Ka=10   1301 sec (17) 21 × 21 × 9 per node 390 sec (25.5) 11 × 11 × 9 per node 135 sec (68) 6 × 6 × 9 per node 354294 Ka=20     1302 sec (40) 21 × 21 × 9 per node 390 sec (75) 11 × 11 × 9 per node • For Ka = 20, a one-CPU C-90 takes 425 seconds running at 550 MFLOPS • (·) represents set up time taken by the host • Ka = 20 case runs at 1.5 MFLOPS/node on the nCUBE (60% of peak performance) Figure 5.2 Scalable performance on a massively parallel architecture. Some sample results are shown here to illustrate the present capability. The scalable performance of the structured grid CEM code is shown in Figure 5.2 for the nCUBE architecture. As the number of nodes and the problem size increase, the turnaround clock time is maintained, demonstrating one of the scalability measures. Similar scalable performances are achieved on the Intel Paragon and the Cray T3D. Figure 5.3 shows the results for a square inlet for both a CW case and a pulse case. The comparison of monostatic RCS with experimental data is very good. Figure 5.4 shows an application of the code for predicting the RCS of a complete fighter. Some of the other applications of the CEM code are shown in Figure 5.5. Figure 5.6 shows an application of the CEM code for bioelectromagnetics to study the effects of microwave heating of cancer tissues (hyperthermia treatment). Currently work is continuing to further develop the unstructured grid-based CEM code for massively parallel architectures with special emphasis on domain decomposition techniques and applications to problems involving hundreds of millions of grid points. Future efforts will include coupling of the CEM codes with CFD and Computational Structural Mechanics for multidisciplinary optimization studies. ACKNOWLEDGMENT This work is funded by Rockwell IR&D, AFOSR, Army Research Laboratory at Aberdeen Proving Ground, NASA Ames Research Center, and the Electromagnetic Code Consortium. REFERENCE Mohammadian, A., V. Shankar, and W.F. Hall, 1991, ''Computation of electromagnetic scattering and radiation using a time-domain finite-volume discretization procedure,''Comput. Phys. Commun.68, 175-196.

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Large-Scale Structures in Acoustics and Electromagnetics: Proceedings of a Symposium Figure 5.3 Monostatic RCS for a square inlet.

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Large-Scale Structures in Acoustics and Electromagnetics: Proceedings of a Symposium Figure 5.4 RCS for a complete fighter

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Large-Scale Structures in Acoustics and Electromagnetics: Proceedings of a Symposium Figure 5.5 Different applications of the CEM code.

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Large-Scale Structures in Acoustics and Electromagnetics: Proceedings of a Symposium Figure 5.6 Bioelectromagnetic application of the CEM code.