Mahadevan Ganesh

A high-order tangential basis algorithm for electromagnetic scattering by curved surfaces

We describe, analyze, and demonstrate a high-order spectrally accurate surface integral algorithm for simulating time-harmonic electromagnetic waves scattered by a class of deterministic and stochastic perfectly conducting three-dimensional obstacles. A key feature of our method is spectrally accurate approximation of the tangential surface current using a new set of tangential basis functions. The construction of spectrally accurate tangential basis functions allows a one-third reduction in the number of unknowns required compared with algorithms using non-tangential basis functions. The spectral accuracy of the algorithm leads to discretized systems with substantially fewer unknowns than required by many industrial standard algorithms, which use, for example, the method of moments combined with fast solvers based on the fast multipole method. We demonstrate our algorithm by simulating electromagnetic waves scattered by medium-sized obstacles (diameter up to 50 times the incident wavelength) using a direct solver (in a small parallel cluster computing environment). The ability to use a direct solver is a tremendous advantage for monostatic radar cross section computations, where thousands of linear systems, with one electromagnetic scattering matrix but many right hand sides (induced by many transmitters) must be solved.

A spectrally accurate algorithm for electromagnetic scattering in three dimensions

In this work we develop, implement and analyze a high-order spectrally accurate algorithm for computation of the echo area, and monostatic and bistatic radar cross-section (RCS) of a three dimensional perfectly conducting obstacle through simulation of the time-harmonic electromagnetic waves scattered by the conductor. Our scheme is based on a modified boundary integral formulation (of the Maxwell equations) that is tolerant to basis functions that are not tangential on the conductor surface. We test our algorithm with extensive computational experiments using a variety of three dimensional perfect conductors described in spherical coordinates, including benchmark radar targets such as the metallic NASA almond and ogive. The monostatic RCS measurements for non-convex conductors require hundreds of incident waves (boundary conditions). We demonstrate that the monostatic RCS of small (to medium) sized conductors can be computed using over one thousand incident waves within a few minutes (to a few hours) of CPU time. We compare our results with those obtained using method of moments based industrial standard three dimensional electromagnetic codes CARLOS, CICERO, FE-IE, FERM, and FISC. Finally, we prove the spectrally accurate convergence of our algorithm for computing the surface current, far-field, and RCS values of a class of conductors described globally in spherical coordinates.

A reduced basis method for electromagnetic scattering by multiple particles in three dimensions

We consider the development of efficient and fast computational methods for parametrized electromagnetic scattering problems involving many scattering three dimensional bodies. The parametrization may describe the location, orientation, size, shape and number of scattering bodies as well as properties of the source field such as frequency, polarization and incident direction. The emphasis is on problems that need to be solved rapidly to accurately simulate the interaction of scattered fields under parametric variation, e.g., for design, detection, or uncertainty quantification. For such problems, the use of a brute force approach is often ruled out due to the computational cost associated with solving the problem for each parameter value. In this work, we propose an iterative reduced basis method based on a boundary element discretization of few reference scatterers to resolve the computationally challenging large scale problem. The approach includes (i) a computationally intensive offline procedure to create a selection of a set of snapshot parameters and the construction of an associated reduced basis for each reference scatterer and (ii) an inexpensive online algorithm to generate the surface current and scattered field of the parametrized configuration, for any choice of parameters within the parameter domains used in the offline procedure. Comparison of our numerical results with directly measured results for some benchmark configurations demonstrate the power of our method to rapidly simulate the interacting electromagnetic fields under parametric variation of the overall multiple particle configuration.

A high-order algorithm for multiple electromagnetic scattering in three dimensions

We describe a fully discrete high-order algorithm for simulating multiple scattering of electromagnetic waves in three dimensions by an ensemble of perfectly conducting scattering objects. A key component of our surface integral algorithm is high-order tangential approximation of the surface current on each obstacle in the ensemble. The high-order nature of the algorithm leads to relatively small numbers of unknowns, which allows us to use either a direct method or an iterative boundary decomposition method for simulations of multiple scattering. We demonstrate the algorithm using both of these techniques for near and well separated obstacles. Using a small computing cluster (with 20 processors), we simulate multiple scattering by up to 125 objects for frequencies in the resonance region, and by paired obstacles of diameter 20 to 30 times the incident wavelength. Many physically important problems, such as scattering by atmospheric aerosols or ice crystals, involve multiple scattering by ensembles of particles, each particle having its own distinct shape, but with all particles fitting a stochastic description with a small number of fixed parameters in local spherical coordinates. We demonstrate our algorithm for multiple scattering by ensembles of such unique particles, whose stochastic description corresponds to computer models of ice crystals and dust particles.

A Hybrid High-Order Algorithm for Radar Cross Section Computations

We describe a high-order method for computing the monostatic and bistatic radar cross section (RCS) of a class of three-dimensional targets. Our method is based on an electric field surface integral equation reformulation of the Maxwell equations. The hybrid nature of the scheme is due to approximations based on a combination of tangential and nontangential basis functions on a parametric reference spherical surface. A principal feature of the high-order algorithm is that it requires solutions of linear systems with substantially fewer unknowns than existing methods. We demonstrate that very accurate RCS values for medium (electromagnetic-) sized scatterers can be computed using a few tens of thousands of unknowns. Thus linear systems arising in the high-order method for low to medium frequency scattering can be solved using direct solves. This is extremely advantageous in monostatic RCS computations, for which transmitters and receivers are co-located and hence the discretized electromagnetic linear system must be solved for hundreds of right-hand sides corresponding to receiver locations. We demonstrate the high-order convergence of our method for several three-dimensional targets. We prove the high-order spectral accuracy of our approximations to the RCS for a class of perfect conductors described globally in spherical coordinates.

Three dimensional electromagnetic scattering T-matrix computations

The infinite T-matrix method is a powerful tool for electromagnetic scattering simulations, particularly when one is interested in changes in orientation of the scatterer with respect to the incident wave or changes of configuration of multiple scatterers and random particles, because it avoids the need to solve the fully reconfigured systems. The truncated T-matrix (for each scatterer in an ensemble) is often computed using the null-field method. The main disadvantage of the null-field based T-matrix computation is its numerical instability for particles that deviate from a sphere. For large and/or highly non-spherical particles, the null-field method based truncated T-matrix computations can become slowly convergent or even divergent. In this work, we describe an electromagnetic scattering surface integral formulation for T-matrix computations that avoids the numerical instability. The new method is based on a recently developed high-order surface integral equation algorithm for far field computations using basis functions that are tangential on a chosen non-spherical obstacle. The main focus of this work is on the mathematical details required to apply the high-order algorithm to compute a truncated T-matrix that describes the scattering properties of a chosen perfect conductor in a homogeneous medium. We numerically demonstrate the stability and convergence of the T-matrix computations for various perfect conductors using plane wave incident radiation at several low to medium frequencies and simulation of the associated radar cross of the obstacles.

A fully discrete H 1 -Galerkin method with quadrature for nonlinear advection–diffusion–reaction equations

We propose and analyze a fully discrete H1-Galerkin method with quadrature for nonlinear parabolic advection–diffusion–reaction equations that requires only linear algebraic solvers. Our scheme applied to the special case heat equation is a fully discrete quadrature version of the least-squares method. We prove second order convergence in time and optimal H1 convergence in space for the computer implementable method. The results of numerical computations demonstrate optimal order convergence of scheme in Hk for k = 0, 1, 2.

A pseudospectral quadrature method for Navier-Stokes equations on rotating spheres

In this work, we describe, analyze, and implement a pseudospectral quadrature method for a global computer modeling of the incompressible surface Navier-Stokes equations on the rotating unit sphere. Our spectrally accurate numerical error analysis is based on the Gevrey regularity of the solutions of the Navier-Stokes equations on the sphere. The scheme is designed for convenient application of fast evaluation techniques such as the fast Fourier transform (FFT), and the implementation is based on a stable adaptive time discretization.

A High Performance Computing and Sensitivity Analysis Algorithm for Stochastic Many-Particle Wave Scattering

We present a high performance computing framework for quantifying uncertainties in the propagation of acoustic waves through a stochastic media comprising a large number of three-dimensional particles. We subsequently describe an efficient postprocessing approach using our framework to statistically quantify the sensitivity of the uncertainties with respect to input parameters that govern the stochasticity in the model. The stochasticity arises through the random positions and orientations of the component particles in the media. Simulation even for a single deterministic three-dimensional configuration is inherently difficult because of the large number of particles; the stochasticity leads to a larger dimensional model involving three spatial variables and additional stochastic variables, and accounting for uncertainty in key parameters of the input probability distributions leads to prohibitive computational complexity. In the first part of our paper we describe a high performance computing framework f...

High-order FEM-BEM computer models for wave propagation in unbounded and heterogeneous media

Efficient computational models that retain essential physics of the associated continuous mathematical models are important for several applications including acoustic horn optimization. For heterogeneous wave propagation models that are naturally posed on unbounded domains, a crucial physical requirement is that the scattered fields are radiating and satisfy a radiation condition at infinity. We describe and implement an efficient high-order coupled computer model for acoustic wave propagation in an unbounded region comprising bounded heterogeneous media with several obstacles. Our unbounded and heterogeneous media computer model retains the radiation condition exactly and hence is readily applicable for the celebrated acoustic horn problem. This approach is more suitable than using a standard low-order approximation of the radiation condition. Using parallel computing environments, we demonstrate the high-order algorithm with extensive numerical experiments and computational analysis, including the model horn problem with several material property parameters. Our efficient computer models and validation in this work lead to some interesting mathematical and numerical analysis problems for the acoustic system defined on unbounded and heterogeneous media comprising smooth, non-smooth, horn, impenetrable, and penetrable obstacles.