Generalized Method of Moments: A Novel Discretization Technique for Integral Equations

Nair, N. V.; Shanker, B.

doi:10.1109/tap.2011.2143652

Cited by 32 publications

(28 citation statements)

References 40 publications

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“…The solution to the EFIE is typically effected by using method of moments (MoM) wherein surface current is represented by a set of vector basis functions, say the Rao-Wilton-Glisson basis functions [30] which are equivalent to the lowest order Raviart-Thomas functions [31]. Alternatives to this approach has been a topic of significant recent interest; these include using generalized method of moments (GMM) [32], [33], subdivision surfaces [27], discontinuous basis set [34] , and more recently, Debye sources [19]. All the aforementioned methods try to bring features into modeling electromagnetic scattering; but a common thread that ties GMM, MoM on subdivision surfaces, and Debye sources is the use of surface Helmholtz decomposition.…”

Section: Formulations a Electric Field Integral Equationmentioning

confidence: 99%

Generalized Debye Sources-Based EFIE Solver on Subdivision Surfaces

Jiang

et al. 2017

IEEE Trans. Antennas Propagat.

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Abstract-The electric field integral equation is a well known workhorse for obtaining fields scattered by a perfect electric conducting (PEC) object. As a result, the nuances and challenges of solving this equation have been examined for a while. Two recent papers motivate the effort presented in this paper. Unlike traditional work that uses equivalent currents defined on surfaces, recent research proposes a technique that results in well conditioned systems by employing generalized Debye sources (GDS) as unknowns. In a complementary effort, some of us developed a method that exploits the same representation for both the geometry (subdivision surface representations) and functions defined on the geometry, also known as isogeometric analysis (IGA). The challenge in generalizing GDS method to a discretized geometry is the complexity of the intermediate operators. However, thanks to our earlier work on subdivision surfaces, the additional smoothness of geometric representation permits discretizing these intermediate operations. In this paper, we employ both ideas to present a well conditioned GDS-EFIE. Here, the intermediate surface Laplacian is well discretized by using subdivision basis. Likewise, using subdivision basis to represent the sources, results in an efficient and accurate IGA framework. Numerous results are presented to demonstrate the efficacy of the approach.

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Section: Formulations a Electric Field Integral Equationmentioning

confidence: 99%

Generalized Debye Sources-Based EFIE Solver on Subdivision Surfaces

Jiang

et al. 2017

IEEE Trans. Antennas Propagat.

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“…However, generating conformal discretizations for engineering system-level simulations is far from trivial, as the complexity of modern engineering applications increases at a fast pace. Among the previous works addressing the above mentioned deficiencies, we mention recent works [28][29][30][31][32].…”

Section: Discontinuous Galerkin Formulationmentioning

confidence: 99%

ADAPTIVE AND PARALLEL SURFACE INTEGRAL EQUATION SOLVERS FOR VERY LARGE-SCALE ELECTROMAGNETIC MODELING AND SIMULATION (Invited Paper)

MacKie-Mason¹,

Greenwood²,

Peng³

2015

PIER

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Abstract-This work investigates an adaptive, parallel and scalable integral equation solver for very large-scale electromagnetic modeling and simulation. A complicated surface model is decomposed into a collection of components, all of which are discretized independently and concurrently using a discontinuous Galerkin boundary element method. An additive Schwarz domain decomposition method is proposed next for the efficient and robust solution of linear systems resulting from discontinuous Galerkin discretizations. The work leads to a rapidly-convergent integral equation solver that is scalable for large multi-scale objects. Furthermore, it serves as a basis for parallel and scalable computational algorithms to reduce the time complexity via advanced distributed computing systems. Numerical experiments are performed on large computer clusters to characterize the performance of the proposed method. Finally, the capability and benefits of the resulting algorithms are exploited and illustrated through different types of real-world applications on high performance computing systems.

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“…Consistent with the central theme of the GMM framework, we develop a scheme that permits different orders of polynomials or different functions to be defined on adjacent patches. It has been shown, for integral equation based solvers 41,42 , that this can be achieved using a …”

Section: Definition Of Gmm Basis Functionsmentioning

confidence: 99%

“…The basis functions developed within this framework are continuous across domains and, as a result, do not need additional constraints to ensure continuity. The authors have recently developed methods that extend this idea to surface integral equations as applied to electromagnetics 41,42 , and have demonstrated convergence, well conditioned properties as well as application analysis of scattering from the range of targets. Unfortunately, this method still relies on an underlying tessellation.…”

Section: Introductionmentioning

confidence: 99%

Generalized Method of Moments

2007

Encyclopedia of Measurement and Statistics

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The analysis of scattering from complex objects using surface integral equations is a challenging problem. Its resolution has wide ranging applications-from crack propagation to diagnostic medicine. The two ingredients of any integral equation methodology is the representation of the domain and the design of approximation spaces to represent physical quantities on the domain. The order of convergence depends on both the surface and geometry representation. For instance, most surface models are restricted to piecewise flat or second order tessellations. Similarly, the most commonly known basis spaces for acoustics are piecewise constant functions. What is desirable is a framework that permits adaptivity (of size and order) in both geometry and function representations. Unlike volumetric, differential equation solvers, such as the finite element method, developing an hpadaptive framework for surface integral equations is very difficult. This papers proposes a resolution to this problem by developing a novel framework that relies on reconstruction of the surface using locally smooth parameterizations, and defining partition of unity functions and higher order basis spaces on overlapping domains. This permits easy refinement of both the geometry and function representation. This capabilities of the proposed framework are shown via a number of numerical examples.

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Generalized Method of Moments: A Novel Discretization Technique for Integral Equations

Cited by 32 publications

References 40 publications

Generalized Debye Sources-Based EFIE Solver on Subdivision Surfaces

Generalized Debye Sources-Based EFIE Solver on Subdivision Surfaces

ADAPTIVE AND PARALLEL SURFACE INTEGRAL EQUATION SOLVERS FOR VERY LARGE-SCALE ELECTROMAGNETIC MODELING AND SIMULATION (Invited Paper)

Generalized Method of Moments

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