Generalized Debye Sources-Based EFIE Solver on Subdivision Surfaces

Fu, Xin; Li, Jie; Jiang, Li Jun; Shanker, B.

doi:10.1109/tap.2017.2740976

Cited by 16 publications

(12 citation statements)

References 31 publications

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“…In order to numerically compute the eigenfunctions of the LBO or MHs, we discretize (14) using the Loop Subdivision FEM Galerkin method. This is akin to similar efforts using Lagrangian surface descriptions [55,32] that have shown both h− and p− convergence [56,55,24,32].…”

Section: Subdivision Fem For Computing Eigenfunctions Of Lbomentioning

confidence: 90%

“…Isogeometric methods use the same underlying basis sets to represent both the geometry and physics on the geometry. This class of techniques was pioneered by Hughes [20], and has since been adapted for a number of different types of problems in structural mechanics [8,9], electromagnetics [21,22,23,24] and acoustics [25,26]. The research in using isogeometric analysis has treaded along two paths; use of non-unifrom B-splines and subdivision surfaces; our focus is on subdivision.…”

Section: Introductionmentioning

confidence: 99%

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Laplace-Beltrami based Multi-Resolution Shape Reconstruction on Subdivision Surfaces

Alsnayyan,

Shanker

2021

Preprint

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The eigenfunctions of the Laplace-Beltrami operator have widespread applications in a number of disciplines of engineering, computer vision/graphics, machine learning, etc. These eigenfunctions or manifold harmonics, provide the means to smoothly interpolate data on a manifold. They are highly effective, specifically as it relates to geometry representation and editing; manifold harmonics form a natural basis for multi-resolution representation (and editing) of complex surfaces and functioned defined therein. In this paper, we seek to develop the framework to exploit the benefits of manifold harmonics for shape reconstruction. To this end, we develop a highly compressible, multi-resolution shape reconstruction scheme using manifold harmonics. The method relies on subdivision basis sets to construct both boundary element isogeometric methods for analysis and surface finite elements to construct manifold harmonics. We pair this technique with volumetric source reconstruction method to determine an initial starting point. Examples presented highlight efficacy of the approach in the presence of noisy data, including significant reduction in the number of degrees of freedom for complex objects, accuracy of reconstruction, and multi-resolution capabilities.

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Section: Subdivision Fem For Computing Eigenfunctions Of Lbomentioning

confidence: 90%

Section: Introductionmentioning

confidence: 99%

Laplace-Beltrami based Multi-Resolution Shape Reconstruction on Subdivision Surfaces

Alsnayyan,

Shanker

2021

Preprint

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“…Due to the fact that the formulation is nonsingular, Gaussian quadrature can be used on all elements (including the previous singular ones) and the implementation of higher order elements is straightforward. Although the desingularized surface integral framework is not essential to solve the considered electromagnetic scattering problem, it does greatly improve the ease of implementation of the matrix systems G and H , see (10) and Appendix B in [14], and it also improves the accuracy of the solution.…”

Section: Appendix B Nonsingular Surface Integral Equationmentioning

confidence: 99%

Field-only surface integral equations: scattering from a dielectric body

Sun¹,

Klaseboer²,

Yuffa³

et al. 2020

J. Opt. Soc. Am. A

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A robust and efficient field-only nonsingular surface integral method to solve Maxwell's equations for the components of the electric field on the surface of a dielectric scatterer is introduced. In this method, both the vector Helmholtz equation and the divergence-free constraint are satisfied inside and outside the scatterer. The divergence-free condition is replaced by an equivalent boundary condition that relates the normal derivatives of the electric field across the surface of the scatterer. Also, the continuity and jump conditions on the electric and magnetic fields are expressed in terms of the electric field across the surface of the scatterer. Together with these boundary conditions, the scalar Helmholtz equation for the components of the electric field inside and outside the scatterer is solved by a fully desingularized surface integral method. Comparing with the most popular surface integral methods based on the Stratton-Chu formulation or the PMCHWT formulation, our method is conceptually simpler and numerically straightforward because there is no need to introduce intermediate quantities such as surface currents and the use of complicated vector basis functions can be avoided altogether. Also, our method is not affected by numerical issues such as the zero frequency catastrophe and does not contain integrals with (strong) singularities. To illustrate the robustness and versatility of our method, we show examples in the Rayleigh, Mie, and geometrical optics scattering regimes. Given the symmetry between the electric field and the magnetic field, our theoretical framework can also be used to solve for the magnetic field.

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“…More importantly, the surface representation is C 2 almost everywhere making it an attractive candidate for defining physical basis sets as it avoids the requirement of defining additional mathematical framework that is commonplace in other low order basis set [2], [14], [25]. There has been a concerted effort to develop IGA methods on subdivision surfaces in a number of fields, including electromagnetics [21], [26], [27], acoustics [20], [28] and shape reconstruction/optimization [29]- [33].…”

Section: Introductionmentioning

confidence: 99%

“…This paper builds on our earlier body of work on Loop subdivision based IGA for the electric field integral equations [21] and construction of Debye sources [26]. In both these cases, the objects analyzed were simply connected and electrically small.…”

Section: Introductionmentioning

confidence: 99%

Iso-geometric Integral Equation Solvers and their Compression via Manifold Harmonics

Alsnayyan,

Shanker

2021

Preprint

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The state of art of electromagnetic integral equations has seen significant growth over the past few decades, overcoming some of the fundamental bottlenecks: computational complexity, low frequency and dense discretization breakdown, preconditioning, and so on. Likewise, the community has seen extensive investment in development of methods for higher order analysis, in both geometry and physics. Unfortunately, these standard geometric descriptors are C 0 at the boundary between patches with a few exceptions; as a result, one needs to define additional mathematical infrastructure to define physical basis sets for vector problems. In stark contrast, the geometric representation used for design is higher order differentiable over entire surface. Geometric descriptions that have C 2 -continuity almost everywhere on the surfaces are common in computer graphics. Using these description for analysis opens the door to several possibilities, and is the area we explore in this paper. Our focus is on Loop subdivision based isogeometric methods. In this paper, our goals are two fold: (i) development of computational infrastructure necessary to effect efficient methods for isogeometric analysis of electrically large simply connected objects, and (ii) to introduce the notion of manifold harmonics transforms and its utility in computational electromagnetics. Several results highlighting the efficacy of these two methods are presented.

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Generalized Debye Sources-Based EFIE Solver on Subdivision Surfaces

Cited by 16 publications

References 31 publications

Laplace-Beltrami based Multi-Resolution Shape Reconstruction on Subdivision Surfaces

Laplace-Beltrami based Multi-Resolution Shape Reconstruction on Subdivision Surfaces

Field-only surface integral equations: scattering from a dielectric body

Iso-geometric Integral Equation Solvers and their Compression via Manifold Harmonics

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