Accurate calculation of spherical and vector spherical harmonic expansions via spectral element grids

Wang, Bo; Wang, Li-Lian; Xie, Zhongliang

doi:10.1007/s10444-017-9569-1

Cited by 11 publications

(13 citation statements)

References 38 publications

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“…From the lower bounds shown in (3.30) and P (r) < 0 proved before, as r ∈ 88) as H 2 (r) > 0, P (r) < 0.…”

Section: Error Estimatementioning

confidence: 58%

“…[39,93,94]), but the extension to three dimensions is highly non-trivial (cf. [88]). It is also noteworthy that with a good tradeoff between accuracy and efficiency, the high order ABCs without high-order derivatives become appealing [46].…”

Section: Existing Techniques For Domain Reductionmentioning

confidence: 99%

“…where we used the fact: B = T t CT , due to (2.72) and the property: 86) we have We next evaluate the terms involving B, n, and w. The key is to show that the singular coefficients in (2.83) can be fully absorbed by w. Although the derivation appears a bit lengthy and tedious, we strive to present the formulation in an accessible manner, which depends only on the configuration of the layer: R 1 (θ), , and the coordinate transformation: T and σ 0 , σ 1 . However, to make the derivation relatively concise, we shall arrange the powers of 88) and use following regular functions in r:…”

Section: Substitution and Implementationmentioning

confidence: 99%

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Perfect absorbing layers for wave scattering problems and related topics

Gao¹

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List of Figures 2.3 Profiles of the PML solution U p with regular function (2.14) (σ 0 = 1, n = 2) and singular function (2.15) vs PAL solution U a in Ω and V a in Ω d with k = 99.99, (L, d) = (1, 0.2) along the x-axis at y = π/2. Left: real part solution; Right: imaginary part solution.. .. .. .. .. . 2.4 Left: illustration of the circular PAL from radial direction. Middle: schematic illustration of the rectangular PAL Ω d with four trapezoidal patches. Right: illustration of the elliptic PAL under elliptic coordinate. 2.5 Left: errors against various N for different layers (width d = 0.1) with boundary condition g(y) = sin(5y) − sin(30y); Right: errors against various N for different layers (width d = 0.5) with boundary condition g(y) = sin(5y) − sin(30y).

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“…From the lower bounds shown in (3.30) and P (r) < 0 proved before, as r ∈ 88) as H 2 (r) > 0, P (r) < 0.…”

Section: Error Estimatementioning

confidence: 58%

Section: Existing Techniques For Domain Reductionmentioning

confidence: 99%

Section: Substitution and Implementationmentioning

confidence: 99%

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Perfect absorbing layers for wave scattering problems and related topics

Gao¹

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“…We show in [110] that given the nodal values of U on a longitude-latitude θφ partition of S, and using the trigonometric form of the associated Legendre functions {P m l (cos θ)}, we can evaluate the double integral on S (the counterpart of (3.23)) by products of two one-dimensional integrals, and notably, similar analytic formulas can be used. Note that such a partition of S can be seamlessly integrated with spectral elements (but with some non-standard elements near two poles).…”

Section: Extension To Three-dimensional Simulationsmentioning

confidence: 99%

“…The expansion coefficients {φ m T,l , φ m Y,l } are associated with the expansion: 110) and given by…”

Section: )mentioning

confidence: 99%

Efficient spectral and spectral-element methods for transformation electromagnetics

Yang¹

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This thesis is devoted to the computation and analysis of electromagnetic wave scattering problems, with particular applications in accurate simulation of invisibility cloaks arisen from the field of transformation electromagnetics (TE). An efficient spectral-element method (SEM) is proposed for solving general two-dimensional Helmholtz equations in anisotropic media. In practice, we adopt a transparent boundary condition (TBC) characterized by the Dirichletto-Neumann (DtN) map to reduce wave propagation in an unbounded domain to a bounded domain. We then introduce a semi-analytic technique to integrate the global TBC with local curvilinear elements seamlessly, which is accomplished by using a novel elemental mapping and analytic formulas for evaluating global Fourier coefficients on spectral-element grids exactly. From the perspective of TE, an invisibility cloak is devised by a singular coordinate transformation of Maxwell's equations that leads to anisotropic materials coating the cloaked region to render any object inside invisible to observers outside. An important issue resides in the imposition of appropriate boundary conditions, i.e., cloaking boundary conditions (CBCs), in order to achieve perfect invisibility. Based upon the principle that a well-behaved electromagnetic field in the original space must be well-behaved in the transformed space as well, we propose new CBCs for circular, elliptic and polygonal invisibility cloaks from the essential "pole" conditions related to singular transformations. We emphasize that our proposal of CBCs is different from any existing ones.

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