Improvements in the computational efficiency and convergence of the Invariant Imbedding T-matrix method for spheroids and hexagonal prisms

Siyao Zhai; R. Lee Panetta; R. Lee Panetta; Ping Yang; Ping Yang

doi:10.1364/OE.27.0A1441

Improvements in the computational efficiency and convergence of the Invariant Imbedding T-matrix method for spheroids and hexagonal prisms

Siyao Zhai, R. Lee Panetta, and Ping Yang

Optics Express
Vol. 27,
Issue 20,
pp. A1441-A1457
(2019)
•https://doi.org/10.1364/OE.27.0A1441

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Siyao Zhai, R. Lee Panetta, and Ping Yang, "Improvements in the computational efficiency and convergence of the Invariant Imbedding T-matrix method for spheroids and hexagonal prisms," Opt. Express 27, A1441-A1457 (2019)

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Table of Contents Category
- Atmospheric and Oceanic Optics
Optics & Photonics Topics
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About this Article
History
- Original Manuscript: June 7, 2019
- Revised Manuscript: July 15, 2019
- Manuscript Accepted: July 16, 2019
- Published: September 12, 2019

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Open Access

Abstract

The invariant-imbedding T-matrix (II-TM) method is a numerical method for accurately computing the single-scattering properties of dielectric particles. Because of the linearity of Maxwell’s equations, the incident electric field and the scattered electric field can be related through a transition matrix (T-Matrix). The II-TM method computes the T-matrix through a matrix recurrence formula which stems from an electromagnetic volume integral equation. The recurrence starts with an inscribed sphere within the particle and ends with the circumscribed sphere of the particle. For each iteration, a matrix known as the U-matrix is computed with the Gauss Legendre (GL) quadrature, and matrix inversion is subsequently performed to obtain the T-matrix corresponding to the portion of the particle enclosed by the spherical shell. We modify a commonly used scheme to avoid applying the quadrature scheme to discontinuities. Moreover, we apply a new scheme to generate nodes and weights in conjunction with the GL quadrature. This leads to a considerable improvement on convergence and computational efficiency in the cases of hexagonal prisms and spheroids. The basic shapes of ice crystals in the atmosphere are hexagonal columns and plates. The single-scattering properties of hexagonal ice prisms are important to atmospheric optics, radiative transfer, and remote sensing. We demonstrate that the present approach can significantly accelerate the convergence in simulating light scattering by hexagonal ice crystals.

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References

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|
Year
|
Author
|
Publication

C. F. Bohren and D. R. Huffman, Absorption and Scattering of Light by Small Particles (John Wiley & Sons, 1983).
H. C. van de Hulst, Light Scattering by Small Particles. (Dover, 1981).
P. C. Waterman, “Matrix formulation of electromagnetic scattering,” Proc. IEEE 53(8), 805–812 (1965).
[Crossref]
P. C. Waterman, “The T-matrix revisited,” J. Opt. Soc. Am. A 24(8), 2257–2267 (2007).
[Crossref] [PubMed]
N. G. Khlebtsov, “Orientational averaging of light-scattering observables in the J-matrix approach,” Appl. Opt. 31(25), 5359–5365 (1992).
[Crossref] [PubMed]
M. I. Mishchenko, “Light scattering by randomly oriented axially symmetric particles,” J. Opt. Soc. Am. A 8(6), 871–882 (1991).
[Crossref]
T. Wriedt, “Review of the null-field method with discrete sources,” J. Quant. Spectrosc. Radiat. Transf. 106(1-3), 535–545 (2007).
[Crossref]
M. I. Mishchenko and L. D. Travis, “Capabilities and limitations of a current FORTRAN implementation of the T-matrix method for randomly oriented, rotationally symmetric scatterers,” J. Quant. Spectrosc. Radiat. Transf. 60(3), 309–324 (1998).
[Crossref]
L. Bi, P. Yang, G. W. Kattawar, and M. I. Mishchenko, “Efficient implementation of the invariant imbedding T-matrix method and the separation of variables method applied to large nonspherical inhomogeneous particles,” J. Quant. Spectrosc. Radiat. Transf. 116, 169–183 (2013).
[Crossref]
L. Bi and P. Yang, “Accurate simulation of the optical properties of atmospheric ice crystals with the invariant imbedding T-matrix method,” J. Quant. Spectrosc. Radiat. Transf. 138, 17–35 (2014).
[Crossref]
T. Wriedt and Y. Eremin, The Generalized Multipole Technique for Light Scattering. (Springer, 2018).
I. Bogaert, “Iteration-Free Computation of Gauss–Legendre Quadrature Nodes and Weights,” SIAM J. Sci. Comput. 36(3), A1008–A1026 (2014).
[Crossref]
B. R. Johnson, “Invariant imbedding T matrix approach to electromagnetic scattering,” Appl. Opt. 27(23), 4861–4873 (1988).
[Crossref] [PubMed]
M. Mischenko, L. Travis, and A. Lacis, Scattering, absorption, and emission of light by small particles, (Cambridge University Press,1970).
W. R. C. Somerville, B. Auguié, and E. C. Le Ru, “Simplified expressions of the T-matrix integrals for electromagnetic scattering,” Opt. Lett. 36(17), 3482–3484 (2011).
[Crossref] [PubMed]
W. J. Wiscombe, “Improved Mie scattering algorithms,” Appl. Opt. 19(9), 1505–1509 (1980).
[Crossref] [PubMed]
F. M. Kahnert, J. J. Stamnes, and K. Stamnes, “Application of the extended boundary condition method to particles with sharp edges: a comparison of two surface integration approaches,” Appl. Opt. 40(18), 3101–3109 (2001).
[Crossref] [PubMed]
A. Glaser, X. Liu, and V. Rokhlin, “A Fast Algorithm for the Calculation of the Roots of Special Functions,” SIAM J. Sci. Comput. 29(4), 1420–1438 (2007).
[Crossref]
G. H. Golub and J. H. Welsch, “Calculation of Gauss Quadrature Rules,” Math. Comput. 23(106), 221–230 (1969).
[Crossref]
N. Hale and A. Townsend, “Fast and Accurate Computation of Gauss–Legendre and Gauss–Jacobi Quadrature Nodes and Weights,” SIAM J. Sci. Comput. 35(2), A652–A674 (2013).
[Crossref]
A. Townsend, T. Trogdon, and S. Olver, “Fast computation of Gauss quadrature nodes and weights on the whole real line,” IMA J. Numer. Anal. 36(1), 337–358 (2014).
A. Townsend, “The race to compute high-order Gauss-Legendre quadature,” SIAM J. Sci. Comput. (2015).
B. Sun, P. Yang, G. W. Kattawar, and X. Zhang, “Physical-geometric optics method for large size faceted particles,” Opt. Express 25(20), 24044–24060 (2017).
[Crossref] [PubMed]

2017 (1)

B. Sun, P. Yang, G. W. Kattawar, and X. Zhang, “Physical-geometric optics method for large size faceted particles,” Opt. Express 25(20), 24044–24060 (2017).
[Crossref] [PubMed]

2014 (3)

A. Townsend, T. Trogdon, and S. Olver, “Fast computation of Gauss quadrature nodes and weights on the whole real line,” IMA J. Numer. Anal. 36(1), 337–358 (2014).

L. Bi and P. Yang, “Accurate simulation of the optical properties of atmospheric ice crystals with the invariant imbedding T-matrix method,” J. Quant. Spectrosc. Radiat. Transf. 138, 17–35 (2014).
[Crossref]

I. Bogaert, “Iteration-Free Computation of Gauss–Legendre Quadrature Nodes and Weights,” SIAM J. Sci. Comput. 36(3), A1008–A1026 (2014).
[Crossref]

2013 (2)

L. Bi, P. Yang, G. W. Kattawar, and M. I. Mishchenko, “Efficient implementation of the invariant imbedding T-matrix method and the separation of variables method applied to large nonspherical inhomogeneous particles,” J. Quant. Spectrosc. Radiat. Transf. 116, 169–183 (2013).
[Crossref]

N. Hale and A. Townsend, “Fast and Accurate Computation of Gauss–Legendre and Gauss–Jacobi Quadrature Nodes and Weights,” SIAM J. Sci. Comput. 35(2), A652–A674 (2013).
[Crossref]

2011 (1)

W. R. C. Somerville, B. Auguié, and E. C. Le Ru, “Simplified expressions of the T-matrix integrals for electromagnetic scattering,” Opt. Lett. 36(17), 3482–3484 (2011).
[Crossref] [PubMed]

2007 (3)

A. Glaser, X. Liu, and V. Rokhlin, “A Fast Algorithm for the Calculation of the Roots of Special Functions,” SIAM J. Sci. Comput. 29(4), 1420–1438 (2007).
[Crossref]

P. C. Waterman, “The T-matrix revisited,” J. Opt. Soc. Am. A 24(8), 2257–2267 (2007).
[Crossref] [PubMed]

T. Wriedt, “Review of the null-field method with discrete sources,” J. Quant. Spectrosc. Radiat. Transf. 106(1-3), 535–545 (2007).
[Crossref]

2001 (1)

F. M. Kahnert, J. J. Stamnes, and K. Stamnes, “Application of the extended boundary condition method to particles with sharp edges: a comparison of two surface integration approaches,” Appl. Opt. 40(18), 3101–3109 (2001).
[Crossref] [PubMed]

1998 (1)

M. I. Mishchenko and L. D. Travis, “Capabilities and limitations of a current FORTRAN implementation of the T-matrix method for randomly oriented, rotationally symmetric scatterers,” J. Quant. Spectrosc. Radiat. Transf. 60(3), 309–324 (1998).
[Crossref]

1969 (1)

G. H. Golub and J. H. Welsch, “Calculation of Gauss Quadrature Rules,” Math. Comput. 23(106), 221–230 (1969).
[Crossref]

1965 (1)

P. C. Waterman, “Matrix formulation of electromagnetic scattering,” Proc. IEEE 53(8), 805–812 (1965).
[Crossref]

Auguié, B.

W. R. C. Somerville, B. Auguié, and E. C. Le Ru, “Simplified expressions of the T-matrix integrals for electromagnetic scattering,” Opt. Lett. 36(17), 3482–3484 (2011).
[Crossref] [PubMed]

Bi, L.

Bogaert, I.

I. Bogaert, “Iteration-Free Computation of Gauss–Legendre Quadrature Nodes and Weights,” SIAM J. Sci. Comput. 36(3), A1008–A1026 (2014).
[Crossref]

Glaser, A.

A. Glaser, X. Liu, and V. Rokhlin, “A Fast Algorithm for the Calculation of the Roots of Special Functions,” SIAM J. Sci. Comput. 29(4), 1420–1438 (2007).
[Crossref]

Golub, G. H.

G. H. Golub and J. H. Welsch, “Calculation of Gauss Quadrature Rules,” Math. Comput. 23(106), 221–230 (1969).
[Crossref]

Hale, N.

N. Hale and A. Townsend, “Fast and Accurate Computation of Gauss–Legendre and Gauss–Jacobi Quadrature Nodes and Weights,” SIAM J. Sci. Comput. 35(2), A652–A674 (2013).
[Crossref]

Johnson, B. R.

B. R. Johnson, “Invariant imbedding T matrix approach to electromagnetic scattering,” Appl. Opt. 27(23), 4861–4873 (1988).
[Crossref] [PubMed]

Kahnert, F. M.

Kattawar, G. W.

B. Sun, P. Yang, G. W. Kattawar, and X. Zhang, “Physical-geometric optics method for large size faceted particles,” Opt. Express 25(20), 24044–24060 (2017).
[Crossref] [PubMed]

Khlebtsov, N. G.

N. G. Khlebtsov, “Orientational averaging of light-scattering observables in the J-matrix approach,” Appl. Opt. 31(25), 5359–5365 (1992).
[Crossref] [PubMed]

Le Ru, E. C.

W. R. C. Somerville, B. Auguié, and E. C. Le Ru, “Simplified expressions of the T-matrix integrals for electromagnetic scattering,” Opt. Lett. 36(17), 3482–3484 (2011).
[Crossref] [PubMed]

Liu, X.

A. Glaser, X. Liu, and V. Rokhlin, “A Fast Algorithm for the Calculation of the Roots of Special Functions,” SIAM J. Sci. Comput. 29(4), 1420–1438 (2007).
[Crossref]

Mishchenko, M. I.

M. I. Mishchenko, “Light scattering by randomly oriented axially symmetric particles,” J. Opt. Soc. Am. A 8(6), 871–882 (1991).
[Crossref]

Olver, S.

A. Townsend, T. Trogdon, and S. Olver, “Fast computation of Gauss quadrature nodes and weights on the whole real line,” IMA J. Numer. Anal. 36(1), 337–358 (2014).

Rokhlin, V.

A. Glaser, X. Liu, and V. Rokhlin, “A Fast Algorithm for the Calculation of the Roots of Special Functions,” SIAM J. Sci. Comput. 29(4), 1420–1438 (2007).
[Crossref]

Somerville, W. R. C.

W. R. C. Somerville, B. Auguié, and E. C. Le Ru, “Simplified expressions of the T-matrix integrals for electromagnetic scattering,” Opt. Lett. 36(17), 3482–3484 (2011).
[Crossref] [PubMed]

Stamnes, J. J.

Stamnes, K.

Sun, B.

B. Sun, P. Yang, G. W. Kattawar, and X. Zhang, “Physical-geometric optics method for large size faceted particles,” Opt. Express 25(20), 24044–24060 (2017).
[Crossref] [PubMed]

Townsend, A.

A. Townsend, T. Trogdon, and S. Olver, “Fast computation of Gauss quadrature nodes and weights on the whole real line,” IMA J. Numer. Anal. 36(1), 337–358 (2014).

N. Hale and A. Townsend, “Fast and Accurate Computation of Gauss–Legendre and Gauss–Jacobi Quadrature Nodes and Weights,” SIAM J. Sci. Comput. 35(2), A652–A674 (2013).
[Crossref]

A. Townsend, “The race to compute high-order Gauss-Legendre quadature,” SIAM J. Sci. Comput. (2015).

Travis, L. D.

Trogdon, T.

A. Townsend, T. Trogdon, and S. Olver, “Fast computation of Gauss quadrature nodes and weights on the whole real line,” IMA J. Numer. Anal. 36(1), 337–358 (2014).

Waterman, P. C.

P. C. Waterman, “The T-matrix revisited,” J. Opt. Soc. Am. A 24(8), 2257–2267 (2007).
[Crossref] [PubMed]

P. C. Waterman, “Matrix formulation of electromagnetic scattering,” Proc. IEEE 53(8), 805–812 (1965).
[Crossref]

Welsch, J. H.

G. H. Golub and J. H. Welsch, “Calculation of Gauss Quadrature Rules,” Math. Comput. 23(106), 221–230 (1969).
[Crossref]

Wiscombe, W. J.

W. J. Wiscombe, “Improved Mie scattering algorithms,” Appl. Opt. 19(9), 1505–1509 (1980).
[Crossref] [PubMed]

Wriedt, T.

T. Wriedt, “Review of the null-field method with discrete sources,” J. Quant. Spectrosc. Radiat. Transf. 106(1-3), 535–545 (2007).
[Crossref]

Yang, P.

B. Sun, P. Yang, G. W. Kattawar, and X. Zhang, “Physical-geometric optics method for large size faceted particles,” Opt. Express 25(20), 24044–24060 (2017).
[Crossref] [PubMed]

Zhang, X.

B. Sun, P. Yang, G. W. Kattawar, and X. Zhang, “Physical-geometric optics method for large size faceted particles,” Opt. Express 25(20), 24044–24060 (2017).
[Crossref] [PubMed]

Appl. Opt. (4)

N. G. Khlebtsov, “Orientational averaging of light-scattering observables in the J-matrix approach,” Appl. Opt. 31(25), 5359–5365 (1992).
[Crossref] [PubMed]

B. R. Johnson, “Invariant imbedding T matrix approach to electromagnetic scattering,” Appl. Opt. 27(23), 4861–4873 (1988).
[Crossref] [PubMed]

W. J. Wiscombe, “Improved Mie scattering algorithms,” Appl. Opt. 19(9), 1505–1509 (1980).
[Crossref] [PubMed]

IMA J. Numer. Anal. (1)

A. Townsend, T. Trogdon, and S. Olver, “Fast computation of Gauss quadrature nodes and weights on the whole real line,” IMA J. Numer. Anal. 36(1), 337–358 (2014).

J. Opt. Soc. Am. A (2)

M. I. Mishchenko, “Light scattering by randomly oriented axially symmetric particles,” J. Opt. Soc. Am. A 8(6), 871–882 (1991).
[Crossref]

P. C. Waterman, “The T-matrix revisited,” J. Opt. Soc. Am. A 24(8), 2257–2267 (2007).
[Crossref] [PubMed]

J. Quant. Spectrosc. Radiat. Transf. (4)

T. Wriedt, “Review of the null-field method with discrete sources,” J. Quant. Spectrosc. Radiat. Transf. 106(1-3), 535–545 (2007).
[Crossref]

Math. Comput. (1)

G. H. Golub and J. H. Welsch, “Calculation of Gauss Quadrature Rules,” Math. Comput. 23(106), 221–230 (1969).
[Crossref]

Opt. Express (1)

B. Sun, P. Yang, G. W. Kattawar, and X. Zhang, “Physical-geometric optics method for large size faceted particles,” Opt. Express 25(20), 24044–24060 (2017).
[Crossref] [PubMed]

Opt. Lett. (1)

W. R. C. Somerville, B. Auguié, and E. C. Le Ru, “Simplified expressions of the T-matrix integrals for electromagnetic scattering,” Opt. Lett. 36(17), 3482–3484 (2011).
[Crossref] [PubMed]

Proc. IEEE (1)

P. C. Waterman, “Matrix formulation of electromagnetic scattering,” Proc. IEEE 53(8), 805–812 (1965).
[Crossref]

SIAM J. Sci. Comput. (3)

I. Bogaert, “Iteration-Free Computation of Gauss–Legendre Quadrature Nodes and Weights,” SIAM J. Sci. Comput. 36(3), A1008–A1026 (2014).
[Crossref]

N. Hale and A. Townsend, “Fast and Accurate Computation of Gauss–Legendre and Gauss–Jacobi Quadrature Nodes and Weights,” SIAM J. Sci. Comput. 35(2), A652–A674 (2013).
[Crossref]

A. Glaser, X. Liu, and V. Rokhlin, “A Fast Algorithm for the Calculation of the Roots of Special Functions,” SIAM J. Sci. Comput. 29(4), 1420–1438 (2007).
[Crossref]

Other (5)

M. Mischenko, L. Travis, and A. Lacis, Scattering, absorption, and emission of light by small particles, (Cambridge University Press,1970).

C. F. Bohren and D. R. Huffman, Absorption and Scattering of Light by Small Particles (John Wiley & Sons, 1983).

H. C. van de Hulst, Light Scattering by Small Particles. (Dover, 1981).

T. Wriedt and Y. Eremin, The Generalized Multipole Technique for Light Scattering. (Springer, 2018).

A. Townsend, “The race to compute high-order Gauss-Legendre quadature,” SIAM J. Sci. Comput. (2015).

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Fig. 1 Schematic of the radial recurrence. Particle in this schematic configuration is a particle with arbitrary geometry marked in green. Coordinate origin is on the geometric center of the particle. Two spherical shells are indicated by dashed circles. Arrows indicate the two radii

r_{p}

and

r_{p + 1}

Download Full Size | PPT Slide | PDF

Fig. 2 Procedures in the sequence of expanding spherical shells in the recurrence. In the case of the spheroid,

θ_{1} = 0

θ_{2}

is the angle where the spherical shell intersects the spheroid.

Download Full Size | PPT Slide | PDF

Fig. 3 Procedures in the sequence of expanding spherical shells in the recurrence for a hexagonal column.

θ_{2}

is the angle where the spherical shell intersects the column.

Download Full Size | PPT Slide | PDF

Fig. 4 An example

F_{m m'} (r, θ)

as function of

θ

for hexagonal column.

Download Full Size | PPT Slide | PDF

Fig. 5

f_{1} (x)

and

f_{2} (x)

Download Full Size | PPT Slide | PDF

Fig. 6 Convergence rate of the outputs (

Q_{e x t}

, asymmetry factor (g factor),

P_{11} (180^{\circ})

) with respect to number of quadrature nodes

N_{q}

. Six panels correspond to 6 different spheroidal shapes indicated with size parameter x and aspect ratio a/b.

Download Full Size | PPT Slide | PDF

Fig. 7 Convergence rate of the outputs (

Q_{e x t}

, asymmetry factor (g factor),

P_{11} (180^{\circ})

) with respect to number of quadrature nodes

N_{q}

. Six panels correspond to 6 different hexagonal column shapes indicated with size parameter x and aspect ratio a/b.

Download Full Size | PPT Slide | PDF

Fig. 8 Convergence rate of the outputs (

Q_{e x t}

, asymmetry factor (g factor),

P_{11} (180^{\circ})

) with respect to number of quadrature nodes

N_{q}

. Six panels correspond to 6 different hexagonal column shapes indicated with size parameter x and aspect ratio a/b. Unlike Fig. 7, the integration range is split into 2 smooth intervals so the convergence rate is accelerated (see the text).

Download Full Size | PPT Slide | PDF

Fig. 9 Phase matrix elements for hexagonal column with size parameter x = 225 (defined with the circumscribed sphere radius). Comparison is between PGOM (blue) and II-TM (red dashed). Particle refractive index is

m = 1.308 + i 1.43 \times 10^{- 9}

Download Full Size | PPT Slide | PDF

Fig. 10 Phase matrix elements for hexagonal column with size parameter x = 225 (defined with the circumscribed sphere radius). Comparison is between PGOM (blue) and II-TM (red dashed). Particle refractive index is

m = 1.276 + i 0.413

Download Full Size | PPT Slide | PDF

Tables (3)

Table 1 Convergence rate of Gaussian quadrature for 2 different integrands.

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Table 2 CPU time required to run the same code section with Newton and Bogaert algorithms.

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Table 3 Acceleration for hexagonal column and spheroids of various sizes and aspect ratios.

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Equations (39)

Equations on this page are rendered with MathJax. Learn more.

{\overset{⇀}{E}}_{i n c} (\overset{⇀}{r}) = E_{0} \sum_{n = 0}^{\infty} \sum_{m = - n}^{n} a_{m n} {\overset{⇀}{M}}_{m n}^{(1)} (k \overset{⇀}{r}) + b_{m n} {\overset{⇀}{N}}_{m n}^{(1)} (k \overset{⇀}{r}),

{\overset{⇀}{E}}_{s c a} (\overset{⇀}{r}) = E_{0} \sum_{n = 0}^{\infty} \sum_{m = - n}^{n} p_{m n} {\overset{⇀}{M}}_{m n}^{(3)} (k \overset{⇀}{r}) + q_{m n} {\overset{⇀}{N}}_{m n}^{(3)} (k \overset{⇀}{r}),

(\begin{matrix} p_{m n} \\ q_{m n} \end{matrix}) = E_{0} \sum_{n^{'} = 0}^{\infty} \sum_{m^{'} = - n^{'}}^{n^{'}} (\begin{matrix} T_{m n m^{'} n^{'}}^{11} & T_{m n m^{'} n^{'}}^{12} \\ T_{m n m^{'} n^{'}}^{21} & T_{m n m^{'} n^{'}}^{22} \end{matrix}) (\begin{matrix} a_{m^{'} n^{'}} \\ b_{m^{'} n^{'}} \end{matrix}) .

p = T a .

\begin{array}{l} T (r_{p}) = Q_{11} (r_{p}) + [I + Q_{12} (r_{p})] {[I - T (r_{p - 1}) Q_{22} (r_{p})]}^{- 1} \cdot \\ T (r_{p}) [I + Q_{21} (r_{p})] . \end{array}

Q_{11} (r_{p}) = i k J^{T} (r_{p}) Q (r_{p}) J (r_{p}),

Q_{12} (r_{p}) = i k J^{T} (r_{p}) Q (r_{p}) H (r_{p}),

Q_{21} (r_{p}) = i k H^{T} (r_{p}) Q (r_{p}) J (r_{p}),

Q_{22} (r_{p}) = i k H^{T} (r_{p}) Q (r_{p}) H (r_{p}),

{\bar{\bar{J}}}_{n} (r_{p}) = (\begin{matrix} j_{n} (k r_{p}) \\ 0 \\ 0 \end{matrix} \begin{matrix} 0 \\ \frac{1}{k r} \frac{\partial}{\partial r} r j_{n} (k r_{p}) \\ \sqrt{n (n + 1)} \frac{j_{n} (k r_{p})}{k r_{p}} \end{matrix}),

{\bar{\bar{H}}}_{n} (r_{p}) = (\begin{matrix} h_{n}^{(1)} (k r_{p}) \\ 0 \\ 0 \end{matrix} \begin{matrix} 0 \\ \frac{1}{k r} \frac{\partial}{\partial r} r h_{n}^{(1)} (k r_{p}) \\ \sqrt{n (n + 1)} \frac{h_{n}^{(1)} (k r_{p})}{k r_{p}} \end{matrix}),

Q (r_{p}) = Δ r {[I - Δ r U (r_{p}) g (r_{p}, r_{p})]}^{- 1} U (r_{p}) .

U_{m n m^{'} n^{'}}^{i j} = A_{m n m^{'} n^{'}} \int_{0}^{π} d θ \sin θ F_{m m^{'}} (r, θ) K_{m n m^{'} n^{'}}^{i j} (θ), i, j = 1, 2,

U_{m n m^{'} n^{'}}^{33} = A^{'}_{m n m^{'} n^{'}} \int_{0}^{π} d θ \sin θ F^{'}_{m m^{'}} (r, θ) d_{0 m}^{n} (θ) d_{0 m^{'}}^{n^{'}} (θ),

F_{m m^{'}} (r, θ) = \int_{0}^{2 π} d φ e^{- i (m - m^{'}) φ} [ε (r, θ, φ) - 1],

F^{'}_{m m^{'}} (r, θ) = \int_{0}^{2 π} d φ e^{- i (m - m^{'}) φ} \frac{[ε (r, θ, φ) - 1]}{ε (r, θ, φ)},

A_{m n m^{'} n^{'}} = {(k r)}^{2} {(- 1)}^{m + m^{'}} {[\frac{2 n + 1}{4 π n (n + 1)}]}^{1 / 2} {[\frac{2 n^{'} + 1}{4 π n^{'} (n^{'} + 1)}]}^{1 / 2},

A^{'}_{m n m^{'} n^{'}} = \sqrt{n n^{'} (n + 1) (n^{'} + 1)} A_{m n m^{'} n^{'}} .

\begin{array}{l} K_{m n m^{'} n^{'}}^{i j} (θ) = \\ (\begin{matrix} π_{m n} (θ) π_{m^{'} n^{'}} (θ) + τ_{m n} (θ) τ_{m^{'} n^{'}} (θ) & - i [π_{m n} (θ) τ_{m^{'} n^{'}} (θ) + τ_{m n} (θ) π_{m^{'} n^{'}} (θ)] \\ i [π_{m n} (θ) τ_{m^{'} n^{'}} (θ) + τ_{m n} (θ) π_{m^{'} n^{'}} (θ)] & π_{m n} (θ) π_{m^{'} n^{'}} (θ) + τ_{m n} (θ) τ_{m^{'} n^{'}} (θ) \end{matrix}), \end{array}

π_{m n} (θ) = \frac{m}{\sin θ} d_{0 m}^{n} (θ),

τ_{m n} (θ) = \frac{d}{d θ} d_{0 m}^{n} (θ) .

ε (r, θ) - 1 = {\begin{matrix} 0, θ_{2} < θ, \\ ε - 1, 0 < θ < θ_{2}, \end{matrix}

\begin{array}{l} [π_{m n} (θ) π_{m n^{'}} (θ) + τ_{m n} (θ) τ_{m n^{'}} (θ)] \sin θ \\ = \frac{d}{d θ} (τ_{m n'} (θ) d_{0 m}^{n} (θ) \sin θ) + n' (n' + 1) \sin θ d_{0 m}^{n} (θ) d_{0 m}^{n'} (θ) \\ = \frac{d}{d θ} (τ_{m n} (θ) d_{0 m}^{n'} (θ) \sin θ) + n (n + 1) \sin θ d_{0 m}^{n} (θ) d_{0 m}^{n'} (θ), \end{array}

[π_{m n} (θ) τ_{m n^{'}} (θ) + τ_{m n} (θ) π_{m n^{'}} (θ)] \sin θ = m \frac{d}{d θ} (d_{0 m}^{n} (θ) d_{0 m}^{n'} (θ)) .

\begin{array}{l} U_{m n m' n'}^{11} = U_{m n m' n'}^{22} \\ = A_{m n m' n'} 2 π δ_{m m'} [ε - 1] \cdot \\ [c_{n n'} τ_{n n'} (θ_{2}) d_{0 m}^{n} (θ_{2}) \sin θ_{2} + n' (n' + 1) \int_{0}^{θ_{2}} d θ \sin θ d_{0 m}^{n} (θ) d_{0 m}^{n'} (θ)], \\ c_{n n'} = 1 + (^{- 1) n + n'}, \end{array}

U_{m n m' n'}^{33} = A'_{m n m' n'} 2 π δ_{m m'} \frac{ε - 1}{ε} c_{n n'} \int_{0}^{θ_{2}} d θ \sin θ d_{0 m}^{n} (θ) d_{0 m}^{n'} (θ),

\begin{array}{l} U_{m n m' n'}^{12} = - U_{m n m' n'}^{21} \\ = A_{m n m' n'} 2 π δ_{m m'} [ε - 1] c'_{n n'} (- i m) [d_{0 m}^{n} (θ_{2}) d_{0 m}^{n'} (θ_{2}) - d_{0 m}^{n} (0) d_{0 m}^{n'} (0)], \\ c'_{n n'} = 1 + (^{- 1) n + n' + 1}, \end{array}

\begin{array}{l} U_{m n m' n'}^{11} = U_{m n m' n'}^{22} = A_{m n m' n'} c_{n n'} \int_{θ_{1}}^{θ_{2}} d θ \sin θ F_{m m'} (r, θ) K_{m n m' n'}^{i i} (θ), \\ c_{n n'} = 1 + (^{- 1) n + n'}, \end{array}

U_{m n m' n'}^{33} = A'_{m n m' n'} c_{n n'} \int_{θ_{1}}^{θ_{2}} d θ \sin θ F'_{m m'} (r, θ) d_{0 m}^{n} (θ) d_{0 m'}^{n'} (θ),

\begin{array}{l} U_{m n m' n'}^{12} = - U_{m n m' n'}^{21} \\ = A_{m n m' n'} c'_{n n'} \int_{θ_{1}}^{θ_{2}} d θ \sin θ F_{m m'} (r, θ) K_{m n m' n'}^{i j} (θ), \\ c'_{n n'} = 1 + (^{- 1) n + n' + 1} . \end{array}

\int_{0}^{π} (d θ \sin θ) (F_{m m'} (r, θ) K_{m n m' n'}^{i j} (θ)) \approx \sum_{l = 1}^{N_{q}} w_{l} F {}_{m m'}{(r, θ_{l})} K_{m n m' n'}^{i j} (θ_{l}) .

\int_{0}^{1} f_{1} (x) d x \approx \sum_{n = 1}^{N_{q}} w_{n} f_{1} (x_{n}), f_{1} (x) = {\begin{matrix} 1, 0 \leq x \leq 0.5, \\ 2 (1 - x), 0.5 < x \leq 1. \end{matrix}

\int_{0}^{1} f_{2} (x) d x \approx \sum_{n = 1}^{N_{q}} w_{n} f_{2} (x_{n}), f_{2} (x) = {\begin{matrix} 1, 0 \leq x \leq 0.5, \\ 0.5 [1 + \cos 2 π (x - 0.5)], 0.5 < x \leq 1. \end{matrix}

\int_{0}^{θ_{2}} d θ \sin θ d_{0 m}^{n} (θ) d_{0 m}^{n'} (θ),

N = x + 4.05 x^{1 / 3} + 5,

N_{M I E} = x + 4.05 x^{1 / 3} + 2,

\frac{Δ r}{R - r_{0}} = 0.1,

\int_{θ_{1}}^{θ_{2}} d θ \sin θ F_{m m'} (r, θ) K_{m n m' n'}^{i j} (θ) .

\begin{array}{l} \int_{θ_{1}}^{θ_{2}} d θ \sin θ F_{m m'} (r, θ) K_{m n m' n'}^{i j} (θ) = \\ \int_{θ_{1}}^{θ_{j u m p}} d θ \sin θ F_{m m'} (r, θ) K_{m n m' n'}^{i j} (θ) + \int_{θ_{j u m p}}^{θ_{2}} d θ \sin θ F_{m m'} (r, θ) K_{m n m' n'}^{i j} (θ) . \end{array}

	$f_{1} (x)$	$f_{2} (x)$
	Relative error = (Estimate – Exact)/Exact
$N_{q}$	order of magnitude of relative error	order of magnitude of relative error
10	$10^{- 3}$	$10^{- 9}$
100	$10^{- 5}$	$10^{- 9}$
1000	$10^{- 7}$	$10^{- 9}$
10000	$10^{- 9}$	$10^{- 9}$

$N_{q}$	$f_{1} (x)$		CPU time acceleration(Newton/Bogaert)
$N_{q}$	CPU time(sec) Newton	CPU time(sec) Bogaert	CPU time acceleration(Newton/Bogaert)
10	<0.001	<0.001	–
100	<0.001	<0.001	–
1000	0.026	<0.001	–
10000	1.643	0.001	1643

Shape	Size parameter	Aspect ratio	CPU time acceleration(Newton/Bogaert)	Wall clock time acceleration (Newton/Bogaert)
Hexagonal columns	x = 30	0.5	7.9	283.7
	x = 30	0.75	9.5	432
	x = 65	0.5	2.2	34.1
	x = 65	0.75	2.3	34.6
	x = 100	0.5	1.7	8.6
	x = 100	0.75	1.6	9.1

spheroids	x = 30	0.5	1.1	0.9
	x = 30	0.75	1.03	1.1
	x = 65	0.5	1.03	1.0
	x = 65	0.75	1.2	1.1
	x = 100	0.5	1.1	0.8
	x = 100	0.75	1.03	1.0

	$f_{1} (x)$	$f_{2} (x)$
	Relative error = (Estimate – Exact)/Exact
$N_{q}$	order of magnitude of relative error	order of magnitude of relative error
10	$10^{- 3}$	$10^{- 9}$
100	$10^{- 5}$	$10^{- 9}$
1000	$10^{- 7}$	$10^{- 9}$
10000	$10^{- 9}$	$10^{- 9}$

$N_{q}$	$f_{1} (x)$		CPU time acceleration(Newton/Bogaert)
$N_{q}$	CPU time(sec) Newton	CPU time(sec) Bogaert	CPU time acceleration(Newton/Bogaert)
10	<0.001	<0.001	–
100	<0.001	<0.001	–
1000	0.026	<0.001	–
10000	1.643	0.001	1643

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