Abstract

In this paper, we consider acoustic or electromagnetic scattering in two dimensions from an infinite three-layer medium with thousands of wavelength-size dielectric particles embedded in the middle layer. Such geometries are typical of microstructured composite materials, and the evaluation of the scattered field requires a suitable fast solver for either a single configuration or for a sequence of configurations as part of a design or optimization process. We have developed an algorithm for problems of this type by combining the Sommerfeld integral representation, high order integral equation discretization, the fast multipole method and classical multiple scattering theory. The efficiency of the solver is illustrated with several numerical experiments.

© 2014 Optical Society of America

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  1. G. Bao and J. Lai, “Radar cross section reduction of a cavity in the ground plane,” Commun. Comput. Phys. 15, 895–910 (2014).
  2. W. J. Parnell, I. D. Abrahams, and P. R. Brazier-Smith, “Effective properties of a composite half-space: Exploring the relationship between homogenization and multiple-scattering theories,” Quantum J. Mech. Appl. Math. 63, 145–175 (2010).
    [CrossRef]
  3. Y. Wu and Z.-Q. Zhang, “Dispersion relations and their symmetry properties of electromagnetic and elastic metamaterials in two dimensions,” Phys. Rev. B 79, 195111 (2009).
    [CrossRef]
  4. Z. Gimbutas and L. Greengard, “Fast multi-particle scattering: A hybrid solver for the Maxwell equations in microstructured materials,” J. Comput. Phys. 232, 22–32 (2013).
    [CrossRef]
  5. D. Colton and R. Kress, Integral Equation Method in Scattering Theory (Wiley-Interscience, 1983).
  6. W. C. Chew, Waves and Fields in Inhomogeneous Media (IEEE, 1995).
  7. M. O’Neil, L. Greengard, and A. Pataki, “On the efficient representation of the half-space impedance green’s function for the helmholtz equation,” Wave Motion 51, 1–13 (2014).
    [CrossRef]
  8. A. Barnett and L. Greengard, “A new integral representation for quasi-periodic scattering problems in two dimensions,” BIT Numer. Math. 51, 67–90 (2011).
    [CrossRef]
  9. D. Colton and R. Kress, Inverse Acoustic and Electromagnetic Scattering Theory, Applied Mathematical Sciences 93 (Springer-Verlag, 1998).
    [CrossRef]
  10. F. W. J. Olver, D. W. Lozier, R. F. Boisvert, and C. W. Clark, NIST Handbook of Mathematical Functions (Cambridge University, 2010).
  11. R. Kress and G. F. Roach, “Transmission problems for the helmholtz equation,” J. Math. Phys. 19, 1433–1437 (1978).
    [CrossRef]
  12. V. Rokhlin, “Rapid solution of integral equations of scattering theory in two dimensions,” J. Comput. Phys. 86, 414–439 (1990).
    [CrossRef]
  13. Y. Saad and M. Schultz, “Gmres: A generalized minimal residual algorithm for solving nonsymmetric linear-systems,” SIAM J. Sci. Stat. Comput. 7, 856–869 (1986).
  14. H. Cheng, W. Crutchfield, Z. Gimbutas, L. Greengard, J. Huang, V. Rokhlin, N. Yarvin, and J. Zhao, “Remarks on the implementation of the wideband fmm for the helmholtz equation in two dimensions,” Contemp. Math. 408, 99–110 (2006).
    [CrossRef]
  15. L. L. Foldy, “The multiple scattering of waves. i. general theory of isotropic scattering by randomly distributed scatterers,” Phys. Rev. 67, 107–119 (1945).
    [CrossRef]
  16. N. A. Gumerov and R. Duraiswami, “A scalar potential formulation and translation theory for the time-harmonic maxwell equations,” J. Comput. Phys. 225, 206–236 (2007).
    [CrossRef]
  17. K. Huang, P. Li, and H. Zhao, “An efficient algorithm for the generalized Foldy–Lax formulation,” J. Comput. Phys. 234, 376–398 (2013).
    [CrossRef] [PubMed]
  18. B. K. Alpert, “Hybrid Gauss-trapezoidal quadrature rules,” SIAM J. Sci. Comput. 20, 1551–1584 (1999).
    [CrossRef]
  19. C. Müller, Foundations of the Mathematical Theory of Electromagnetic Waves (Springer-Verlag, 1969).
    [CrossRef]
  20. M. Haider, S. Shipman, and S. Venakides, “Boundary-integral calculations of two-dimensional electromagnetic scattering in infinite photonic crystal slabs: Channel defects and resonances,” SIAM J. Appl. Math. 62, 2129–2148 (2002).
    [CrossRef]
  21. V. Rokhlin, “Solution of acoustic scattering problems by means of second kind integral equations,” Wave Motion 5, 257–272 (1983).
    [CrossRef]
  22. A. A. Lacis, L. D. Travis, and M. I. Mishchenko, Scattering, Absorption, and Emission of Light by Small Particles (Cambridge University, 2002).
  23. A. Dutt and V. Rokhlin, “Fast fourier transforms for nonequispaced data,” SIAM J. Sci. Comput. 14, 1368–1393 (1993).
    [CrossRef]
  24. A. Dutt and V. Rokhlin, “Fast fourier transforms for nonequispaced data, II,” Appl. Comput. Harmon. Anal. 2, 85–100 (1995).
    [CrossRef]
  25. L. Greengard and J. Lee, “Accelerating the nonuniform fast fourier transform,” SIAM Rev. 46, 443–454 (2004).
    [CrossRef]
  26. J. Lee and L. Greengard, “The type 3 nonuniform FFT and its applications,” J. Comput. Phys. 206, 1–5 (2005).
    [CrossRef]
  27. J.-P. Berrut and L. N. Trefethen, “Barycentric lagrange interpolation,” SIAM Rev. 46, 501–517 (2004).
    [CrossRef]
  28. H. Cheng, J. Huang, and T. J. Leiterman, “An adaptive fast solver for the modified helmholtz equation in two dimensions,” J. Comput. Phys. 211, 616–637 (2006).
    [CrossRef]
  29. J. Bremer, V. Rokhlin, and I. Sammis, “Universal quadratures for boundary integral equations on two-dimensional domains with corners,” J. Comput. Phys. 229, 8259–8280 (2010).
    [CrossRef]
  30. J. Helsing and R. Ojala, “Corner singularities for elliptic problems: Integral equations, graded meshes, quadrature, and compressed inverse preconditioning,” J. Comput. Phy. 227, 8820–8840 (2008).
    [CrossRef]
  31. M. H. Cho and W. Cai, “A parallel fast algorithm for computing the Helmholtz integral operator in 3-d layered media,” J. Comput. Phys. 231, 5910–5925 (2012).
    [CrossRef]
  32. K. L. Greengard, L. Ho, and J.-Y. Lee, “A fast direct solver for scattering from periodic structures with multiple material interfaces in two dimensions,” J. Comput. Phys. 258, 738–751 (2014).
    [CrossRef]

2014

G. Bao and J. Lai, “Radar cross section reduction of a cavity in the ground plane,” Commun. Comput. Phys. 15, 895–910 (2014).

M. O’Neil, L. Greengard, and A. Pataki, “On the efficient representation of the half-space impedance green’s function for the helmholtz equation,” Wave Motion 51, 1–13 (2014).
[CrossRef]

K. L. Greengard, L. Ho, and J.-Y. Lee, “A fast direct solver for scattering from periodic structures with multiple material interfaces in two dimensions,” J. Comput. Phys. 258, 738–751 (2014).
[CrossRef]

2013

Z. Gimbutas and L. Greengard, “Fast multi-particle scattering: A hybrid solver for the Maxwell equations in microstructured materials,” J. Comput. Phys. 232, 22–32 (2013).
[CrossRef]

K. Huang, P. Li, and H. Zhao, “An efficient algorithm for the generalized Foldy–Lax formulation,” J. Comput. Phys. 234, 376–398 (2013).
[CrossRef] [PubMed]

2012

M. H. Cho and W. Cai, “A parallel fast algorithm for computing the Helmholtz integral operator in 3-d layered media,” J. Comput. Phys. 231, 5910–5925 (2012).
[CrossRef]

2011

A. Barnett and L. Greengard, “A new integral representation for quasi-periodic scattering problems in two dimensions,” BIT Numer. Math. 51, 67–90 (2011).
[CrossRef]

2010

W. J. Parnell, I. D. Abrahams, and P. R. Brazier-Smith, “Effective properties of a composite half-space: Exploring the relationship between homogenization and multiple-scattering theories,” Quantum J. Mech. Appl. Math. 63, 145–175 (2010).
[CrossRef]

J. Bremer, V. Rokhlin, and I. Sammis, “Universal quadratures for boundary integral equations on two-dimensional domains with corners,” J. Comput. Phys. 229, 8259–8280 (2010).
[CrossRef]

2009

Y. Wu and Z.-Q. Zhang, “Dispersion relations and their symmetry properties of electromagnetic and elastic metamaterials in two dimensions,” Phys. Rev. B 79, 195111 (2009).
[CrossRef]

2008

J. Helsing and R. Ojala, “Corner singularities for elliptic problems: Integral equations, graded meshes, quadrature, and compressed inverse preconditioning,” J. Comput. Phy. 227, 8820–8840 (2008).
[CrossRef]

2007

N. A. Gumerov and R. Duraiswami, “A scalar potential formulation and translation theory for the time-harmonic maxwell equations,” J. Comput. Phys. 225, 206–236 (2007).
[CrossRef]

2006

H. Cheng, W. Crutchfield, Z. Gimbutas, L. Greengard, J. Huang, V. Rokhlin, N. Yarvin, and J. Zhao, “Remarks on the implementation of the wideband fmm for the helmholtz equation in two dimensions,” Contemp. Math. 408, 99–110 (2006).
[CrossRef]

H. Cheng, J. Huang, and T. J. Leiterman, “An adaptive fast solver for the modified helmholtz equation in two dimensions,” J. Comput. Phys. 211, 616–637 (2006).
[CrossRef]

2005

J. Lee and L. Greengard, “The type 3 nonuniform FFT and its applications,” J. Comput. Phys. 206, 1–5 (2005).
[CrossRef]

2004

J.-P. Berrut and L. N. Trefethen, “Barycentric lagrange interpolation,” SIAM Rev. 46, 501–517 (2004).
[CrossRef]

L. Greengard and J. Lee, “Accelerating the nonuniform fast fourier transform,” SIAM Rev. 46, 443–454 (2004).
[CrossRef]

2002

M. Haider, S. Shipman, and S. Venakides, “Boundary-integral calculations of two-dimensional electromagnetic scattering in infinite photonic crystal slabs: Channel defects and resonances,” SIAM J. Appl. Math. 62, 2129–2148 (2002).
[CrossRef]

1999

B. K. Alpert, “Hybrid Gauss-trapezoidal quadrature rules,” SIAM J. Sci. Comput. 20, 1551–1584 (1999).
[CrossRef]

1995

A. Dutt and V. Rokhlin, “Fast fourier transforms for nonequispaced data, II,” Appl. Comput. Harmon. Anal. 2, 85–100 (1995).
[CrossRef]

1993

A. Dutt and V. Rokhlin, “Fast fourier transforms for nonequispaced data,” SIAM J. Sci. Comput. 14, 1368–1393 (1993).
[CrossRef]

1990

V. Rokhlin, “Rapid solution of integral equations of scattering theory in two dimensions,” J. Comput. Phys. 86, 414–439 (1990).
[CrossRef]

1986

Y. Saad and M. Schultz, “Gmres: A generalized minimal residual algorithm for solving nonsymmetric linear-systems,” SIAM J. Sci. Stat. Comput. 7, 856–869 (1986).

1983

V. Rokhlin, “Solution of acoustic scattering problems by means of second kind integral equations,” Wave Motion 5, 257–272 (1983).
[CrossRef]

1978

R. Kress and G. F. Roach, “Transmission problems for the helmholtz equation,” J. Math. Phys. 19, 1433–1437 (1978).
[CrossRef]

1945

L. L. Foldy, “The multiple scattering of waves. i. general theory of isotropic scattering by randomly distributed scatterers,” Phys. Rev. 67, 107–119 (1945).
[CrossRef]

Abrahams, I. D.

W. J. Parnell, I. D. Abrahams, and P. R. Brazier-Smith, “Effective properties of a composite half-space: Exploring the relationship between homogenization and multiple-scattering theories,” Quantum J. Mech. Appl. Math. 63, 145–175 (2010).
[CrossRef]

Alpert, B. K.

B. K. Alpert, “Hybrid Gauss-trapezoidal quadrature rules,” SIAM J. Sci. Comput. 20, 1551–1584 (1999).
[CrossRef]

Bao, G.

G. Bao and J. Lai, “Radar cross section reduction of a cavity in the ground plane,” Commun. Comput. Phys. 15, 895–910 (2014).

Barnett, A.

A. Barnett and L. Greengard, “A new integral representation for quasi-periodic scattering problems in two dimensions,” BIT Numer. Math. 51, 67–90 (2011).
[CrossRef]

Berrut, J.-P.

J.-P. Berrut and L. N. Trefethen, “Barycentric lagrange interpolation,” SIAM Rev. 46, 501–517 (2004).
[CrossRef]

Boisvert, R. F.

F. W. J. Olver, D. W. Lozier, R. F. Boisvert, and C. W. Clark, NIST Handbook of Mathematical Functions (Cambridge University, 2010).

Brazier-Smith, P. R.

W. J. Parnell, I. D. Abrahams, and P. R. Brazier-Smith, “Effective properties of a composite half-space: Exploring the relationship between homogenization and multiple-scattering theories,” Quantum J. Mech. Appl. Math. 63, 145–175 (2010).
[CrossRef]

Bremer, J.

J. Bremer, V. Rokhlin, and I. Sammis, “Universal quadratures for boundary integral equations on two-dimensional domains with corners,” J. Comput. Phys. 229, 8259–8280 (2010).
[CrossRef]

Cai, W.

M. H. Cho and W. Cai, “A parallel fast algorithm for computing the Helmholtz integral operator in 3-d layered media,” J. Comput. Phys. 231, 5910–5925 (2012).
[CrossRef]

Cheng, H.

H. Cheng, J. Huang, and T. J. Leiterman, “An adaptive fast solver for the modified helmholtz equation in two dimensions,” J. Comput. Phys. 211, 616–637 (2006).
[CrossRef]

H. Cheng, W. Crutchfield, Z. Gimbutas, L. Greengard, J. Huang, V. Rokhlin, N. Yarvin, and J. Zhao, “Remarks on the implementation of the wideband fmm for the helmholtz equation in two dimensions,” Contemp. Math. 408, 99–110 (2006).
[CrossRef]

Chew, W. C.

W. C. Chew, Waves and Fields in Inhomogeneous Media (IEEE, 1995).

Cho, M. H.

M. H. Cho and W. Cai, “A parallel fast algorithm for computing the Helmholtz integral operator in 3-d layered media,” J. Comput. Phys. 231, 5910–5925 (2012).
[CrossRef]

Clark, C. W.

F. W. J. Olver, D. W. Lozier, R. F. Boisvert, and C. W. Clark, NIST Handbook of Mathematical Functions (Cambridge University, 2010).

Colton, D.

D. Colton and R. Kress, Inverse Acoustic and Electromagnetic Scattering Theory, Applied Mathematical Sciences 93 (Springer-Verlag, 1998).
[CrossRef]

D. Colton and R. Kress, Integral Equation Method in Scattering Theory (Wiley-Interscience, 1983).

Crutchfield, W.

H. Cheng, W. Crutchfield, Z. Gimbutas, L. Greengard, J. Huang, V. Rokhlin, N. Yarvin, and J. Zhao, “Remarks on the implementation of the wideband fmm for the helmholtz equation in two dimensions,” Contemp. Math. 408, 99–110 (2006).
[CrossRef]

Duraiswami, R.

N. A. Gumerov and R. Duraiswami, “A scalar potential formulation and translation theory for the time-harmonic maxwell equations,” J. Comput. Phys. 225, 206–236 (2007).
[CrossRef]

Dutt, A.

A. Dutt and V. Rokhlin, “Fast fourier transforms for nonequispaced data, II,” Appl. Comput. Harmon. Anal. 2, 85–100 (1995).
[CrossRef]

A. Dutt and V. Rokhlin, “Fast fourier transforms for nonequispaced data,” SIAM J. Sci. Comput. 14, 1368–1393 (1993).
[CrossRef]

Foldy, L. L.

L. L. Foldy, “The multiple scattering of waves. i. general theory of isotropic scattering by randomly distributed scatterers,” Phys. Rev. 67, 107–119 (1945).
[CrossRef]

Gimbutas, Z.

Z. Gimbutas and L. Greengard, “Fast multi-particle scattering: A hybrid solver for the Maxwell equations in microstructured materials,” J. Comput. Phys. 232, 22–32 (2013).
[CrossRef]

H. Cheng, W. Crutchfield, Z. Gimbutas, L. Greengard, J. Huang, V. Rokhlin, N. Yarvin, and J. Zhao, “Remarks on the implementation of the wideband fmm for the helmholtz equation in two dimensions,” Contemp. Math. 408, 99–110 (2006).
[CrossRef]

Greengard, K. L.

K. L. Greengard, L. Ho, and J.-Y. Lee, “A fast direct solver for scattering from periodic structures with multiple material interfaces in two dimensions,” J. Comput. Phys. 258, 738–751 (2014).
[CrossRef]

Greengard, L.

M. O’Neil, L. Greengard, and A. Pataki, “On the efficient representation of the half-space impedance green’s function for the helmholtz equation,” Wave Motion 51, 1–13 (2014).
[CrossRef]

Z. Gimbutas and L. Greengard, “Fast multi-particle scattering: A hybrid solver for the Maxwell equations in microstructured materials,” J. Comput. Phys. 232, 22–32 (2013).
[CrossRef]

A. Barnett and L. Greengard, “A new integral representation for quasi-periodic scattering problems in two dimensions,” BIT Numer. Math. 51, 67–90 (2011).
[CrossRef]

H. Cheng, W. Crutchfield, Z. Gimbutas, L. Greengard, J. Huang, V. Rokhlin, N. Yarvin, and J. Zhao, “Remarks on the implementation of the wideband fmm for the helmholtz equation in two dimensions,” Contemp. Math. 408, 99–110 (2006).
[CrossRef]

J. Lee and L. Greengard, “The type 3 nonuniform FFT and its applications,” J. Comput. Phys. 206, 1–5 (2005).
[CrossRef]

L. Greengard and J. Lee, “Accelerating the nonuniform fast fourier transform,” SIAM Rev. 46, 443–454 (2004).
[CrossRef]

Gumerov, N. A.

N. A. Gumerov and R. Duraiswami, “A scalar potential formulation and translation theory for the time-harmonic maxwell equations,” J. Comput. Phys. 225, 206–236 (2007).
[CrossRef]

Haider, M.

M. Haider, S. Shipman, and S. Venakides, “Boundary-integral calculations of two-dimensional electromagnetic scattering in infinite photonic crystal slabs: Channel defects and resonances,” SIAM J. Appl. Math. 62, 2129–2148 (2002).
[CrossRef]

Helsing, J.

J. Helsing and R. Ojala, “Corner singularities for elliptic problems: Integral equations, graded meshes, quadrature, and compressed inverse preconditioning,” J. Comput. Phy. 227, 8820–8840 (2008).
[CrossRef]

Ho, L.

K. L. Greengard, L. Ho, and J.-Y. Lee, “A fast direct solver for scattering from periodic structures with multiple material interfaces in two dimensions,” J. Comput. Phys. 258, 738–751 (2014).
[CrossRef]

Huang, J.

H. Cheng, J. Huang, and T. J. Leiterman, “An adaptive fast solver for the modified helmholtz equation in two dimensions,” J. Comput. Phys. 211, 616–637 (2006).
[CrossRef]

H. Cheng, W. Crutchfield, Z. Gimbutas, L. Greengard, J. Huang, V. Rokhlin, N. Yarvin, and J. Zhao, “Remarks on the implementation of the wideband fmm for the helmholtz equation in two dimensions,” Contemp. Math. 408, 99–110 (2006).
[CrossRef]

Huang, K.

K. Huang, P. Li, and H. Zhao, “An efficient algorithm for the generalized Foldy–Lax formulation,” J. Comput. Phys. 234, 376–398 (2013).
[CrossRef] [PubMed]

Kress, R.

R. Kress and G. F. Roach, “Transmission problems for the helmholtz equation,” J. Math. Phys. 19, 1433–1437 (1978).
[CrossRef]

D. Colton and R. Kress, Inverse Acoustic and Electromagnetic Scattering Theory, Applied Mathematical Sciences 93 (Springer-Verlag, 1998).
[CrossRef]

D. Colton and R. Kress, Integral Equation Method in Scattering Theory (Wiley-Interscience, 1983).

Lacis, A. A.

A. A. Lacis, L. D. Travis, and M. I. Mishchenko, Scattering, Absorption, and Emission of Light by Small Particles (Cambridge University, 2002).

Lai, J.

G. Bao and J. Lai, “Radar cross section reduction of a cavity in the ground plane,” Commun. Comput. Phys. 15, 895–910 (2014).

Lee, J.

J. Lee and L. Greengard, “The type 3 nonuniform FFT and its applications,” J. Comput. Phys. 206, 1–5 (2005).
[CrossRef]

L. Greengard and J. Lee, “Accelerating the nonuniform fast fourier transform,” SIAM Rev. 46, 443–454 (2004).
[CrossRef]

Lee, J.-Y.

K. L. Greengard, L. Ho, and J.-Y. Lee, “A fast direct solver for scattering from periodic structures with multiple material interfaces in two dimensions,” J. Comput. Phys. 258, 738–751 (2014).
[CrossRef]

Leiterman, T. J.

H. Cheng, J. Huang, and T. J. Leiterman, “An adaptive fast solver for the modified helmholtz equation in two dimensions,” J. Comput. Phys. 211, 616–637 (2006).
[CrossRef]

Li, P.

K. Huang, P. Li, and H. Zhao, “An efficient algorithm for the generalized Foldy–Lax formulation,” J. Comput. Phys. 234, 376–398 (2013).
[CrossRef] [PubMed]

Lozier, D. W.

F. W. J. Olver, D. W. Lozier, R. F. Boisvert, and C. W. Clark, NIST Handbook of Mathematical Functions (Cambridge University, 2010).

Mishchenko, M. I.

A. A. Lacis, L. D. Travis, and M. I. Mishchenko, Scattering, Absorption, and Emission of Light by Small Particles (Cambridge University, 2002).

Müller, C.

C. Müller, Foundations of the Mathematical Theory of Electromagnetic Waves (Springer-Verlag, 1969).
[CrossRef]

O’Neil, M.

M. O’Neil, L. Greengard, and A. Pataki, “On the efficient representation of the half-space impedance green’s function for the helmholtz equation,” Wave Motion 51, 1–13 (2014).
[CrossRef]

Ojala, R.

J. Helsing and R. Ojala, “Corner singularities for elliptic problems: Integral equations, graded meshes, quadrature, and compressed inverse preconditioning,” J. Comput. Phy. 227, 8820–8840 (2008).
[CrossRef]

Olver, F. W. J.

F. W. J. Olver, D. W. Lozier, R. F. Boisvert, and C. W. Clark, NIST Handbook of Mathematical Functions (Cambridge University, 2010).

Parnell, W. J.

W. J. Parnell, I. D. Abrahams, and P. R. Brazier-Smith, “Effective properties of a composite half-space: Exploring the relationship between homogenization and multiple-scattering theories,” Quantum J. Mech. Appl. Math. 63, 145–175 (2010).
[CrossRef]

Pataki, A.

M. O’Neil, L. Greengard, and A. Pataki, “On the efficient representation of the half-space impedance green’s function for the helmholtz equation,” Wave Motion 51, 1–13 (2014).
[CrossRef]

Roach, G. F.

R. Kress and G. F. Roach, “Transmission problems for the helmholtz equation,” J. Math. Phys. 19, 1433–1437 (1978).
[CrossRef]

Rokhlin, V.

J. Bremer, V. Rokhlin, and I. Sammis, “Universal quadratures for boundary integral equations on two-dimensional domains with corners,” J. Comput. Phys. 229, 8259–8280 (2010).
[CrossRef]

H. Cheng, W. Crutchfield, Z. Gimbutas, L. Greengard, J. Huang, V. Rokhlin, N. Yarvin, and J. Zhao, “Remarks on the implementation of the wideband fmm for the helmholtz equation in two dimensions,” Contemp. Math. 408, 99–110 (2006).
[CrossRef]

A. Dutt and V. Rokhlin, “Fast fourier transforms for nonequispaced data, II,” Appl. Comput. Harmon. Anal. 2, 85–100 (1995).
[CrossRef]

A. Dutt and V. Rokhlin, “Fast fourier transforms for nonequispaced data,” SIAM J. Sci. Comput. 14, 1368–1393 (1993).
[CrossRef]

V. Rokhlin, “Rapid solution of integral equations of scattering theory in two dimensions,” J. Comput. Phys. 86, 414–439 (1990).
[CrossRef]

V. Rokhlin, “Solution of acoustic scattering problems by means of second kind integral equations,” Wave Motion 5, 257–272 (1983).
[CrossRef]

Saad, Y.

Y. Saad and M. Schultz, “Gmres: A generalized minimal residual algorithm for solving nonsymmetric linear-systems,” SIAM J. Sci. Stat. Comput. 7, 856–869 (1986).

Sammis, I.

J. Bremer, V. Rokhlin, and I. Sammis, “Universal quadratures for boundary integral equations on two-dimensional domains with corners,” J. Comput. Phys. 229, 8259–8280 (2010).
[CrossRef]

Schultz, M.

Y. Saad and M. Schultz, “Gmres: A generalized minimal residual algorithm for solving nonsymmetric linear-systems,” SIAM J. Sci. Stat. Comput. 7, 856–869 (1986).

Shipman, S.

M. Haider, S. Shipman, and S. Venakides, “Boundary-integral calculations of two-dimensional electromagnetic scattering in infinite photonic crystal slabs: Channel defects and resonances,” SIAM J. Appl. Math. 62, 2129–2148 (2002).
[CrossRef]

Travis, L. D.

A. A. Lacis, L. D. Travis, and M. I. Mishchenko, Scattering, Absorption, and Emission of Light by Small Particles (Cambridge University, 2002).

Trefethen, L. N.

J.-P. Berrut and L. N. Trefethen, “Barycentric lagrange interpolation,” SIAM Rev. 46, 501–517 (2004).
[CrossRef]

Venakides, S.

M. Haider, S. Shipman, and S. Venakides, “Boundary-integral calculations of two-dimensional electromagnetic scattering in infinite photonic crystal slabs: Channel defects and resonances,” SIAM J. Appl. Math. 62, 2129–2148 (2002).
[CrossRef]

Wu, Y.

Y. Wu and Z.-Q. Zhang, “Dispersion relations and their symmetry properties of electromagnetic and elastic metamaterials in two dimensions,” Phys. Rev. B 79, 195111 (2009).
[CrossRef]

Yarvin, N.

H. Cheng, W. Crutchfield, Z. Gimbutas, L. Greengard, J. Huang, V. Rokhlin, N. Yarvin, and J. Zhao, “Remarks on the implementation of the wideband fmm for the helmholtz equation in two dimensions,” Contemp. Math. 408, 99–110 (2006).
[CrossRef]

Zhang, Z.-Q.

Y. Wu and Z.-Q. Zhang, “Dispersion relations and their symmetry properties of electromagnetic and elastic metamaterials in two dimensions,” Phys. Rev. B 79, 195111 (2009).
[CrossRef]

Zhao, H.

K. Huang, P. Li, and H. Zhao, “An efficient algorithm for the generalized Foldy–Lax formulation,” J. Comput. Phys. 234, 376–398 (2013).
[CrossRef] [PubMed]

Zhao, J.

H. Cheng, W. Crutchfield, Z. Gimbutas, L. Greengard, J. Huang, V. Rokhlin, N. Yarvin, and J. Zhao, “Remarks on the implementation of the wideband fmm for the helmholtz equation in two dimensions,” Contemp. Math. 408, 99–110 (2006).
[CrossRef]

Appl. Comput. Harmon. Anal.

A. Dutt and V. Rokhlin, “Fast fourier transforms for nonequispaced data, II,” Appl. Comput. Harmon. Anal. 2, 85–100 (1995).
[CrossRef]

BIT Numer. Math.

A. Barnett and L. Greengard, “A new integral representation for quasi-periodic scattering problems in two dimensions,” BIT Numer. Math. 51, 67–90 (2011).
[CrossRef]

Commun. Comput. Phys.

G. Bao and J. Lai, “Radar cross section reduction of a cavity in the ground plane,” Commun. Comput. Phys. 15, 895–910 (2014).

Contemp. Math.

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Figures (9)

Fig. 1
Fig. 1

Geometry of the three-layered medium, with a large number of dielectric particles embedded in the middle layer.

Fig. 2
Fig. 2

Two inclusions and their enclosing disks. The scattering matrix Si for each inclusion Ωi with wavenumber kp is defined as the map from an incoming field on Di to the corresponding outgoing field. It is computed by solving a sequence of boundary value problems on the inclusion itself in a precomputation phase (see text). In this paper, we assume that all the inclusions are identical but may be rotated, as drawn here.

Fig. 3
Fig. 3

The Sommerfeld contour in the complex λ plane: Each segment in the contour is discretized using Gauss-Legendre quadrature. The branch cut (shown in red) points upward from k and downward from −k.

Fig. 4
Fig. 4

Real part of the total field with 5, 000 dielectric inclusions randomly distributed in a three-layered medium. The wavenumber for each particle is kp = 2.0 and the wavenumbers for the three layers are k1 = 1.0, k2 = 3.0, k3 = 1.0. The size of each particle is approximately 0.1 wavelength for the wavenumber k2.

Fig. 5
Fig. 5

Convergence behavior of GMRES and the CPU time required for various numbers of inclusions embedded in either (a) free space or (b)a three layered medium. For (a), we set k1 = k2 = k3 = 3.0 and for (b), we set k1 = 1.0, k2 = 3.0, k3 = 1.0.

Fig. 6
Fig. 6

Real part of the total field for 200 dielectric inclusions distributed in a three layer medium with wavenumbers k1 = 1.0, k2 = 10.0 + 0.01i, k3 = 1.0. For each inclusion, the wavenumber is kp = 2.0 + 1.0i. The inclusions are approximately 0.3 wavelength in size for the wavenumber k2.

Fig. 7
Fig. 7

Convergence behavior of GMRES iteration and the CPU time required for 200 inclusions embedded in the central layer, where k2 is allowed to vary from 1.0 + 0.01i to 20.0 + 0.01i. In (a), we create a homogeneous background by setting k1 = k2 = k3, while in (b), k1 and k3 are fixed at 1.0, and k2 varies.

Fig. 8
Fig. 8

Real part of the total field when 1, 000 inclusions are embedded in a three-layered medium with k1 = 1.0, k2 = 3.0, k3 = 2.0. Each inclusion is a smoothed five-pointed star, approximately 0.2 wavelengths in size.

Fig. 9
Fig. 9

Convergence behavior of GMRES for a tolerance of 10−6 and the CPU time required as the number of inclusions embedded in the central layer varies. In (a), we create a homogeneous background by setting k1 = k2 = k3 = 3.0, while in (b), k1 = 1.0, k2 = 3.0, k3 = 2.0

Tables (1)

Tables Icon

Table 1 Comparison of CPU time in seconds for the Sommerfeld-to-local and multipole-to-Sommerfeld operators, using both the direct and NUFFT-based schemes (see text). The Sommerfeld contour is discretized with 500 Gauss-Legendre points (240 points for Γ1 and Γ3, with 20 points for Γ2).

Equations (53)

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Δ u s + k 2 u s = 0 ,
lim r r ( u s n i k u s ) = 0 ,
G k 1 ( x , x 0 ) = 1 4 π e λ 2 k 1 2 | y y 0 | λ 2 k 1 2 e i λ ( x x 0 ) d λ .
u 1 s = 1 4 π e λ 2 k 1 2 y λ 2 k 1 2 e i λ ( x x 0 ) σ 1 ( λ ) d λ ,
u 2 t = 1 4 π e λ 2 k 2 2 y λ 2 k 2 2 e i λ ( x x 0 ) σ 2 + ( λ ) d λ ,
u 2 b = 1 4 π e λ 2 k 2 2 ( y + d ) λ 2 k 2 2 e i λ ( x x 0 ) σ 2 ( λ ) d λ ,
u 3 s = 1 4 π e λ 2 k 3 2 ( y + d ) λ 2 k 3 2 e i λ ( x x 0 ) σ 3 ( λ ) d λ .
[ u ] = 0 ,
[ u n ] = 0 ,
( 1 λ 2 k 1 2 1 λ 2 k 2 2 e λ 2 k 2 2 d λ 2 k 2 2 0 0 e λ 2 k 2 2 d λ 2 k 2 2 1 λ 2 k 2 2 1 λ 2 k 3 2 1 1 e λ 2 k 2 2 d 0 0 e λ 2 k 2 2 d 1 1 ) ( σ 1 ( λ ) σ 2 + ( λ ) σ 2 ( λ ) σ 3 ( λ ) ) = ( e λ 2 k 1 2 y 0 λ 2 k 1 2 0 e λ 2 k 1 2 y 0 0 )
Δ u + k 2 2 u = 0 .
lim r r ( u s n i k 2 u s ) = 0 ,
Δ u + k p 2 u = 0 .
u s = n = β n H n ( k 2 r ) e i n θ
u = n = γ n J n ( k p r ) e i n θ
u inc = n = α n J n ( k 2 r ) e in θ , u inc r = n = α n k 2 J n ( k 2 r ) e in θ .
[ H n ( k 2 R ) J n ( k p R ) k 2 H n ( k 2 R ) k p J n ( k p R ) ] [ β n γ n ] = [ α n J n ( k 2 R ) α n k 2 J n ( k 2 R ) ] ,
β n = [ k p J n ( k 2 R ) J n ( k p R ) k 2 J n ( k 2 R ) J n ( k p R ) k 2 H n ( k 2 R ) J n ( k R ) k p J n ( k p R ) H n ( k 2 R ) ] α n ,
γ n = [ k 2 J n ( k 2 R ) H n ( k 2 R ) k 2 J n ( k 2 R ) H n ( k 2 R ) k 2 H n ( k 2 R ) J n ( k p R ) k p J n ( k p R ) H n ( k 2 R ) ] α n .
β n = J n ( k 2 R ) H n ( k 2 R ) α n .
β m = S p [ α m ] , for m = 1 , , M .
n = β n m H n ( k 2 r m ) e in θ m
u = n = α n l J n ( k 2 r l ) e in θ l
α n l = n = e in ( θ l m π ) β n n m H n ( k 2 x m x l ) .
α m = a m + j = 1 j m M T j m β j ,
( 𝒮 1 𝒯 ) [ β 1 β 2 β M ] = [ a 1 a 2 a M ] ,
𝒮 = [ S p S p S p ] , 𝒯 = [ 0 T 21 T M 1 T 12 0 T M 2 T 1 M T 2 M 0 ] .
u s = S k 2 σ + D k 2 μ , for x Ω c ,
u = S k p σ + D k p μ , for x Ω ,
S k σ = Ω G k ( x , y ) σ ( y ) d s y ,
D k μ = Ω G k ( x , y ) n ( y ) μ ( y ) d s y .
N k σ = Ω G k ( x , y ) n ( x ) σ ( y ) d s y , T k μ = Ω 2 G k ( x , y ) n ( x ) n ( y ) μ ( y ) d s y .
μ + [ S k 2 S k p ] σ + [ D k 2 D k p ] μ = u inc ,
σ + [ N k 2 N k p ] σ + [ T k 2 T k p ] μ = u inc n .
u = n = p p α n J n ( k 2 r ) e in θ ,
u = l = p p β l n H l ( k 2 r ) e i l θ ,
β l n = Ω j [ J l ( k 2 | y | ) e i l θ j ( y ) σ n ( y ) ] + n [ J l ( w | y | ) e i l θ j ( y ) μ n ( y ) ] d s y .
u 1 ( x ) = G k 1 ( x , x 0 ) + u 1 s u 2 ( x ) = u 2 t + u 2 b + j = 1 M n = p p β n m H n ( k 2 r m ) e in θ m u 3 ( x ) = u 3 s
{ Γ 1 : t i b , t ( 0 , ) , Γ 2 : i t , t [ b , b ] , Γ 3 : t + i b , t ( , 0 ) .
[ A B C D ] [ σ β ] = [ b 0 ] .
e i k r cos θ = n = i n J n ( k r ) e in θ .
e λ j 2 k 2 2 y + i λ j ( x x 0 ) = e λ j 2 k 2 2 y 1 + i λ j ( x 1 x 0 ) n = i n J n ( k 2 r ) e in ( ϕ + θ ) ,
1 4 π Γ 1 e λ 2 k 2 2 y λ 2 k 2 2 e i λ ( x x 0 ) σ 2 + ( λ ) d λ = 1 4 π e b ( x x 0 ) 0 t max g ( t ) e i t x d t ,
g ( t ) = e ( t i b ) 2 k 2 2 y ( t i b ) 2 k 2 2 e i t x 0 σ 2 + ( t i b ) .
a n = u n J n ( k 2 R ) .
a n = u n J n ( k 2 R ) + u n k J n ( k 2 R ) J n 2 ( k 2 R ) + ( k J n ( k 2 R ) ) 2 , for j = p , , p .
u j t = 1 4 π 1 λ 2 k 2 2 e i λ ( x x 0 ) σ m p + ( λ ) d λ ,
u j b = 1 4 π 1 λ 2 k 2 2 e i λ ( x x 0 ) σ m p ( λ ) d λ ,
H n ( k r ) e in θ = ( 1 ) n 4 π e λ 2 k 2 y j λ 2 k 2 e i λ ( x x j ) ( λ 2 k 2 + k 2 k 2 ) n d λ ,
H n ( k r ) e in θ = ( 1 ) n 4 π e λ 2 k 2 ( d + y j ) λ 2 k 2 e i λ ( x x j ) ( λ 2 k 2 k 2 k 2 ) n d λ .
{ σ m p + ( λ j ) } n e λ j 2 k 2 2 y j ( λ j 2 k 2 2 + k 2 2 k 2 2 ) n l = 1 n 1 a n l e i λ j ( x j x 0 ) { σ m p ( λ j ) } n e λ j 2 k 2 2 ( d + y j ) ( λ j 2 k 2 2 k 2 2 k 2 2 ) n l = 1 n 1 a n l e i λ j ( x j x 0 )
[ D CA 1 B ] [ β ] = CA 1 b
{ x = ( a 1 + a 2 cos ( a 3 t ) ) cos ( t ) , y = ( a 1 + a 2 cos ( a 3 t ) ) sin ( t ) , for 0 t < 2 π .

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