### Abstract

A periodic fast multipole algorithm (P-FMA) is devised for evaluating 3-D periodic Green's function (PGF) for a 2-D lattice which can be used to solve scattering by a structure with 2-D periodicity. The introduction of periodicity in the Green's function formulation produces image sources at each lattice site. Like multilevel FMA (ML-FMA), P-FMA takes advantage of the distance between image sources and observation points to factorize the field using multipoles. By substituting known factorizations of the free-space Green's function into the expression for PGF, one can isolate the summation over the lattice into the translation phase of the FMA. For both plane wave and multipole factorizations, a common term known as lattice constant appears. The lattice constant is an infinite sum over the lattice which does not converge absolutely when expressed as a spatial sum. Using the Ewald summation technique, the lattice constants can be evaluated with exponential convergence and high accuracy. The resulting P-FMA is between O(N) and O(N log N) in memory use and computational complexity, depending on the object size relative to the wavelength.

Original language | English (US) |
---|---|

Article number | 7891536 |

Pages (from-to) | 3134-3145 |

Number of pages | 12 |

Journal | IEEE Transactions on Antennas and Propagation |

Volume | 65 |

Issue number | 6 |

DOIs | |

State | Published - Jan 1 2017 |

### Fingerprint

### Keywords

- Ewald summation
- Fast multipole method (ML-FMA)
- Lattice sum
- Method of moments (MoM)
- Multilevel
- Periodic Green's function (PGF)
- Periodic scattering
- Periodic structures

### ASJC Scopus subject areas

- Electrical and Electronic Engineering

### Cite this

*IEEE Transactions on Antennas and Propagation*,

*65*(6), 3134-3145. [7891536]. https://doi.org/10.1109/TAP.2017.2690533

**A broadband ML-FMA for 3-D periodic green's function in 2-D lattice using ewald summation.** / Wei, Michael; Chew, Weng C.

Research output: Contribution to journal › Article

*IEEE Transactions on Antennas and Propagation*, vol. 65, no. 6, 7891536, pp. 3134-3145. https://doi.org/10.1109/TAP.2017.2690533

}

TY - JOUR

T1 - A broadband ML-FMA for 3-D periodic green's function in 2-D lattice using ewald summation

AU - Wei, Michael

AU - Chew, Weng C.

PY - 2017/1/1

Y1 - 2017/1/1

N2 - A periodic fast multipole algorithm (P-FMA) is devised for evaluating 3-D periodic Green's function (PGF) for a 2-D lattice which can be used to solve scattering by a structure with 2-D periodicity. The introduction of periodicity in the Green's function formulation produces image sources at each lattice site. Like multilevel FMA (ML-FMA), P-FMA takes advantage of the distance between image sources and observation points to factorize the field using multipoles. By substituting known factorizations of the free-space Green's function into the expression for PGF, one can isolate the summation over the lattice into the translation phase of the FMA. For both plane wave and multipole factorizations, a common term known as lattice constant appears. The lattice constant is an infinite sum over the lattice which does not converge absolutely when expressed as a spatial sum. Using the Ewald summation technique, the lattice constants can be evaluated with exponential convergence and high accuracy. The resulting P-FMA is between O(N) and O(N log N) in memory use and computational complexity, depending on the object size relative to the wavelength.

AB - A periodic fast multipole algorithm (P-FMA) is devised for evaluating 3-D periodic Green's function (PGF) for a 2-D lattice which can be used to solve scattering by a structure with 2-D periodicity. The introduction of periodicity in the Green's function formulation produces image sources at each lattice site. Like multilevel FMA (ML-FMA), P-FMA takes advantage of the distance between image sources and observation points to factorize the field using multipoles. By substituting known factorizations of the free-space Green's function into the expression for PGF, one can isolate the summation over the lattice into the translation phase of the FMA. For both plane wave and multipole factorizations, a common term known as lattice constant appears. The lattice constant is an infinite sum over the lattice which does not converge absolutely when expressed as a spatial sum. Using the Ewald summation technique, the lattice constants can be evaluated with exponential convergence and high accuracy. The resulting P-FMA is between O(N) and O(N log N) in memory use and computational complexity, depending on the object size relative to the wavelength.

KW - Ewald summation

KW - Fast multipole method (ML-FMA)

KW - Lattice sum

KW - Method of moments (MoM)

KW - Multilevel

KW - Periodic Green's function (PGF)

KW - Periodic scattering

KW - Periodic structures

UR - http://www.scopus.com/inward/record.url?scp=85021722658&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=85021722658&partnerID=8YFLogxK

U2 - 10.1109/TAP.2017.2690533

DO - 10.1109/TAP.2017.2690533

M3 - Article

AN - SCOPUS:85021722658

VL - 65

SP - 3134

EP - 3145

JO - IEEE Transactions on Antennas and Propagation

JF - IEEE Transactions on Antennas and Propagation

SN - 0018-926X

IS - 6

M1 - 7891536

ER -