## An Introduction to the Locally-corrected Nyström MethodThis lecture provides a tutorial introduction to the Nystrom and locally-corrected Nystrom methods when used for the numerical solutions of the common integral equations of two-dimensional electromagnetic fields. These equations exhibit kernel singularities that complicate their numerical solution. Classical and generalized Gaussian quadrature rules are reviewed. The traditional Nystrom method is summarized, and applied to the magnetic field equation for illustration. To obtain high order accuracy in the numerical results, the locally-corrected Nystrom method is developed and applied to both the electric field and magnetic field equations. In the presence of target edges, where current or charge density singularities occur, the method must be extended through the use of appropriate singular basis functions and special quadrature rules. This extension is also described. About Synthesis This volume is a printed version of a work that appears in the Synthesis Digital Library of Engineering and Computer Science. Synthesis Lectures provide concise, original presentations of important research and development topics, published quickly, in digital and print formats. For more information visit www.morganclaypool.com |

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### Contents

1 | |

2 | |

REFERENCES | 3 |

Classical Quadrature Rules | 7 |

21 TRAPEZOID RULE 23 | 8 |

22 ROMBERG INTEGRATION RULES | 9 |

23 GAUSSLEGENDRE QUADRATURE RULES | 10 |

24 GAUSSLOBATTO QUADRATURE | 11 |

Generalized Gaussian Quadrature | 59 |

53 HIGH ORDER REPRESENTATION OF CURRENT DENSITY AT EDGES IN TWODIMENSIONS | 63 |

54 QUADRATURE RULES FOR THE SINGULAR DEGREES OF FREEDOM IN TABLE 51 | 65 |

55 SUMMARY | 67 |

LCN Treatment of Edge Singularities | 69 |

62 TM SCATTERING FROM A SQUARE CYLINDER | 71 |

63 TE SCATTERING FROM A SQUARE CYLINDER | 74 |

64 INPUT IMPEDANCE OF A HOLLOW LINEAR DIPOLE ANTENNA | 78 |

25 RELATIVE PERFORMANCE OF QUADRATURE RULES | 12 |

REFERENCES | 16 |

The Classical NyströmMethod | 19 |

32 FLATFACETED DISCRETIZATION | 20 |

33 DISCRETIZATION USING EXACT MODELS OF A CIRCULAR CYLINDER | 26 |

34 NYSTRÖM DISCRETIZATIONS USING CLOSED QUADRATURE RULES | 30 |

35 SUMMARY | 32 |

REFERENCES | 33 |

The LocallyCorrected Nyström Method | 35 |

42 APPLICATION OF THE LCN TO THE MFIE | 37 |

43 ALTERNATE INTERPRETATION OF THE LCN | 43 |

44 APPLICATION OF THE LCN TO THE TM EFIE | 44 |

45 APPLICATION OF THE LCN TO THE TE EFIE | 48 |

46 ALTERNATE LCN IMPLEMENTATION OF THE TE EFIE USING GAUSSLOBATTO QUADRATURE | 53 |

47 INITIAL APPLICATION OF THE LCN TO CYLINDRICAL STRUCTURES WITH EDGE SINGULARITIES | 55 |

48 SUMMARY | 56 |

REFERENCES | 57 |

65 SUMMARY | 81 |

REFERENCES | 82 |

Parametric Description of Curved Cell Models | 83 |

BEZIER MAPPING FOR CELLS ON A CIRCLE | 84 |

A3 JACOBIAN RELATIONSHIPS FOR THE INTEGRALS IN SECTION 45 | 87 |

Basis Functions and Quadrature Rules forEdge Cells | 89 |

B2 TE CASE WEDGE ANGLE 0 DEGREES | 91 |

B3 TM CASE WEDGE ANGLE 30 DEGREES | 92 |

B4 TE CASE WEDGE ANGLE 30 DEGREES | 93 |

B5 TM CASE WEDGE ANGLE 60 DEGREES | 94 |

B6 TE CASE WEDGE ANGLE 60 DEGREES | 95 |

B7 TM CASE WEDGE ANGLE 90 DEGREES | 96 |

B8 TE CASEWEDGE ANGLE 90 DEGREES | 97 |

Reference Data for Square Cylinder | 99 |

Authors Biographies | 103 |

### Other editions - View all

An Introduction to the Locally-Corrected Nystrom Method Andrew Peterson,Malcolm Bibby Limited preview - 2009 |

### Common terms and phrases

3-point 60 degree A. F. Peterson accuracy Antennas Propagat approximation basis functions Bk(t Chapter circular cylinder Computational Computational Electromagnetics conducting cylinder corner cells current density produced curved cell deﬁned degrees of freedom domain edge singularities error curves error levels evaluated exactly integrate ﬁrst Gauss-Legendre quadrature rules Gauss-Legendre rule Gauss-Lobatto Gaussian quadrature rules integral operator integrand Jacobian Jz(t kernel LCN approach LCN discretization LCN implementation LCN method LCN procedure Legendre polynomials linear dipole locally-corrected Nyström method matrix entries method of moments MFIE mj,ni nodes and weights nodes associated nodes located number of cells number of unknowns Nyström samples observer location observer nodes obtained order q parametric plane wave polarization polynomial basis quadrature order radius reference solution representation Romberg Romberg integration singular basis functions singular functions source cell square cylinder surface current density synthesize Table theTM trapezoid rule ture rule underlying representation Weights and nodes