An Introduction to the Locally-corrected Nyström Method

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Morgan & Claypool Publishers, 2010 - Mathematics - 103 pages
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This 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.

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Contents

Introduction
1
12 THE LOCALLYCORRECTED NYSTRÖM METHOD
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
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