Introduction to the Finite Element Method in Electromagnetics

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Morgan & Claypool Publishers, 2006 - Technology & Engineering - 115 pages
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This series lecture is an introduction to the finite element method with applications in electromagnetics. The finite element method is a numerical method that is used to solve boundary-value problems characterized by a partial differential equation and a set of boundary conditions. The geometrical domain of a boundary-value problem is discretized using sub-domain elements, called the finite elements, and the differential equation is applied to a single element after it is brought to a weak integro-differential form. A set of shape functions is used to represent the primary unknown variable in the element domain. A set of linear equations is obtained for each element in the discretized domain. A global matrix system is formed after the assembly of all elements. This lecture is divided into two chapters. Chapter 1 describes one-dimensional boundary-value problems with applications to electrostatic problems described by the Poisson's equation. The accuracy of the finite element method is evaluated for linear and higher order elements by computing the numerical error based on two different definitions. Chapter 2 describes two-dimensional boundary-value problems in the areas of electrostatics and electrodynamics (time-harmonic problems). For the second category, an absorbing boundary condition was imposed at the exterior boundary to simulate undisturbed wave propagation toward infinity. Computations of the numerical error were performed in order to evaluate the accuracy and effectiveness of the method in solving electromagnetic problems. Both chapters are accompanied by a number of Matlab codes which can be used by the reader to solve one- and two-dimensional boundary-value problems. These codes can be downloaded from the publisher's URL: www.morganclaypool.com/page/polycarpou"
 

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Contents

OneDimensional BoundaryValue Problems
1
13 THE FINITE ELEMENT METHOD
3
14 DOMAIN DISCRETIZATION
5
THE GALERKIN APPROACH
7
17 ASSEMBLY OF ELEMENTS
13
18 IMPOSITION OF BOUNDARY CONDITIONS
19
182 Mixed Boundary Conditions
22
110 ONEDIMENSIONAL HIGHER ORDER INTERPOLATION FUNCTIONS
29
232 Bilinear Quadrilateral Element
59
THE GALERKIN APPROACH
61
25 EVALUATION OF ELEMENT MATRICES AND VECTORS
66
251 Linear Triangular Elements
67
252 Bilinear Quadrilateral Elements
75
26 ASSEMBLY OF THE GLOBAL MATRIX SYSTEM
86
27 IMPOSITION OF BOUNDARY CONDITIONS
90
29 POSTPROCESSING OF THE RESULTS
91

1101 Quadratic Elements
30
1102 Cubic Elements
33
111 ELEMENT MATRIX AND RIGHTHANDSIDE VECTOR USING QUADRATIC ELEMENTS
36
112 ELEMENT MATRIX AND RIGHTHANDSIDE VECTOR USING CUBIC ELEMENTS
39
CUBIC ELEMENTS
47
115 SOFTWARE
48
TwoDimensional BoundaryValue Problems
51
22 DOMAIN DISCRETIZATION
52
23 INTERPOLATION FUNCTIONS
54
210 APPLICATION PROBLEMS
92
2102 TwoDimensional Scattering Problem
97
211 HIGHER ORDER ELEMENTS
105
2111 A NineNode Quadratic Quadrilateral Element
106
2112 A SixNode Quadratic Triangular Element
108
2113 A TenNode Cubic Triangular Element
110
212 SOFTWARE
111
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Page 112 - M. Abramowitz and IA Stegun, Handbook of Mathematical Functions, New York: Dover Publications, 1972, pp.

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