## Introduction to the Finite Element Method in ElectromagneticsThis 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

1 | |

3 | |

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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### Common terms and phrases

assembly process bilinear quadrilateral element boundary-value problems BVP at hand charge distribution computed constant corresponds cubic elements cubic shape functions cubic triangular derivative Dirichlet boundary conditions dx dy electric field electric potential electrostatic potential element coefficient matrix element matrix element right-hand-side vector entries of matrix evaluated exact analytical solution expressed finite element mesh finite element method finite element solution Galerkin approach Gauss quadrature given global coefficient matrix global matrix system global right-hand-side vector higher order elements imposed inside an element Jacobi transformation Jacobian matrix L2 norm Lagrange polynomials linear elements linear interpolation functions linear triangular elements master element Matlab matrices and vectors matrix and right-hand-side mixed boundary condition natural coordinate system nodal number of nodes numerical error numerical percent error numerical solution obtained partial differential equation polynomials primary unknown quantity quadratic elements quadratic shape functions scattering problem Section shown in Figure solve speciﬁc weak formulation x-coordinate zero

### Popular passages

Page 112 - M. Abramowitz and IA Stegun, Handbook of Mathematical Functions, New York: Dover Publications, 1972, pp.

### References to this book

Linear and Nonlinear Inverse Scattering Algorithms Applied in 2-D ... Jinghong Miao Limited preview - 2008 |