Matrix or Vector | Geometry | Shape Functions | Integration Points |
---|---|---|---|

Magnetic Potential Coefficient Matrix; and Permanent Magnet and Applied Current Load Vector | Quad | Equation 11–123 | 2 x 2 |

Triangle | Equation 11–103 | 1 if planar | |

Thermal Conductivity Matrix | Quad | Equation 11–128 | Same as coefficient matrix |

Triangle | Equation 11–108 | ||

Stiffness Matrix; and Thermal and Magnetic Force Load Vector | Quad | Equation 11–120 and Equation 11–121 and, if modified extra shapes are included (KEYOPT(2) = 0) and element has 4 unique nodes) Equation 11–132 and Equation 11–133. | Same as coefficient matrix |

Triangle | Equation 11–100 and Equation 11–101 | ||

Mass and Stress Stiffness Matrices | Quad | Equation 11–120 and Equation 11–121 | Same as coefficient matrix |

Triangle | Equation 11–100 and Equation 11–101 | ||

Specific Heat Matrix | Same as conductivity matrix. Matrix is diagonalized as described in Lumped Matrices | Same as coefficient matrix | |

Damping (Eddy Current) Matrix | Quad | Equation 11–123 and Equation 11–129 | Same as coefficient matrix |

Triangle | Equation 11–103 and Equation 11–109 | ||

Convection Surface Matrix and Load Vector | Same as conductivity matrix, specialized to the surface | 2 | |

Pressure Load Vector | Same as mass matrix specialized to the face | 2 |

Load Type | Distribution |
---|---|

Current Density | Bilinear across element |

Current Phase Angle | Bilinear across element |

Heat Generation | Bilinear across element |

Pressure | Linear along each face |

References: Wilson([38]), Taylor, et al.([49]), Silvester, et al.([72]),Weiss, et al.([94]), Garg, et al.([95])

Structures describes the derivation of structural element matrices and load vectors as well as stress evaluations. Heat Flow describes the derivation of thermal element matrices and load vectors as well as heat flux evaluations. Derivation of Electromagnetic Matrices and Electromagnetic Field Evaluations discuss the magnetic vector potential method, which is used by this element. The diagonalization of the specific heat matrix is described in Lumped Matrices.