1 Tensors Under Rotations. Multipole Expansion.

2 Combination of ECFIE with multilevel fast multipole algorithm (MLFMA) can easily solve electromagnetic scattering from conductive target with open cavity.

3 Electric multipole moment method is a simple and applied method of solving problems in electrostatic field.

4 Fast multipole expansion Lagragian scheme is used to improve computational speed.

5 Only magnetic dipole and higher multipole configuratons occur in nature.

6 At last, the multipole and relativistic effects in radiative recombination process have been studied systematically.

7 The model of fractional multipole is presented for calculating the characteristic impedance of coaxial transmission line with circular outer conductor and complex inner conductor.

8 By applying fast multipole expansion, the boundary integral equation about 3 D potential problem is made discrete.

9 The multipole expansion algorithm greatly reduces the memory requirements without losing accuracy.

10 The fast multipole method(FMM) was used with the boundary element method(BEM) to predict fractures in large castings.

11 Based on the theory of static magnetic field, the surface magnetic field on the high-resolution multipole magnetic drum for a magnetic rotary encoder was analyzed.

12 The three-dimensional boundary element method for elastic materials was extended to develop a fast multipole expansion formulation of the fundamental solutions for quadratic elements.

13 By comparison with ANSYS, it was clear that BEM could solve the problem not only effectively but also precisely. In chapter 3, the theory of Multipole Expansion Method is introduced.

14 It is necessary to choose the coordinate gauge and take BD multipole moments as astronomical constants.

15 A method for determining Hall sensor positions in three-phase multipole BLDC with concentrated winding is introduced and analyzed.