2 . Continuous Formulation of Micromagnetics

Computational micromagnetics plays an important role in design and control of magnetostrictive actuators. A systematic approach to calculate magnetic dynamics and magnetostriction is presented. A finite difference method is developed to solve the coupled Landau-Lifshitz-Gilbert(LLG) equation for dynamics of magnetization and a one dimensional elastic motion equation. The effective field in the LLG equation consists of the external field, the demagnetizing field, the exchange field, and the anisotropy field. A hierarchical algorithm using multipole approximation speeds up the evaluation of the demagnetizing field, reducing computational cost from O(N) to O(NlogN). A hybrid 3D/1D rod model is adopted to compute the magnetostriction: a 3D model is used in solving the LLG equation for the dynamics of magnetization; then assuming that the rod is along z-direction, we take all cells with same z-cordinate as a new cell. The values of the magnetization and the effective field of the new cell are obtained from averaging those of the original cells that the new cell contains. Each new cell is represented as a mass-spring in solving the motion equation. Numerical results include: 1. domain wall dynamics, including domain wall formation and motion; 2. effects of physical parameters, grid geometry, grid refinement and field step on H −M hysteresis curves; 3. magnetostriction curve.