µMAG Standard Problem #2 Results: Donahue et al.

Computed demagnetization energy as a function of cell size for a uniformly magnetized 5:1:0.1 rectangular particle. Energy is scaled so that 1.0 is the correct value. Cell size is expressed relative to particle thickness t. Each trace corresponds to uniform magnetization in a different direction, as indicated. The energy is calculated using either sampled demagnetizing fields (OOMMF 1.0; open symbols) or averaged demagnetizing fields based on the analytic formulae of Newell et al. (OOMMF 1.1; solid red diamonds).

Since the magnetization is uniform, all cell discretizations accurately represent the particle magnetization; the demagnetizing field is not uniform, however. OOMMF 1.0 samples the demagnetizing field in the center of the cell, and uses that value as representative of the field throughout the cell. The variation in the demagnetizing field (H_demag) in the xy-plane can be captured by reducing the size of the calculation cell, but variation perpendicular to this plane cannot be tracked because OOMMF 1.0 supports only 2D meshes. As seen above, for particles which are uniformly magnetized in the xy-plane (m=(1,0,0) and m=(0,1,0)), the demagnetization energy calculated by OOMMF 1.0 is higher than the true value (for sufficiently small cells).

Conversely, demagnetization energy calculations based on the analytic formulae (OOMMF 1.1) are completely accurate for uniformly magnetized particles across all discretizations (even though it too supports only 2D meshes). The formulae are complex, and a technique such as doubly compensated summation (N. J. Higham, in Accuracy and Stability of Numerical Algorithms, SIAM, 1996, pp. 96-97) is recommended for increased accuracy. Once kernel initialization is complete, calculation time using the averaged H_demag model is the same as using the sampled H_demag form.

Coercive (H_c) and switching (H_s) fields for µMAG Standard Problem #2 as a function of particle size. Switching field H_s is strictly larger than H_c for d/l_ex<15. Note that H_c decreases as the particle size increases, but H_s goes through a maximum at approximately d/l_ex=5.