7.1.1 Fixpoint iteration

This method is suitable for static energy minimization. If one is only interested in finding a minimum of the Landau free energy (), but does not want to consider domain wall motion for example, the modified fixpoint iteration provides a simple algorithm.

Brown's equations () can be written in an algebraic form as

$$\{ H \} \mathbf{m} = 0 \quad,$$

where $\{ H \}$ stands for a matrix. However, this matrix is not constant, because the effective field depends on the magnetization. Therefore, we cannot apply a standard method, like the Jacobi or Gauß-Seidel method. A modified iterative technique has been proposed by LaBonte [43], which is used in a simplified form.

First, the effective field () in each subdivision of the magnetic body is calculated. Then the magnetization vector in each subdivision is rotated to the direction of the effective field at that position. After all subdivisions have been updated, the maximum angle of this rotation in any one of them is compared with a preset tolerance. Unless the maximum angle is smaller than the tolerance, the effective field is recalculated, the magnetization updated again, and so on.

This ``LaBonte-like'' method can be extended by an under- or overrelaxation factor. In the former case, the magnetization vectors are not fully rotated into the direction of the effective field (fig. ), whereas in the latter, the magnetization vector ``overtakes'' the vector of the effective field (fig. ). For this method, the damping term of the dynamic Landau-Lifshitz equation () can be used. It has been applied for the simulation of the $\mu$mag standard problem #3, which is described in section .

Figure: In the fixpoint iteration method the magnetization vectors are rotated towards the effective field.

[Underrelaxation] [Overrelaxation]