Numerical Micromagnetics

1. Numerical Micromagnetics Xiaobo Tan John S. Baras P. S. Krishnaprasad University of Maryland CDCSS Program Review Harvard University Oct. 24-25, 2000 Presentation to Dr. Randy Zachery, ARO May 25, 2004 at Harvard University

2. Sectional view of the Etrema magnetostrictive actuator

3. Magnetic Dynamics: LLG Equation Mechanical Dynamics where T is the stress tensor, TM is a tensor dependent on M accounting for magneto-elasticity.

4. Effective Field Heff

5. Computation challenges • Non-local nature of Hdemag leads to complexity of O(N2) in its evaluation. • Any off-the-shelf integration scheme will not preserve the magnitude of M. • Goals • Development of a mathematical model for micromagnetics and magnetostriction from first principles to capture hysteresis and other important properties • Fast evaluation of Hdemag to make computation on large grids feasible • Development of high order integration schemes which preserve the geometric structure

6. Fast Evaluation of the Demagnetizing Field — The Hierarchical Evaluation Algorithm • We use a tree structured hierarchical algorithm to evaluate the demagnetizing field, and reduce the complexity to O(N log N). • The aggregate contribution from remote cells is approximated by multipole expansion, and the field contribution from adjacent ones is calculated directly. This achieves good performance in both speed and accuracy.

7. Fast Evaluation of the Demagnetizing Field — The Hierarchical Evaluation Algorithm A An 8 x 8 grid of cells divided into boxes suitable for evaluation of the demagnetizing field at the center of cell A (2D, for illustrative purpose)

8. Fast Evaluation of the Demagnetizing Field — Computation Results Computation time of Integral Algorithm(IA) and Fast Algorithm(FA) for different grid sizes. IA does direct pair-wise evaluation using the integral formula, while FA uses the multipole approximation.

9. Geometric Integration — Motivation and Examples • Natural dynamical systems often display a variety of geometric structures and symmetries together with various conserved quantities or invariants. • One would like to preserve the geometric structures in the simulation study of these systems for good prediction of long term behaviors. • Examples: • The simple mid-point rule preserves the Lie-Poisson structure to the second order and it preserves original conserved quantities having only linear and quadratic terms. (See, Almost Poisson Integration of Rigid Body Systems, by Austin, Krishnaprasad and Wang, 1993, J. Comp. Phys, 107 (1): 105-117). • Symplectic integrators.

10. Mid-Point Rule for LLG Integration For a grid of N cells, let Mi be the magnetization of i-th cell, and Mall be [M1 M2 … MN]. Discretized LLG can be written in the following form: where F is skew. The mid-point rule is: We can use contraction mapping to solve the coupled, implicit equations.

11. Mid-Point Rule for LLG Integration • The solution of mid-point rule is • Mid-point rule is a Cayley transform, as one can see from above, so the magnitude of M will be preserved along the trajectory of evolution. • Disadvantages of mid-point rule • Mid-point rule is of second order accuracy, and it’s difficult to generalize to higher order. • It requires solving implicit equations and that takes time.

12. Higher Order Geometric Integrator— Cayley Transform and Lifting of LLG • Derive dynamics of underlying transition matrices (t) (which belong to SO(3)) from dynamics of M(t). • By Cayley transform, derive dynamics of (t) in the Lie algebra so(3) from that of (t). • Apply high order integration schemes to the equation of (t), then substitute the solution back to get (t). The numerical counterpart of (t) will stay in SO(3) and thus the magnitude of M will be preserved to machine accuracy. See, e.g., On Cayley-transform methods for the discretization of Lie-group equations, by Arieh Iserles, 1999, preprint, Cambridge University

13. Higher Order Geometric Integrator— Cayley Transform and Lifting of LLG For simplicity of presentation, we work with single cell case. Generalization to case of N cells is straightforward. LLG equation: with F skew. Let From (1), Let

14. Higher Order Geometric Integrator— Cayley Transform and Lifting of LLG Similarly we get a(t) from A(t), then (3) is equivalent to the following vector equation: Equation (4) is what we are going to integrate with explicit 4th order Runge-Kutta method, which we call Cay-RK4 scheme.

15. Higher Order Geometric Integrator— Performance Comparision Comparison of integration schemes on a 2 by 2 by 4 grid, using the result of RK4 with much smaller stepsize as the benchmark.RK4: Runge-Kutta 4-th order, MP: Mid-point rule. Cay_RK4: Cayley transform with RK4.

16. Higher Order Geometric Integrator— Performance Comparision Comparison of performance on norm preserving

17. Higher Order Geometric Integrator— Summary of Features • Fast • Explicit • On the right track • Accurate due to high order • Norm preserving

18. Future Work • Work with NIST on benchmark problems • Incorporation of our code in the public domain tool OOMMF(Object-Oriented MicroMagnetics Framework) • Potential impact on broader community of micromagnetics researchers