We present a steepest descent energy minimization scheme for micromagnetics. The method searches on a curve that lies on the sphere which keeps the magnitude of the magnetization vector constant. The step size is selected according to a modified Barzilai-Borwein method. Standard linear tetrahedral finite elements are used for space discretization. For the computation of static hysteresis loops the steepest descent minimizer is faster than a Landau-Lifshitz micromagnetic solver by more than a factor of two. The speed up on a graphic processor is 4.8 as compared to the fastest single-core CPU implementation.

Different numerical approaches for the stray-field calculation in the context of micromagnetic simulations are investigated. We compare finite difference based fast Fourier transform methods, tensor grid methods and the finite-element method with shell transformation in terms of computational complexity, storage requirements and accuracy tested on several benchmark problems. These methods can be subdivided into integral methods (fast Fourier transform methods, tensor-grid method) which solve the stray field directly and in differential equation methods (finite-element method), which compute the stray field as the solution of a partial differential equation. It turns out that for cuboid structures the integral methods, which work on cuboid grids (fast Fourier transform methods and tensor grid methods) outperform the finite-element method in terms of the ratio of computational effort to accuracy. Among these three methods the tensor grid method is the fastest. However, the use * of the tensor grid method in the context of full micromagnetic codes is not well investigated yet. The finite-element method performs best for computations on curved structures. Stray-Field ProblemConsider a magnetization configuration M that is defined on a finite region Ω = {r : M (r) = 0}. In order to perform minimization of the full micromagnetic energy functional or solve the Landau-Lifshitz-Gilbert (LLG) equation it is necessary to compute the stray field within the finite region Ω. The stray-field energy is given by(1)

The development of permanent magnets containing less or no rareearth elements is linked to profound knowledge of the coercivity mechanism. Prerequisites for a promising permanent magnet material are a high spontaneous magnetization and a sufficiently high magnetic anisotropy. In addition to the intrinsic magnetic properties the microstructure of the magnet plays a significant role in establishing coercivity. The influence of the microstructure on coercivity, remanence, and energy density product can be understood by using micromagnetic simulations. With advances in computer hardware and numerical methods, hysteresis curves of magnets can be computed quickly so that the simulations can readily provide guidance for the development of permanent magnets. The potential of rare-earth reduced and free permanent magnets is investigated using micromagnetic simulations. The results show excellent hard magnetic properties can be achieved in grain boundary engineered NdFeB, rare-earth magnets with a ThMn 12 structure, Co-based nano-wires, and L1 0 -FeNi provided that the magnet's microstructure is optimized.

We have developed a finite-element micromagnetic simulation code based on the FEniCS package called magnum.fe. Here we describe the numerical methods that are applied as well as their implementation with FEniCS. We apply a transformation method for the solution of the demagnetization-field problem. A semi-implicit weak formulation is used for the integration of the Landau-Lifshitz-Gilbert equation. Numerical experiments show the validity of simulation results. magnum.fe is open source and well documented. The broad feature range of the FEniCS package makes magnum.fe a good choice for the implementation of novel micromagnetic finite-element algorithms.

Conjugate gradient methods for energy minimization in micromagnetics are compared. When the step length in the line search is controlled, conjugate gradient techniques are a fast and reliable way to compute the hysteresis properties of permanent magnets. The method is applied to investigate demagnetizing effects in NdFe12 based permanent magnets. The reduction of the coercive field by demagnetizing effects is µ0H = 1.4 T at 450 K. I. INTRODUCTIONThe computation of hysteresis properties of large ferromagnetic systems such as sensor elements or permanent magnets require fast and reliable solvers. Hysteresis simulations are based on the theory of micromagnetics "Brown (1963)". The primary purpose of these simulations is to understand the influence of the microstructure on magnetization reversal. In this work we are focusing on the role of demagnetizing fields in platelet shaped grains of permanent magnets. We also describe the key elements of a micromagnetic solver suitable for simulating large magnetic systems.After discretization of the total Gibbs free energy with finite elements or finite differences the states along the demagnetization curve can be computed by subsequent minimization of the energy for decreasing applied field as outlined in "Kinderlehrer (1997)". The system is in a metastable state. A change of the applied field shifts the position of the local energy minimum. At a critical field, the magnetization becomes unstable. An irreversible switching occurs which is seen as a kink in the demagnetization curve. Then the system either accesses a different metastable state or if fully reversed the magnetization is in a stable state. A reliable numerical method for energy minimization must track all local minima along the demagnetization curve. The resulting algebraic minimization problem is large. Typically the number of unknowns is in the order of 10 to 50 million for a model magnet consisting of around 10 grains. Therefore fast numerical methods are required to obtain results in a Electronic

We introduce an accurate and efficient method for a class of nonlocal potential evaluations with free boundary condition, including the 3D/2D Coulomb, 2D Poisson and 3D dipolar potentials. Our method is based on a Gaussian-sum approximation of the singular convolution kernel and Taylor expansion of the density. Starting from the convolution formulation, for smooth and fast decaying densities, we make a full use of the Fourier pseudospectral (plane wave) approximation of the density and a separable Gaussian-sum approximation of the kernel in an interval where the singularity (the origin) is excluded. Hence, the potential is separated into a regular integral and a near-field singular correction integral, where the first integral is computed with the Fourier pseudospectral method and the latter singular one can be well resolved utilizing a low-order Taylor expansion of the density. Both evaluations can be accelerated by fast Fourier transforms (FFT). The new method is accurate (14-16 digits), efficient (O(N log N ) complexity), low in storage, easily adaptable to other different kernels, applicable for anisotropic densities and highly parallelable.

A direct integration algorithm is described to compute the magnetostatic field and energy for given magnetization distributions on not necessarily uniform tensor grids. We use an analytically-based tensor approximation approach for function-related tensors, which reduces calculations to multilinear algebra operations. The algorithm scales with N4/3 for N computational cells used and with N2/3 (sublinear) when magnetization is given in canonical tensor format. In the final section we confirm our theoretical results concerning computing times and accuracy by means of numerical examples.

The coercive field of permanent magnets decreases with increasing grain size. The grain size dependence of coercivity is explained by a size dependent demagnetizing factor. In Dy free Nd$_2$Fe$_{14}$B magnets the size dependent demagnetizing factor ranges from 0.2 for a grain size of 55 nm to 1.22 for a grain size of 8300 nm. The comparison of experimental data with micromagnetic simulations suggests that the grain size dependence of the coercive field in hard magnets is due to the non-uniform magnetostatic field in polyhedral grains