Lukas Exl

Grain-size dependent demagnetizing factors in permanent magnets

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 Nd2Fe14B 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.

Numerical methods for the stray-field calculation: A comparison of recently developed algorithms

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 for a given spatial discretization. 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.

LaBonte's method revisited: An effective steepest descent method for micromagnetic energy minimization

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 quasistatic 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 central processing unit (CPU) implementation.

Fast stray field computation on tensor grids

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.

magnum.fe: A micromagnetic finite-element simulation code based on FEniCS

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.

Nonlinear conjugate gradient methods in micromagnetics

Conjugate gradient methods for energy minimization in micromagnetics are compared. The comparison of analytic results with numerical simulation shows that standard conjugate gradient method may fail to produce correct results. A method that restricts the step length in the line search is introduced, in order to avoid this problem. 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 μ0ΔH = 1.4 T at 450 K.

On the limits of coercivity in permanent magnets

The maximum coercivity that can be achieved for a given hard magnetic alloy is estimated by computing the energy barrier for the nucleation of a reversed domain in an idealized microstructure without any structural defects and without any soft magnetic secondary phases. For Sm1–zZrz(Fe1–yCoy)12–xTix based alloys, which are considered an alternative to Nd2Fe14B magnets with a lower rare-earth content, the coercive field of a small magnetic cube is reduced to 60% of the anisotropy field at room temperature and to 50% of the anisotropy field at elevated temperature (473 K). This decrease of the coercive field is caused by misorientation, demagnetizing fields, and thermal fluctuations.

FFT-based Kronecker product approximation to micromagnetic long-range interactions

We derive a Kronecker product approximation for the micromagnetic long-range interactions in a collocation framework by means of separable sinc quadrature. Evaluation of this operator for structured tensors (Canonical format, Tucker format, Tensor Trains) scales below linear in the volume size. Based on efficient usage of FFT for structured tensors, we are able to accelerate computations to quasi-linear complexity in the number of collocation points used in one dimension. Quadratic convergence of the underlying collocation scheme as well as exponential convergence in the separation rank of the approximations is proved. Numerical experiments on accuracy and complexity confirm the theoretical results.

Accurate and efficient computation of nonlocal potentials based on Gaussian-sum approximation

Abstract We introduce an accurate and efficient method for the numerical evaluation of nonlocal potentials, including the 3D/2D Coulomb, 2D Poisson and 3D dipole–dipole potentials. Our method is based on a Gaussian-sum approximation of the singular convolution kernel combined with a Taylor expansion of the density. Starting from the convolution formulation of the nonlocal potential, 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. The potential is separated into a regular integral and a near-field singular correction integral. The first is computed with the Fourier pseudospectral method, while the latter is well resolved utilizing a low-order Taylor expansion of the density. Both parts are accelerated by fast Fourier transforms (FFT). The 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 parallelizable.

Guided self-assembly of magnetic beads for biomedical applications

Abstract Micromagnetic beads are widely used in biomedical applications for cell separation, drug delivery, and hyperthermia cancer treatment. Here we propose to use self-organized magnetic bead structures which accumulate on fixed magnetic seeding points to isolate circulating tumor cells. The analysis of circulating tumor cells is an emerging tool for cancer biology research and clinical cancer management including the detection, diagnosis and monitoring of cancer. Microfluidic chips for isolating circulating tumor cells use either affinity, size or density capturing methods. We combine multiphysics simulation techniques to understand the microscopic behavior of magnetic beads interacting with soft magnetic accumulation points used in lab-on-chip technologies. Our proposed chip technology offers the possibility to combine affinity and size capturing with special antibody-coated bead arrangements using a magnetic gradient field created by Neodymium Iron Boron permanent magnets. The multiscale simulation environment combines magnetic field computation, fluid dynamics and discrete particle dynamics.