Micromagnetics requires calculation of the demagnetization field, anisotropy field, applied field and exchange field. With the MagTense framework, demagnetization is calculated analytically [1] and both anisotropy and applied field are local, leaving only the exchange field in the form of a second order partial derivative. Standard micromagnetics are limited to finite element methods with large, extraneous simulation volumes or finite difference methods with homogeneous simulation grids, as the demagnetization is calculated using a fast Fourier transform [2], but with MagTense, no such limitation exists. However, here an optimal method of calculating second order partial derivatives on arbitrary meshes in micromagnetics must be determined, continuing previous research [3]. We present solutions to the mumag standard micromagnetic problems (mumag) 3[4] and 4[5] using a direct second order partial derivate technique on four different meshes: Prismal, tetrahedral, grained prismal and grained tetrahedral. The grained meshes consist of voronoi generated and Lloyd iterated grain regions with lower resolution towards the center and higher towards the edges, along with high resolution intergrain regions. In fig. 1a) is shown an example of a grained prismal mesh with 9 grains and an intergrain region in gray. Of course, in the mumag standard problems considered, all regions have the same material properties, but it is easy to envision scenarios where they do not. In fig. 1b) is shown the exchange crossover length of mumag 3 as a function of resolution for three different meshes compared to published solutions. All mesh types converge to the correct value, with the tetrahedral meshes converging faster. In fig. 2 is shown the error compared to published solutions of mumag 4 as a function of resolution using regular-, 4 grained- and 9 grained prismal meshes. The unstructured meshes converge to the correct result in a way similar to a regular grid. In both cases it is apparent that while for these problems grained meshes offer few advantages, they are nonetheless correctly implemented and open up a slew of possibilities for accurately simulating complex geometries with far fewer elements in irregular meshes.

![](https://assets.underline.io/uploads/markdown_image/1/image/caeb1548a619f79eb0339cd8844da2f3.jpg)

Fig. 1

![](https://assets.underline.io/uploads/markdown_image/1/image/0298bb3974ce262b20277aca1c2d6b4d.jpg)

Fig. 2

Solving the Standard Micromagnetic Problems using Unstructured Meshes with MagTense

With the increasing focus on energy efficiency for the green transition, there is a desire to identify and reduce all losses possible. For rotary machines losses occur as friction at the bearings, and therefore frictionless magnetic bearings can be an important upgrade to high-speed machinery. However, as proven by Samuel Earnshaw in 1842 magnetic bearings are inherently unstable. Typically, magnetic bearings are stable in the lift (often vertical) direction. This results in instability in the radial direction and therefore some kind of stabilizing force is required in this direction [1]. For a magnetic bearing, the radial stiffness determines the active force needed to keep the bearing stable. Thus there is a compromise in designing a magnetic bearing which can both lift as high a load as possible and at the same time have a low and constant radial instability [2-5]. Here we describe a new magnetic bearing aiming to fulfill these requirements [6]. The bearing consists of two nested conical axi-symmetric permanent magnet rings with quadratic cross-sections, as shown in Fig. 1. This configuration is a superposition between a radially magnetized bearing and an axially magnetized bearing. The rotor and stator are centered on the same axis, and when load is applied on the rotor bearing this will move along the z-axis. The stator has a magnetization direction that is along the inwards normal to its surface, while the rotor has a magnetization direction outwards normal to its surface. Thus, there is repulsion between the rings. The idea behind this design is that the magnetic forces will both have a component providing a lifting force along the z-axis, but also a component along the radial direction, which might allow the stiffness along this direction to be controlled. Such a design of tilted conical rings of permanents magnets can easily be made of cubes of permanent magnets placed in a ring. We consider a design where there is a minimum distance between rotor and stator of 0.2 mm and where the conical rings that make up both the rotor and the stator have quadratic cross-section side lengths of 3 mm. The radius of the center of the tilted ring in the stator is fixed at 24 mm, while it is varied for the rotor. We consider different tilt angles of the stator/rotor rings in the bearing and we also consider ring geometries with twice the cross-section and different magnetization directions. The “single” bearing consists of two tilted concentric rings with opposite magnetization as described above, the “double” bearing has two magnets in each tilted ring, and finally the “alternating” bearing has the magnetization of one part of the ring reversed in both the stator and rotor. All configurations are illustrated in Fig. 1. We assume that the permanent magnets in the bearing have a remanence of 1.4 T and a relative permeability of 1. We determine the force and stiffness of the bearing using a magnetostatic finite element (FEM) axi-symmetric model in Comsol Multiphysics. As the simulations are 2D axi-symmetric a very high mesh resolution can be used, eliminating the known convergence issues with bearing models in 3D [7]. We have conducted a variation study to determine the optimal geometry of the bearing as function of the tilt angle and the bearing type. Tilt angles from 0 to 90 degrees in steps of 15 degrees were modelled. Combined with the three bearing types studied, this gives a total of 21 bearings modelled. For each of these we use an adaptive routine to determine the optimal rotor radius to a precision of 0.1 mm. We show that the optimal value of the rotor radius is almost identical for the different bearing geometries considered, with decreasing rotor radius for increasing tilt angle. Following this we determine which bearing would be easiest to control, i.e. has a constant stiffness value, but which also provides a high lifting force. We investigate this by calculating the relative standard deviation for both the axial force and stiffness. This is shown in Fig. 2 for all studied bearings which are stable in the axial direction. Choosing the best bearing depends on the desired characteristics of the bearing. If one desires a bearing that is easy to control, i.e. has a mostly constant stiffness, for a 3 mm axial displacement the double geometry with a tilt angle between 60-75 degrees is clearly the best. When a smaller axial displacement of 1 mm is considered, all bearings with a tilt angle of 60 degrees display almost constant stiffness. This is a remarkable property, as such a bearing is very easy to control with an active bearing. In this bearing the exact load on the rotor is not of importance, as long as it is between the minimum and maximum lift, and at the same time the stiffness is constant regardless of the load. We thus show that by varying the tilt angle and the bearing geometry it is possible to tailor a bearing to the desired characteristics and achieve an easily controllable bearing with a high lifting force.

**References:**

[1] J. P. Yonnet, IEEE Transactions on Magnetics, Vol. 17, p.1169-1173 (1981) [2] J. G. Detoni, Proceedings of the Institution of Mechanical Engineers, Part C: Journal of Mechanical Engineering Science, Vol. 228, p.1829-1844 (2014) [3] J. Sun, D. Chen and Y. Ren. , IEEE Transactions on Instrumentation and Measurement, Vol. 62, p.2960-2965 (2013) [4] L. Zhang, H. Wu, P. Li, Y. Hu and C. Song, Mathematical Problems in Engineering, Vol. 2019, p.4265698 (2019) [5] K. K. Nielsen, C. R. H. Bahl, N. A. Dagnaes, I. F. Santos and R. Bjørk, IEEE Transactions on Magnetics, 10.1109/TMAG.2020.3042957 (2020) [6] R. Bjørk and C. R. H. Bahl, “A conical passive magnetic bearing with constant stiffness”, submitted for publication (2021) [7] N. A. Dagnæs-Hansen and I. F. Santos., Proceedings of the Institution of Mechanical Engineers, Part C: Journal of Mechanical Engineering Science, Vol. 233, p.5280-5293 (2019)

**Images:**

![](https://assets.underline.io/uploads/markdown_image/1/image/da90a1119a35e4bf7055dcd0b30f5edb.jpg)
Fig.1: The bearing geometry considered. The stator, shown in green, and the rotor, shown in grey, each consists of two tilted rings of permanent magnets with a quadratic cross-section. Their direction of magnetization is also indicated. On the right is shown a cross-section of the bearing, with the three types of magnet configurations considered. The bearing illustrated here has a tilt angle of 30 degrees.

![](https://assets.underline.io/uploads/markdown_image/1/image/015e67cee587135f544e04ac077b1ccd.jpg)
Fig.2: The relative standard deviation of the stiffness in the z-direction in percent as function of the relative standard deviation of the force in the z-direction in percent.

 A constant stiffness magnetic bearing

– Electric motors, magnetic resonance imaging and many other modern applications are built out of systems of multiple permanent magnets, which are arranged so that the magnetic field that they produce has the required properties. Here, starting from a given magnetic field, we invert the design procedure to predict a matching magnetic structure using deep learning. A convolutional neural network (CNN) learns the mapping from a desired magnetic field to a simple magnetic structure with an error of 0.22%. We further extend the method to work with multiple permanent magnets. Introduction Systems of multiple magnets are the basis for many modern applications like magnetic resonance imaging, frictionless bearings in flywheels or electric motors. In these applications permanent magnets of different shapes and magnetic properties are assembled to produce a magnetic field, whose field strength and homogeneity match the requirements for the given application. In order to provide better medical devices, store energy or pave the way to green mobility with electric vehicles, optimized structures of multiple magnets are essential. Traditionally, the design procedure for producing optimized structures of permanent magnets is by selecting a known design that produces a field close to the desired field and then optimizing the structure through modeling. As an example, consider the Halbach cylinder [1] where in real-world applications manufacturing tolerances mean that its actual field will not be exactly equal to the analytically calculated field. In this case one can e.g. use a genetic algorithm to incorporate the manufacturing variations of the single magnetic segments in a traditional design optimization approach [2]. However, this optimization approach is limited in scope, as it relies on a design structure to start from and is computationally intense. Problem Here we consider the general inverse problem of determining a magnetic structure that generates a desired magnetic field. We investigate this through the use of a deep neural network, with the network learning the mapping of Maxwell’s equations in an inverse manner. We want to predict the properties of a system of multiple permanent magnets from a part of the magnetic field, as illustrated in Fig. 1. We consider complicated structures of permanent magnets formed out of different simple geometries, e.g. prisms, tetrahedrons or cylindrical tiles. Previously, artificial intelligence has been used to approximate the solution of Maxwell’s equations for electromagnetic structures [3], in order to calculate the magnetic field in a computationally cheap manner. Here, however we invert this problem to use deep learning to predict magnet structures from the magnetic field. A major challenge is to find an invariant representation for systems of a varying number of magnets. Each of the permanent magnets is specified by its geometric appearance, location, and magnetic properties. The output space easily explodes with an increasing number of magnets, when a poor representation is chosen. Additionally, the problem becomes complex when predicting the properties of the magnetic system from a magnetic field, which does not contain the structure itself. There may exist several solutions for the given desired field. Method The magnetic field in a so-called “design area” is calculated point-wise, which results in a matrix of field values in each of the spatial dimensions. That tensor is fed into a neural network of multiple convolutional layers. As baseline model, we use a convolutional neural network (CNN) inspired by the ResNet architecture [4]. The output shape of the structural design depends heavily on the chosen magnetic system representation. The model parameters are then updated by backpropagation of the difference between the ground-truth and the predicted target structure. To train such models of millions of tunable parameters artificial neural networks require a large amount of data. We produce accurate field-structure-pairs fast and efficiently with MagTense [5], an open-source magnetostatic and a micromagnetism calculation framework. The learned network parameters are then able to infer magnetic structures of unseen magnetic fields. Initially we limit the input space to be a two-dimensional magnetic field by setting the z-component of the single magnet magnetizations to 0 T. We further only use prisms with a constant height and omit rotations of the geometries. The primarily two-dimensional “design area” lies in the plane, which cuts the single tiles in their center point. Results We show in [6] that the prediction of the properties of a single permanent magnet from its magnetic field has an average error rate of 0.22%, which is much below magnet manufacturing tolerances. Predictions of the trained CNN are hardly distinguishable from its ground-truth, when given a previously unseen magnetic field, as exemplified in Fig. 2. We discuss the error rates in extending this to multiple magnets and finally discuss how to represent magnetic structures with multiple permanent magnets of different shapes.

**References:**

[1] K. Halbach, Nuclear Instruments and Methods, Vol. 169, pp. 1-10 (1980) [2] K.K. Nielsen et al., Journal of Magnetism and Magnetic Materials, Vol. 514, p. 167175 (2020) [3] A. Khan, V. Ghorbanian and D. Lowther, IEEE Transaction on Magnetics, Vol. 55, no. 6 (2019) [4] K. He et al., CVPR, ”Deep Residual Learning for Image Recognition” (2016) [5] R. Bjørk and K.K. Nielsen, www.magtense.org, https://doi.org/10.11581/DTU:00000071 (2019) [6] S. Pollok, R. Bjørk and P.S. Jørgensen, “Inverse Design of Magnetic Fields using Deep Learning”, Under review (2021)

**Images:**

![](https://assets.underline.io/uploads/markdown_image/1/image/67b4dca71c7e6adebff4b6367ae16294.jpg)
Fig. 1. Deep learning pipeline for inverse design of magnetic structures. The magnetic field in a so-called "design area" serves as input to the neural network, which predicts a systems of permanent magnets required to produce such a field.

![](https://assets.underline.io/uploads/markdown_image/1/image/f82853dcf9e607e8bc8bf12268bbdd30.jpg)
Fig. 2. Prediction of a trained convolutional neural network from an unseen magnetic field. When plotting the predicted single permanent magnet it is visually indistinguishable from its ground-truth.

 Convolutional Neural Networks for Inverse Design of Magnetic Structures

At the microscopic scale, magnetic systems can be described by the formalism of micromagnetics, which is used for e.g. micromagnetics of rare-earth efficient permanent magnets [1]. In a micromagnetic model, the governing equations must be discretized and represented by a number of interactions taking place between the discretized moments in the system. These interactions can be described by different magnetic fields, such as an exchange- and an anisotropy field. Of these, it is the calculation of the demagnetization field that is the most time consuming part [2]. There are in general three methods that have been applied to calculate the demagnetization field: finite difference-based fast Fourier transform methods, tensor-grid methods and finite-element methods. All methods have inherent advantages and shortcomings, e.g. the finite difference Fourier-based approaches are extremely fast but unfortunately require that the magnetic moments lie on a regular grid [3], while the finite element method allows easy meshing of complex geometries but also requires that an environment outside the object of interest is modelled. Here we present a novel open-source micromagnetism framework called MagTense that uses an analytically exact approach to calculate the full demagnetization tensor in real space for computations [4]. This ensures that the calculated magnetic field is correct, and at the same time allows for easy meshing of complex geometries. In MagTense the discretization approach used subdivides the region Ω into N "tiles", adjacent to each other. The magnetization field is then represented by a vector of 3N components, three for each tile. The demagnetization field generated by each tile is computed analytically by assuming that the tile is uniformly magnetized. The analytical expression of the demagnetization tensor is known for many different tiles, such as the rectangular cuboid (rectangular prism) [5] and the tetrahedron [6], thus allowing great flexibility to build meshes for any geometry. The total demagnetization field at any point is obtained by summing the individual contributions from all the tiles in the magnet domain Ω. Besides the calculation of the demagnetization field, the other components of the effective field must also be calculated, i.e. the exchange-, external- and anisotropy field. For the case of a Cartesian mesh of rectangular cuboids the exchange interaction field can easily be computed by applying standard finite difference schemes. However, methods are also available for the case of unstructured polyhedral meshes. Using MagTense we have calculated the solution to the μmag standard micromagnetic problems 2, 3 and 4 and find that the MagTense framework accurately predicts the solution to each of these [7]. For example for standard problem 3, we get side length at the crossover energy between the vortex and flower state of 8.477, which is in line with two of the published μmag results. We have also implemented the MagTense framework to support both CPU and GPU calculations, where in the latter the demagnetization calculations are performed on the GPU using the CUDA framework. On μmag standard problem 4, we see a decrease in total simulation time of a factor of 100 for the GPU implementation. To demonstrate the use of MagTense on a sample with multiple grains we show in Fig. 1 the magnetization evolution of a thin slice of a multi-grain material starting from a random state. The structure has been generated using a centroidal Voronoi tessellation. The hard magnetic phase in the five grains is characterized by a strong anisotropy, corresponding to the direction indicated by the thick black arrows, while the grain-boundary is soft-phase with isotropic properties. Inside the grains we observe the emergence of magnetic domains aligned to the easy axis, while at the grain-boundary we see the formation of magnetic vortexes. To simulate realistic grain samples, we have also utilized a sintering modeling framework based on kinetic Monte Carlo [8] to generate fully realistic microstructures of sintered powder compacts. We then subdivide these grain structures into magnetic tiles for simulating the magnetic properties of the complete sample. Shown in Fig. 2 (a) is an example of multi-grain sample generated using the sintering modeling framework. The colours indicate different grains, gray corresponding to the grain boundary soft-phase. The thickness of the grain boundary phase can be specified as an input parameter. Fig. 2 (b) and 2(c) show the tetrahedral and Cartesian meshes, respectively, applied to this geometry. The colours indicate the distance of a point from the nearest grain boundary, which in both cases was used to control the resolution of the mesh. In this way the mesh is finer at the boundary between grains which is where the nucleation is expected to occur during a field-reversal process.

**References:**

[1] J. Fischbacher, A. Kovacs, M. Gusenbauer, H. Oezelt, L. Exl, S. Bance, and T. Schrefl., J. Phys. D: Appl. Phys, 51(19):193002, (2018). [2] C. Abert, L. Exl, G. Selke, A. Drews, T. Schrefl, Journal of Magnetism and Magnetic Materials,326:176–185, (2013). [3] A. Vansteenkiste, Ben Van de Wiele, Journal of Magnetism and Magnetic Materials, 323(21):2585–2591, (2011). [4] R. Bjørk and K. K. Nielsen, Magtense - a micromagnetism and magnetostatic framework,doi.org/10.11581/DTU:00000071, https://www.magtense.org, (2019). [5] R. Bjørk, E. B. Poulsen, K. K. Nielsen, A. R. Insinga, “MagTense: a micromagnetic frameworkusing the analytical demagnetization tensor”, submitted, (2021). [6] A. Smith, K. K. Nielsen, D. V. Christensen, C. R. H. Bahl, R. Bjørk, J. Hattel. Journal ofApplied Physics, 107(10):103910, (2010). [7] K. K. Nielsen, A. R. Insinga, R. Bjørk, IEEE Magnetics Letters, 10:8918242, (2019). [8] V. Tikare, M. Braginsky, D. Bouvard, A. Vagnon, Comp. Mat. Sci., 48:317-25, (2010).

**Images:**

![](https://assets.underline.io/uploads/markdown_image/1/image/21f2425d866e8b1eb830bc6b71276f78.jpg)
Spontaneous evolution of magnetic domains in a multi-grain structure. The color indicate the local direction of magnetization.

![](https://assets.underline.io/uploads/markdown_image/1/image/368893183811e0823acd0ef821e19e69.jpg)
Simulated sintered powder sample and corresponding adaptive meshes.

 Simulating sintered magnets using the full demagnetization tensor with MagTense

**Session chair:** Yasushi Kanai, Niigata Institute of Technology


HOM - Micromagnetic and Hysteresis Modeling

technical paper

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Presentations

Solving the Standard Micromagnetic Problems using Unstructured Meshes with MagTense

A constant stiffness magnetic bearing

Convolutional Neural Networks for Inverse Design of Magnetic Structures

Simulating sintered magnets using the full demagnetization tensor with MagTense

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