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The sintering behavior of ellipsoidally-shaped particles, particularly spheroids, were studied using a numerical kinetic Monte Carlo model for solid state sintering. Compact packings of spheroids with five different aspect ratios from 0.5 to 2.0, thus comprising both oblate, spherical and prolate particles were generated by simulating the pouring of such particles into a cubic container. For each spheroid particle aspect ratio five different packings were generated to provide statistics on the sintering properties. The sintering behavior is quantified by the densification rate, relative density, anisotropic strain, grain size and grain coordination number, as determined from the Monte Carlo model. The spherical particles were found to sinter to a relative density of 0.87 before grain growth occurs, while the oblate and prolate particles reach a relative density of 0.91 before grain growth sets in, with the prolate particles sintering slightly better compared to oblate particles. The more extreme the particle aspect ratio, the more anisotropic the strain is. Finally, the oblate and prolate spheroids have a slightly higher mean grain coordination number and a slightly higher initial relative density compared to the spherical particles.

In deep learning, neural networks consisting of trainable parameters are designed to model unknown functions based on available data. When the underlying physics of the system at hand are known, e.g., Maxwell’s equation in electromagnetism, then these can be embedded into the deep learning architecture to obtain better function approximations.

We present the results of a systematic investigation into how the coercivity and maximum energy product of NdFeB permanent magnets are affected by magnetic and geometric microscopic properties. The results are based on numerical micromagnetic simulations carried out with the open source numerical framework MagTense. We considered artificially generated realistic microstructures for which we can control the number of crystal grains as well as the thickness of the intergrain region. Based on variations of the exchange constant, easy axis orientation, grain boundary width and intergrain material properties, the results indicate that while all of these can contribute to a reduction of coercivity, the easy axis orientation has the largest influence. For this, if the easy axis orientation is distributed within a cone with an opening angle of 15°, that is enough to reduce the coercivity by 1 T. Regarding the maximum energy product, the width of the grain boundary layer as well as the easy axis orientation were seen to have the largest influence, with the exchange constant only very weakly influencing the maximum energy product. Our analysis thus methodically clarifies the different factors contributing to the reduction of the value of the coercivity and maximum energy product when compared to the theoretical limit, a discrepancy known as Brown’s paradox.

Four different magnet bearing configurations, with varying number of combined radially and axially magnetized rings, are studied both experimentally and using the modeling framework MagTense. First, the optimal vertical position of the radially magnetized ring is determined using the numerical model. Then, the model is validated using experimental data. Finally, we show that a bearing where the rotor and the stator each consist of a single ring of axially magnetized magnets and a single ring of radially magnetized magnets has the smallest radial force per axial lift force at an axial air gap of 1 mm. For this bearing, the ratio of the radial force per axial lift here is 0.383 times that of the same bearing without the radially magnetized ring.

The present challenge with all-oxide thermoelectric modules is their poor durability at high temperatures caused by the instability of the metal-oxide interfaces at the hot side. This work explains a new module concept based on a hybrid p-n junction, fabricated in one step by spark plasma co-sintering of Ca3Co4-xO9+δ (CCO, p-type) and CaMnO3-δ/CaMn2O4 (CMO, n-type). Different module (unicouple) designs were studied to obtain a thorough understanding of the role of the in situ formed hybrid p-n junction of Ca3CoMnO6 (CCMO, p-type) and Co-oxide rich phases (p-type) at the p-n junction (>700 °C) in the module performance. A time-enhanced performance of the modules attributed to this p-n junction formation was observed due to the unique electrical properties of the hybrid p-n junction being sufficiently conductive at high temperatures (>700 °C) and nonconductive at moderate and low temperatures. The alteration of module design resulted in a variation of the power density from 12.4 (3.1) to 28.9 mW/cm2 (7.2 mW) at ΔT ∼ 650 °C after 2 days of isothermal hold (900 °C hot side). This new concept provides a facile method for the fabrication of easily processable, cheap, and high-performance high-temperature modules.

We present the processing of functionally graded multi-material freeze-cast structures, i.e. structures with varying material properties but continuous and homogeneous microchannels. We demonstrate this for stepwise, continuous, highly viscous and gelation freezing to achieve freeze-cast structures of graded La0.66Ca0.24Sr0.09Mn1.05O3 (LCSM9) and La0.67Ca0.27Sr0.06Mn1.05O3 (LCSM6) ceramics. The two phases are successfully distinguished from one another by the addition of 10 wt% Ce0.9Gd0.1O2 (CGO) to the LCSM6 phase. The relative Ce-content facilitates tracking of the LCSM6 phase using energy-dispersive X-ray (EDS) elemental analysis. Coupled with scanning electron microscopy, we show well-defined interfaces with continuous microchannels between the two LCSM phases in both green and sintered samples. By implementing constant freezing rates of −1.5 K/min a consistent freezing front velocity of ∼13 μm/s is maintained, and we found that a structural continuity can be maintained across the LCSM9–LCSM6/CGO interface, however, with varying pore morphology depending on the various freezing procedures.

In micromagnetic simulations it is necessary to compute the effective field associated to the exchange interaction, which corresponds to applying the Laplace operator to the magnetization field. The most widely used approach for computing the exchange field is to apply finite-difference schemes to homogeneous Cartesian meshes. In many situations it would be computationally advantageous to perform micromagnetic simulation on inhomogeneous meshes, but doing so presents a challenge with regard to the calculation of the exchange field. Here we present the construction of an exchange field calculation method for inhomogeneous meshes. A novel second order differential operator method is introduced and applied to a series of analytical functions on irregular meshes and compared to existing methods, finding an accuracy improvement of up to 81% relative to the second order finite difference method. The method is then used in combination with the MagTense framework and tested on the μmag standard micromagnetics problems 3 and 4. Here we find that especially for tetrahedral meshes, the proposed method results in much faster convergence as function of mesh size. We demonstrate that the method converges stably even for meshes where some mesh elements have a volume as small as 1/8 compared to the base resolution. Finally, we show that the method works equally well for meshes with spatially varying material parameters.

This work investigates the performance of an electromagnetic vibration harvester for two-dimensional vibrations with variable magnetic stiffness and electromagnetic damping. The device consists of a free-to-move cylindrical magnetic structure with a set of bearings located on top and bottom, a couple of coils located on top and bottom of the device and finally a fixed system of disk-shaped magnets placed inside a ring holder. The number of disk-shaped magnets in the ring holder can be varied to change the magnetic stiffness of the system. The performance of the device is characterized experimentally for nine different configurations of disk-shaped magnets, exploring both symmetric and asymmetric designs. Using an XY-shaker to vibrate the system in two dimensions in frequencies from 1 Hz to 10 Hz and with motion amplitude of 2 mm on both axes, a maximum power of 27 mW was harvested. This occurs for an asymmetric device, i.e. with different magnetic stiffnesses along its two axis. For symmetric devices the power is lower by a factor of two. Finally, varying the electromagnetic damping, which is controlled by varying the coil dimensions, can further increase the power to 42 mW.

An elliptically shaped electromagnetic vibration energy harvester is presented for two- and one-dimensional motions with the easy tuning of the resonance frequency. The harvester consists of a free-to-move ring-shaped permanent magnet with radial magnetization, a set of cube magnets distributed elliptically in a fixed holder, and two coil windings located above and below the harvester. When the device is exposed to vibrations, the free-to-move ring magnet moves until the magnetic restoring force from the fixed cube magnets pushes it back while inducing an electromagnetic force on the fixed coils. The performance of the device is characterized using an XY-shaker, whose frequency is swept from 1 Hz to 10 Hz with motion amplitudes of 2 mm and 4 mm on both axes simultaneously. Unlike 1-D harvesters, the resulting output power shows two resonant frequencies at 4.5 Hz and 7 Hz, at which the harvester can generate around 1.5 mW. The device also displays a nonlinear hardening resonator behaviour. Finally, 1D vibration experimental results show that the output power of the prototype depends on its angular position with respect to the motion direction of the vibration source, reaching a maximum at an angular position of 45 degree.

Detecting weak magnetic fields is paramount in areas such as scanning magnetometers and manipulation of magnetic nanoparticles, thus rendering it crucial to increase the weak-field sensitivity for developing next-generation magnetic sensors. The current approaches for high-sensitivity sensors, such as superconducting quantum interference devices, are complex and expensive. By contrast, magnetoresistive sensors and particularly extraordinary magnetoresistive sensors offer a simple operation at room temperature but, to date, at inferior sensitivity. To overcome these challenges, we induce device symmetry breaking to enhance the weak magnetic field sensitivity in semiconductor-metal hybrid structures exhibiting extraordinary magnetoresistance. Retaining the device mirror symmetry yields symmetric magnetoresistance curves with R(B) = R(-B), which results in inferior detection of weak magnetic fields as [dR/dB]B→0 = 0. Using finite element modeling, we study the change in device behavior as the symmetry is broken by varying the device geometry by spatially varying the constituent material properties or both. We show that symmetry breaking has three important implications: First, breaking the mirror symmetry causes an asymmetric sensor response with R(B) ≠ R(-B), benefiting from a largely enhanced sensitivity to weak magnetic fields and detection of the magnetic field direction. Second, an interplay with the Hall effect causes a large negative magnetoresistance exceeding 79% at B = 1 T and room temperature without magnetic constituents or explicit optimization of this property. Third, the asymmetric geometries can be used as a key ingredient toward designing on-demand magnetoresistive characteristics such as linearity at low magnetic fields, step functions, and magnetic switchlike behavior. These implications pave the way for asymmetric topology optimization of magnetoresistive devices with unparalleled performance.

We use the Nobel Foundations Scientific Background material to determine which journals have published Nobel Prize-awarded papers in physics since 1995. Analysing all references in the Nobel Prize Scientific Background material reveals that the journal Physical Review Letters published 28.5% of the Nobel Prize-awarded papers. It is followed by the Astrophysical journal, which accounts for 11.2%, Science, accounting 5.6% and Nature, accounting 4.7%. This is contrary to the journals respective Impact Factors, where Physical Review Letters and the Astrophysical Journal have much lower impact factors than Nature and Science. If works cited for background by the Nobel Foundation in the Scientific Background material are included in the analysis, the most referenced journal is still Physical Review Letters, now followed by Physical Review. The conclusion is that the most ground-breaking scientific work in physics is not necessarily published in the journals with the highest Impact Factor.

Freeze‐casting is a technique used to produce structures with anisotropic porosity in the form of well‐defined microchannels throughout a sample. Here, this technique is used on the magnetocaloric ceramic La0.66Ca0.26Sr0.07 Mn1.05O3. We show that a dynamic freezing profile, where the temperature is decreased continuously at −10 K/min, results in homogeneous, lamellar channels with widths of ∼15 µm, while static freezing, where the temperature is kept constant at 177 K, results in channels of increasing size away from the initial ice crystal nucleation site. The effect of gelation before freeze‐casting is also investigated. Gelation inhibits ice crystal growth, which significantly changes the morphology by making channel cross sections less elongated, while additionally introducing more dendrites and ceramic bridges in the structure. The latter significantly dominates the flow path through the gelated structures, affecting the calculated tortuosity, which increases to τ ≈ 4 when compared to non‐gelated samples where calculated tortuosities are in the range of ∼1.3 to ∼3. Finally, we present a systematic and automatic approach for evaluating channel and wall sizes and calculating tortuosities. This is based on analysis of images obtained by scanning electron microscopy using a continuous particle size distribution method and the TauFactor application in MATLAB®.

The stray- and demagnetization tensor field for a homogeneously magnetized tetrahedron is found analytically. The tetrahedron is a special case of four triangular faces with constant magnetization-charge surface density, for which we also determine the tensor field. The tensor field is implemented in the open source micromagnetic and magnetostatic simulation framework MagTense and compared with the obtained magnetic field from an FEM solution, showing excellent agreement. This result is important for modeling magnetostatics in general and for micromagnetism in particular as the demagnetizing field of an arbitrary body discretized using conventional meshing techniques is significantly simplified with this approach., Comment: 5 pages, 4 figures

We consider the motion of an electromagnetic vibrational energy harvester (EMVEH) as function of the initial position and velocity and show that this displays a classical chaotic dynamical behavior. The EMVEH considered consists of three coaxial cylindrical permanent magnets and two coaxial coils. The polarities of the three magnets are chosen in such a way that the central magnet floats, with its lateral motion being prevented by enclosion in a hollow plastic tube. The motion of the floating magnet, caused by e.g. environmental vibrations, induces a current in the coils allowing electrical energy to be harvested. We analyze the behavior of the system using a numerical model employing experimentally verified expressions of the force between the magnets and the damping force between the floating magnet and the coils. We map out the phase space of the motion of the system with and without gravity, and show that this displays a fractal-like behavior and that certain driving frequencies and initial conditions allow a large power to be harvested, and that more stable states than two exists. Finally, we show that at leasts fifth order polynomial approximation is necessary to approximate the magnet-magnet force and correctly predict the system behavior.

We here present a novel deep learning (DL) approach for designing structures of permanent magnets. The challenge for the DL method in this kind of problem is to learn the mapping from a desired magnetic field to a simple magnetic structure, i.e. an inverse design approach. We demonstrate this approach by training six different standard convolutional neural network (CNN) structures previously used in the ImageNet Large Scale Visual Recognition Challenge (ILSVRC) to inversely predict the properties of a single hard magnet (magnetization, size and location) from a given two-dimensional magnetic field. We show that the best network, ResNeXt-50, can perform this prediction with an error of 0.22% in the properties of the magnet.

The design of a rotary active magnetic regenerator heat pump device with a multi-bed concept is presented. Important design features are the rotating two-pole magnet assembly, the laminated iron ring, the 13 fixed tapered regenerator beds, and the dynamically adjustable parallel flow circuit. The optimized magnet design was developed with optimally shaped segments and optimum remanence for the desired magnetic field distribution oscillating between 0 and 1.44 T in the air gap. The iron ring was laminated to reduce the eddy currents, allowing the device to run at cycle frequencies up to 3 Hz. The design of the regenerator housing was optimized with respect to parasitic losses and even flow distribution in both directions. Employing 3.4 kg of La(Fe,Mn,Si)13Hy(CALORIVAC HS) refrigerant and at a hot reservoir temperature of 295 K and a cycle frequency of 0.5Hz, the heat pump achieved a maximum second-law efficiency of 20.6%, while providing a heating load of 340 W with a heating COP of 6.7 at a 10.3 K span. The COP values presented only consider the magnetic power and ideal pump power delivered to the AMR, neglecting the pump efficiency. At 1.2 Hz, the device produced a maximum heating power of 950 W while maintaining a 5.6K span, resulting in a heating coefficient of performance and second-law efficiency of 7.0 and 11.6%, respectively. The performance demonstrated in this paper could be an important milestone in the development of future magnetocaloric devices.

The efficiency of the magnetic refrigeration process strongly depends on the heat transfer performance of the regenerator. As a potential way to improve the heat transfer performance of a regenerator, the design of sub-millimeter hydraulic diameter porous structures is realized by freeze-cast structures. Four freeze-cast regenerators with different pore widths are characterized experimentally and numerically. Empirical parameters are determined for the correlations of heat transfer and flow resistance via a 1D model. Thermal effectiveness and pressure drop are measured for thermal-hydraulic evaluations. Temperature span and specific cooling capacity are obtained to compare the magnetocaloric potential based on the material La0.66Ca0.27Sr0.06Mn1.05O3. The stability of freeze-cast regenerators is validated by comparing the performance during, before and after oscillatory flow and periodic magnetic field tests. Smaller pore design obtain the better heat transfer performance and required mechanical strength, while pore design with significant dendrites provides the worst tradeoff between heat transfer performance and flow resistance.

A vibration energy harvester (VEH) converts the kinetic energy of a moving source into electrical energy. In this article, we consider a 1-D electromagnetic vibration energy harvester (1-D EMVEH) that consists of three coaxial cylindrical permanent magnets enclosed in a tube such that the middle magnet is levitating. The resulting movement of the middle magnet can then induce an electromotive force (EMF) in one or more surrounding coils. Using an analytical model, we derive expressions for the 1-D-EMVEHs characteristic frequency and output power by using the Fourier space approach. First, the magnetostatic energy of the system as a function of the position of the levitating magnet is calculated. Its spatial gradient gives the force acting on a magnet, which drives its dynamics. Next, more accurate magnetic flux and EMF expressions are obtained. The results are compared with experimental measurements, revealing an excellent agreement.

In this chapter, thermoelectrics is introduced, with special focus on oxide materials. The history of thermoelectrics is briefly described, after which the thermoelectric effect and different thermoelectric materials are discussed in detail. The relevant material parameters for various oxide thermoelectric materials are also given. After this, thermoelectric devices, such as leg, unicouples, and modules are discussed, along with their efficiencies. A numerical model for calculating the efficiency of a thermoelectric system, if material properties are known, is also presented. Advanced ways to improve the performance of thermoelectric devices, such as segmentation of legs or cascaded designs are also discussed in detail. Finally, the joining of thermoelectric materials and the associated detrimental effect of contact resistance on thermoelectric performance is discussed.

A novel freeze-casting device utilizing a thermoelectric element for high precision temperature control allowing for dynamic freezing conditions of freeze-cast materials is presented. Freeze-casting is a processing route for producing materials of anisotropic porosity in the form of aligned and well-defined microchannels. In freeze-casting, particulates of a material are suspended in a fluid and a thermal gradient is applied across for directional freezing. Controlling the thermal gradient across the suspension amounts to controlling the kinetics and freezing direction in the suspension and thus the resulting structural features and dimensions of the microchannels. The performance of the device presented here was evaluated by directional freezing of both water and aqueous ceramic suspension samples using both linear and exponential freezing profiles. The freezing front was successfully tracked by continuously measuring the temperature gradient along the sample using thermocouples directly mounted on the freeze-casting mold. The current minimum operational temperature of the freeze-caster is ∼220 K, with freezing front velocities in the range of ∼5 μm/s to 30 μm/s for sample lengths of 5 mm-25 mm.

We present the open source micromagnetic framework, MagTense, which utilizes a novel discretization approach of rectangular cuboid or tetrahedron geometry "tiles" to analytically calculate the demagnetization field. Each tile is assumed to be uniformly magnetized, and from this assumption only, the demagnetization field can be analytically calculated. Using this novel approach we calculate the solution to the mmag standard micromagnetic problems 2, 3 and 4 and find that the MagTense framework accurately predicts the solution to each of these. Finally, we show that simulation time can be significantly improved by performing the dense demagnetization tensor matrix multiplications using NVIDIA CUDA., 10 pages, 7 figures

We reply to Dr. Xu's comment on our paper ‘Performance of Halbach magnet arrays with finite coercivity’ (JMMM 407 (2016), 369–376). Contrary to Dr. Xu's objections we show that the procedure employed by us correctly accounts for the shape effects of the magnet elements. We show that the partial differential equation for the magnetic vector potential, derived from the Maxwell equations, incorporates all shape effects. On the other hand, the local constitutive relations express a point-wise link between, e.g., magnetic field and magnetic flux density, and are as such independent of geometry. We confirm that the results of our computations are perfectly consistent with the constitutive relation which is assumed as starting point.

The optimal segmentation of three-dimensional permanent-magnet systems into uniformly magnetized blocks to generate a desired field is a question that has been studied extensively in recent years. We present a procedure that generates the theoretically optimal shape and magnetization direction of all the magnet segments in the system, given only the total number of blocks as a constraint. Materials of arbitrary magnetic permeability can be included in this optimization framework. As an optimization objective we consider any functional that is linear with respect to the magnetic field distribution. We furthermore assume that all magnetic materials obey a linear constitutive relation. We show that with these assumptions calculating an optimal three-dimensional segmentation is equivalent to determining a centroidal Voronoi tessellation. We use the most-well-known and most-straightforward algorithm for the generation of centroidal Voronoi tessellations, known as "Lloyd's method," which can also be thought of as a fixed-point iteration. We show that the procedure is guaranteed to lead to a configuration that is at least locally optimal with respect to the assumed linear objective functional. However, we present results providing a strong indication that for most design problems applying our technique a few times starting from different random configurations is very likely to lead to the globally optimal solution.

We investigate if the demagnetization factor for a randomly packed powder of magnetic spheroidal particles depend on the shape of the spheroidal particles and what the internal variation in magnetization is within such a powder. A spheroid is an ellipsoid of revolution, i.e. an ellipsoid with two semi-major axis being equal. The demagnetization factor is calculated as function of particle aspect ratio using two independent numerical models for several different packings, and assuming a relative permeability of 2. The calculated demagnetization factor is shown to depend on particle aspect ratio, not because of direct magnetic interaction but because the particle packing depend on the aspect ratio of the particles. The relative standard deviation of the magnetization across the powder was 3\%-8\%, increasing as the particle shape deviates from spherical, while the relative standard deviation within each particle was relatively constant around 5\%., 7 pages, 9 figures

The design of a magnetic field source that can switch from a high field to a low field configuration by rotation by $90^\circ$ of a set of iron pieces is investigated using topology optimization. A Halbach cylinder is considered as the magnetic field source and iron inserts are placed in the air gap of the Halbach cylinder. The ideal shape of these iron inserts is determined as function of the field generated by the Halbach cylinder and as function of the size of the iron segments. The topology optimized structures are parabolic shaped pieces and have a difference in flux density between the high and low positions that is on average 1.29 times higher than optimized regular pole pieces. The maximum increase is a factor of 2.08 times higher than the regular pole pieces., 10 pages, 13 figures

We present an optimization approach that can be employed to calculate the globally optimal segmentation of a 2-D magnetic system into uniformly magnetized pieces. For each segment, the algorithm calculates the optimal shape and the optimal direction of the remanent flux density vector, with respect to a linear objective functional. We illustrate the approach with results for magnet design problems from different areas, such as a permanent magnet electric motor, a beam-focusing quadrupole magnet for particle accelerators, and a rotary device for magnetic refrigeration.

In Part I of this study, the performance of an experimental integrated thermoelectric generator (TEG)-heat exchanger was presented. In the current study, Part II, the obtained experimental results are compared with those predicted by a finite element (FE) model. In the simulation of the integrated TEG-heat exchanger, the thermal contact resistance between the TEG and the heat exchanger is modeled assuming either an ideal thermal contact or using a combined Cooper–Mikic–Yovanovich (CMY) and parallel plate gap formulation, which takes into account the contact pressure, roughness and hardness of the interface surfaces as well as the air gap thermal resistance at the interface. The combined CMY and parallel plate gap model is then further developed to simulate the thermal contact resistance for the case of an interface material. The numerical results show good agreement with the experimental data with an average deviation of 17% for the case without interface material and 12% in the case of including additional material at the interfaces. The model is then employed to evaluate the power production of the integrated system using different interface materials, including graphite, aluminum (Al), tin (Sn) and lead (Pb) in a form of thin foils. The numerical results show that lead foil at the interface has the best performance, with an improvement in power production of 34% compared to graphite foil. Finally, the model predicts that for a certain flow rate, increasing the parallel TEG channels for the integrated systems with 4, 8, and 12 TEGs enhances the net power per TEG with average values of 2.5%, 3% and 5%, respectively.

A numerical method to study the effect of finite coercivity on the Halbach cylinder geometry is presented. Despite the fact that the analytical solution available for this geometry does not set any limit to the maximum air gap flux density achievable, in real life the non-linear response of the magnetic material and the fact that the coercivity is not infinite will limit the attainable field. The presented method is able to predict when and where demagnetization will occur, and these predictions are compared with the analytical solution for the case of infinite coercivity. However, the approach presented here also allows quantification of the decrease in flux density and homogeneity for a partially demagnetized magnet. Moreover, the problem of how to realize a Halbach cylinder geometry using a mix of materials with different coercivities without altering the overall performance is addressed. Being based on a numerical approach, the presented method can be employed to analyze the demagnetization effects due to coercivity for any geometry, even when the analytical solution is not available.

Experiments with a recently constructed rotary multi-bed active magnetic regnenerator (AMR) prototype have revealed strong impacts on the temperature span from variations in the resistances of the flow channels carrying heat transfer fluid in and out of the regenerator beds. In this paper we show through numerical modeling how unbalanced flow in the beds decreases the cooling power and COP for a dual bed device. Furthermore, it is shown how resistance variations in multi-bed devices give rise to unbalanced flow in the individual beds and how this decreases cooling powers and COPs of the machines by approximately 30% and 50%, respectively.

We investigate whether field homogeneity of a magnetic assembly can be optimized by varying the remanence of its constituting magnetic segments. We specifically study this hypothesis for a Halbach cylinder using a numerical model, MagTense. We consider a Halbach cylinder consisting of six layers of three concentric rings, each ring made from 16 segments. We show that ideally, the homogeneity can reach close to 1 ppm for a finite magnet. We then proceed to consider a real world set of magnet segments, i.e. non-ideal magnets with a variation in their remanence. This reduces the field homogeneity to about 1000 ppm when considering a Gaussian perturbation of the remanence with a standard deviation of 1 %. However, we also show that the reduction in homogeneity may be countered by organizing the magnet pieces found through optimization, which is possible if each magnet segment is well characterized experimentally. We note that the presented method is applicable to any case where homogeneity of the field is important. The results we present are considered for the specific case of nuclear magnetic resonance for concretenes.