Introduction

Non-invasive brain stimulation techniques (NIBS) have been widely used as tools for clinical evaluation and treatment of various neurological and psychiatric conditions1,2,3,4. Within these techniques, transcranial magnetic stimulation (TMS) and transcranial direct current stimulation (tDCS) have been applied to the treatment of epilepsy using different protocols and different etiologies of patients, leading to non-systematic results5,6. In contrast, the transcranial static magnetic stimulation (tSMS) has shown the capacity to decrease systematically the cortical excitability in different experimental models (from mice, rats and monkeys to humans). McLean et al.7 found that a combination of static magnetic fields (SMF) and a pharmacological pretreatment elevated the seizure threshold and had anticonvulsant properties in Black Swiss mice. Rivadulla et al.8 tested the effectivity of a magnetic flux density of 0.50 T generated by a neodymium nickel-plated cylinder over the skull for rats and monkeys. The results show that the SMF can delay the appearance of signs of epilepsy in rats and reduce the intensity and frequency of spontaneous and sensory-triggered seizures in monkeys. Oliviero et al.9 tested the safeness of this therapy with a 2 h application, detecting no cognitive alteration and concluding that the treatment is safe in healthy human subjects, at least within the parameters declared by Aguila et al.1, in the range 0.15–0.20 T.

Experimental evidence suggests that static magnetic fields, even at low intensities, may exert measurable effects on neural activity and behaviour. De Vocht et al.10 reported deficits in visual and motor performance following exposure to a 1.5 T static field, while Nojima et al.11 observed a transient suppression of the corticospinal pathway after stimulation with a compact magnet. Recent clinical studies have reinforced the potential of static magnetic stimulation devices in the treatment of various neurological conditions: they have shown therapeutic promise in amyotrophic lateral sclerosis (ALS) through long-term self-administration12, demonstrated enhanced modulation of reaction times using multi-magnet configurations13, and proven effective in reducing cortical excitability after a single 30-min session in healthy individuals14. In parallel, numerous clinical trials highlight tSMS as a promising technique for certain pathologies, but also underscore the need for further research to establish its therapeutic efficacy15,16,17.

Although these findings might appear to contradict studies focused on MRI safety, primarily concerned with high field intensities, they nonetheless underscore the potential for biologically meaningful effects at lower magnetic strengths and illustrate the growing interest in static magnetic fields within the field of non-invasive neuromodulation18. Experimental evidence19,20 suggests a functional relationship between magnetic flux density and neuronal modulation, with exposure duration significantly influencing the magnitude of the effect. While longer stimulation times generally enhance and prolong neuromodulatory outcomes, achieving a minimum effective magnetic flux density at the targeted brain region remains indispensable, as duration alone cannot elicit substantial modulation without adequate field strength. The conclusions of these studies suggest the possibility of applying it in models of epilepsy, a pathology where neuronal excitability is abnormally augmented, and a pilot program is being developed with promising results21.

Besides the favourable signs translating the results to patients is now limited to focal, superficial epilepsies, since static magnetic field intensity decays quickly with the distance22 and current designs are mainly limited to single magnets affecting the area below the magnet. This aspect, that could be considered as a strength of the technique since it would allow to personalise the optimal area of stimulation to each patient, it is a weakness when a larger area needs to be reached. In this regard those epilepsies classified as generalised epilepsy, including those with abnormal patterns detected at several areas, isolated or simultaneously23, including the group of child epileptic encephalopathies (Dravet, Lennox-Gastaut, West, Landau-Kleffner, etc.24) that generate motor and cognitive limitations, eventually significantly elevating the risk of unexpected death would not be treatable with the existing tools. Keeping in mind that about 30% of epilepsies are classified as not responding to the pharmacological treatments and their low efficacy on encephalopathies, promoting the research on new therapeutic alternatives is necessary, such as the previously mentioned tSMS.

To evaluate the penetration of the generated magnetic fields, the use of modelling and numerical techniques has recently increased. In this field, several researchers performed realistic calculations of the electric field induced by transcranial magnetic and electric stimulation using the open source software SimNIBS25,26, coupling it in same cases with neuronal models27. As an alternative, other authors used COMETS, a MATLAB based toolbox which simulates the magnitude of the current density and electric field as a result of tDCS administration, taking advantage of the finite element method (FEM)28. The capabilities of the FEM simulations were also explored by Wang et al.29, evaluating different methods for the repetitive-TMS, concluding the smoothed method as an interesting option. Nevertheless, none of these studies address the reproduction of static magnetic fields. In this regard, Park et al.30 simulated the magnetic field generated by a single magnet (they tested five different models of permanent magnets) using an axisymmetric 2D finite element solver embedded in the software FEMM. An analogous configuration with a single magnet was used by Rivadulla et al.31. In this case, 3D simulations were conducted in the software ANSYS to reproduce the magnetic field generated in the experimental tests on mouse brains, allowing for the assessment of the penetration and distribution of the magnetic field. Similarly, Tharayil et al.32 performed calculations in COMSOL placing the single magnet in several locations of a human head and characterising the spatial distribution of the magnetic field. The use of FEM in this context is supported both by its solid mathematical foundations and its proven suitability for modelling complex magnetic field interactions in realistic anatomical settings. While simplified analytical models may suffice for single-magnet configurations in ideal conditions, they fall short when addressing systems involving multiple interacting sources and irregular geometries. FEM, by contrast, facilitates the evaluation of complex magnetic configurations with enhanced accuracy, making it an essential tool for personalized neuromodulation strategies. tSMF represents a simple, portable and affordable NIBS technique when compared to TMS or tDCS which highlights it as a suitable option for an outpatient or even on house treatment. From a clinical perspective, tSMS allows for a variable spatial arrangement that can be tailored to individual patients to optimise the influence of the magnetic field on the region of interest (ROI). Moreover, tSMF can be employed for deep brain stimulation, in contrast to TMS, which is rarely utilized for targeting such regions. However, current single magnets configuration (although there is a triple-magnets proposal33) it is only useful if the ROI is well localized and superficial, limiting its possible use as a treatment in pathologies such as non-focal epilepsy where multiple brain regions should be reached31, or on those pathologies with a deep ROI.

This revision of the state of the art shows that little literature was found in the evaluation of the distribution of static transcranial magnetic fields and the capacity to reach deep areas of the brain. Furthermore, these studies are mainly conducted using a focal configuration in which the magnetic field is generated at a specific location of the head, which does not allow for the treatment of non-focal epilepsies. Accordingly, the objective of this study is to numerically evaluate the spatial distribution of static magnetic fields generated by multi-magnet configurations, assessing their potential to achieve therapeutically relevant magnetic flux densities across extensive cortical and deep brain regions. This approach aims to provide critical insights and methodological tools for the development of personalized static magnetic field-based therapies, particularly for non-focal epilepsies. To this end, various configurations with up to ten magnets have been numerically modelled to optimize the magnetic field distribution across the cortical surface and to maximize deep penetration.

Results

The results of this study are firstly shown individually according to the configuration: Case A (single magnet), Case B (three magnets) and Case C (ten magnets. The contours included in this analysis represent the magnetic flux density in Tesla. They are limited to a maximum value of 0.15 T, considered an appropriate threshold based on the bibliography. In this regard, the minimum magnetic field value shown to produce changes in the cerebral cortex is 0.038 T34, while more recent clinical trials with animals pointed that the threshold value for effective magnetic therapy is on the order of 0.10 T8. To date, no single definitive biophysical explanation has been confirmed to account for how magnetic fields alter neuronal depolarization or the physical properties that underlie the observed empirical results. Although several parameters could be considered valid, magnetic flux density was chosen due to its direct connection to stimulation strength and the effects reported in previous experimental studies.

Case A

The contours for a single magnet configuration (Fig. 1) illustrates not just the diversity of magnetic flux density across the brain’s exterior, but also the pronounced effect that a single magnet’s position has on field concentration. High-intensity areas are distinctly visible, suggesting a focused influence of the magnetic field in the region closest to the magnet’s pole. This local peak suggests a highly concentrated area of influence, potentially useful for very targeted brain stimulation applications (focal).

Fig. 1
figure 1

Contours of magnetic flux density for Case A, (1 magnet), Case B (3 magnets) and Case B (10 magnets).

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In contrast, the regions far from the magnet exhibit a significantly weaker field, underlining the stark gradients that a single magnet creates. This indicates that while the use of one magnet can create a strong localised effect, it may also result in a sharp fall-off in field strength, potentially limiting the area of therapeutic influence. Observing the contours on the base of the brain we gain a deeper understanding of the penetration of the magnetic field. The low intensity shown in this area suggests that the influence of a single magnet, while potent at the surface, may not extend deeply into the brain, thereby delineating its potential effectiveness for surface-level stimulation rather than deep brain structures.

Moreover, the visualisations from this case provide valuable insights into the spatial characteristics of the magnetic field produced by a single magnet. The observed patterns of field distribution and the extent of penetration underscore the necessity of strategic placement and orientation of the magnet when designing interventions for neural modulation. This simplified model emphasises the need for precision in the application of single-magnet configurations to achieve the desired therapeutic outcomes while ensuring safety and effectiveness.

Case B

The contours for the configuration with three magnets (Fig. 1) depict the uneven distribution of the magnetic field across the brain’s exterior as well; and highlight the influence of the magnets’ specific arrangement and orientation. The visual representation showcases a broader area high magnetic flux density implying the convergence of the magnetic field lines and the influence of magnet poles or constructive interactions between field lines from neighbouring magnets. In contrast, we also observe areas far from the magnets where the magnetic field strength is considerably weaker. Understanding these areas of minimal magnetic influence is just as vital as identifying the peaks, as they signify regions less affected by the magnetic field, which is a critical consideration for the application of this technology in neurological treatments.

The inspection of the contours in the brains´ base reveals the variations in the field’s strength within the brain. The data indicates that the magnetic field with three magnets not only impacts the surface but also extends into the brain, with specific areas suggesting a more profound reach potentially due to the magnets’ orientation. Such regions of deep field penetration are essential to identify areas where neural activity could be significantly influenced, highlighting them as potential sites for stimulation or therapeutic endeavours.

Case C

The contours shown in Fig. 1 reveal how the number and position of the magnets can lead to areas of both high and low magnetic field concentration. In this case, the higher number of magnets leads to larger regions of strong magnetic field, suggesting the influence of magnet poles or synergistic interactions between field lines from adjacent magnets. Conversely, there are also zones where the magnetic field is notably weaker, likely resulting from the juxtaposition of similarly oriented magnets causing a cancellation effect. These null zones are as critical to understand as the peaks, as they represent areas of minimal magnetic influence, which could have significant implications for the application of this technology in neurological therapies.

When examining the contours in the base of the brain, it is noticeable that the distribution of magnets around the entire surface of the brain favours the penetration of the magnetic field and the reach of deeper areas. The intensity of the magnetic field is considerably higher than in the previous cases, with magnitudes comparable to the ones reached in the regions directly affected by the magnets. These areas of deep field penetration suggest regions where neural activity could be most affected, marking them as targets for potential stimulation or therapeutic intervention.

Furthermore, the simulations indicate that when magnets are positioned in such a way that their fields are aligned in the same direction, the resulting interaction can lead to reduced magnetic flux density. The interplay between the magnetic fluxes can create areas where the influence on neural tissue is diminished, providing a nuanced understanding of how spatial arrangement and polar orientation impact the overall magnetic field’s effectiveness and safety. This insight is essential for optimising the design of magnetic arrays for precise and controlled application in magnetic stimulation therapies.

Discussion

To extend the evaluation of the generated magnetic field and to highlight e the differences between the simulated configurations, three main approaches have been used: effect on the surface, effect on the volume, and effect on the depth.

The effect on the surface is described in the top of Fig. 2. The histogram shows the magnetic field distribution on the brain’s surface, ranging from 0 to 0.335 T in 0.005 T intervals. For the case with a single magnet (Case A), the histogram shows a peak at 0.0125 T, representing 27.16% of the brain surface. For the case with three magnets (Case B), the peak ascends to 0.0225 T over 16.58% of the surface, whereas for the case with ten magnets (Case C), the most frequent class is 0.0825 T, covering only 6.34% of the surface. These results prove that multiple magnets create a broader magnetic flux density distribution on the brain’s external tissue, with a higher percentage of the surface above significant field intensity values. This is consistent with the colourmaps in the figure. Colourmaps depict the region of the brain´s surface above 0.10 T. The contours of Cases A and B show maximum intensity values up to 0.30 T under the magnets, with areas above 0.10 T coincides with the magnet position (12.20% and 27.65% of the surface, respectively). In contrast, the inclusion of more magnets in Case C reduces the extension of highly intense regions but increases the surface area affected by fields above 0.10 T, representing 71.68% of the surface. This suggests that more magnets do not necessarily increase magnetic intensity due to field interactions creating regions of positive and negative interaction. Thus, the number, position, and orientation of the magnets are crucial, and the child’s head size also affects the generated magnetic field characteristics. Boxplots provide additional statistical information, in which the maximum and minimum values are obtained applying a distance from the box of 1.5 times the interquartile range. Boxplots for Case A show a narrow interquartile range of 0.025 T, with all the significant results below 0.07 T. Above this maximum value, 15.38% of the data is identified as atypical. Thus, the region near the magnet that receives a high field density is an exception with respect to the entire surface brain, which is coherent to a therapy for focal epilepsy, but inefficient for treatments of non-focal epilepsies. For Case B, the interquartile range is 0.09 T, the broadest of the three cases; Q3 increases to 0.2475 T, and atypical values ascend to 5.17% of the surface. This option generates a more intense field over more surface, however, the median is 0.0325 T, and only 27.65% of the surface exceeds 0.10 T, indicating an unbalanced distribution so adequate for focalized therapy. Conversely, the median for Case C is significantly higher (0.12 T), with the interquartile range mainly above 0.10 T, the efficacy threshold. Only 0.18% of the data is atypical, ensuring proper magnetic field density in over 70% of the brain surface, suggesting this option is more suitable for global epilepsy treatment.

Fig. 2
figure 2

Histogram boxplot analyses and colourmaps of the magnetic field distribution in the surface and volume of the brain.

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Secondly, the effect on the volume is described in bottom of Fig. 2. Similarly, the histogram shows the distribution of the magnetic field on the volume of the brain, ranging from 0 to 0.335 T in intervals of 0.005 T represented by the mean value of each class. In this case, the colourmaps represent surfaces of constant value of the magnetic field density (0.075, 0.15 and 0.225 T) within the brain´s volume. The distribution of magnetic field density on the volume follows a pattern similar to that on the brain’s surface. The most frequent class in each case is nearly the same as in the surface analysis (0.0125, 0.0275, and 0.0775 T for Cases A, B, and C respectively), but the percentage of volume in this class is lower (17.47%, 8.85%, and 6.86% for Cases A, B, and C respectively). This is coherent with the therapy’s nature, applied externally and more intense in the outer layers of the brain. The colourmaps provide penetration information. For Case A, the three isosurfaces are close to the cortex and limited to the region below the magnet. Case B shows a similar trend, with the isosurface of 0.075 T reaching deeper regions but balanced towards the magnets’ position. Case C exhibits a smaller extension of the 0.225 T isosurface due to magnet interaction, but a larger extension of the 0.075 T isosurface, covering a significant region of the child’s brain volume and reaching areas near the brain base. The behaviour of the boxplots is similar to the previous ones. The volume of the brain with a magnetic field density above 0.10 T is 8.68% for a single magnet, 21.81% for three magnets, and 31.85% for ten magnets. The effect of using multiple magnets is less significant in the volume analysis than in the surface one, as expected. However, this option shows the most uniform distribution and the highest median (0.08 T), making it the most suitable arrangement to reach deeper brain areas associated with global epilepsy abnormalities.

The results of the magnetic field distribution in the brain volume provide information about field penetration. However, this is limited regarding the localization of values. In cases of non-focal epilepsy, quantifying penetration and distribution in deep brain regions with abnormalities is crucial. For this detailed analysis, illustrated in Fig. 3, cross-sectional planes and lines towards the brain’s interior were created, allowing visualization of magnetic field intensity along them for the three simulated cases. Colourmaps shown in the cross-sectional planes of Case A (Fig. 3a), a gradient of field intensity from the magnet (magnet 2) is evident in both the sagittal (S–S) and coronal (C–C) planes. This gradient is also observable in Case B (Fig. 3b), but the transition to lower values is slower, indicating greater depth penetration. This is particularly notable in the axial (A-A) plane, which traverses the deepest brain region. In contrast, the colour maps for Case C (Fig. 3c) do not show such a progressive gradient towards the brain’s interior, but a more uniform magnetic field with values above 0.05 T in profound areas. Here, areas with minimal intensity values are near the brain’s outer surface between magnets, indicating negative interactions between generated magnetic fields.

Fig. 3
figure 3

Magnetic flux density along different detailed paths (a) Case A, (b) Case B, (c) Case C.

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In this context, the results obtained along the three analysed lines offer more specific quantitative insights. The darker solid lines depict values along a line perpendicular to magnet number 2, present in all three configurations, extending from the magnet’s centre to the brain’s interior. The highest values correspond in the three cases to the position closest to the magnet, in the cerebral cortex, decrease with the addition of more magnets to the configuration. Specifically, in Case A, the highest value is 0.28 T, followed by 0.25 T in Case B, and 0.18 T in Case C. However, values at the line’s end, in deeper brain regions, exhibit an inverse trend, increasing as the number of magnets rises (Case A < Case B < Case C).

The lighter solid lines correspond to a line connecting the centres of magnets 1 and 4, traversing the brain longitudinally from the forehead to the back through the deepest brain region. The graphs for cases A and B are quite similar, reaching a maximum under magnet 2 and decreasing when separating from it. Magnitudes are slightly higher when using three magnets. The presence of magnets 1 and 4 in Case C modifies the trend significantly, generating peaks of maximum intensity at the beginning and end of the line. Besides, the value in the intermediate section of the line, corresponding to the inner part of the brain, is higher than the one measured in cases A and B.

Lastly, dashed lines represent the intersection of the cerebral cortex with the sagittal plane, with values displayed angularly according to the upper horizontal axis of the graphs. For Case A and Case B, results follow a bell-shaped trend in which the peak agrees with the position of magnet 2, at an angle θ around 60º. Despite similarities, Case B exhibits a slightly wider bell due to interaction between the three magnets. Conversely, the values for Case C are very different, as this line captures the highs and lows generated by the position of the different magnets and the interaction in the region between them. Thus, the peaks do not reach the maximum value of a single magnet, but the magnetic field is above an effective threshold in almost the entire line.

Numerical simulation has been employed to model the magnetic field generated by various arrangements of static magnets on a child’s head, evaluating their potential as a clinical therapy for global epilepsy. The obtained magnetic field intensity results from simulations with single and multiple magnets, considering the interaction between the generated fields, were experimentally validated. Simulation results were analysed to assess their ability to cover the brain cortex surface and penetrate deep brain areas, quantifying the field intensity near the brain base. Numerical simulation has proven to be a flexible and advantageous tool for this assessment. Based on the targeted areas of the brain obtained by image diagnosis and considering the characteristics of the child´s morphology, it is possible to define a personalising procedure and develop the most suitable treatment device for an effective and efficient tSMF therapy, selecting the appropriate number, location, and orientation of the magnets to generate a magnetic field that penetrates sufficiently into the brain´s volume. The results obtained in this research can be considered essential for a proper understanding of the complexity of the magnetic field, and the consequences of selecting a certain magnet configuration. Future research on this field might consider the extension of the evaluation to adult morphologies that challenge the capacity of penetration of the magnetic field due to the larger brain´s sizes.

Methods

Geometrical model and mesh sensibility

The geometrical model (see Fig. 4a) employed in the simulations has been created using ANSYS Desing Modeler module. It consists of a child´s brain model obtained in the database BrainWeb35; and ten magnets distributed along the surface of the brain, looking for an appropriate coverage of the entire head. These are neodymium magnets: 2, 3, 4, 9 and 10 with a diameter of 70 mm (grade N52), and 1, 5, 6, 7 and 8 with a diameter of 50 mm (grade N45). Neodymium was chosen because of the powerful and depth of the generated magnetic fields8,33.

Fig. 4
figure 4

(a) Description of the distribution of the magnets, and (b) study of mesh sensibility.

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The distance of the magnets with respect to the brain have been set based on the anatomical measurements of the brain-scalp distance by Beauchamp et al.36, ranging from 12.2 to 5.7 mm. These values correspond to the published data for the age group between 19 months and 5 years. The geometry of the skull has not been included in the model since its effect on the generated magnetic field is considered negligible due to the low magnetic permeability.

Regarding the meshing parameters, a study of sensitivity based on the size of the surface elements was conducted. For this purpose, five tetrahedral meshes were created using 2D sizes of 1.5, 2, 3, 5 and 10 mm; and simulated using the configuration with ten magnets. The performance of each mesh was evaluated by measuring the magnetic field density on the brain´s surface. The results, shown in Fig. 4b, indicate a significant effect of the size of the surface elements on the generated magnetic field. Thus, the mesh of 2 mm is revealed as the best option for an appropriate balance between accuracy and number of cells.

Numerical methodology and boundary conditions

The magnetic field generated by a single, isolated magnet can be approximated using analytical models. However, given the complex geometry of the human head and the demands of clinical applications, it becomes essential to account for the intricate interactions among multiple magnetic sources, with magnets varying in type, position, and orientation. This level of complexity surpasses the capabilities of analytical approaches, whereas numerical simulations offer the spatial precision necessary for accurate modelling and clinical applicability. The numerical methodology consisted in steady simulations in which the field is generated due to the magnetic properties of the cylindrical magnets. The magnetisation direction was imposed in the normal direction of the flat magnet surface, associating the positive pole with the face far from the brain, and without considering any external magnetic field. The properties of the Neodymium material were taken from the supplier37, and they are described in Table 1.

Table 1 Main gaussmeter and magnets characteristics.
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Because of the magnetic characteristics, both the brain volume and the rest of the computational domain are considered with the properties of standard air. The solver automatically ensures continuity of the magnetic field at the interface between the magnet and the surrounding materials by accounting for the material properties, without requiring additional boundary conditions on these surfaces. Magnetic insulation boundary conditions were imposed at the outer surfaces of the air domain. Within the magnet volume, a uniform remanent magnetization vector is assigned to represent its intrinsic magnetic field. Convergence criteria were set with a relative residual threshold of 10−6, and a maximum of 200 iterations.

To account for the evaluation of the different situations included in this paper, the following cases were simulated:

  • Case A: simulation of a single magnet (magnet number 2). This setup represents the most simple and straightforward magnetic field. The simplicity of this design enables a focused study on the direct effects of a magnetic field on brain tissue, minimising the interference from additional magnetic sources.

  • Case B: simulation with three magnets (numbers 2, 7 and 8). This setup allows for the studying of the combined effects of multiple magnetic fields.

  • Case C: simulation with ten magnets. This setup allows for the evaluation of the combined effects of the entire distribution of magnets around the brain.

The simulations described in this study have been carried out using the Magnetostatic tool included in the Mechanical module of the ANSYS® Workbench 2021R1 software, and performed on an Intel® Xeon Processor E5-2697 2.6 Hz cluster with 1280 GB of RAM.

Experimental validation

Experimental validation is crucial for any modelling research to ensure the accuracy and reliability in real-world applications. In this case, empirical evidence was provided from experimental setups to verify the magnetic field modelled with ANSYS software. More specifically, two different approaches were followed: a single magnet arrangement, corresponding to “Case A”, to ensure a proper modelling of the intensity decay; and a three-magnet arrangement, corresponding to “Case B”, to verify the interaction between the magnetic fields generated from several magnetic sources. All magnet used on this research correspond to the properties and dimensions explained in the previous section, in Table 1. For simplicity and accessibility, and also to share the same boundary conditions on both the experimental and the simulation configuration, all tests were conducted with magnets surrounded by air.

To collect the experimental data, a Hirst Magnetic GM0838 gaussmeter was used. This gaussmeter has a Hall gauge on his probe to catch the magnetic field strength that crosses its surface perpendicularly. Additionally, it can measure both DC or AC (between 15 Hz and 10 kHz) directional magnetic field strengths. The most relevant characteristics of this gaussmeter are detailed in Table 1, being the measurements of this study primarily conducted within ranges 3 and 4.

The process of measuring the magnetic field intensity was carried out identically in both analysed configurations. A continuous line was defined in the intersection of the plane formed by the centres of the three magnets (2, 7, and 8) used in the ‘Case B’ arrangement, with the magnets support. This same line and measurement points were maintained for the ‘Case A’ configuration, as shown on the right side of Fig. 5. This line ensures the positional correspondence of the measurement points between numerical simulation and experimental data collection, as these points coincide with solid elements of the support used in both arrangements and are easily identifiable.

Fig. 5
figure 5

Results of experimental validation tests (a) Case A setup, and (b) Case B setup.

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The measurement sequence was conducted at intervals of 5 mm along the line indicated in Fig. 5, from left to right, maintaining the sensor in each position around 5 s to ensure measurement stabilisation before recording it. This sequence was repeated ten times for each case. The average values recorded at each position are depicted in Fig. 5 using the squared symbols, and the experimental dispersion is indicated by an error line with whisker. The values obtained in the numerical simulations are shown in orange dashed lines. The results obtained with the single magnet configuration, ‘Case A’ (Fig. 5a), exhibit a nearly identical trend. Although there is relevant experimental uncertainty in the areas near the edges of the magnet, an average deviation of only 3.5% was obtained, ensuring that the model adequately captures the decay of the magnetic field with respect to the distance from the magnet. Regarding the ‘Case B’ setup (Fig. 5b) with three interacting magnets, the results obtained from the simulation faithfully follow the behaviour measured experimentally with an average deviation of 2.8%, even though the previously mentioned uncertainty in the edges of the magnets is as well observed. In this case, the agreement in the valleys between magnets is especially remarkable, capturing appropriately the interacting areas. Therefore, based on the obtained results, it is concluded that both main effects are validated: the decay of the magnetic field from the magnet and the interaction of magnetic fields generated by multiple magnets.