Exact and Computationally Robust Solutions for Cylindrical Magnets Systems with Programmable Magnetization

Abstract Magnetic systems based on permanent magnets are receiving growing attention, in particular for micro/millirobotics and biomedical applications. Their design landscape is expanded by the possibility to program magnetization, yet enabling analytical results, crucial for containing computational costs, are lacking. The dipole approximation is systematically used (and often strained), because exact and computationally robust solutions are to be unveiled even for common geometries such as cylindrical magnets, which are ubiquitously used in fundamental research and applications. In this study, exact solutions are disclosed for magnetic field and gradient of a cylindrical magnet with generic uniform magnetization, which can be robustly computed everywhere within and outside the magnet, and directly extend to magnets systems of arbitrary complexity. Based on them, exact and computationally robust solutions are unveiled for force and torque between coaxial magnets. The obtained analytical solutions overstep the dipole approximation, thus filling a long‐standing gap, and offer strong computational gains versus numerical simulations (up to 106, for the considered test‐cases). Moreover, they bridge to a variety of applications, as illustrated through a compact magnets array that could be used to advance state‐of‐the‐art biomedical tools, by creating, based on programmable magnetization patterns, circumferential and helical force traps for magnetoresponsive diagnostic/therapeutic agents.

Note: This Supporting Information fully details the procedure leading to the exact solutions reported in the main text.For completeness, deep-level passages are also reported, for each step of the solution procedure.Moreover, in order to foster reproducibility, contextual details are added when citing some references, as appropriate for pointing to specific expressions therein.For instance, ([SR1] 19.2.11) points to expression 19.2.11 in reference [SR1].Furthermore, equations are sequentially numbered starting from the last number appearing in the main text, for ease of readability.
S1.The function that permits to compute all the solutions: C All the solutions achieved in the study are computed based on the so-called Bulirsch integral C, which is defined as follows: with k c , p, a, b ∈ R, k c ̸ = 0 and p ̸ = 0 ([SR1] 19.2.11).C, which is commonly implemented in software libraries [SR2], is the only function that is needed to implement all the analytical solutions obtained for magnetic field, gradient, force and torque.
C permits to conveniently compute the complete elliptic integrals of the first, second and third kind, respectively denoted in literature by K, E and Π [SR3], as follows: The following relations, which were used in the derivations, are thus immediately verified: In order to avoid some representation singularities at ρ = 0 (e.g., for H ∥ρ /ρ when computing grad (H) in cylindrical coordinates), we also exploited the following relation: (which can be derived by applying the Gauss transformation to the underlying elliptic integrals [SR3]), in particular through the introduction of f 3 in Table 4 (from which even k ci and d i are recalled).For the sake of illustration, Figure S1a shows some contour plots of C(k c , p, a, b), for selected values of p, a and b.
In order to determine the solution for magnetic field and gradient, we also used the so-called normalized Heuman Lambda function Λ, since its introduction allowed us to circumvent singularities that arise, in particular, when using Π.Λ can be defined as follows ([SR3] 150.02, up to a sign amendment consistent with 410.04):Λ(σ 2 , k) ) and ω := 1− p. (For reference, let us observe that σ 2 corresponds to sin 2 (β) in [SR3]).Λ is well-behaved over its domain (0 , and Λ(σ 2 , 1) = arcsin( √ σ 2 ) (see Figure S1b).However, by observing that ([SR3] 339.01) ), we defined Λ via C as in Table 4, thus making it possible to compute all the solutions by means of a single algorithmic building block, namely C. While enabling compact and robust implementations, this can also foster computational efficiency, e.g., by algorithmic optimization, in particular when addressing complex magnets systems.

Figure
Figure S1.a) Illustrative contour plots of C(k c , p, a, b), for selected values of p, a and b. b) Contour plot of Λ(σ 2 , k).