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.

The Magnetic Field from Cylindrical Arc Coils and Magnets: A Compendium with New Analytic Solutions for Radial Magnetization and Azimuthal Current

“…Recently, the complete and singularity‐free solution for arbitrary magnetization was given by ref. [43] for an axisymmetric geometry, also including the field gradient, by using Bulirsh's integrals and the Heuman lambda function. Arbitrary magnetization, however, does not describe the cylindrical magnetizations in the radial and azimuthal directions, which are also included in the solutions presented in this article.…”

Stray magnetic fields from elliptical-shaped and stadium-shaped ferromagnets

“…Computation of magnetic field from a ferromagnetic body, or more generally solving a Possion equation, has been an attractive problem in both mathematics and physics, [1][2][3][4][5][6][7][8][9][10][11][12] and still prompts publications recently. [13][14][15][16][17][18] Even after great development of numerical solvers, [19][20][21][22][23][24] deriving a solution of potential, or field, in terms of analytical functions or in forms of some integrals with simplified approximations, such as macrospin or rigid-vortex assumption, is highly demanded, particularly when many-body problems are of interest. [25][26][27][28][29] This is because it enables us to estimate various parameters, such as coercive and stray fields, with adequate calculation cost.…”

“…The past works have mainly focused on the internal magnetic field and derived the solutions of the demagnetization coefficients for various shapes of ferromagnets such as ellipsoid, cylinder, and cuboid, [1][2][3][4][5][6][8][9][10] while the stray magnetic field, originated from magnetostatic interaction, for vortex, cylinder, and so on has also been investigated. 7,[11][12][13][14][15][16][17][18] An interesting target nowadays related to this is to compare stray magnetic fields from two kinds of uniformly magnetized ferromagnets, namely elliptical-shaped and stadium-shaped ferromagnets, 30,31) which are schematically shown in Figs. 1(a) and 1(b), respectively.…”