Coming out of highschool, I chose to study engineering and spent the next decade building a vast wealth of knowledge in science and mathematics. This page lists my qualifications and details my PhD.
Qualifications
- PhD in Engineering
- Bachelor of Engineering (Mechatronics) with First Class Honours
- Bachelor of Mathematical and Computer Sciences (Applied Mathematics)
My PhD
At the beginning of 2018, I began my doctoral degree, with my thesis entitled Interactions between polyhedral permanent magnets. This research was focused on deriving expressions and formulating methods to calculate magnetic fields, forces, and torques related to permanent magnets with generalised polyhedral geometries.
Thesis
A pdf of my thesis can be found here.Publications
- A Non-Iterative Method to Solve for Magnetic Fields, Forces, and Torques Due to Permanent Magnets with Non-Unity Relative Permeability
- Optimization of the Magnetic Field Produced by Frustum Permanent Magnets for Single Magnet and Planar Halbach Array Configurations
- Simplified equations for the magnetic field due to an arbitrarily-shaped polyhedral permanent magnet
- Analytic Magnetic Fields and Semi-Analytic Forces and Torques Due to General Polyhedral Permanent Magnets
Background
Permanent magnets are widely found in modern society, especially with the prevalence of electromechanical devices such as motors, speakers, and hard drives. As such, it is beneficial to accurately model magnetic materials for more optimal design and reduced waste.
Magnetic fields due to polyhedral magnets
The first section of my PhD was to find expressions for the field produced by polyhedral permanent magnets. I chose to use the magnetic charge model, in which magnets are represented as a collection of magnetic charges. While these magnetic charges don't actually exist, they simplify the mathematics quite nicely. Under the common assumption of a spatially constant, uniform, and field-independent magnetisation vector, the volume integrals of the magnetic charge model disappear, leaving only surface integrals. With a polyhedral geometry, the magnet can now be represented as a collection of magnetically charged triangular surfaces.
With some clever coordinate system manipulation, the surface integrals from the magnetic charge model become double integrals for each triangle. While difficult, I was able to solve these integral expressions, producing a magnetic field equation for each triangular surface. Since the magnetisation vector is field-independent, these field equations can be summed together to give the total magnetic field produced by the magnet.
Although the integral equations were challenging to solve, I spent a considerable amount of time greatly simplifying the solutions to reduce computation complexity. Compared to the solutions of similar problems found in literature, my solutions were far simpler with reduced complexity and singularity treatment.
Forces and torques between permanent magnets
With computationally efficient magnetic field solutions found, I could tackle forces and torques. Unfortunately, I was unable to analytically find expressions for these. However, that was expected, since analytic forces and torques require at least a quadruple integral with non-constant bounds. Rather, I chose to estimate the solutions by numerically integrating. Here, a mesh was defined on one magnet, and the magnetic field produced by another was calculated at each of the mesh nodes before being multiplied by the magnetic charge density. This was summed over the whole magent to give an accurate estimation of the force on the first magnet due to the second.
A similar approach can be taken to estimate the torque on a magnet. Here, cross product of the force on each element and the vector from the point of rotation is computed before summation over all elements.
With a strong focus on computational efficiency while deriving the field equations, the force and torque equations could be solved extremely quickly. It is possible that low-fidelity real-time simulations are possible with this methodology. The results were validated against finite element analysis simulations, giving more accurate results while taking orders of magnitude less time per computation. In fact, the results from a 40 minute FEA simulation were less accurate than a 100 millisecond force computation. It must, of course, be noted that FEA is extremely general and can solve many types of problems, whereas my methodology is fairly rigid in what it can do.
Permeability
Permeability is the effect of a magnet having its magnetisation state change as the surrounding magnetic field changes. In fact, the field produced by a magnet has an effect on its own magnetisation vector. Permeability is extremely complicated and an entire career could be spent modelling and understanding it. For instance, the permeability of a material can change with temperature, surrounding magnetic field, space, and many other factors. In addition, it is often not linear, requiring complicated models to accurately predict.
While permeability is very difficult to model, I wanted to have a basic implementation in my methodology. Luckily for me, most modern permanent magnet materials have a relatively low permeability, meaning the effect of permeability is small. I assumed constant permeability and reconsidered the previously-derived field equations.
With a non-unity relative permeability (i.e. permeability has an effect), the magnetisation and hence magnetic charge density varies over a magnet. However, with the assumption of constant permeability, the magnetic charges remain on the surface, meaning the magnetic charge model remains only a surface integral. Unfortunately, since the charge density varies over the magnet surface, a fully analytic solution is not feasible. Instead, a triangular mesh was applied to a magnet, and each surface element assumed a constant charge density.
In much literature, the challenge of permeability requires an iterative solution, with higher permeability requiring a larger number of iterations before convergence is achieved. However, I was able to form my problem into a single matrix equation. Upon inverting a large matrix, the solution could be found without iteration.