Recent Posts

Our paper Programming evolution of geometry in shape-morphing sheets via spatio-temporal activation has just been published in Proceedings of the Royal Society A. It was selected to be the cover feature, which is nice!

Our paper Lifting, Loading, and Buckling in Conical Shells has been published as an “Editor’s Suggestion” in Physical Review Letters. It’s been featured in articles on Phys.org and in the Physics magazine, which we’re chuffed about.

The coolest thing about slender objects is that they can undergo large shape changes at very little energy cost (and hence only small stresses and strains). As illustration, bending a plate of thickness $t$ into a cylinder of radius $R$ incurs strains only $\sim t/R \ll 1$.

Every object has a scalar field associated with it called a distance function (DF), which equals the shortest distance from each point in space to the object. I think they’re quite beautiful things, and they’re extremely powerful tools in simulations (meshing, tracking interfaces) and graphics (ray marching, defining complex objects with simple expressions).

There’s a distinction between an active transformation and a passive one. In an active transformation, the actual value of a field at a given point in space or spacetime will change in general — you actually transform the field into a different field.

Here’s a subtlety that I only noticed embarrassingly late: Let’s say we’re doing magnetostatics, and trying to find $\mathbf B$ given a current density $\mathbf J$, with some BCs involving $\mathbf B$.

People sometimes mention (correctly) that $\nabla \times \mathbf E = \mathbf 0$ only implies $\mathbf E = -\nabla \phi$ in a region that is simply connected. A simply connected domain is one in which any loop can be contracted to a point without any part of the loop leaving the domain.

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Maxwell’s equations seem to have more equations than degrees of freedom (‘unknowns’) in many reasonable problems, forming an overdetermined system. Writing the equations as $$\begin{align} \nabla \cdot \mathbf {E} &= \rho/\varepsilon_0 , \label{eq:m1}\tag{1}\\ \nabla \cdot \mathbf {B} &= 0 , \label{eq:m2}\tag{2}\\ \nabla \times \mathbf{E} &= -\dot{\mathbf{B}} , \label{eq:m3}\tag{3}\\ \nabla \times \mathbf{B} &= \mu_0 \mathbf{J} + \mu_0 \varepsilon_0 \dot{\mathbf{E} } , \label{eq:m4}\tag{4} \end{align}$$ we see that we have 8 scalar equations in total, since (\ref{eq:m3}) and (\ref{eq:m4}) each have 3 components.

I’ve been thinking a bit about complex numbers and why we use them. Complex numbers are just arrows in the 2D plane. They are thus essentially just 2D vectors in the 2D Euclidean plane, with some extra rules operations on them defined on top of the usual ones (addition, dot product etc), which turn out to be very useful.

Let’s think of integration in a Riemann-like way: We have a function $f(x)$, and you take a particular interval $[a,b]$ of the variable $x$, with $a<b$, and split that interval into $N$ subintervals labelled by $i$, each having width $\Delta x = (b-a)/N$ and (say) central $x$ value of $x_i$, and then the integral $$\int_{[a,b]} f(x) \, \mathrm{d}x = \lim_{N \to \infty} \, \sum_{i=1}^N f(x_i) \, \Delta x .

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I gave a talk on this paper at the 2021 SIAM Mathematical Aspects of Materials Science conference, which you can find (and watch!) under the talks tab on the navigation bar (top of this page).

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Hi, I just made this website. It’s built with Wowchemy, using the Hugo academic-pages template. This is my first post, partly by way of testing, partly by way of introduction.