A natural question is modeling the optimal geometry of a waffle. Specifically, the layout of chambers and walls to optimize for conflicting goals:
Syrup coverage: how quickly and thoroughly syrup, poured at one or more locations, spreads through the domain
Syrup retention: syrup saturation depending on the waffle pattern
Crunchiness: modeled as a field that decays with distance from the nearest wall (i.e., edges provide crisp texture)
Let $\Omega \subset \mathbb{R}^2$ be the waffle domain, and let $\Gamma \subset \Omega$ denote the wall subdomain.
Define the following crunchiness field: $$ \kappa(x) = \sum_i K_0 e^{-d_i(x)/\lambda} $$
where $d_i(x)$ is the distance from point $x$ to wall $i$, $K_0$ is the base crunch contribution, and $\lambda$ is a decay parameter.
The syrup wetting time field $T(x)$ is the time it takes syrup to reach $x$, modeled either via flood-fill dynamics or a PDE such as porous medium flow. The syrup saturation function $\sigma(x,t)$ is the amount of syrup accumulated at $x$ at time $t$.
Different users may prioritize different aspects of the waffle: Some value crunchiness and prefer maximal proximity to edges. Others dislike waiting and want rapid syrup distribution across the surface. Some prefer deep saturation and want the chambers to hold syrup even if coverage is uneven.
To model this, define preference weights $w_\kappa$, $w_T$, and $w_S \in [0,1]$, and a utility functional: $$ U(\Gamma) = \int_{\Omega} \left[ w_\kappa \kappa(x) + w_S \sigma(x, t_f) - w_T T(x) \right] dx $$ where $t_f$ is a fixed evaluation time such as a few seconds after pouring.
This framework provides a toy model for multi-objective shape optimization, with links to porous media, PDE-constrained optimization, optimal transport, geometry, and graph design.
