Level set shape optimisation tasks itself with determining the optimal boundary of a domain by solving the so-called level set equation – also known as a Hamilton-Jacobi equation. The method can be used to track the evolution of the boundaries of structures undergoing structural optimization by representing the boundary as an interface between a zero-plane and a level set function known as the level set. Such systems have typically been solved by using finite difference methods but recent years has seen an increase in the use of Finite Element Methods to solve the level set equation. A major issue with the use of standard conforming finite element methods is the difficulty of representing the boundary of the level set on a given mesh. To this end, we are designing a novel finite element method using polytopic elements operating under a Discontinuous Galerkin scheme which will be able to more accurately track the boundary of the level set as well as improving computation speeds due to the reduced number of elements.