This paper presents a topology optimisation approach that combines an adjoint-based sensitivity analysis, [1], with level set methods (LSM), [2], for front propagation, and the discontinuous Galerkin (DG) symmetric interior penalty (SIP) method, [3]. The problems considered in this paper will be limited to the minimum compliance design of two-dimensional linear elastic structures.

The boundary of the interior 'voids' will be defined implicitly using a level set function and modelled as an 'ersatz material'. The optimal shape of the boundary is found by the propagation of the boundary, which occurs through the evolution of the level set function in pseudo-time which is governed by a Hamilton-Jacobi equation. The advection velocity of the Hamilton-Jacobi equation is defined as the topological derivative of the objective functional, which is computed using an adjoint problem. The main novelty of this paper is that both the physical model and the boundary propagation model will be discretised using the DGSIP method.

References
[1] Gregoire Allaire, Francois Jouve, and Anca-Maria Toader. Structural optimization using sensitivity analysis and a level-set method. Journal of computational physics, 194(1):363-393, 2004.
[2] James Albert Sethian. Level set methods and fast marching methods: evolving interfaces in computational geometry, fluid mechanics, computer vision, and materials science, volume 3. Cambridge university press, 1999.
[3] Douglas N Arnold, Franco Brezzi, Bernardo Cockburn, and L Donatella Marini. Unified analysis of discontinuous galerkin methods for elliptic problems. SIAM journal on numerical analysis, 39(5):1749-1779, 2002.