Minimum Thickness Constraints in Shape and Topology Optimisation using GridapTopOpt.jl

Supervisors:  Assoc Prof Vivien Challis & Zach Wegert

Shape and topology optimisation aims to find an ‘optimum’ shape for a given problem. One popular application is in structural mechanics – designing components that achieve a minimum compliance while keeping the volume of material used below a certain limit. The ‘objectives’ such as compliance and volume are represented by functionals defined over the shape, and in order to optimise a shape we must find the derivative of these objective functionals with respect to the boundary of the shape.
We can then perturb the boundary of the shape in the direction that will decrease the objective functional. In the level set method of topology optimisation, the zero contour of a level set function is used to track the boundary of a shape. This allows for the boundary of the shape to be explicitly defined while using a structured mesh (e.g., a Cartesian grid). A key step in level set-based topology optimisation is reinitialisation of the level set function to the signed distance function. This makes the level set method particularly suited to the implementation of geometric functionals involving the signed distance function. GridapTopOpt is a framework for level set-based topology optimisation written in the programming language Julia that utilises the Gridap ecosystem to solve weak formulations of partial differential equations via the finite element method. This investigation presents – in GridapTopOpt, using Gridap – an implementation of the 2020 work by Feppon et. al. considering a variational formulation for computing line integrals along flow fields. When the signed distance function is used as the flow field in question, this allows for simple computation of geometric functionals along rays normal to the boundary of the shape. In particular, we have investigated implementing minimum thickness constraints for linear minimum compliance problems.