Conventionally, topology optimisation is formulated as a non-linear optimisation problem, where the material is distributed in a manner which maximises the stiffness of the structure. Due to the nature of non-linear, non-convex optimisation problems, a multitude of local optima will exist and the solution will depend on the starting point. Moreover, while stress is an essential consideration in topology optimisation, accounting for the stress locally requires a large number of constraints to be considered in the optimisation problem; therefore, global methods are often deployed to alleviate this with less control of the stress field as a consequence. In the present work, a strength-based formulation with stress-based elements is introduced for plastic isotropic von Mises materials. The formulation results in a convex optimisation problem which ensures that any local optimum is the global optimum, and the problems can be solved efficiently using interior point methods. Four plane stress elements are introduced and several examples illustrate the strength of the convex stress-based formulation including mesh independence, rapid convergence and near-linear time complexity.
- Topology optimisation
- Stress constraints
- Von Mises criterion
- Second-order cone programming