On the variational formulation of stress constraints in isogeometric design

The isogeometric paradigm is aimed at unifying the geometric and analysis descriptions of engineering problems. This unification is brought about by employing the same basis functions to describe exact geometry and to approximate the physical response. The isogeometric framework has recently been extended to include design optimisation. A novel variational stress constraint formulation is proposed in this paper that further extends recent efforts in isogeometric design. The weak form of the constraint is obtained by utilising the same basis functions not only to represent geometry, response, and design variables, but also Lagrange multipliers. In this form, the proposed variational formulation fits seamlessly with the underlying analysis and design environment. The variational constraint formulation is compared to the widely utilised stress averaging and collocation approaches. To this end, volume minimisation of elastic arches subjected to stress constraints is considered. Results indicate that the proposed variational approach has the best convergence rates and utilises a much smaller number of discrete constraints than competing methods.