On multi-material topology optimisation problems under inhomogeneous Neumann–Dirichlet boundary conditions

Abstract

This work deals with the topology optimisation of structures made of multiple material phases. The proposed approach is based on non-uniform rational basis spline (NURBS) hyper-surfaces to represent the geometric descriptor related to each material phase composing the continuum, and an improved multiphase material interpolation (MMI) scheme to penalise the stiffness tensor of the structure. In this context, the problem is formulated in the most general case by considering inhomogeneous Neumann–Dirichlet boundary conditions and by highlighting the differences between two different problem formulations. The first one uses the work of applied forces and displacements as cost function and the resulting optimisation problem is not self-adjoint. The second one considers the generalised compliance (related to the total potential energy), and the resulting optimisation problem is self-adjoint. Moreover, the improved MMI scheme proposed here does not require the introduction of artificial filtering techniques to smooth the boundary of the topological descriptors of the material phases composing the structure. The effectiveness of the method is proven on both 2D and 3D problems. Specifically, an extensive campaign of numerical analyses is conducted to investigate the influence of the type of geometric descriptor, of the integer parameters involved in the definition of the NURBS entity, of the type of cost function, of the type of lightness requirement, of the number and type of material phases, of the applied boundary conditions on the optimised topology. • Topology optimisation of multi-material media under mixed boundary conditions • The generalised compliance is used as a measure of the structural stiffness • The work of external actions is not a self-adjoint function (unlike the compliance) • The geometrical descriptor of each material phase is a NURBS hyper-surface • Influence of NURBS parameters and number of materials on the optimal solution