On the structural stiffness maximisation of anisotropic continua under inhomogeneous Neumann-Dirichlet boundary conditions
Résumé
This work discusses three aspects of topology optimisation (TO) problems dealing with structural stiffness maximisation of anisotropic continua under mixed inhomogeneous Neumann–Dirichlet boundary conditions (BCs). Firstly, the total potential energy (TPE) is introduced as intuitive measure of the structural stiffness, instead of the work of applied forces and displacements (WAFD). Secondly, it is proven that the WAFD under mixed BCs is not a self-adjoint functional, while the one related to the TPE is always a self-adjoint functional, regardless of the BCs nature. Thirdly, the influence of the anisotropy, of the applied BCs and of the design requirement on the volume fraction on the optimised topology is investigated: depending on these features, the optimal solutions of the two problem formulations, i.e., minimisation of the functional involving the TPE or minimisation of the WAFD subject to a constraint on the volume fraction, can coincide. The problem is formulated in the context of a special density-based TO approach wherein a Non-Uniform Rational Basis Spline (NURBS) hyper-surface is used to represent the topological descriptor, i.e., the pseudo-density field. The properties of NURBS entities are exploited to derive the gradient of the physical responses involved in the problem formulation and to easily satisfy the minimum length scale requirement (related to manufacturing needs). The differences between TPE-based and WAFD-based formulations and the effectiveness of the proposed method are shown on 2D and 3D problems.