论文库

原文摘要

We present an automated framework for the shape optimisation of a geometrically nonlinear, five-parameter Naghdi model on faceted triangulations. By recovering directors numerically, we remove the requirement for an analytic or spline parameterisation of the mid-surface, whereas partial selective reduced integration alleviates locking. The residual and tangent of symbolic energies are generated in an automated fashion building on Firedrake's and UFL's symbolic differentiation capabilities. Additionally, UFL's and pyadjoint's algorithmic differentiation enables automated first and second shape derivative evaluations that are verifiably correct via Taylor tests. The forward model agrees with the Sze/Abaqus semi-cylinder reference. A three-model mesh-refinement study indicates that geometry recovery and surface faceting have small effects on the predicted deflection. Numerical experiments show that 100 trust-region iterations suffice to reduce the squared-displacement functional of a semi-cylinder by 98.4% and the elastic strain energy of a bracket by 94.6%. The latter result is validated against a COMSOL benchmark. Non-uniqueness of optimal shapes is demonstrated by optimising initial designs subject to small perturbations. Comparisons with linearised designs distinguish optimisation performance from the accuracy of the predicted nonlinear response.