Stiffness design of geometrically nonlinear structures using topology optimization

The paper deals with topology optimization of structures undergoing large deformations. The geometrically nonlinear behaviour of the structures are modelled using a total Lagrangian finite element formulation and the equilibrium is found using a Newton-Raphson iterative scheme. The sensitivities of the objective functions are found with the adjoint method and the optimization problem is solved using the Method of Moving Asymptotes. A filtering scheme is used to obtain checkerboard-free and mesh-independent designs and a continuation approach improves convergence to efficient designs.

Different objective functions are tested. Minimizing compliance for a fixed load results in degenerated topologies which are very inefficient for smaller or larger loads. The problem of obtaining degenerated "optimal" topologies which only can support the design load is even more pronounced than for structures with linear response. The problem is circumvented by optimizing the structures for multiple loading conditions or by minimizing the complementary elastic work. Examples show that differences in stiffnesses of structures optimized using linear and nonlinear modelling are generally small but they can be large in certain cases involving buckling or snap-through effects.