Sensitivity computation and shape optimization for a non-linear arch model with limit-points instabilities

A shape optimization method for geometrically non-linear structural mechanics based on a sensitivity gradient is proposed. This gradient is computed by means of an adjoint state equation and the structure is analysed with a total Lagrangian formulation. This classical method is well understood for regular cases, but standard equations1 have to be modi ed for limit points and simple bifurcation points. These modi cations introduce numerical problems which occur at limit points.2 Numerical systems are very sti and the quadratic conver- gence of Newton{Raphson algorithm vanishes, then higher-order derivatives have to be computed with respect to state variables.3 A geometrically non-linear curved arch is implemented with a nite element method via a formal calculus approach. Thickness and=or shape for di erentiable costs under linear and non-linear con- straints are optimized. Numerical results are given for linear and non-linear examples and are compared with analytic solutions.