International Journal of applied mathematics and computer science

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Paper details

Number 2 - June 1996
Volume 6 - 1996

Thickness optimization of a geometrically non-linear arch at a limit point

Pierre Aubert

An optimization method for geometrically non-linear mechanical structures based on a sensitivity gradient is proposed. This gradient is computed by using an adjoint state equation and the structure is analysed by means of a total Lagrangian formulation. This classical method is well-understood for regular cases, but standard equations (see e.g. Rousselet et al., 1995) have to be modified for the limit-point case. The case of sensitivity of a bifurcation point is under development (see (Mróz and Haftka, 1994) for more details). An arc-length algorithm embedded in the optimization algorithm is built. These modifications introduce numerical problems which occur at limit points (Doedel et al., 1991). All systems are very stiff and the quadratic convergence of the Newton-Raphson algorithm is lost, so higher-order derivatives with respect to state variables have to be computed (Wriggers and Simo, 1990). The thickness distribution of the arch is optimized for differentiable costs under linear and non-linear constraints. Numerical results of optimal design of arches undergoing small and large displacements are given and compared with analytic solutions. Related topics of shape optimization can be found in (Aubert and Rousselet, 1996), and theoretical results with details in (Aubert, 1996).