A cubic Hermite finite element-continuation method for numerical solutions of the von Kármán equations

We study a cubic Hermite finite element method for numerical solutions of the von Kármán equations defined in a rectangular domain with totally clamped boundary conditions. A novel iterative method combined with a predictor-corrector continuation algorithm is exploited to trace solution curves of the von Kármán equations. The fourth order finite element approximations compute accurate numerical solutions for the deformation and the Airy stress function as well as their first order partial derivatives and the mixed second order partial derivatives. In this regard, the classical predictor-corrector continuation method is interpreted in a different way. Our numerical results show that the bifurcation scenario of the von Kármán equations with totally clamped boundary conditions is different from those with simply supported and partially clamped boundary conditions.