Computation of homoclinic orbits in partial differential equations: An application to cylindrical shell buckling

This paper concerns numerical computation of localized solutions in partial differential equations (PDEs) on unbounded domains. The application is to the von Karman-Donnell equations, a coupled system of elliptic equations de scribing the equilibrium of an axially compressed cylindrical shell. Earlie r work suggests that axially localized solutions are the physically preferr ed buckling modes. Hence the problem is posed on a cylindrical domain that is unbounded axially and solutions are sought which are homoclinic in the a xial variable and periodic circumferentially. The numerical method is based on a Galerkin spectral decomposition circumfe rentially to pose ordinary differential equations (ODEs) in the unbounded c oordinate. Methods for location and parameter continuation of homoclinic so lutions of ODEs are then adapted, making special use of the symmetry and re versibility properties of solutions observed experimentally. Thus a formall y well-posed continuation problem is reduced to a rotational subgroup circu mferentially and posed over a truncation of the half-interval axially. The method for location of solutions makes use of asymptotic approximations whe re available. More generally, an adaptation of the "successive continuation " shooting method is used in the lowest possible number of circumferential modes, followed by additional homotopies to add more modes by continuation in the strength of nonlinear mode-interaction terms. The method is illustrated step by step to produce a variety of homoclinic s olutions of the equations and compute their bifurcation diagrams as the loa ding parameter varies. Good agreement with experimental data is found. All computations are performed using auto. The techniques illustrated here for the von Karman-Donnell equations are applicable to a wider class of PDEs.