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

This paper concerns numerical computation of localised solutions in partial differential equations (PDEs) on unbounded domains. The application is to the von Karaman Donnell equations, a coupled system of elliptic equations describing the equilibrium of an axially compressed cylindrical shell. Earlier work suggests that axially-localised solutions are the physically prefered buckling modes. Hence the problem is posed on a cylindrical domain that is unbounded axially and solutions are sought which are homoclinic in the axial variable and periodic circumferentially. The numerical method is based on a Galerkin spectral decomposition circumferentially to pose ordinary differential equations (ODEs) in the unbounded co-ordinate. Methods for location and parameter continuation of homoclinic solutions of ODEs are then adapted, making special use of the symmetry and reversibility properties of solutions observed experimentally. Thus a formally well-posed continuation problem is reduced to a rotational subgroup circumferentially and posed over a truncation of the half-interval axially. The method for location of solutions makes use of asymptotic approximations where 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 solutions of the equations and compute their bifurcation diagrams as the loading parameter varies. All computations are performed using AUTO. The techniques illustrated here for the von Karaman Donnell equations are applicable to a wider class of PDEs.