A New Method for Nonlinear Two-Point Boundary Value Problems in Solid Mechanics

A local and conditional linearization of vector fields, referred to as locally transversal linearization (LTL), is developed for accurately solving nonlinear and/or nonintegrable boundary value problems governed by ordinary differential equations. The locally linearized vector field is such that solution manifolds of the linearized equation transversally intersect those of the nonlinear BVP at a set of chosen points along the axis of the only independent variable. Within the framework of the LTL method, a BVP is treated as a constrained dynamical system, which in turn is posed as an initial value problem. (IVP) In the process, the LTL method replaces the discretized solution of a given system of nonlinear ODEs by that of a system of coupled nonlinear algebraic equations in terms of certain unknown solution parameters at these chosen points. A higher order version of the LTL method, with improved path sensitivity, is also considered wherein the dimension of the linearized equation needs to be increased. Finally, the procedure is used to determine post-buckling equilibrium paths of a geometrically nonlinear column with and without imperfections. Moreover, deflections of a tip-loaded nonlinear cantilever beam are also obtained. Comparisons with exact solutions, whenever available, and other approximate solutions demonstrate the remarkable accuracy of the proposed LTL method.