Multiple Solutions to Non-convex Variational Problems with Implications for Phase Transitions and Numerical Computation

Non-convex variational/boundary-value problems are studied using a modified version of the Ericksen bar model in nonlinear elasticity. The strain-energy function is a general fourth-order polynomial in a suitable measure of strain that provides a convenient model for the study of, for example, phase transitions. On the basis of a canonical duality theory, the nonlinear differential equation for the non-convex, non-homogeneous variational problem, here with either mixed or Dirichlet boundary conditions, is converted into an algebraic equation, which can, in principle, be solved to obtain a complete set of solutions. It should be emphasized that one important outcome of the theory is the identification and characterization of the local energy extrema and the global energy minimizer. For the soft loading device criteria for the existence, uniqueness, smoothness and multiplicity of solutions are presented and discussed. The iterative finite-difference method (FDM) is used to illustrate the difficulty of capturing non-smooth solutions with traditional FDMs. The results illustrate the important fact that smooth analytic or numerical solutions of a nonlinear mixed boundary-value problem might not be minimizers of the associated potential variational problem. From a ‘dual’ perspective, the convergence (or non-convergence) of the FDM is explained and numerical examples are provided.