Complementary Principle, Algorithm, and Complete Solutions to Phase Transitions in Solids Governed by Landau-Ginzburg Equation

Abstract

The complementary variational principles and associated algonrthm are presented in this paper for solving a class of nonconvex mechanics problems governed by the Landau-Ginzburg equation. The method used here is the canonical dual transfonnation developed recently in nonconvex analysis by the author. It is shown that, by this method, certain very difficult nonconvex potential variational problems can be converted into a coupled quadratic mixed variational problem. The extremality conditions of these complementary variational problems are controlled by the triality theory discovered recently. Therefore, a complementary energy principle (i.e. the canonical duality theorem with zero duality gap) is established, which leads to a potentially useful primal-dual algorithm. Applications in finite-dimensional problems are discussed. Results show that the discretized nonconvex potential variational problems in n-dimensional space can be converted into certain simple canonical (either convex or concave) dual problems. Therefore, a complete set of solutions is obtained for the Landau-Ginzburg equation in finite-dimensional space.