2010 Mathematics subject classification: primary 90C46; secondary 90C26. Keywords and phrases: canonical duality theory, canonical penalty-duality, 3D topology optimisation problem, large deformation, postbuckling problems, nonconvex functional, finite element method. Canonical duality theory (CDT) is a newly developed, potentially powerful methodological theory which can transfer general multiscale nonconvex/discrete problems in R n to a unified convex dual problem in continuous space R m with

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... m ≤ n and without a duality gap. The associated triality theory provides extremality criteria for both global and local optimal solutions, which can be used to develop powerful algorithms for solving the general nonconvex variational problem. First, this thesis presents a detailed study of large deformation problems in a twodimensional structural system. Based on canonical duality theory, a canonical dual finite element method is applied to find a global minimisation to the general nonconvex optimisation problem using a new primal-dual semidefinite programming algorithm. Applications are illustrated by numerical examples with different structural designs and different external loads. Next, a new methodology and algorithm for solving post-buckling problems of a large deformed elastic beam is investigated. The total potential energy of this beam is a nonconvex functional, which can be used to model both pre-and post-buckling phenomena. By using the canonical dual finite element method, a new primal-dual semidefinite programming algorithm is presented, which can be used to obtain all possible post-buckled solutions. In order to verify the triality theory, mixed meshes of different dual stress interpolations are applied to obtain the closed dimensions between discretised displacement and discretised stress. Applications are illustrated by several numerical examples with different boundary conditions. We find that the global minimum solution of the nonconvex potential leads to a stable configuration of