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This paper demonstrates a mathematically correct and computationally powerful method for solving 3D topology optimization problems. This method is based on canonical duality theory (CDT) developed by Gao in nonconvex mechanics and global optimization. It shows that the so-called NP-hard knapsack problem in topology optimization can be solved deterministically in polynomial time via a canonical penalty-duality (CPD) method to obtain precise 0-1 global optimal solution at each volume evolution. The relation between this CPD method and Gao's pure complementary energy principle is revealed for the first time. A CPD algorithm is proposed for 3-D topology optimization of linear elastic structures. Its novelty is demonstrated by benchmark problems. Results show that without using any artificial technique, the CPD method can provide mechanically sound optimal design, also it is much more powerful than the well-known BESO and SIMP methods. Additionally, computational complexity and conceptual/mathematical mistakes in topology optimization modeling and popular methods are explicitly addressed. usually suffer from having different intrinsic disadvantages, such as slow convergence, the gray scale elements and checkerboards patterns, etc [6,44,45].Canonical duality theory (CDT) is a methodological theory, which was developed from Gao and Strang's original work in 1989 on finite deformation mechanics [28]. The key feature of this theory is that by using certain canonical strain measure, general nonconvex/nonsmooth potential variational problems can be equivalently reformulated as a pure (stress-based only) complementary energy variational principle [11]. The associated triality theory provides extremality criteria for both global and local optimal solutions, which can be used to develop powerful algorithms for solving general nonconvex variational problems [12]. This pure complementary energy variational principle solved a well-known open problem in nonlinear elasticity and is known as the Gao principle in literature [35]. Based on this principle, a canonical dual finite element method was proposed in 1996 for large deformation nonconvex/nonsmooth mechanics [9]. Applications have been given to post-buckling problems of large deformed beams [1], nonconvex variational problems [24], and phase transitions in solids [29]. It was discovered by Gao in 2007 that by simply using a canonical measure (x) = x(x − 1) = 0, the 0-1 integer constraint x ∈ {0, 1} in general nonconvex minimization problems can be equivalently converted to a unified concave maximization problem in continuous space, which can be solved deterministically to obtain global optimal solution in polynomial time [14]. Therefore, this pure complementary energy principle plays a fundamental role not only in computational nonlinear mechanics, but also in discrete optimization [25]. Most recently, Gao proved that the topology optimization should be formulated as a bi-level mixed integer nonlinear programming problem (BL-MINLP) [18,20]. The upper-level optimization of this ...

This paper presents a new methodology and algorithm for solving post buckling problems of a large deformed elastic beam. 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 a canonical dual finite element method, a new primal-dual semi-definite programming (PD-SDP) algorithm is presented, which can be used to obtain all possible post-buckled solutions. 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 the buckled beam, the local maximum solution leads to the unbuckled state, and both of these two solutions are numerically stable. However, the local minimum solution leads to an unstable buckled state, which is very sensitive to axial compressive forces, thickness of beam, numerical precision, and the size of finite elements. The method and algorithm proposed in this paper can be used for solving general nonconvex variational problems in engineering and sciences.

The goal of this paper is to solve the post buckling phenomena of a large deformed elastic beam by a canonical dual mixed finite element method (CD-FEM). The total potential energy of this beam is a nonconvex functional which can be used to model both pre-and post-buckling problems. Different types of dual stress interpolations are used in order to verify the triality theory. Applications are illustrated with different boundary conditions and external loads by using semi-definite programming (SDP) algorithm. The results show that the global minimum of the total potential energy is stable buckled configuration, the local maximum solution leads to the unbuckled state, and both of these two solutions are numerically stable. While the local minimum is unstable buckled configuration and very sensitive to both stress interpolations and the external loads.