  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.
