  This paper presents a canonical duality approach for solving a general
topology optimization problem of nonlinear elastic structures. By using finite
element method, this most challenging problem can be formulated as a mixed
integer nonlinear programming problem (MINLP), i.e. for a given deformation,
the first-level optimization is a typical linear constrained 0-1 programming
problem, while for a given structure, the second-level optimization is a
general nonlinear continuous minimization problem in computational nonlinear
elasticity. It is discovered that for linear elastic structures, first-level
optimization is a typical Knapsack problem, which is considered to be
NP-complete in computer science. However, by using canonical duality theory,
this well-known problem can be solved analytically to obtain exact integer
solution. A perturbed canonical dual algorithm (CDT) is proposed and
illustrated by benchmark problems in topology optimization. Numerical results
show that the proposed CDT method produces desired optimal structure without
any gray elements. The checkerboard issue in traditional methods is much
reduced.
