The general problem in topology optimization is correctly formulated as a double-\min mixed integer nonlinear programming (MINLP) problembased on the minimum total potential energy principle. It is proved that for linear elastic structures, the alternative iteration leads to a Knapsack problem, which is considered to be NP-hard in  computer science. However, by using canonical duality theory (CDT) developed recently by the author, this challenging 0-1 integer programming problem can be solved analytically to obtain global optimal solution at each design iteration.  The novel CDT method for general topology optimization is refined and tested mainly by both 2-D benchmark problems in topology optimization . Numerical results show that without using filter  , the CDT method can obtain exactly 0-1 optimal density distribution with almost no checkerboard pattern. Its performance and novelty are compared with the popular SIMP and BESO approaches.  A brief review on the canonical duality theory for solving a unified problem in multi-scale  systems is given in Appendix.
Topology optimization for general materials is correctly formulated as a bi-level knapsack problem, which is considered to be NP-hard in global optimization and computer science. By using canonical duality theory (CDT) developed by the author, the linear knapsack problem can be solved analytically to obtain global optimal solution at each design iteration. Both uniqueness, existence, and NP-hardness are discussed. The novel CDT method for general topology optimization is refined and tested by both 2-D and 3-D benchmark problems . Numerical results show that without using filter and any other artificial technique , the CDT method can produce exactly 0-1 optimal density distribution with almost no checkerboard pattern. Its performance and novelty are compared with the popular SIMP and BESO approaches. Additionally, some mathematical and conceptual mistakes in literature are explicitly pointed out. A brief review on the canonical duality theory for solving a unified problem in multi-scale nonconvex/discrete systems is given in Appendix.
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The general problem in topology optimization
Topology optimization for general materials
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double-\min mixed integer nonlinear programming (MINLP) problembased on the minimum total potential energy principle. It is proved that for linear elastic structures, the alternative iteration leads to a Knapsack problem,
bi-level knapsack problem,
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global optimization and
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However, by
By
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recently
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this challenging 0-1 integer programming
the linear knapsack
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Both uniqueness, existence, and NP-hardness are discussed.
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mainly
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benchmark problems in topology optimization
and 3-D benchmark problems
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and any other artificial technique
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obtain
produce
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Additionally, some mathematical and conceptual mistakes in literature are explicitly pointed out.
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nonconvex/discrete