Optimal design of periodic frame structures with negative thermal expansion via mixed integer programming

When structures and microstructures consisting of two or more materials with positive thermal expansion have specific configurations, they are able to have negative thermal expansion coefficients, i.e., they contract when heated. This paper proposes a topology optimization methodology of frame structures for designing a planar periodic structure that exhibits negative thermal expansion property. Provided that beam section of each existing member is chosen from a set of finitely many predetermined candidates, we show that this topology optimization problem with multiple material phases can be formulated as a mixed-integer linear programming problem. A global optimal solution can hence be found with a readily available software package. Since the proposed method treats frame structures and addresses local stress constraints, the optimal solution contains neither thin members nor hinge-like regions. To avoid too complicated structural designs realized as assemblage of many small pieces, this paper develops the constraints that separate distributions of two different materials. Numerical experiments are performed to show that structures with negative or near zero thermal expansion can be obtained by the proposed method.