Extensions of optimal layout design using the homogenization method.

Abstract

In this thesis, the concept of optimal layout design of continuum structures using the homogenization method is applied to several fields: in heat transfer problems, layouts for high conductivity; in linear elastic problems, layouts consisting of full material and void regions ('clear layouts'), layouts consisting of fiber reinforced composite, and layouts considering local stress constraints. Optimal layout design has become an important field in structural optimization. In this thesis, spatially periodic microstructures are introduced for partial relaxation of the optimal problem and the shapes of the microstructures are controlled to obtain the optimal material distribution. The material properties of this composite are calculated using the homogenization method. The optimal material distributions can be used as an initial topology for detailed design. For high conductivity, optimal layouts are obtained for 2D, shell and 3D problems, in which conduction and convection heat transfers are considered with several boundary conditions. The effects of different microstructures on the optimal layout are studied in this problem. For clear layouts, various microstructures are introduced: microstructures whose shapes provide isotropic homogenized properties, and microstructures whose stiffness are the same as the weighted values of the rectangular type microstructure. Criteria are suggested to determine whether the optimal layouts can be regarded as the clear layout. For fiber reinforced composite, the optimal layout for the stiff structure is obtained using the triangular microstructure which is composed of fiber reinforced composite and void. In this layout, the amount and the direction of the fiber can be determined: the layout of the fiber can be interpreted as both continuous and discontinuous. For the local stress, the Von Mises equivalent stress is used and the local constraints are approximated by one global constraint. The Von Mises stress for yielding is assumed to be proportional to the material density in the porous material. The concept of the optimal layout is successfully extended to the above four fields, giving results which can be applied in detailed design, except for the layout considering local stress constraints in which the Von Mises stress for yielding should be studied for porous material.