An improved approach for determining the optimal orientation of orthotropic material

1 I n t r o d u c t i o n The optimal topology design of structures has become a subject that calls for more and more attention of researchers and engineers in the f ie ldof structural optimization recently. Bends¢e and Kikuchi (1988) introduced a Homogenization Based Optimization (HBO) method for finding optimal structures without the requirement of conjecturing the initial topologies. This method transforms the topological optimization problem into an Optimal Material Distribution (OMD) problem, where the composite material with many microscopical voids is introduced into the design problem (Fig. 1) so as to relax the original problem which was restricted to the use of an isotropic solid material. As was pointed out by Kohn (1986), the relaxed formulation has the advantage of obtaining fewer local minima, thus the global minima can be reached relatively easily. Such an idea is employed by Bends0e and Kikuchi (1988), who employed a microstructure model for relaxing the problem as shown in Fig. 2, where three design variables, the sizes a, b of the rectangular cavity and the orientation 0 of the microstructure, are employed for a plane stress problem. Then the stiffness coefficients as well as the mass density of the structure are evaluated as the functions of these three design variables using the homogenization method. The optimization process is then used to obtain the OMD, which gives the optimal structure with the optimal topology. There may exist a question: how important is the use of an orientational design variable for relaxing the problem? Also, one might ask how a microstructure that forms an orthotropie material can be optimally oriented by a mathematical or physical consideration. This kind of problem can be generalized as an optimal orientation problem of orthotropic material, and was first discussed by Pedersen (1988). However, the discussion by Pedersen (1988) is limited to a unit cell case, in which the orientational variable is separated from the whole design domain to obtain the extreme strain energy density. This consideration is not sufficient for the real design problem, in which the orientational variable is a distributed parameter in the design domain, and the coupling exists among the orientational variables in two different spatial points of the design domain. Alternatively, Suzuki and Kikuchi (1991) indicated that the optimal orientation of the microstructure can be determined by the principal stress directions of the structure. This idea has been implemented in the HBO algorithm for solving the layout optimization problem. Later, Dfaz and Bendsee (1992) presented another approach dealing with the layout optimization problem under multi-loading. This approach is similar to that of Pedersen (1988), but instead of using the strain energy density, it employs the stress based formulation to calculate the optimal orientation. Apparently, the stress based approach can provide much better results than the strain based approach (see also Olhoff e~ al. 1992). Hence, it is very interesting to study the distinctions between these different approaches, essentially between the stress based and strain based approaches, and their rationalism, in determining the optimal orientation. The goals of this paper are not only to clarify the questionable points stated above, but also to derive a general approach for determining the optimal orientation in dealing with various optimization problems. It will be shown that the stress field is less sensitive than the strain field with respect to the variation of the orientational variable. Therefore, the coupling between the discrete orientational variables is relatively weak when the stress field is used. This explains why the stress based approach, as employed by Suzuki and Kikuchi (1991) and Diaz and Bendsee (1992), is more efficient than the strain based method. This kind of approach will be generalized in this paper to deal with more general optimization problems. A typical eigenvalue optimization problem is Considered, but the approach can be applied to other optimization problems. It will be shown that the new approach is also more accurate and efficient in determining the optimal orientation than the previous methods. To substantiate the issues discussed herein, the results obtained by the new approach will be compared with those of the previous methods,