Shape and layout optimization using homogenization method.

Abstract

A generalized shape and layout optimization technique based on the homogenization method is described and applied to the design of two and three dimensional solids and to three dimensional shells. This technique provides a new approach to structural optimization that is free of apriori restrictions on the topology of final design, one that possesses the capability of optimizing topology as well as shape. The optimization problem is defined as minimize the mean compliance while total material to be used is constrained. The same technique is extended to cover cases where multiple load is applied within the given design domain. To determine elastic coefficients of structural material with microstructure, homogenization theory is used. A resizing rule is formulated on the basis of the optimality criteria method. The ability to predict optimal layout as well as shape has been demonstrated via several examples. Convergence of the design with respect to mesh refinement is examined. Stability with respect to mesh distortion is demonstrated as well. It is shown that different initial material distributions can lead to different final distributions. It is shown that the transition of topology is gradual rather than rapid when topology changes with some factors. Several examples are shown for the optimal design of three dimensional shells. These results can be interpreted as the design of optimal layout of stiffener as well as optimal layout of frame on the predetermined surface. The homogenization theory for shells with rapidly changing rib height is presented. When the design surface is flat, structures with hinge lines are shown to be generated, while when the surface is curved hinge lines do not appear. Finally, the current optimization technique is applied to problems in a three dimensional domain using three dimensional solid finite elements. Several cases confirm that the procedure produces designs that appear to be consistent with common engineering practice. Sandwich structure for bending load and box type structure for torsional load are generated. Also it has been demonstrated that the prediction of topology is sensitive to the choice of model for the mechanics.