均質化法による最適設計理論

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After pioneering work of Schmit and Fox in the early sixties using mathematical programming methods with sensitivity analysis, the optimal design problem becomes a research subject not only for structural engineers but also for applied mathematicians. Various optimization algorithms have been introduced to solve structural optimization problems, while existence of the global optimum has been examined by using the theory of convex analysis. In the seventies and early eighties, the shape optimization is introduced and is solved by defining the shape of a structure using a set of appropriate spline functions without changing the topology of the structure. Mostly for plane and folded plate/shell structures, the shape optimization problem is solved after integrating an automatic mesh generation method so that remeshing is possible to follow large change of the shape of a structure to define a discrete finite element model. As shown in many mathematical works on structural optimization if the number of parametric design variables to define the size or shape of a structure is finite, existence of the optimum might be shown by using the compactness argument. However, if it becomes infinite, the standard compactness argument cannot be applied to derive existence of the optimum. This leads a more general setting of structural optimization, namely a relaxed optimal design problem involving possibly fine scale microstructures in the medium of a structure as shown in Kohn, Strang, Murat, and Tartar for the shape optimization as well as Cheng and Olhoff for the sizing optimization. Existence of the optimum in a relaxed design problem has been shown by Cherkaev and Lurie in the early eighties using the notion of G-convergence, and is then extensively studied by French and Italian mathematicians using the theory of homogenization. Based on the notion of relaxed design with homogenization, Bendsoe and Kikuchi solved the shape optimization problem without fixing the topology of a structure in 1988. With this new approach, the size, shape, and topology of a structure can be optimized at once without defining them by using appropriate functions. Here these developments in the theory of structural optimization will be described both from engineering and mathematical points of view. Some computational results are also described to illustrate the new approach of structural optimization.
日本応用数理学会の論文
1993-03-15

均質化法による最適設計理論

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