Mathematical Sciences: Theoretical and Computational Methods in Optimal Design of Elastic Structures

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9625129 Cherkaev A program of research is proposed for structural optimization of elastic and conducting bodies which combines modern variational techniques, homogenization, and new computational methods. Since structural optimization problems are typically ill-posed as stated, the efficiency and stability of numerical algorithms for these problems depends upon theoretical formulation of a well-posed problem. The expected results include advances in relaxation of non-convex variational problems, discovery of new microstructures with extremal properties, and developing effective algorithms for highly sensitive problems. The applications include real-time design of practical optimal structures, such as rotors, cylindrical shells, wheels, domes, etc. Various efficiency functionals such as the overall stiffness, principal eigenfrequency, weighted deflection, etc. will be examined. Systematic selection of suboptimal projects will also be considered. The research involves improving upon and integrating the best current theoretical and computational methods for systematically designing structures with optimal performance characteristics. This permits the reduction of the overall cost of materials, and their weight, which greatly influences the ultimate cost in aerospace and similar applications. The theoretical aspect involves understanding the nature of optimal microstruatures, while the computational aspect involves distributing and modifying these composites to best adapt to practical working conditions. The efficiency of both the design methods and the designs themselves should prove advantageous in industrial applications.