Optimal Design from Noether’s Viewpoint

A theorem formulated by Emmy Noether in 1918 can be summarized in a few words: “To every continuous symmetry of a physical system belongs a conservation law”. The inverse statement reads: “To every conservation law of a physical system belongs a continuous symmetry”. The Noether theorem provides a powerful method for the establishing of the conservation laws form the corresponding symmetries. Structural optimization has over the past decades qualified as an important tool in the design process. The method can be clustered into shape, topology, and anisotropy optimization. The objective of the optimization is commonly the minimization of the equivalent stresses, mass or compliance for a given amount of material and boundary conditions. The correspondence between Noether’s theorem and methods of optimal design is the following. The “Lagrangian of optimal design” plays the role of “conservation law of a physical system”. The “continuous symmetry” corresponds to “necessary optimality condition.” Thus, Noether’s theorem can be resembled for the structural optimization: “To every necessary optimality condition belongs a “Lagrangian of optimal design.” Inversely, “To every Lagrangian of optimal design belongs a necessary optimality condition.”