Second-order shape derivatives along normal trajectories, governed by Hamilton-Jacobi equations

• Published in: Structural and multidisciplinary optimization Vol. 54; no. 5; pp. 1245 - 1266
• Main Authors: Allaire, G., Cancès, E., Vié, J.-L.
• Format: Journal Article
• Language: English
• Published: Berlin/Heidelberg Springer Berlin Heidelberg 01.11.2016; Springer Nature B.V; Springer Verlag
• Subjects: Algorithms; Computational Mathematics and Numerical Analysis; Deformation; Differential equations; Differentiation; Engineering; Engineering Design; Hamilton-Jacobi equation; Linear equations; Mathematical analysis; Mathematics; Numerical Analysis; Optimization and Control; Ordinary differential equations; Research Paper; Shape optimization; Theoretical and Applied Mechanics; Level-set method; Shape and topology optimization; Second-order shape derivative; Newton method; second-order shape derivative; shape and topology optimization; level-set method
• ISSN: 1615-147X; 1615-1488
• DOI: 10.1007/s00158-016-1514-2

In this paper we introduce a new variant of shape differentiation which is adapted to the deformation of shapes along their normal direction. This is typically the case in the level-set method for shape optimization where the shape evolves with a normal velocity. As all other variants of the original Hadamard method of shape differentiation, our approach yields the same first order derivative. However, the Hessian or second-order derivative is different and somehow simpler since only normal movements are allowed. The applications of this new Hessian formula are twofold. First, it leads to a novel extension method for the normal velocity, used in the Hamilton-Jacobi equation of front propagation. Second, as could be expected, it is at the basis of a Newton optimization algorithm which is conceptually simpler since no tangential displacements have to be considered. Numerical examples are given to illustrate the potentiality of these two applications. The key technical tool for our approach is the method of bicharacteristics for solving Hamilton-Jacobi equations. Our new idea is to differentiate the shape along these bicharacteristics (a system of two ordinary differential equations).