Nonlinear Shape-Manifold Learning Approach: Concepts, Tools and Applications

In this paper, we present the concept of a “shape manifold” designed for reduced order representation of complex “shapes” encountered in mechanical problems, such as design optimization, springback or image correlation. The overall idea is to define the shape space within which evolves the boundary of the structure. The reduced representation is obtained by means of determining the intrinsic dimensionality of the problem, independently of the original design parameters, and by approximating a hyper surface, i.e. a shape manifold, connecting all admissible shapes represented using level set functions. Also, an optimal parameterization may be obtained for arbitrary shapes, where the parameters have to be defined a posteriori. We also developed the predictor-corrector optimization manifold walking algorithms in a reduced shape space that guarantee the admissibility of the solution with no additional constraints. We illustrate the approach on three diverse examples drawn from the field of computational and applied mechanics.