Speaker
Description
Many applications in the field of nonlinear elasticity aim at finding a global minimizer of functionals of the form
I(u) = ∫_Ω W(∇u(x))dx
over a domain Ω ⊂ ℝᵈ in spatial dimension d ∈ {2, 3} for a suitable weak class of deformations u : Ω → ℝᵈ. In many relevant cases, the density W : ℝᵈ × ᵈ → ℝ̅ ≔ ℝ ∪ {∞} does not satisfy a suitable notion of convexity, and the existence of minimizers cannot be guaranteed. In fact, the infimum may not be reached, and nonconvexity, e.g. multiwell structure of the energy density, may lead to the emergence of increasingly fine microstructures within the minimising sequences. Moreover, the application of standard discretisation methods for the minimisation of I typically leads to mesh-dependent results with oscillations in the discrete deformation gradient at the length scale of the mesh size. Therefore, alternative approaches are introduced for both mathematical analysis and numerical simulation using relaxed formulations, e.g.~based on polyconvexification of the energy density, that focus on macroscopic features responsible for global behaviour by extracting the relevant information from the unresolved microstructures. Different algorithms have been developed in the literature to perform numerically the polyconvexification. However, the computational time of these algorithms remains high due to the high-dimensionality of the problem. In this talk, by combining the recent singular value polyconvexification and Fully Input Convex Neural Network (FICCN), we accelarate the prediction of the polyconvex envelope. The significant speed-up associated with the neural network compression is demonstrated in a series of numerical experiments.
