Numerical Methods for Nonlinear Bending: From Multiscale Models to Neural Approximations

Abstract

This thesis examines numerical methods for approximating various nonlinear bending models for elastic plates and shells. The numerical treatment of these models is challenging because they involve fourth-order partial differential equation problems that cannot be solved using standard methods and are often formulated with nonlinear constraints. This thesis contributes to the development of reliable and efficient numerical schemes tailored to several nonlinear bending problems. <br/> Specifically, four different models are considered. The first contribution concerns the numerical approximation of bending deformations in elastic shells described by parametrized surfaces. Here, a finite element method based on the discrete Kirchhoff triangle is developed that enables the computation of bending deformations under an isometry constraint. <br/> Moreover, this thesis addresses the numerical approximation of a nonlinear homogenized bending model for elastic plates featuring microstructures in the material. To this end, a multiscale finite element method is proposed, coupling a linear three-dimensional microscopic problem with a nonlinear two-dimensional macroscopic problem. <br/> Furthermore, a model is introduced to represent the nonlinear bending of elastic plates with prescribed folds. These folds are described by a phase field function. The model is discretized via finite elements and applied to compute elastic deformations of folded plates. Furthermore, this model serves as the foundation for a fold-optimization framework based on the same phase field formulation. <br/> Finally, a novel numerical method is presented for simulating phase field-based Willmore flow, a time-dependent problem governed by bending energies of shells. Here, a minimizing movement formulation is combined with a neural network approach for approximating the mean curvature flow. This method demonstrates potential applications in geometry processing and computer graphics.