Numerical analysis and computation of nonlinear variational problems in materials science

Abstract

This dissertation focuses on the numerical analysis and scientific computation of two classes of nonlinear variational problems that originate from materials science: the large deformation of plates with metric constraint and constrained energy minimizations for nematic liquid crystals (LCs).For the former, we design a local discontinuous Galerkin method (LDG) finite element approach for prestrained and bilayer plates, and the LDG hinges on the notion of reconstructed Hessian. We consider both Dirichlet and free boundary conditions, the former imposed on part of the boundary. In order to solve the ensuing discrete minimization problems subject to nonconvex metric constraints, we propose discrete gradient flow schemes. We prove $\Gamma$-convergence of the discrete energy to the continuous energy for each problem. Then we prove that the discrete gradient flow decreases the energy at each step and computes discrete minimizers with controllable discrete metric constraint violation. We present several insightful numerical experiments for each problem, some of practical interest, and assess various computational aspects of the approximation process. For LCs we focus on the one-constant Ericksen model that couples a director field with a scalar degree of orientation variable, and allows the formation of various defects with finite energy. We propose a simple but novel finite element approximation of the problem that can be implemented easily within standard finite element packages. Our scheme is projection-free and thus circumvents the use of weakly acute meshes, which are quite restrictive in 3d but are required by recent algorithms for convergence. We prove stability and $\Gamma$-convergence properties of the new FEM in the presence of defects. We also design an effective nested gradient flow algorithm for computing minimizers that in turn controls the violation of the unit-length constraint of the director. We present several simulations in 2d and 3d that document the performance of the proposed scheme and its ability to capture quite intriguing defects.