LDG approximation of large deformations of prestrained plates

• Published in: Journal of computational physics Vol. 448; p. 110719
• Main Authors: Bonito, Andrea, Guignard, Diane, Nochetto, Ricardo H., Yang, Shuo
• Format: Journal Article
• Language: English
• Published: Cambridge Elsevier Inc 01.01.2022; Elsevier Science Ltd
• Subjects: Approximation; Computational physics; Deformation; Discontinuous Galerkin; Gradient flow; Mathematical analysis; Metric constraint; Nonlinear elasticity; Plate bending; Plates; Prestrained materials; Reconstructed Hessian; Discontinuous Galerkin; Reconstructed Hessian; Prestrained materials; Nonlinear elasticity; Plate bending; Metric constraint
• ISSN: 0021-9991; 1090-2716
• DOI: 10.1016/j.jcp.2021.110719

•Formal derivation of prestrained hyper-elastic plates model.•Novel reformulation amenable to variational approximation methods.•Novel finite element methods based on reconstructed Hessians.•Insightful numerical experiments. A reduced model for large deformations of prestrained plates consists of minimizing a second order bending energy subject to a nonconvex metric constraint. The former involves the second fundamental form of the middle plate and the latter is a restriction on its first fundamental form. We discuss a formal derivation of this reduced model along with an equivalent formulation that makes it amenable computationally. We propose a local discontinuous Galerkin (LDG) finite element approach that hinges on the notion of reconstructed Hessian. We design discrete gradient flows to minimize the ensuing nonconvex problem and to find a suitable initial deformation. We present several insightful numerical experiments, some of practical interest, and assess various computational aspects of the approximation process.