Numerical analysis of the LDG method for large deformations of prestrained plates. (31st January 2022)

• Record Type: Journal Article
• Authors: Bonito, Andrea; Guignard, Diane; Nochetto, Ricardo H; Yang, Shuo
• Is Part Of: IMA journal of numerical analysis. Volume 43:Number 2(2023)
• Page Start: 627
• Page End: 662
• Publication Date: 2022-01-31
• Subjects: prestrained materials -- metric constraint -- local discontinuous Galerkin -- reconstructed Hessian -- discrete gradient flow -- free boundary conditions; Numerical analysis -- Periodicals; 519.405
• DOI: 10.1093/imanum/drab103
• Languages: English
• ISSNs: 0272-4979

Abstract

Abstract: A local discontinuous Galerkin (LDG) method for approximating large deformations of prestrained plates is introduced and tested on several insightful numerical examples in Bonito et al. (2022, LDG approximation of large deformations of prestrained plates. J. Comput. Phys., 448, 110719). This paper presents a numerical analysis of this LDG method, focusing on the free boundary case. The problem consists of minimizing a fourth-order bending energy subject to a nonlinear and nonconvex metric constraint. The energy is discretized using LDG and a discrete gradient flow is used for computing discrete minimizers. We first show $\varGamma $ -convergence of the discrete energy to the continuous one. Then we prove that the discrete gradient flow decreases the energy at each step and computes discrete minimizers with control of the metric constraint defect. We also present a numerical scheme for initialization of the gradient flow and discuss the conditional stability of it.