[Paper Review] Numerical analysis of the LDG method for large deformations of prestrained plates

Andrea Bonito, Diane Guignard+2 more|arXiv (Cornell University)|Jun 25, 2021

Advanced Numerical Methods in Computational Mathematics22 references4 citations

TL;DR

This paper presents a numerical analysis of the local discontinuous Galerkin (LDG) method for simulating large deformations in prestrained plates governed by a nonlinear, nonconvex metric constraint. It establishes $γ$-convergence of the discrete energy, proves energy decay in the discrete gradient flow, and ensures controlled metric constraint violation, enabling stable and accurate computation of minimizers for complex plate shapes.

ABSTRACT

A local discontinuous Galerkin (LDG) method for approximating large deformations of prestrained plates is introduced and tested on several insightful numerical examples in our previous computational work. 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 $Γ$-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.

Motivation & Objective

• To develop a robust numerical method for simulating large deformations in prestrained plates with complex, non-isometric geometries.
• To address the challenge of minimizing a fourth-order bending energy subject to a nonlinear, nonconvex metric constraint.
• To ensure the discrete energy converges to the continuous energy via $γ$-convergence.
• To prove that the discrete gradient flow decreases energy at each step and controls metric constraint violation.
• To provide a stable initialization scheme for the gradient flow using a preprocessing step based on the bi-Laplacian equation.

Proposed method

• The energy is discretized using the local discontinuous Galerkin (LDG) method with reconstructed Hessian and stabilization terms.
• A discrete gradient flow is employed to compute minimizers, ensuring energy decrease at each iteration.
• The metric constraint is enforced weakly via Nitsche's method, with stabilization parameters chosen to ensure stability.
• A preprocessing step solves a bi-Laplacian problem to generate an initial deformation with small prestrain defect.
• The method handles both free and Dirichlet boundary conditions, with boundary conditions imposed weakly using Nitsche's formulation.
• The analysis includes error control for the metric constraint defect, showing it remains bounded under time step restrictions.

Experimental results

Research questions

• RQ1 Does the discrete LDG energy $γ$-converge to the continuous energy as mesh size tends to zero?
• RQ2 Can the discrete gradient flow be proven to decrease energy monotonically and compute minimizers with controlled metric constraint violation?
• RQ3 How can a stable and accurate initial deformation be constructed to minimize prestrain defect before gradient flow?
• RQ4 What conditions ensure the stability of the discrete gradient flow scheme under time stepping?
• RQ5 How do the boundary conditions affect the convergence and stability of the discrete scheme?

Key findings

• The discrete energy constructed via the LDG method $γ$-converges to the continuous energy as the mesh size tends to zero.
• The discrete gradient flow strictly decreases the energy at each step, ensuring convergence to a discrete minimizer.
• The metric constraint defect is controlled by $D_{h}(\mathbf{y}_{h}^{n}) \leq \varepsilon_{0} + c\tau(E_{h}(\mathbf{y}_{h}^{0}) + \widetilde{c})$, with $c$ depending only on the domain and boundary.
• The preprocessing step using the bi-Laplacian equation produces an initial deformation $\widehat{\mathbf{y}}_{h}$ that approximately satisfies the boundary conditions and reduces prestrain defect.
• For Dirichlet boundary conditions, the energy and constraint defect bounds are preserved with modified stability constants, ensuring robustness.
• The method achieves $E^{b}_{h}(\widetilde{\mathbf{y}}_{h}^{n_{h}}) \lesssim 1$ and $D_{h}(\widetilde{\mathbf{y}}_{h}^{n_{h}}) \lesssim \sigma_{h}^{1/2}$ when $E^{p}_{h}(\widetilde{\mathbf{y}}_{h}^{n_{h}}) \lesssim \sigma_{h}$, confirming effective constraint control.