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 $\Gamma$-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.
翻译:在先前的计算工作中,我们采用并测试了一种局部不连续的加列金(LDG)方法,以近似于事先加工过的板块的大规模变形。本文介绍了对这一LDG方法的数值分析,重点是自由边界案例。问题在于尽量减少受非线性和非节点约束的第四顺序弯曲能量。使用LDG将能量分解,而计算离散最小化器则使用离散梯度流。我们首先显示离散能量的 $\Gamma$-converggation 和 连续的。然后,我们证明离散梯度流会减少每一步的能量,并计算离散最小化器,以控制参数限制缺陷。我们还提出了一个初始化梯度流的数值方案,并讨论其有条件的稳定性。