We examine a reduced membrane model of liquid crystal polymer networks (LCNs) via asymptotics and computation. This model requires solving a minimization problem for a non-convex stretching energy. We show a formal asymptotic derivation of the membrane model from rubber elasticity. We construct approximate solutions with point defects. We design a finite element method with regularization, and propose a nonlinear gradient flow with Newton inner iteration to solve the non-convex discrete minimization problem. We present numerical simulations of practical interests to illustrate the ability of the model and our method to capture rich physical phenomena.

中文翻译：

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液晶聚合物网络的简化膜模型：渐近学和计算

我们通过渐进和计算研究了液晶聚合物网络（LCN）的简化膜模型。该模型需要解决非凸拉伸能量的最小化问题。我们展示了橡胶弹性膜模型的形式渐近推导。我们构建具有点缺陷的近似解。我们设计了正则化的有限元方法，并提出了采用牛顿内迭代的非线性梯度流来解决非凸离散最小化问题。我们提出了实际兴趣的数值模拟，以说明模型和我们的方法捕获丰富物理现象的能力。