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 2D membrane model from 3D 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.

翻译：通过渐近分析与数值计算，我们研究了液晶聚合物网络的简化薄膜模型。该模型需要求解一个非凸拉伸能量的极小化问题。我们展示了从三维橡胶弹性理论形式推导二维薄膜模型的渐近过程，并构造了含点缺陷的近似解。我们设计了带有正则化的有限元方法，提出了一种结合牛顿内迭代的非线性梯度流方法，以求解该非凸离散极小化问题。最后，我们给出了具有实际意义的数值模拟，验证了模型及方法捕捉丰富物理现象的能力。