Bilayer plates are slender structures made of two thin layers of different materials. They react to environmental stimuli and undergo large bending deformations with relatively small actuation. The reduced model is a constrained minimization problem for the second fundamental form, with a given spontaneous curvature that encodes material properties, subject to an isometry constraint. We design a local discontinuous Galerkin (LDG) method, which imposes a relaxed discrete isometry constraint and controls deformation gradients at barycenters of elements. We prove $\varGamma $-convergence of LDG, design a fully practical gradient flow, which gives rise to a linear scheme at every step, and show energy stability and control of the isometry defect. We extend the $\varGamma $-convergence analysis to piecewise quadratic creases. We also illustrate the performance of the LDG method with several insightful simulations of large deformations, one including a curved crease.

中文翻译：

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双层板大弯曲变形的伽马收敛LDG方法

双层板是由不同材料的两层薄层制成的细长结构。它们对环境刺激做出反应，并在相对较小的驱动下经历较大的弯曲变形。简化模型是第二基本形式的约束最小化问题，具有编码材料属性的给定自发曲率，并受到等距约束。我们设计了一种局部不连续伽辽金（LDG）方法，该方法施加宽松的离散等距约束并控制单元重心处的变形梯度。我们证明了 LDG 的 $\varGamma $ 收敛性，设计了一个完全实用的梯度流，它在每一步都产生了线性方案，并展示了能量稳定性和等距缺陷的控制。我们将 $\varGamma $ 收敛分析扩展到分段二次折痕。我们还通过对大变形（其中包括弯曲折痕）的几次富有洞察力的模拟来说明 LDG 方法的性能。