$Γ$-收敛的 LDG 方法用于双层板的大弯曲变形
$Γ$-convergent LDG method for large bending deformations of bilayer plates
AI总结：
针对双层板大弯曲变形的约束最小化问题，提出了一种施加松弛离散等距约束的局部间断伽辽金（LDG）方法，证明了其 $Γ$-收敛性并设计了能量稳定的线性梯度流格式。
摘要：
双层板是由两层不同材料制成的细长结构。它们对环境刺激产生反应，在相对较小的驱动下发生大弯曲变形。该简化模型是关于第二基本形式的约束最小化问题，具有给定的自发曲率（编码材料属性），并受等距约束。我们设计了一种局部间断伽辽金（LDG）方法，该方法施加松弛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 $Γ$-收敛性，设计了完全实用的梯度流（每步产生线性格式），并展示了能量稳定性和等距缺陷控制。我们将 $Γ$-收敛分析扩展到分段二次折痕。我们还通过几个有意义的大变形模拟（包括一个弯曲折痕）展示了 LDG 方法的性能。
英文摘要：
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 $Γ$-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 $Γ$-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.
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