We discuss a discretization of the Euler–Bernoulli bending energy and of Euler elasticae under clamped boundary conditions by polygonal lines. We show Hausdorff convergence of the set of almost minimizers of the discrete bending energy to the set of smooth Euler elasticae under mesh refinement in (i) the |$W^{1,\infty }$|-topology for piecewise-linear interpolation; and in (ii) the |$W^{2,p}$|-topology, |$p \in [2,\infty [$|⁠, using a suitable smoothing operator to create |$W^{2,p}$|-curves from polygons.

中文翻译：

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离散弹性的变分收敛

我们通过多边形线讨论了在夹紧边界条件下的Euler–Bernoulli弯曲能和Euler弹性体的离散化。我们在（i）| $ W ^ {1，\ infty} $ |中，在网格细化下显示了离散弯曲能量的几乎最小化集合到光滑的Euler弹性集合的Hausdorff收敛性。-分段线性插值的拓扑；以及（ii）| $ W ^ {2，p} $ | -topology，| $ p \ in [2，\ infty [$ |⁠]中，使用合适的平滑运算符创建| $ W ^ {2，p} $ | -来自多边形的曲线。