弹性体与障碍和粘附效应的数值分析
Numerical analysis of elastica with obstacle and adhesion effects
AI总结：
本文研究了受障碍和粘附影响的弹性体变分问题的数值计算，通过离散化和正则化能量函数，结合Γ收敛分析，探讨了离散极小值的收敛性及数值实验中奇异局部极小值的存在性。
摘要：
我们考虑了一个来自材料科学的变分问题的数值计算。目标泛函是一种受障碍和粘附影响的弹性能量。由于其强非线性和不连续性，欧拉-拉格朗日方程非常复杂，其临界点的数值计算困难。本文将目标能量离散化并正则化为定义在多边形曲线空间上的泛函。此外，我们在此Γ收敛框架下进行了离散极小值的收敛性分析。我们首先证明了离散能量泛函Γ收敛于原始泛函。然后，我们建立了离散极小值序列的紧性性质。这两个结果使得可以从离散极小值中提取收敛子序列。我们还在论文的最后一部分展示了某些数值例子。数值实验暗示了奇异局部极小值的存在性。
英文摘要：
We consider the numerical computation of a variational problem that arises from materials science. The target functional is a type of elastic energy that is influenced by obstacles and adhesion. Owing to its strong nonlinearity and discontinuity, the Euler-Lagrange equation is very complicated, and numerical computation of its critical points is difficult. In this paper, we discretize and regularize the target energy as a functional defined on a space of polygonal curves. Moreover, we develop convergence analysis for discrete minimizers in the framework of $Γ$-convergence. We first show that the discrete energy functional $Γ$-converges to the original one. Then, we establish the compactness property for the sequence of discrete minimizers. These two results allow us to extract a convergent subsequence from the discrete minimizers. We also present some numerical examples in the last section of the paper. Existence of singular local minimizers is suggested by numerical experiments.
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