有限元法中的CFL条件
A CFL condition for the finite cell method
AI总结：
本文提出了一种修改的CFL条件，用于有限元法中临界时间步长的估计，通过分析α参数对临界时间步长的影响，确定了最小临界时间步长的下限。
摘要：
浸入边界有限元方法允许用户避免潜在的麻烦任务，即生成符合边界条件的网格。当与显式时间积分结合时， poorly cut elements with little support in the physical domain导致严重的临界时间步长减小，对浸入波传播模拟构成重大挑战。有限元法通过在虚域中定义问题的弱形式来稳定切元素，但该形式被一个小值α缩放。本文研究了有限元法对显式时间积分的临界时间步长的影响。从一个分析的一度自由度模型开始，我们系统地研究了α稳定化对角点和薄片切割的最大学生值的影响，从而影响临界时间步长。分析通过一个具有一个元素和增加多项式次数的例子的数值研究得到补充，证实了临界时间步长不会低于某个极限，即使切分比例趋于零。这个下限由α的选择控制。在更高维度中，薄片切割比角点切割更有害，因此决定了最小临界时间步长。增加多项式次数对这种退化影响很小。基于这些观察，我们推导出临界时间步长的最小值作为α函数的估计，我们用它来提出有限元法的修改CFL条件。该条件的有效性在二维孔洞板示例中得到验证。
英文摘要：
Immersed boundary finite element methods allow the user to bypass the potentially troublesome task of boundary-conforming mesh generation. When combined with explicit time integration, poorly cut elements with little support in the physical domain lead to a severely reduced critical time step size, posing a major challenge for immersed wave propagation simulations. The finite cell method stabilizes cut elements by defining the weak form of the problem also in the fictitious domain, but scaled by a small value $α$. This paper investigates the effect of the finite cell method on the critical time step size for explicit time integration. Starting with an analytical one-degree-of-freedom model, we systematically study the influence of $α$-stabilization on the maximum eigenvalue, and thus on the critical time step size, for corner and sliver cuts. The analysis is complemented by a numerical study of an example with one element and increasing polynomial degree, confirming that the critical time step size does not decrease below a certain limit, even as the cut fraction tends to zero. This lower bound is controlled by the choice of $α$. In higher dimensions, sliver cuts are found to be more detrimental than corner cuts, thus determining the minimum critical time step size. Increasing the polynomial degree has only little effect on this degradation. Based on these observations, we derive an estimate of the minimum critical time step size as a function of $α$, which we use to propose a modified CFL condition for the finite cell method. The validity of this condition is demonstrated on a two-dimensional perforated plate example.
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