In this article, we study the effect of small-cut elements on the critical time-step size in an immersogeometric context. We analyze different formulations for second-order (membrane) and fourth-order (shell-type) equations, and derive scaling relations between the critical time-step size and the cut-element size for various types of cuts. In particular, we focus on different approaches for the weak imposition of Dirichlet conditions: by penalty enforcement and with Nitsche's method. The stability requirement for Nitsche's method necessitates either a cut-size dependent penalty parameter, or an additional ghost-penalty stabilization term is necessary. Our findings show that both techniques suffer from cut-size dependent critical time-step sizes, but the addition of a ghost-penalty term to the mass matrix serves to mitigate this issue. We confirm that this form of `mass-scaling' does not adversely affect error and convergence characteristics for a transient membrane example, and has the potential to increase the critical time-step size by orders of magnitude. Finally, for a prototypical simulation of a Kirchhoff-Love shell, our stabilized Nitsche formulation reduces the solution error by well over an order of magnitude compared to a penalty formulation at equal time-step size.

翻译：本文研究了在浸入几何背景下小切割单元对临界时间步长的影响。我们分析了二阶（薄膜型）和四阶（壳型）方程的不同公式，并推导了各类切割情况下临界时间步长与切割单元尺寸之间的缩放关系。特别地，我们重点关注弱施加狄利克雷条件的两种方法：罚函数法和尼采法。尼采法的稳定性要求要么采用与切割尺寸相关的罚参数，要么需要附加鬼罚稳定项。研究结果表明，两种技术均受到与切割尺寸相关的临界时间步长问题的影响，但在质量矩阵中添加鬼罚项有助于缓解该问题。我们证实，这种"质量缩放"形式不会对瞬态薄膜示例的误差和收敛特性产生不利影响，并有望将临界时间步长提升数个数量级。最后，在基尔霍夫-洛夫壳的原型仿真中，当采用相同的时间步长时，我们的稳定化尼采公式相比罚函数公式可将求解误差降低一个数量级以上。