The penalty method has proven to be a well-suited approach for the application of coupling and boundary conditions on (trimmed) multi-patch NURBS shell structures within isogeometric analysis. Beside its favorable simplicity and efficiency, the main challenge is the appropriate choice of the underlying penalty factor — choosing the penalty factor too low yields a poor constraint accuracy, while choosing it too high causes numerical issues like ill-conditioned system matrices or a small infeasible time step size in explicit dynamics. Although recommendations for penalty values exist, profound methods for its determination are still an active field of research.

We address this issue and provide formulas allowing an a priori determination of penalty factors for NURBS-based Reissner–Mindlin shells with penalty-based coupling and boundary conditions. The underlying approach is inspired by a methodology previously used for conventional finite elements, for which penalty factors are derived through a comparison with exact Lagrange multiplier solutions. In that way, penalty formulas consisting of a problem-dependent factor and a problem-independent intensity factor are obtained. The fact that the latter is a direct measure of the penalty-induced error is the main advantage of this approach and enables a problem-independent definition of the penalty factor as a function of the desired accuracy.

We demonstrate the validity of the derived formulas and the corresponding error measure with benchmark problems in linear elasticity including trimmed non-matching NURBS shells. Furthermore we show that the mesh-adaptivity of the penalty formulas improves the convergence behavior and conditioning of penalty methods.

罚分法已被证明是在等几何分析中在（修剪的）多面体NURBS壳体结构上应用耦合和边界条件的一种非常合适的方法。除了其优越的简单性和效率，主要的挑战是潜在的惩罚因子的适当选择 - 选择惩罚因子过低的收益率差约束的准确性，而选择它过高的原因数值的问题，如不良的空调系统矩阵或小不可行显式动力学中的时间步长。尽管存在惩罚值的建议，但确定惩罚值的深刻方法仍然是研究的活跃领域。

我们解决了这个问题，并提供了公式，可以基于惩罚性耦合和边界条件，预先确定基于NURBS的Reissner-Mindlin壳的惩罚因子。底层方法是从以前用于常规有限元的方法中获得启发的，为此，惩罚因子是通过与精确的拉格朗日乘数解进行比较得出的。以此方式，获得了由与问题相关的因子和与问题无关的强度因子组成的惩罚公式。后者是惩罚导致的误差的直接量度这一事实是此方法的主要优势，并且可以根据期望的精度对惩罚因子进行独立于问题的定义。

我们用线性弹性基准问题（包括修剪的不匹配NURBS壳）证明了导出公式的正确性和相应的误差度量。此外，我们表明惩罚公式的网格适应性改善了惩罚方法的收敛性和条件。