In this contribution, we provide a new mass lumping scheme for explicit dynamics in isogeometric analysis (IGA). To this end, an element formulation based on the idea of dual functionals is developed. Non-Uniform Rational B-splines (NURBS) are applied as shape functions and their corresponding dual basis functions are applied as test functions in the variational form, where two kinds of dual basis functions are compared. The first type are approximate dual basis functions (AD) with varying degree of reproduction, resulting in banded mass matrices. Dual basis functions derived from the inversion of the Gram matrix (IG) are the second type and already yield diagonal mass matrices. We will show that it is possible to apply the dual scheme as a transformation of the resulting system of equations based on NURBS as shape and test functions. Hence, it can be easily implemented into existing IGA routines. Treating the application of dual test functions as preconditioner reduces the additional computational effort, but it cannot entirely erase it and the density of the stiffness matrix still remains higher than in standard Bubnov-Galerkin formulations. In return applying additional row-sum lumping to the mass matrices is either not necessary for IG or the caused loss of accuracy is lowered to a reasonable magnitude in the case of AD. Numerical examples show a significantly better approximation of the dynamic behavior for the dual lumping scheme compared to standard NURBS approaches making use of row-sum lumping. Applying IG yields accurate numerical results without additional lumping. But as result of the global support of the IG dual basis functions, fully populated stiffness matrices occur, which are entirely unsuitable for explicit dynamic simulations. Combining AD and row-sum lumping leads to an efficient computation regarding effort and accuracy.

翻译：本文为等几何分析（IGA）中的显式动力学提出了一种新的质量集中方案。为此，我们基于对偶泛函的思想开发了一种单元公式。采用非均匀有理B样条（NURBS）作为形函数，并在变分形式中将其对应的对偶基函数作为测试函数，比较了两种类型的对偶基函数。第一种是具有不同再生程度的近似对偶基函数（AD），可产生带状质量矩阵。第二种是由Gram矩阵（IG）求逆导出的对偶基函数，直接生成对角质量矩阵。我们将证明，可以将对偶方案作为基于NURBS作为形函数和测试函数的所得方程组的变换来应用。因此，它可以轻松地集成到现有的IGA程序中。将对偶测试函数的应用视为预处理可以减少额外的计算量，但无法完全消除，刚度矩阵的密度仍然高于标准Bubnov-Galerkin公式。作为回报，对质量矩阵应用额外的行和集中，对于IG来说要么不是必要的，要么在AD情况下将导致的精度损失降低到合理的水平。数值算例表明，与使用行和集中的标准NURBS方法相比，对偶集中方案对动态行为的近似效果显著更好。应用IG无需额外集中即可获得准确的数值结果。但由于IG对偶基函数的全局支持，会出现完全填充的刚度矩阵，这完全不适合显式动力学模拟。结合AD和行和集中可以在计算量和精度方面实现高效计算。