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：本文提出了一种新的用于等几何分析（IGA）显式动力学的质量集中方案。为此，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，并将其对应的对偶基函数作为变分形式中的检验函数，比较了两种对偶基函数。第一类是近似对偶（AD）基函数，具有不同程度的再生性，可生成带状质量矩阵。第二类是通过对格拉姆矩阵求逆（IG）导出的对偶基函数，可直接产生对角质量矩阵。我们将证明，可以将对偶方案作为基于NURBS的形函数和检验函数所得方程组的变换来应用，因此可以轻松集成到现有的IGA程序中。将对偶检验函数的应用视为预处理算子可减少额外的计算开销，但无法完全消除，且刚度矩阵的稠密度仍高于标准Bubnov-Galerkin公式。作为回报，对于IG方案，无需额外对质量矩阵进行行求和集中处理，而对于AD方案，由行求和集中引起的精度损失可降低到合理的程度。数值算例表明，与利用行求和集中的标准NURBS方法相比，对偶集中方案能显著改善动态行为的近似效果。应用IG方案无需额外集中即可获得精确的数值结果，但由于IG对偶基函数的全局支撑性，会产生完全稠密的刚度矩阵，这完全不适用于显式动力学模拟。而结合AD方案与行求和集中，可在计算效率和精度上实现高效计算。