An efficient mass lumping scheme for isogeometric analysis based on approximate dual basis functions

TL;DR

This study develops a mass lumping strategy for explicit dynamics in isogeometric analysis by employing dual test functions. It introduces two duals—inverse Gramian (IG) and approximate dual (AD)—and demonstrates that a simple transformation operator can embed these dual formulations into existing Bubnov–Galerkin IGA pipelines. IG provides diagonal mass matrices at the expense of a fully populated stiffness matrix, while AD yields banded, diagonally dominant matrices whose bandwidth is tunable via the reproduction degree q, enabling efficient and accurate explicit dynamics, especially when combined with row-sum lumping. The results show that AD dual lumping outperforms standard row-sum lumping, achieving high-order convergence and improved eigenfrequency accuracy, with IG offering robust mass diagonalization but prohibitive stiffness densification for explicit schemes. The approach is straightforward to implement as a black-box transformation and extends naturally to NURBS, with potential extensions to 2D problems and higher-order PDEs in future work.

Abstract

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.

Figures (32)

• Figure 1: B-spline basis functions $N_i(\xi)$ of order $p=2$ for an open knot vector $\mathbf{\Xi}=[0,0,0,\frac{1}{5},\frac{2}{5},\frac{3}{5},\frac{4}{5},1,1,1]$.
• Figure 2: Inverse Gramian dual basis functions $\lambda_i(\xi)$ corresponding to the B-spline basis functions shown in Fig. \ref{['fig:basisfun']}.
• Figure 3: Approximate dual basis functions $\tilde{\lambda}_i(\xi)$ corresponding to the B-spline basis functions shown in Fig. \ref{['fig:basisfun']}.
• Figure 4: Equilibrium on element level.
• Figure 5: System sketch for truss under uniaxial loading $q(x)$.