Highly Accurate Mass Lumping in Explicit Dynamics: Application to Isogeometric Analysis
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In this contribution, we propose a method to successfully apply mass lumping in the context of isogeometric analysis (IGA). Typically, Non-Uniform Rational B-Splines (NURBS) are chosen as high-order basis functions for IGA. They enable an exact geometry representation and higher continuity of the solution spaces, compared to standard finite element methods (FEM). Although IGA formulations provide highly accurate results, they suffer from increased computational costs, due to the larger support of NURBS, which leads to increased bandwidths of the system matrices. For structural dynamics, where a system of equations has to be solved within every single time step, the reduced sparsity of the system matrices is a real drawback of IGA in comparison to standard FEM. Explicit time integration schemes are particularly costly, as a critical time step size must not be exceeded. To avoid the solution of a system of equations, the global mass matrix should be lumped. However, employing established mass lumping schemes, such as the row-sum technique, significantly impacts the attainable accuracy, leading to the continued use of consistent mass matrices in IGA as a consequence. The novel mass lumping scheme proposed in this contribution is based on approximate dual basis functions. Diagonally-dominant mass matrices are obtained and therefore, the error of additional conventional mass lumping is reduced to reasonable magnitude. The main advantage of our approach is, that it can be applied to existing IGA formulations by means of a transformation operator.