An analysis of high order FEM and IGA for explicit dynamics: Mass lumping and immersed boundaries

Abstract

We investigate the behavior of different shape functions for the discretization of hyperbolic problems. In particular, we consider classical Lagrange polynomials and B-splines. The studies focus on the performance of the these functions as a spatial discretization approach combined with an explicit time marching scheme. In this regard, a major concern is the maximum eigenvalue that imposes restrictions on the critical time step size and suitable lumping techniques that yield a diagonal mass matrix. The accuracy of the discretization methods is assessed in an asymptotic manner in terms of the convergence of eigenvalues and eigenvectors. Further, the global accuracy is investigated in terms of the full spectrum. The results show that B-spline discretization with a consistent mass matrix are more accurate than those based on Lagrange shape functions, which holds true in the boundary-fitted as well as in the immersed setting. On the other hand, Lagrange shape functions are more robust with respect to standard lumping techniques, which cannot be directly applied for B-splines without loss of accuracy. In general, we observe that none of the standard lumping schemes yields optimal results for B-splines, even in the boundary-fitted setting. For the immersed setting, also Lagrange shape functions show a drop in accuracy which depends on the position of the boundary that cuts the element. Several remedies are considered in order to overcome these issues, including interpolatory B-spline bases as well as eigenvalue stabilization methods. While accuracy and stability can be improved using these remedies, we conclude from our study that it is still an open question, how to design a discretization method that achieves large critical time step sizes in combination with a diagonal mass matrix and high accuracy in the immersed setting. We note that these considerations primarily relate to linear structural dynamics applications, such as for example, structural acoustics. In nonlinear problems, such as automotive crash dynamics, other considerations predominate. An example of a one-dimensional elastic-plastic bar impacting a rigid wall is illustrative.